commit f2fb36f646b6e7c0c191922b08fc155a842b7d1e
Author: JAMES RYAN <james.paul9889@gmail.com>
Date: Mon Mar 24 00:43:09 2025 -0400
uploaded all materials
diff --git a/README.md b/README.md
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+notes from matlab seminar
+
+thanks to cat van west for all the knowledge :D
diff --git a/hw1/Ryan_hw1_012424.m b/hw1/Ryan_hw1_012424.m
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+%%%% ECE210-B Matlab Seminar, Homework 1.
+% james ryan, 1/24/24
+
+% preamble
+close all;
+clear;
+clc;
+
+%% Scale-'ers
+% (1)
+% Takes the absolute value of
+% sin(pi/3)
+% PLUS
+% j divided by secant of (-5/3)*pi
+a = abs(sin(pi ./ 3) ...
+ + ((1 * j) ./ (sec((-5 * pi) / 3))) ...
+ );
+
+% (2)
+% n = 3, cubic root
+l = nthroot(8, 3);
+
+% (3)
+% [1, 2, 3, ..., 79, 80]
+u_mat = 1:80;
+
+% Sums all the terms from u_mat, scales by 2 / 6!, and takes the square root
+u = sqrt((2 / factorial(6)) * sum(u_mat));
+
+% q4
+% takes the square
+% of imaginary value
+% of the floor
+% of the log
+% of the square root of 66
+% taken to the power of 7j
+m = (imag(floor(log(sqrt(66).^(7 * j))))).^2;
+
+%% Mother...?
+% (1)
+% Makes a 1x4 matrix from a l u & m,
+% and transposes the formed matrix into a 4x1 matrix
+A = [a; l; u; m];
+
+% (2)
+% makes a 2x2 matrix using a l u & m.
+F = [a l; u m];
+
+% (3)
+% takes the non-conjugate transpose of F
+T = F.';
+
+% (4)
+% takes the inverse of the matrix product of T * F
+B = inv(T * F);
+check = B * (T * F)
+
+% (5)
+% Makes a square matrix comprised of T and F
+C = [T F; F T];
+
+%% Cruelty
+meanB = mean(B, "all"); % sums "all" values in matrix B into one scalar
+ % squashes every row down into the sum of all elements
+ % in a given row
+
+meanC = mean(C, 2); % ... or takes the mean of every value along a
+ % dimension (the 2nd dimension), and squashes
+ % the matrix down into one dimension.
+
+%% Odd Types
+% eval one
+evalOne = T + F;
+%{
+ comments:
+ evalOne =
+
+ 2 5
+ 5 8
+
+ This makes sense, considering T was [1 3; 2 4] and F was [1 2; 3 4]
+ It seems like, at the same index on both matrices, the value occupying
+ that spot was taken from both and summed.
+
+%}
+
+% eval 2
+evalTwo = T + 1;
+%{
+ comments:
+ evalTwo =
+
+ 2 4
+ 3 5
+
+ This makes sense, since it seems like 1 was interpreted as a 2x2 matrix
+ occupied only by 1's. The same summing algorithm was applied, and every
+ value occupying T was incremented by 1.
+
+ This could be confusing if we were multiplying instead of dividing,
+ since it would make more intuitive sense to be referring to the
+ identity matrix as 1, rather than a matrix filled wuth 1's.
+
+%}
+
+% eval kicks
+evalKicks = A + C;
+%{
+ comments:
+ evalKicks =
+
+ 2 5 4 6
+ 3 6 6 8
+ 2 4 4 7
+ 4 6 5 8
+
+ A and C are not the same dimensions. A is 1x4 while C is 4x4.
+ It seems that C was sliced by rows in order to let this
+ operation make sense.
+ so for every Row in C, the value corresponding
+ to the matching column in A was added to that value in C.
+ Put nicely: Every row in C (a 1x4 matrix) was summed with A,
+ and then returned to the row index it previously occupied in C proper.
+
+%}
+
+
+%% Not What it Seems...
+for k = [3 5 10 300 1e6]
+ % sets up a 1xk matrix of `k` evenly spaced vals that are [0, 1]
+ genericMatrix = linspace(0, 1, k);
+
+ % squares every val in the matrix, sums all squared vals together, and divs
+ % every element by k.
+ % Essentially -- average value of x^2 from 0 to 1
+ % Also known as power, according to fred
+ genericScalar = sum(genericMatrix.^2) ./ k;
+end
diff --git a/hw1/feedback/Ryan_hw1_012424.m b/hw1/feedback/Ryan_hw1_012424.m
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+++ b/hw1/feedback/Ryan_hw1_012424.m
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+%%%% ECE210-B Matlab Seminar, Homework 1.
+% james ryan, 1/24/24
+
+ % <-.0> style point: please ensure files are named so that
+ % MATLAB (sorry, i'd rather grade in Octave, but this is not
+ % Octave seminar...) can run them! this one had a space in it.
+
+% preamble
+close all;
+clear;
+clc;
+
+%% Scale-'ers
+% (1)
+% Takes the absolute value of
+% sin(pi/3)
+% PLUS
+% j divided by secant of (-5/3)*pi
+a = abs(sin(pi ./ 3) ...
+ + ((1 * j) ./ (sec((-5 * pi) / 3))) ...
+ );
+ % MATLAB interptes complex literals, so you could also write
+ % `1j`.
+
+% (2)
+% n = 3, cubic root
+l = nthroot(8, 3);
+ % could also do `8^(1/3)`, if you're mathematically inclined.
+
+% (3)
+% [1, 2, 3, ..., 79, 80]
+u_mat = 1:80;
+
+% Sums all the terms from u_mat, scales by 2 / 6!, and takes the square root
+u = sqrt((2 / factorial(6)) * sum(u_mat));
+
+% q4
+% takes the square
+% of imaginary value
+% of the floor
+% of the log
+% of the square root of 66
+% taken to the power of 7j
+m = (imag(floor(log(sqrt(66).^(7 * j))))).^2;
+ % since these are scalars, the `.^` is not strictly necessary,
+ % but i see you have a vector bent...
+
+%% Mother...?
+% (1)
+% Makes a 1x4 matrix from a l u & m,
+% and transposes the formed matrix into a 4x1 matrix
+A = [a; l; u; m];
+
+% (2)
+% makes a 2x2 matrix using a l u & m.
+F = [a l; u m];
+
+% (3)
+% takes the non-conjugate transpose of F
+T = F.';
+
+% (4)
+% takes the inverse of the matrix product of T * F
+B = inv(T * F);
+check = B * (T * F)
+ % aaaaaaa no semicolon
+
+% (5)
+% Makes a square matrix comprised of T and F
+C = [T F; F T];
+
+%% Cruelty
+meanB = mean(B, "all"); % sums "all" values in matrix B into one scalar
+ % squashes every row down into the sum of all elements
+ % in a given row
+
+meanC = mean(C, 2); % ... or takes the mean of every value along a
+ % dimension (the 2nd dimension), and squashes
+ % the matrix down into one dimension.
+
+%% Odd Types
+% eval one
+evalOne = T + F;
+%{
+ comments:
+ evalOne =
+
+ 2 5
+ 5 8
+
+ This makes sense, considering T was [1 3; 2 4] and F was [1 2; 3 4]
+ It seems like, at the same index on both matrices, the value occupying
+ that spot was taken from both and summed.
+
+%}
+
+% eval 2
+evalTwo = T + 1;
+%{
+ comments:
+ evalTwo =
+
+ 2 4
+ 3 5
+
+ This makes sense, since it seems like 1 was interpreted as a 2x2 matrix
+ occupied only by 1's. The same summing algorithm was applied, and every
+ value occupying T was incremented by 1.
+
+ This could be confusing if we were multiplying instead of dividing,
+ since it would make more intuitive sense to be referring to the
+ identity matrix as 1, rather than a matrix filled wuth 1's.
+
+%}
+
+% eval kicks
+evalKicks = A + C;
+%{
+ comments:
+ evalKicks =
+
+ 2 5 4 6
+ 3 6 6 8
+ 2 4 4 7
+ 4 6 5 8
+
+ A and C are not the same dimensions. A is 1x4 while C is 4x4.
+ It seems that C was sliced by rows in order to let this
+ operation make sense.
+ so for every Row in C, the value corresponding
+ to the matching column in A was added to that value in C.
+ Put nicely: Every row in C (a 1x4 matrix) was summed with A,
+ and then returned to the row index it previously occupied in C proper.
+
+%}
+ % <+.06> love the extensive commentary.
+
+
+%% Not What it Seems...
+for k = [3 5 10 300 1e6]
+ % sets up a 1xk matrix of `k` evenly spaced vals that are [0, 1]
+ genericMatrix = linspace(0, 1, k);
+
+ % squares every val in the matrix, sums all squared vals together, and divs
+ % every element by k.
+ % Essentially -- average value of x^2 from 0 to 1
+ % Also known as power, according to fred
+ genericScalar = sum(genericMatrix.^2) ./ k;
+end
+ % right, and what *is* that average, in terms of math objects?
diff --git a/hw2/Ryan_hw2_013124.m b/hw2/Ryan_hw2_013124.m
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+%%%% ECE210-B Matlab Seminar, Homework 2.
+% james kit paul thomas ryan murphy, 1/31/24
+% this is james' full name. -A.
+
+% this should run in octave (7.3.0)!
+close all; clear; clc;
+
+%% Spatial Awareness
+% (1)
+A = reshape(0:63, 8, 8).';
+B = 2 .^ A;
+
+% (2)
+B_colvec = reshape(B, [], 1);
+
+% primes(64) = [2, 3, 5... 61]
+% since we're asking for the prime indices from index 1 to index 64,
+% getting all prime numbers within [0, 64] seems fitting
+B_primeinds = B_colvec(primes(length(B_colvec)));
+
+% (3)
+B_primeinds_geometric_mean = prod(B_primeinds) .^ (1 / length(B_primeinds));
+
+% (4)
+% (1, :) -- Grab ALL of row 1 (aka the top row) in the matrix.
+% fliplr -- flip left to right
+A(1, :) = fliplr(A(1, :)); % :)
+
+% (5)
+% Keep all rows, keep all cols minus the end col
+A_lastcol_del = A(1:8, 1:end-1);
+
+
+%% Smallest Addition
+% (1)
+N = 100;
+t = linspace(0, 6.66, N);
+
+erf_integrand = exp(-t .^ 2);
+
+figure;
+title("Plot of e^{-t^2}, (100pts, [0, 6.66])");
+xlabel("time");
+ylabel("e^{-t^2}");
+plot(t, erf_integrand);
+
+% (2)
+% derivative approximation,
+% form some dt being the distance between points on our time axis
+% form the derivative by dividing the formed erf_integrand over dt
+dt = t(2) - t(1);
+dfdt = diff(erf_integrand) ./ dt;
+
+% padding op
+dfdt = dfdt([1 1:end]);
+
+% mean squared error between approximate taken and real erf deriv
+dfdt_anal = -2 .* t .* exp( -t .^ 2);
+msqerr_dfdt = (1 / N) .* mean((dfdt - dfdt_anal) .^ 2);
+
+figure; hold on;
+title('d/dt[e^{-t^2}] -- analytic vs approximate (100pts, [0, 6.66])');
+xlabel('time');
+ylabel('d/dt[-exp(t^2)]');
+
+plot(t, dfdt, 'g-');
+plot(t, dfdt_anal, 'r-');
+
+% (3)
+
+% integral approx
+% take the area under the curve between t(n) and t(n-1)
+% sum the area and scale by 2/sqrt(pi)
+
+figure; hold on;
+title('erf -- analytic vs approximate (100pts, [0, 6.66])');
+xlabel('time');
+ylabel('erf(t)');
+
+% baseline
+erf_anal = (sqrt(pi) ./ 2) .* erf(t);
+
+% cum sum
+erf_approx_cumsum = cumsum(erf_integrand) .* dt;
+msqerr_erf_cumsum = (1 / N) .* mean((erf_approx_cumsum - erf_anal) .^ 2);
+
+% cum trapz
+erf_approx_cumtrapz = cumtrapz(erf_integrand) .* dt;
+msqerr_erf_cumtrapz = (1 / N) .* mean((erf_approx_cumtrapz - erf_anal) .^ 2);
+
+%{
+ The mean-square error of the cumsum op is 5 orders of magnitude larger
+ than the result from the cumtrapz op.
+ Qualitatively, the cumsum approximation (blue line) is noticably further
+ away than the cumtrapz approximation (red line). the cumsum also has
+ a non-negligible y intercept, which is much further than the cumtrapz
+ assumption. However, the cumtrapz is basically on top of the analytic
+ function (green line), which is especially novel considering both ops
+ used the same data set.
+%}
+
+% check
+plot(t, erf_approx_cumsum, "b-");
+plot(t, erf_approx_cumtrapz, "r-");
+plot(t, erf_anal, "g-");
diff --git a/hw2/feedback/Ryan_hw2_013124.m b/hw2/feedback/Ryan_hw2_013124.m
new file mode 100644
index 0000000..2693c60
--- /dev/null
+++ b/hw2/feedback/Ryan_hw2_013124.m
@@ -0,0 +1,117 @@
+%%%% ECE210-B Matlab Seminar, Homework 2.
+% james kit paul thomas ryan murphy, 1/31/24
+% this is james' full name. -A.
+ % <+.01> i needed this.
+
+% this should run in octave (7.3.0)!
+ % <3, turns out it also runs in MATLAB. love the comment,
+ % though.
+close all; clear; clc;
+
+%% Spatial Awareness
+% (1)
+A = reshape(0:63, 8, 8).';
+B = 2 .^ A;
+
+% (2)
+B_colvec = reshape(B, [], 1);
+
+% primes(64) = [2, 3, 5... 61]
+% since we're asking for the prime indices from index 1 to index 64,
+% getting all prime numbers within [0, 64] seems fitting
+B_primeinds = B_colvec(primes(length(B_colvec)));
+ % very clean, like it.
+
+% (3)
+B_primeinds_geometric_mean = prod(B_primeinds) .^ (1 / length(B_primeinds));
+
+% (4)
+% (1, :) -- Grab ALL of row 1 (aka the top row) in the matrix.
+% fliplr -- flip left to right
+A(1, :) = fliplr(A(1, :)); % :)
+
+% (5)
+% Keep all rows, keep all cols minus the end col
+A_lastcol_del = A(1:8, 1:end-1);
+ % that's one way! another is to assign to A (or a copy) with an
+ % empty matrix:
+ % A(:, end) = [];
+
+
+%% Smallest Addition
+% (1)
+N = 100;
+t = linspace(0, 6.66, N);
+
+erf_integrand = exp(-t .^ 2);
+
+figure;
+title("Plot of e^{-t^2}, (100pts, [0, 6.66])");
+xlabel("time");
+ylabel("e^{-t^2}");
+plot(t, erf_integrand);
+
+% (2)
+% derivative approximation,
+% form some dt being the distance between points on our time axis
+% form the derivative by dividing the formed erf_integrand over dt
+dt = t(2) - t(1);
+dfdt = diff(erf_integrand) ./ dt;
+
+% padding op
+dfdt = dfdt([1 1:end]);
+
+% mean squared error between approximate taken and real erf deriv
+dfdt_anal = -2 .* t .* exp( -t .^ 2);
+msqerr_dfdt = (1 / N) .* mean((dfdt - dfdt_anal) .^ 2);
+ % <-.03> (repeated) this will underreport -- `mean()` already
+ % does a divide by the length of the array, so another divide
+ % by N is not necessary.
+
+figure; hold on;
+title('d/dt[e^{-t^2}] -- analytic vs approximate (100pts, [0, 6.66])');
+xlabel('time');
+ylabel('d/dt[-exp(t^2)]');
+
+plot(t, dfdt, 'g-');
+plot(t, dfdt_anal, 'r-');
+
+% (3)
+
+% integral approx
+% take the area under the curve between t(n) and t(n-1)
+% sum the area and scale by 2/sqrt(pi)
+
+figure; hold on;
+title('erf -- analytic vs approximate (100pts, [0, 6.66])');
+xlabel('time');
+ylabel('erf(t)');
+
+% baseline
+erf_anal = (sqrt(pi) ./ 2) .* erf(t);
+
+% cum sum
+erf_approx_cumsum = cumsum(erf_integrand) .* dt;
+msqerr_erf_cumsum = (1 / N) .* mean((erf_approx_cumsum - erf_anal) .^ 2);
+
+% cum trapz
+erf_approx_cumtrapz = cumtrapz(erf_integrand) .* dt;
+msqerr_erf_cumtrapz = (1 / N) .* mean((erf_approx_cumtrapz - erf_anal) .^ 2);
+
+%{
+ The mean-square error of the cumsum op is 5 orders of magnitude larger
+ than the result from the cumtrapz op.
+ Qualitatively, the cumsum approximation (blue line) is noticably further
+ away than the cumtrapz approximation (red line). the cumsum also has
+ a non-negligible y intercept, which is much further than the cumtrapz
+ assumption. However, the cumtrapz is basically on top of the analytic
+ function (green line), which is especially novel considering both ops
+ used the same data set.
+%}
+ % <+.02> love the explanatory comment. `cumtrapz` is
+ % surprisingly good at what it does!
+
+% check
+plot(t, erf_approx_cumsum, "b-");
+plot(t, erf_approx_cumtrapz, "r-");
+plot(t, erf_anal, "g-");
diff --git a/hw2/notes/lesson02.zip b/hw2/notes/lesson02.zip
new file mode 100644
index 0000000..6439b78
Binary files /dev/null and b/hw2/notes/lesson02.zip differ
diff --git a/hw2/notes/numerical_calculus.m b/hw2/notes/numerical_calculus.m
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+++ b/hw2/notes/numerical_calculus.m
@@ -0,0 +1,75 @@
+%% Lesson 2c: Numerical estimation of integrals and derivatives
+% Numerical integration and differentiation are a staple of numerical
+% computing. We will now see how easy these are in MATLAB!
+clc; clear; close all;
+
+%% The diff and cumsum functions
+% Note the lengths of z, zdiff, and zcumsum! (Fencepost problem)
+z = [0 5 -2 3 4];
+zdiff = diff(z);
+zcumsum = cumsum(z);
+
+%% Setup: A simple function
+% Let's start with a simple example: y = x.^2. The domain is N points
+% linearly sampled from lo to hi.
+lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');
+
+%% Numerical (Approximate) Derivatives
+% We can calculate a difference quotient between each pair of (x,y) points
+% using the diff() function.
+dydx = diff(y)./diff(x); % difference quotient
+
+figure();
+plot(x(1:end-1), dydx);
+
+%% Numerical (Approximate) Integrals
+% Now suppose we want to approximate the cumulative integral (Riemann sum)
+% of a function.
+xdiff = diff(x);
+dx = xdiff(1); % spacing between points
+dx = (hi-lo) / (N-1); % alternative to the above (note: N-1)
+
+Y = cumsum(y) * dx; % Riemann sum
+
+figure();
+plot(x, Y);
+
+%% Error metrics: Check how close we are!
+% dydx should be derivative of x.^2 = 2*x
+dydx_actual = 2 * x;
+
+% Y should be \int_{-2}^{x}{t.^2 dt}
+Y_actual = (x .^3 - (-2)^3) / 3;
+
+% Slightly more accurate -- can you figure out why?
+% Y_actual = (x .^3 - (-2-dx)^3) / 3;
+
+% Error metric: MSE (mean square error) or RMSE (root mean square error)
+% Try changing N and see how the error changes. Try this with both the
+% integral and derivative.
+
+% estimated = dydx;
+% actual = dydx_actual(1:end-1);
+estimated = Y;
+actual = Y_actual;
+mse = mean((estimated - actual) .^ 2);
+rmse = rms(estimated - actual);
+
+%% Fundamental Theorem of Calculus
+% Now, use the approximate derivative to get the original function, y back
+% as yhat and plot it. You may need to use/create another variable for
+% the x axis when plotting.
+figure();
+yhat = diff(Y)./diff(x); % differentiate Y
+plot(x(1:end-1), yhat);
+
+figure();
+yhat2 = cumsum(dydx) * dx; % integrate dydx
+plot(x(1:end-1), yhat2);
\ No newline at end of file
diff --git a/hw2/notes/octave-log.txt b/hw2/notes/octave-log.txt
new file mode 100644
index 0000000..0cb7af8
--- /dev/null
+++ b/hw2/notes/octave-log.txt
@@ -0,0 +1,1378 @@
+[cat@lazarus:~/classes/ece210-materials/2024-van-west/lessons/lesson02]
+$ octave
+GNU Octave, version 7.3.0
+octave:2> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:9> close all
+octave:10>
+octave:10>
+octave:10>
+octave:10>
+octave:10>
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');^[[201~octave:10>
+octave:10>
+octave:10>
+octave:10>
+octave:10> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:17> close all
+octave:18>
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');^[[201~octave:18>
+octave:18>
+octave:18>
+octave:18> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:25> close all
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:33>
+octave:33>
+octave:33>
+octave:33>
+octave:33> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:40> close all
+octave:41>
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+octave:53> x = 1:5:10
+x =
+
+ 1 6
+
+octave:54> x = 1:10
+x =
+
+ 1 2 3 4 5 6 7 8 9 10
+
+octave:55> x(1)
+ans = 1
+octave:56> x(1:3)
+ans =
+
+ 1 2 3
+
+octave:57> x(2:7)
+ans =
+
+ 2 3 4 5 6 7
+
+octave:58> x = 0:9
+x =
+
+ 0 1 2 3 4 5 6 7 8 9
+
+octave:59> x = linspace(0, 1, 10)
+x =
+
+ Columns 1 through 8:
+
+ 0 0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778
+
+ Columns 9 and 10:
+
+ 0.8889 1.0000
+
+octave:60> x = linspace(0, 1, 10);
+octave:61> x = linspace(0, 1, 10).';
+octave:62> x
+x =
+
+ 0
+ 0.1111
+ 0.2222
+ 0.3333
+ 0.4444
+ 0.5556
+ 0.6667
+ 0.7778
+ 0.8889
+ 1.0000
+
+octave:63> x(2:6)
+ans =
+
+ 0.1111
+ 0.2222
+ 0.3333
+ 0.4444
+ 0.5556
+
+octave:64> x([1 3 5 7])
+ans =
+
+ 0
+ 0.2222
+ 0.4444
+ 0.6667
+
+octave:65> x(:)
+ans =
+
+ 0
+ 0.1111
+ 0.2222
+ 0.3333
+ 0.4444
+ 0.5556
+ 0.6667
+ 0.7778
+ 0.8889
+ 1.0000
+
+octave:66> x(end)
+ans = 1
+octave:67> x(end:-1:3)
+ans =
+
+ 1.0000
+ 0.8889
+ 0.7778
+ 0.6667
+ 0.5556
+ 0.4444
+ 0.3333
+ 0.2222
+
+octave:68> x(end:-1:1)
+ans =
+
+ 1.0000
+ 0.8889
+ 0.7778
+ 0.6667
+ 0.5556
+ 0.4444
+ 0.3333
+ 0.2222
+ 0.1111
+ 0
+
+octave:69> x(1:2:end)
+ans =
+
+ 0
+ 0.2222
+ 0.4444
+ 0.6667
+ 0.8889
+
+octave:70> M = 1:100;
+octave:71> size(M)
+ans =
+
+ 1 100
+
+octave:72> help reshape
+'reshape' is a built-in function from the file libinterp/corefcn/data.cc
+
+ -- reshape (A, M, N, ...)
+ -- reshape (A, [M N ...])
+ -- reshape (A, ..., [], ...)
+ -- reshape (A, SIZE)
+ Return a matrix with the specified dimensions (M, N, ...) whose
+ elements are taken from the matrix A.
+
+ The elements of the matrix are accessed in column-major order (like
+ Fortran arrays are stored).
+
+ The following code demonstrates reshaping a 1x4 row vector into a
+ 2x2 square matrix.
+
+ reshape ([1, 2, 3, 4], 2, 2)
+ => 1 3
+ 2 4
+
+ Note that the total number of elements in the original matrix
+ ('prod (size (A))') must match the total number of elements in the
+ new matrix ('prod ([M N ...])').
+
+ A single dimension of the return matrix may be left unspecified and
+ Octave will determine its size automatically. An empty matrix ([])
+ is used to flag the unspecified dimension.
+
+ See also: resize, vec, postpad, cat, squeeze.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:73> N1 = reshape(M, round(sqrt(100)), [])
+N1 =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:74> N1
+N1 =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:75> N1(:, :)
+ans =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:76> N1(1:3, :)
+ans =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+
+octave:77> N1(1:3, 2:6)
+ans =
+
+ 11 21 31 41 51
+ 12 22 32 42 52
+ 13 23 33 43 53
+
+octave:78> N1(1:3, 1:3)
+ans =
+
+ 1 11 21
+ 2 12 22
+ 3 13 23
+
+octave:79> N1(end-3:end)
+ans =
+
+ 97 98 99 100
+octave:83>
+Display all 1790 possibilities? (y or n)
+abs
+accumarray
+__accumarray_max__
+__accumarray_min__
+__accumarray_sum__
+accumdim
+octave:84> x = 3
+x = 3
+octave:85> x = [1 2 3];
+octave:86> x = 0:15
+x =
+
+ Columns 1 through 15:
+
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
+
+ Column 16:
+
+ 15
+
+octave:87> y = x.^2
+y =
+
+ Columns 1 through 13:
+
+ 0 1 4 9 16 25 36 49 64 81 100 121 144
+
+ Columns 14 through 16:
+
+ 169 196 225
+
+octave:88> x
+x =
+
+ Columns 1 through 15:
+
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
+
+ Column 16:
+
+ 15
+
+octave:89> xt = x.'
+xt =
+
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+ 11
+ 12
+ 13
+ 14
+ 15
+
+octave:90> x = (1:9).'
+x =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:91> sin(x)
+ans =
+
+ 0.8415
+ 0.9093
+ 0.1411
+ -0.7568
+ -0.9589
+ -0.2794
+ 0.6570
+ 0.9894
+ 0.4121
+
+octave:92> rad2deg(pi/2)
+ans = 90
+octave:93> length(x)
+ans = 9
+octave:94> mean(x)
+ans = 5
+octave:95> x
+x =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:96> mean(sin(x))
+ans = 0.2172
+octave:97>
+^[[200~T = 1e-6; % Sampling period (s)
+t = 0:T:2e-3; % Time (domain/x-axis)
+f0 = 50; % Initial frequency (Hz)
+b = 10e6; % Chirp rate (Hz/s)
+A = 10; % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);^[[201~octave:97>
+octave:97>
+octave:97>
+octave:97>
+octave:97> T = 1e-6; % Sampling period (s)
+t = 0:T:2e-3; % Time (domain/x-axis)
+f0 = 50; % Initial frequency (Hz)
+b = 10e6; % Chirp rate (Hz/s)
+A = 10; % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);
+octave:105> close all
+octave:109> x = linspace(0, 1, 9)
+x =
+
+ Columns 1 through 8:
+
+ 0 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750
+
+ Column 9:
+
+ 1.0000
+
+octave:110> x = linspace(0, 1, 9).'
+x =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:111> x(1)
+ans = 0
+octave:112> x(9)
+ans = 1
+octave:113> x(10)
+error: x(10): out of bound 9 (dimensions are 9x1)
+octave:114> x(5)
+ans = 0.5000
+octave:115> x([1 2 3])
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+
+octave:116> x([1 3 2])
+ans =
+
+ 0
+ 0.2500
+ 0.1250
+
+octave:117> x(1:9)
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:118> x(1:end)
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:119> x(end)
+ans = 1
+octave:120> x(3:end)
+ans =
+
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:121> x(end:-1:1)
+ans =
+
+ 1.0000
+ 0.8750
+ 0.7500
+ 0.6250
+ 0.5000
+ 0.3750
+ 0.2500
+ 0.1250
+ 0
+
+octave:122> 9:-1:1
+ans =
+
+ 9 8 7 6 5 4 3 2 1
+
+octave:123> 9:-.5:1
+ans =
+
+ Columns 1 through 7:
+
+ 9.0000 8.5000 8.0000 7.5000 7.0000 6.5000 6.0000
+
+ Columns 8 through 14:
+
+ 5.5000 5.0000 4.5000 4.0000 3.5000 3.0000 2.5000
+
+ Columns 15 through 17:
+
+ 2.0000 1.5000 1.0000
+
+octave:124> x(:)
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:125> A = [1 2; 3 4]
+A =
+
+ 1 2
+ 3 4
+
+octave:126> A(:)
+ans =
+
+ 1
+ 3
+ 2
+ 4
+
+octave:127> z = 4
+z = 4
+octave:128> z(1)
+ans = 4
+octave:129> inc
+error: 'inc' undefined near line 1, column 1
+octave:130> a++
+error: in x++ or ++x, x must be defined first
+octave:131> ++x
+ans =
+
+ 1.0000
+ 1.1250
+ 1.2500
+ 1.3750
+ 1.5000
+ 1.6250
+ 1.7500
+ 1.8750
+ 2.0000
+
+octave:132> M = reshape(1:100, 10, 10)
+M =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:133> M(1, 1)
+ans = 1
+octave:134> M(10, 10)
+ans = 100
+octave:135> M(end, end)
+ans = 100
+octave:136> M(end, 1)
+ans = 10
+octave:137> M(1, end)
+ans = 91
+octave:138> M(1:3, :)
+ans =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+
+octave:139> M(1:3, 2:5)
+ans =
+
+ 11 21 31 41
+ 12 22 32 42
+ 13 23 33 43
+
+octave:140> M([1 3 5 7], 2:5)
+ans =
+
+ 11 21 31 41
+ 13 23 33 43
+ 15 25 35 45
+ 17 27 37 47
+
+octave:141> x
+x =
+
+ 1.0000
+ 1.1250
+ 1.2500
+ 1.3750
+ 1.5000
+ 1.6250
+ 1.7500
+ 1.8750
+ 2.0000
+
+octave:142> x(1:5)
+ans =
+
+ 1.0000
+ 1.1250
+ 1.2500
+ 1.3750
+ 1.5000
+
+octave:143> x(1:5) = 1:5
+x =
+
+ 1.0000
+ 2.0000
+ 3.0000
+ 4.0000
+ 5.0000
+ 1.6250
+ 1.7500
+ 1.8750
+ 2.0000
+
+octave:144> x(6:end) = x(6:end).^2
+x =
+
+ 1.0000
+ 2.0000
+ 3.0000
+ 4.0000
+ 5.0000
+ 2.6406
+ 3.0625
+ 3.5156
+ 4.0000
+
+octave:145> M
+M =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:146> M(3:6, 3:6) = 0
+M =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 0 0 0 0 63 73 83 93
+ 4 14 0 0 0 0 64 74 84 94
+ 5 15 0 0 0 0 65 75 85 95
+ 6 16 0 0 0 0 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:147> x
+x =
+
+ 1.0000
+ 2.0000
+ 3.0000
+ 4.0000
+ 5.0000
+ 2.6406
+ 3.0625
+ 3.5156
+ 4.0000
+
+octave:148> x = 1:9.'
+x =
+
+ 1 2 3 4 5 6 7 8 9
+
+octave:149> x = (1:9).'
+x =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:150> reshape(x, 3, 3)
+ans =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:151> x = 1:9
+x =
+
+ 1 2 3 4 5 6 7 8 9
+
+octave:152> reshape(x, 3, 3)
+ans =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:153> x
+x =
+
+ 1 2 3 4 5 6 7 8 9
+
+octave:154> xm = reshape(x, 3, 3)
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:155> xm(1, 1)
+ans = 1
+octave:156> xm(1)
+ans = 1
+octave:157> xm(2)
+ans = 2
+octave:158> xm(3)
+ans = 3
+octave:159> xm(4)
+ans = 4
+octave:160> xm
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:161> reshape(xm, 9, 1)
+ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:162> mx
+error: 'mx' undefined near line 1, column 1
+octave:163> xm
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:164> reshape(xm, 9, 1)
+ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:165> reshape(xm, 9, [])
+ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:166> reshape(xm, 3, [])
+ans =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:167> xm
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:168> xm(2, :) = []
+xm =
+
+ 1 4 7
+ 3 6 9
+
+octave:169> magic(4)
+ans =
+
+ 16 2 3 13
+ 5 11 10 8
+ 9 7 6 12
+ 4 14 15 1
+
+octave:170> sum(magic(4))
+ans =
+
+ 34 34 34 34
+
+octave:171> sum(magic(4), 2)
+ans =
+
+ 34
+ 34
+ 34
+ 34
+
+octave:172>
+^[[200~lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');^[[201~octave:172>
+octave:172>
+octave:172>
+octave:172>
+octave:172> lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');
+octave:179> close all
+octave:180> diff(x)
+ans =
+
+ Columns 1 through 7:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 8 through 14:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 15 through 21:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 22 through 28:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 29 through 35:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 36 through 42:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 43 through 49:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 50 through 56:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 57 through 63:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 64 through 70:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 71 through 77:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 78 through 84:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 85 through 91:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 92 through 98:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Column 99:
+
+ 0.040404
+
+octave:181> diff(y)
+ans =
+
+ Columns 1 through 8:
+
+ -0.1600 -0.1567 -0.1535 -0.1502 -0.1469 -0.1437 -0.1404 -0.1371
+
+ Columns 9 through 16:
+
+ -0.1339 -0.1306 -0.1273 -0.1241 -0.1208 -0.1175 -0.1143 -0.1110
+
+ Columns 17 through 24:
+
+ -0.1077 -0.1045 -0.1012 -0.0979 -0.0947 -0.0914 -0.0882 -0.0849
+
+ Columns 25 through 32:
+
+ -0.0816 -0.0784 -0.0751 -0.0718 -0.0686 -0.0653 -0.0620 -0.0588
+
+ Columns 33 through 40:
+
+ -0.0555 -0.0522 -0.0490 -0.0457 -0.0424 -0.0392 -0.0359 -0.0326
+
+ Columns 41 through 48:
+
+ -0.0294 -0.0261 -0.0229 -0.0196 -0.0163 -0.0131 -0.0098 -0.0065
+
+ Columns 49 through 56:
+
+ -0.0033 0 0.0033 0.0065 0.0098 0.0131 0.0163 0.0196
+
+ Columns 57 through 64:
+
+ 0.0229 0.0261 0.0294 0.0326 0.0359 0.0392 0.0424 0.0457
+
+ Columns 65 through 72:
+
+ 0.0490 0.0522 0.0555 0.0588 0.0620 0.0653 0.0686 0.0718
+
+ Columns 73 through 80:
+
+ 0.0751 0.0784 0.0816 0.0849 0.0882 0.0914 0.0947 0.0979
+
+ Columns 81 through 88:
+
+ 0.1012 0.1045 0.1077 0.1110 0.1143 0.1175 0.1208 0.1241
+
+ Columns 89 through 96:
+
+ 0.1273 0.1306 0.1339 0.1371 0.1404 0.1437 0.1469 0.1502
+
+ Columns 97 through 99:
+
+ 0.1535 0.1567 0.1600
+
+octave:182> diff(y)./diff(x)
+ans =
+
+ Columns 1 through 8:
+
+ -3.9596 -3.8788 -3.7980 -3.7172 -3.6364 -3.5556 -3.4747 -3.3939
+
+ Columns 9 through 16:
+
+ -3.3131 -3.2323 -3.1515 -3.0707 -2.9899 -2.9091 -2.8283 -2.7475
+
+ Columns 17 through 24:
+
+ -2.6667 -2.5859 -2.5051 -2.4242 -2.3434 -2.2626 -2.1818 -2.1010
+
+ Columns 25 through 32:
+
+ -2.0202 -1.9394 -1.8586 -1.7778 -1.6970 -1.6162 -1.5354 -1.4545
+
+ Columns 33 through 40:
+
+ -1.3737 -1.2929 -1.2121 -1.1313 -1.0505 -0.9697 -0.8889 -0.8081
+
+ Columns 41 through 48:
+
+ -0.7273 -0.6465 -0.5657 -0.4848 -0.4040 -0.3232 -0.2424 -0.1616
+
+ Columns 49 through 56:
+
+ -0.0808 0 0.0808 0.1616 0.2424 0.3232 0.4040 0.4848
+
+ Columns 57 through 64:
+
+ 0.5657 0.6465 0.7273 0.8081 0.8889 0.9697 1.0505 1.1313
+
+ Columns 65 through 72:
+
+ 1.2121 1.2929 1.3737 1.4545 1.5354 1.6162 1.6970 1.7778
+
+ Columns 73 through 80:
+
+ 1.8586 1.9394 2.0202 2.1010 2.1818 2.2626 2.3434 2.4242
+
+ Columns 81 through 88:
+
+ 2.5051 2.5859 2.6667 2.7475 2.8283 2.9091 2.9899 3.0707
+
+ Columns 89 through 96:
+
+ 3.1515 3.2323 3.3131 3.3939 3.4747 3.5556 3.6364 3.7172
+
+ Columns 97 through 99:
+
+ 3.7980 3.8788 3.9596
+
+octave:183>
+^[[200~figure();
+plot(x(1:end-1), dydx);^[[201~octave:183>
+octave:183>
+octave:183>
+octave:183>
+octave:183> figure();
+plot(x(1:end-1), dydx);
+error: 'dydx' undefined near line 1, column 18
+octave:201> xdiff = diff(x);
+octave:202> dx = xdiff(1);
+octave:203> dx
+dx = 0.040404
+octave:204> int_of_y = sum(y)*dx
+int_of_y = 5.4960
+octave:205> Y = cumsum(y)*dx
+Y =
+
+ Columns 1 through 8:
+
+ 0.1616 0.3168 0.4656 0.6082 0.7448 0.8754 1.0002 1.1193
+
+ Columns 9 through 16:
+
+ 1.2329 1.3411 1.4440 1.5418 1.6345 1.7224 1.8055 1.8841
+
+ Columns 17 through 24:
+
+ 1.9581 2.0277 2.0932 2.1546 2.2120 2.2655 2.3154 2.3617
+
+ Columns 25 through 32:
+
+ 2.4046 2.4442 2.4806 2.5140 2.5445 2.5722 2.5973 2.6199
+
+ Columns 33 through 40:
+
+ 2.6401 2.6581 2.6739 2.6878 2.6998 2.7101 2.7188 2.7261
+
+ Columns 41 through 48:
+
+ 2.7320 2.7368 2.7405 2.7433 2.7453 2.7466 2.7474 2.7479
+
+ Columns 49 through 56:
+
+ 2.7480 2.7480 2.7480 2.7482 2.7486 2.7494 2.7507 2.7527
+
+ Columns 57 through 64:
+
+ 2.7555 2.7592 2.7640 2.7700 2.7772 2.7859 2.7963 2.8083
+
+ Columns 65 through 72:
+
+ 2.8221 2.8380 2.8559 2.8761 2.8987 2.9238 2.9515 2.9820
+
+ Columns 73 through 80:
+
+ 3.0154 3.0518 3.0914 3.1343 3.1806 3.2305 3.2841 3.3415
+
+ Columns 81 through 88:
+
+ 3.4028 3.4683 3.5380 3.6120 3.6905 3.7736 3.8615 3.9542
+
+ Columns 89 through 96:
+
+ 4.0520 4.1549 4.2631 4.3767 4.4959 4.6207 4.7513 4.8878
+
+ Columns 97 through 100:
+
+ 5.0304 5.1793 5.3344 5.4960
+
+octave:206> figure
+octave:207> hold on
+octave:208> plot(x, y)
+octave:209> plot(x, Y)
+octave:210> close all
+octave:211> help cumtrapz
+'cumtrapz' is a function from the file /usr/share/octave/7.3.0/m/general/cumtr
+apz.m
+
+ -- Q = cumtrapz (Y)
+ -- Q = cumtrapz (X, Y)
+ -- Q = cumtrapz (..., DIM)
+ Cumulative numerical integration of points Y using the trapezoidal
+ method.
+
+ 'cumtrapz (Y)' computes the cumulative integral of Y along the
+ first non-singleton dimension. Where 'trapz' reports only the
+ overall integral sum, 'cumtrapz' reports the current partial sum
+ value at each point of Y.
+
+ When the argument X is omitted an equally spaced X vector with unit
+ spacing (1) is assumed. 'cumtrapz (X, Y)' evaluates the integral
+ with respect to the spacing in X and the values in Y. This is
+ useful if the points in Y have been sampled unevenly.
+
+ If the optional DIM argument is given, operate along this
+ dimension.
+
+ Application Note: If X is not specified then unit spacing will be
+ used. To scale the integral to the correct value you must multiply
+ by the actual spacing value (deltaX).
+
+ See also: trapz, cumsum.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:212> help trapz=
+error: help: 'trapz=' not found
+octave:213> help trapz
+'trapz' is a function from the file /usr/share/octave/7.3.0/m/general/trapz.m
+
+ -- Q = trapz (Y)
+ -- Q = trapz (X, Y)
+ -- Q = trapz (..., DIM)
+
+ Numerically evaluate the integral of points Y using the trapezoidal
+ method.
+
+ 'trapz (Y)' computes the integral of Y along the first
+ non-singleton dimension. When the argument X is omitted an equally
+ spaced X vector with unit spacing (1) is assumed. 'trapz (X, Y)'
+ evaluates the integral with respect to the spacing in X and the
+ values in Y. This is useful if the points in Y have been sampled
+ unevenly.
+
+ If the optional DIM argument is given, operate along this
+ dimension.
+
+ Application Note: If X is not specified then unit spacing will be
+ used. To scale the integral to the correct value you must multiply
+ by the actual spacing value (deltaX). As an example, the integral
+ of x^3 over the range [0, 1] is x^4/4 or 0.25. The following code
+ uses 'trapz' to calculate the integral in three different ways.
+
+ x = 0:0.1:1;
+ y = x.^3;
+ ## No scaling
+ q = trapz (y)
+ => q = 2.5250
+ ## Approximation to integral by scaling
+ q * 0.1
+ => 0.25250
+ ## Same result by specifying X
+ trapz (x, y)
+ => 0.25250
+
+ See also: cumtrapz.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:214> trapz(x, y)
+ans = 5.3344
+octave:215> cmutrapz(x, y)
+error: 'cmutrapz' undefined near line 1, column 1
+octave:216> cumtrapz(x, y)
+ans =
+
+ Columns 1 through 8:
+
+ 0 0.1584 0.3104 0.4561 0.5957 0.7293 0.8570 0.9789
+
+ Columns 9 through 16:
+
+ 1.0953 1.2062 1.3118 1.4121 1.5074 1.5977 1.6832 1.7640
+
+ Columns 17 through 24:
+
+ 1.8403 1.9121 1.9797 2.0431 2.1024 2.1579 2.2097 2.2578
+
+ Columns 25 through 32:
+
+ 2.3024 2.3436 2.3816 2.4165 2.4485 2.4776 2.5040 2.5278
+
+ Columns 33 through 40:
+
+ 2.5492 2.5683 2.5852 2.6000 2.6130 2.6241 2.6336 2.6416
+
+ Columns 41 through 48:
+
+ 2.6483 2.6536 2.6579 2.6611 2.6635 2.6652 2.6662 2.6668
+
+ Columns 49 through 56:
+
+ 2.6671 2.6672 2.6672 2.6673 2.6676 2.6682 2.6693 2.6709
+
+ Columns 57 through 64:
+
+ 2.6733 2.6766 2.6808 2.6862 2.6928 2.7008 2.7103 2.7215
+
+ Columns 65 through 72:
+
+ 2.7344 2.7493 2.7662 2.7852 2.8066 2.8305 2.8569 2.8860
+
+ Columns 73 through 80:
+
+ 2.9179 2.9528 2.9908 3.0321 3.0767 3.1248 3.1765 3.2320
+
+ Columns 81 through 88:
+
+ 3.2914 3.3548 3.4223 3.4942 3.5704 3.6512 3.7367 3.8271
+
+ Columns 89 through 96:
+
+ 3.9223 4.0227 4.1282 4.2391 4.3555 4.4774 4.6052 4.7387
+
+ Columns 97 through 100:
+
+ 4.8783 5.0241 5.1760 5.3344
+
+octave:217> Y = cumtrapz(x, y)
+Y =
+
+ Columns 1 through 8:
+
+ 0 0.1584 0.3104 0.4561 0.5957 0.7293 0.8570 0.9789
+
+ Columns 9 through 16:
+
+ 1.0953 1.2062 1.3118 1.4121 1.5074 1.5977 1.6832 1.7640
+
+ Columns 17 through 24:
+
+ 1.8403 1.9121 1.9797 2.0431 2.1024 2.1579 2.2097 2.2578
+
+ Columns 25 through 32:
+
+ 2.3024 2.3436 2.3816 2.4165 2.4485 2.4776 2.5040 2.5278
+
+ Columns 33 through 40:
+
+ 2.5492 2.5683 2.5852 2.6000 2.6130 2.6241 2.6336 2.6416
+
+ Columns 41 through 48:
+
+ 2.6483 2.6536 2.6579 2.6611 2.6635 2.6652 2.6662 2.6668
+
+ Columns 49 through 56:
+
+ 2.6671 2.6672 2.6672 2.6673 2.6676 2.6682 2.6693 2.6709
+
+ Columns 57 through 64:
+
+ 2.6733 2.6766 2.6808 2.6862 2.6928 2.7008 2.7103 2.7215
+
+ Columns 65 through 72:
+
+ 2.7344 2.7493 2.7662 2.7852 2.8066 2.8305 2.8569 2.8860
+
+ Columns 73 through 80:
+
+ 2.9179 2.9528 2.9908 3.0321 3.0767 3.1248 3.1765 3.2320
+
+ Columns 81 through 88:
+
+ 3.2914 3.3548 3.4223 3.4942 3.5704 3.6512 3.7367 3.8271
+
+ Columns 89 through 96:
+
+ 3.9223 4.0227 4.1282 4.2391 4.3555 4.4774 4.6052 4.7387
+
+ Columns 97 through 100:
+
+ 4.8783 5.0241 5.1760 5.3344
+
+octave:218>
diff --git a/hw2/notes/vectorized_operations.m b/hw2/notes/vectorized_operations.m
new file mode 100644
index 0000000..d538411
--- /dev/null
+++ b/hw2/notes/vectorized_operations.m
@@ -0,0 +1,124 @@
+%% Lesson 2a: More vector and matrix operations
+%
+% Objectives:
+% * Understand how to perform vector operations in MATLAB
+% * Understand arithmetic and basic functions in MATLAB
+
+%% Vector operations
+% In lesson 1, we saw how to create a vector with the colon operator and
+% linspace. Now let's perform some operations on them!
+%
+% There are two common classes of operations that you can perform on vectors:
+% element-wise operations (which produce another vector) and aggregate
+% operations (which produce a scalar value). There are also many functions that
+% don't fall under these categories, but these cover many of the common
+% functions.
+
+%% Element-wise operations
+% Many operations that work on scalars (which are really degenerate matrices)
+% also work element-wise on vectors (or matrices).
+x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+
+%% Aggregate operations
+% Another common class of operations produce a single output or statistic about
+% a vector (or matrix).
+length(x); % number of elements in x
+sum(x); % sum of the elements of x
+mean(x); % average of the elements of x
+min(x); % minimum element of x
+diff(x); % difference between adjacent elements of x
+
+%% Exercise 1 : Vector operations
+T = 1e-6; % Sampling period (s)
+t = 0:T:2e-3; % Time (domain/x-axis)
+f0 = 50; % Initial frequency (Hz)
+b = 10e6; % Chirp rate (Hz/s)
+A = 10; % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);
+
+%% Exercise: Numerical calculus
+% See numerical_calculus.m.
+
+%% Basic indexing in MATLAB
+% The process of extracting values from a vector (or matrix) is called
+% "indexing." In MATLAB, indices start at 1, rather than 0 in most languages
+% (in which it is more of an "offset" than a cardinal index).
+
+%% Exercise 2 : Basic indexing
+% The syntax for indexing is "x(indices)", where x is the variable to index,
+% and indices is a scalar or a vector of indices. There are many variations on
+% this. Note that indices can be any vector
+x(1); % first element of x
+x(1:3); % elements 1, 2 and 3 (inclusive!)
+x(1:length(x)); % all elements in x
+x(1:end); % same as above
+x(:); % same as above
+x(end); % last element of x
+x(3:end); % all elements from 3 onwards
+x([1,3,5]); % elements 1, 3, and 5 from x
+
+x(1:2:end); % all odd-indexed elements of x
+ind = 1:2:length(x);
+x(ind); % same as the previous example
+
+%% Exercises to improve your understanding
+% Take some time to go through these on your own.
+x([1,2,3]); % Will these produce the same result?
+x([3,2,1]);
+
+x2 = 1:5;
+x2(6); % What will this produce?
+x2(0); % What will this produce?
+x2(1:1.5:4); % What will this produce?
+ind = 1:1.5:4;
+x2(ind); % What will this produce?
+
+z = 4;
+z(1); % What will this produce?
+
+%% Matrix operations
+% Matrices is closely related to vectors, and we have also explored some matrix
+% operations last class. This class, we are going to explore functions that are
+% very useful but are hard to grasp for beginners, namely reshape, meshgrid,
+% row-wise and column-wise operations.
+
+%% Reshape
+% Change a matrix from one shape to another. The new shape has to have the same
+% number of elements as the original shape.
+%
+% When you are reshaping an array / matrix, the first dimension is filled
+% first, and then the second dimension, so on and so forth. I.e., elements
+% start filling down columns, then rows, etc.
+M = 1:100;
+N1 = reshape(M,2,2,[]); % It would create a 2*2*25 matrix
+N2 = reshape(M,[2,2,25]); % Same as N1
+N2(:,:,25); % Gives you 97,98,99,100
+N2(:,1,25); % Gives you 97 and 98
+
+%% Row-wise / Column-wise operations
+% Vector operations can also be performed on matrices. We can perform a vector
+% operation on each row/column of a matrix, or on a particular row/column by
+% indexing.
+H = magic(4); % create the magical matrix H
+sum(H,1); % column wise sum, note that this is a row vector(default)
+fliplr(H); % flip H from left to right
+flipud(H); % flip H upside down
+H(1,:) = fliplr(H(1,:)); % flip only ONE row left to right
+H(1,:) = []; % delete the first row
+
+%% Exercise 7 : Matrix Operations
+H2 = randi(20,4,5); % random 4x5 matrix with integers from 1 to 20
+sum(H2(:,2));
+mean(H2(3,:));
+C = reshape(H2,2,2,5);
+C(2,:,:) = [];
diff --git a/hw3/Ryan_HW3_021424.m b/hw3/Ryan_HW3_021424.m
new file mode 100644
index 0000000..9783c56
--- /dev/null
+++ b/hw3/Ryan_HW3_021424.m
@@ -0,0 +1,181 @@
+%%%% ECE-210-B HW 3 - Plotting
+% James Ryan, 02/14/24
+
+% preamble
+close all; clear; clc;
+
+%% Two Dimensions
+% (1)
+x = linspace(0, 2*pi, 50);
+f = sin(x);
+
+figure;
+plot(x, f, "g");
+xlim([0 2*pi]); % plot only within x's range
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+xlabel("x");
+ylabel("sin(x)");
+title('Plot of sin(x) over 50 evenly spaced points between [0, 2\pi]');
+
+% (2)
+x; % 50 evenly spaced pts between [0, 2pi]
+g = cos(2.*x);
+
+figure;
+hold on;
+plot(x, f, "g");
+plot(x, g, "b");
+hold off;
+xlim([0 2*pi]); % plot only within x's range
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+
+xlabel("x");
+ylabel("y");
+legend( 'y = sin(x)', ...
+ 'y = cos(2x)');
+title('knock knock! whose there? sin! sine who? $\pm\sqrt{\frac{1 + cos2\theta}{2}}$', "interpreter", "latex");
+
+% (3)
+
+figure;
+x; % 50 evenly spaced pts between [0, 2pi]
+
+% top plot - f = sin(x)
+subplot(2,1,1);
+stem(x, f, "r*");
+grid on
+xlim([0 2*pi]);
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+xlabel("t [s]"); % we're pretending its time now :)
+ylabel("Position [d]");
+title('Position of a controlled laser pointer, relative to origin location');
+%subtitle('Sampled 50 times over 2\pi seconds.');
+
+
+% bottom plot - g = cos(2x)
+% changing cos(2x)'s amplitude; computing amplitude ratio in d
+rand_amp = 100 * rand;
+g = rand_amp .* g;
+amplitude_ratio_db = 20.*log10(rand_amp);
+
+subplot(2,1,2);
+stem(x, g, "color", "#D95319", "Marker", "+", "MarkerEdgeColor", "#D95319", "MarkerFaceColor", "#D95319");
+grid on
+xlim([0 2*pi]);
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+yticks(linspace(-rand_amp, rand_amp, 5));
+
+xlabel("t [s]");
+ylabel("Position [d]");
+title(sprintf('Pet cats vertical position while seeking laser pointer. \n Position Ratio (vs sin) [dB]: %d' ...
+ ,amplitude_ratio_db));
+%subtitle('Sampled 50 times over 2\pi seconds.');
+
+
+% (4)
+
+pointer = imread("./pointer.jpg");
+%{
+ returns a 720x1280x3 matrix, corresponding to
+ 720 rows
+ 1280 columns
+ 3 color channels per (row,col) pair
+ to form a complete color image, in matrix form.
+ Each entry in the matrix is stored as uint8, or
+ an 8-bit integer, implying colors are specified as
+ hex codes (eg #RRGGBB - specify the Red, Green, and
+ Blue values in HEX, 8 bits per channel, 3 channels).
+%}
+
+% Invert color channels
+new_pointer = 255 - pointer;
+
+figure;
+imshow(pointer);
+figure;
+imshow(new_pointer);
+
+%% Three Dimensions
+% (1)
+x = 0:.01:1;
+y = x;
+z = exp(-x.^2 - y.^2);
+
+figure;
+
+% "Create this plot
+% three times
+% in three different subplots
+% using plot3..., scatter3, surf, *and* mesh (4 tools?)"
+
+% I'm making 4 plots, because you said "and", meaning all the tools.
+
+% plot3
+subplot(2,2,1);
+plot3(x,y,z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (PLOT3)');
+legend('z = exp(-x^2 - y^2)');
+daspect([1 1 1]);
+view([-2 2 1]);
+grid on;
+
+
+% scatter3
+subplot(2,2,2);
+scatter3(x,y,z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (SCATTER)');
+legend('z = exp(-x^2 - y^2)');
+daspect([1 1 1]);
+view([-2 2 1]);
+grid on;
+
+% surf
+[X, Y] = meshgrid(x);
+Z = exp(-X.^2 - Y.^2);
+
+subplot(2,2,3);
+surf(X,Y,Z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (SURF)');
+legend('z = exp(-x^2 - y^2)');
+grid on;
+view([-2 5 1.3]);
+
+% mesh
+subplot(2,2,4);
+mesh(X,Y,Z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (MESH)');
+legend('z = exp(-x^2 - y^2)');
+grid on;
+view([-2 5 1.3]);
+
+% (2)
+t = 0:0.01:1;
+x = t.*cos(27.*t);
+y = t.*sin(27.*t);
+z = t;
+
+figure;
+plot3(x,y,t);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('Wheeee!');
+grid on;
+daspect([1 1 1]);
+view([90 90 20]);
diff --git a/hw3/feedback/Ryan_HW3_021424.m b/hw3/feedback/Ryan_HW3_021424.m
new file mode 100644
index 0000000..265ff92
--- /dev/null
+++ b/hw3/feedback/Ryan_HW3_021424.m
@@ -0,0 +1,195 @@
+%%%% ECE-210-B HW 3 - Plotting
+% James Ryan, 02/14/24
+
+% preamble
+close all; clear; clc;
+
+%% Two Dimensions
+% (1)
+x = linspace(0, 2*pi, 50);
+f = sin(x);
+
+figure;
+plot(x, f, "g");
+xlim([0 2*pi]); % plot only within x's range
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+xlabel("x");
+ylabel("sin(x)");
+title('Plot of sin(x) over 50 evenly spaced points between [0, 2\pi]');
+
+% (2)
+x; % 50 evenly spaced pts between [0, 2pi]
+g = cos(2.*x);
+
+figure;
+hold on;
+plot(x, f, "g");
+plot(x, g, "b");
+hold off;
+xlim([0 2*pi]); % plot only within x's range
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+
+xlabel("x");
+ylabel("y");
+legend( 'y = sin(x)', ...
+ 'y = cos(2x)');
+ % <-.02> (style) one thing about tabs: they don't always line
+ % up the same way on other people's machines. this indentation
+ % looks... strange to me, with my 8-space tabstops.
+title('knock knock! whose there? sin! sine who? $\pm\sqrt{\frac{1 + cos2\theta}{2}}$', "interpreter", "latex");
+ % <+.02> nice.
+
+% (3)
+
+figure;
+x; % 50 evenly spaced pts between [0, 2pi]
+
+% top plot - f = sin(x)
+subplot(2,1,1);
+stem(x, f, "r*");
+grid on
+xlim([0 2*pi]);
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+xlabel("t [s]"); % we're pretending its time now :)
+ylabel("Position [d]");
+title('Position of a controlled laser pointer, relative to origin location');
+subtitle('Sampled 50 times over 2\pi seconds.');
+ % <+.01> the theme here really gets me.
+
+
+% bottom plot - g = cos(2x)
+% changing cos(2x)'s amplitude; computing amplitude ratio in d
+rand_amp = 100 * rand;
+g = rand_amp .* g;
+amplitude_ratio_db = 20.*log10(rand_amp);
+
+subplot(2,1,2);
+stem(x, g, "color", "#D95319", "Marker", "+", "MarkerEdgeColor", "#D95319", "MarkerFaceColor", "#D95319");
+ % (style) these are very long lines...
+grid on
+xlim([0 2*pi]);
+xticks([0:pi/2:2*pi]);
+xticklabels({'0','\pi/2','\pi','3\pi/2','2\pi'})
+yticks(linspace(-rand_amp, rand_amp, 5));
+ % usually ticks are nicely readable numbers for, well,
+ % readability.
+
+xlabel("t [s]");
+ylabel("Position [d]");
+title(sprintf('Pet cats vertical position while seeking laser pointer. \n Position Ratio (vs sin) [dB]: %d' ...
+ ,amplitude_ratio_db));
+ % (interesting place for that comma)
+subtitle('Sampled 50 times over 2\pi seconds.');
+
+
+% (4)
+
+pointer = imread("./pointer.jpg");
+ % <-.05> image was not submitted! i went and found one...
+%{
+ returns a 720x1280x3 matrix, corresponding to
+ 720 rows
+ 1280 columns
+ 3 color channels per (row,col) pair
+ to form a complete color image, in matrix form.
+ Each entry in the matrix is stored as uint8, or
+ an 8-bit integer, implying colors are specified as
+ hex codes (eg #RRGGBB - specify the Red, Green, and
+ Blue values in HEX, 8 bits per channel, 3 channels).
+%}
+
+% Invert color channels
+new_pointer = 255 - pointer;
+
+figure;
+imshow(pointer);
+figure;
+imshow(new_pointer);
+
+%% Three Dimensions
+% (1)
+x = 0:.01:1;
+y = x;
+z = exp(-x.^2 - y.^2);
+
+figure;
+
+% "Create this plot
+% three times
+% in three different subplots
+% using plot3..., scatter3, surf, *and* mesh (4 tools?)"
+
+% I'm making 4 plots, because you said "and", meaning all the tools.
+ % that is indeed what i meant for you to do. good catch.
+
+% plot3
+subplot(2,2,1);
+plot3(x,y,z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (PLOT3)');
+legend('z = exp(-x^2 - y^2)');
+daspect([1 1 1]);
+view([-2 2 1]);
+grid on;
+
+
+% scatter3
+subplot(2,2,2);
+scatter3(x,y,z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (SCATTER)');
+legend('z = exp(-x^2 - y^2)');
+daspect([1 1 1]);
+view([-2 2 1]);
+grid on;
+ % <-.00> i was not totally clear here, so i'll let this slide,
+ % but i meant plot the surface itself using `scatter` and
+ % `plot3`.
+
+% surf
+[X, Y] = meshgrid(x);
+Z = exp(-X.^2 - Y.^2);
+
+subplot(2,2,3);
+surf(X,Y,Z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (SURF)');
+legend('z = exp(-x^2 - y^2)');
+grid on;
+view([-2 5 1.3]);
+
+% mesh
+subplot(2,2,4);
+mesh(X,Y,Z);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('z = exp(-x^2 - y^2) (MESH)');
+legend('z = exp(-x^2 - y^2)');
+grid on;
+view([-2 5 1.3]);
+
+% (2)
+t = 0:0.01:1;
+x = t.*cos(27.*t);
+y = t.*sin(27.*t);
+z = t;
+
+figure;
+plot3(x,y,t);
+xlabel("x");
+ylabel("y");
+zlabel("z");
+title('Wheeee!');
+grid on;
+daspect([1 1 1]);
+view([90 90 20]);
diff --git a/hw3/pointer.jpg b/hw3/pointer.jpg
new file mode 100644
index 0000000..9809854
Binary files /dev/null and b/hw3/pointer.jpg differ
diff --git a/hw4/Ryan_HW4_022124.m b/hw4/Ryan_HW4_022124.m
new file mode 100644
index 0000000..cb2eac9
--- /dev/null
+++ b/hw4/Ryan_HW4_022124.m
@@ -0,0 +1,221 @@
+%%%% ECE-210-B HW 4 - Functions and Objects
+% James Ryan, 02/21/24
+
+% preamble
+close all; clear; clc;
+
+%% My Sinc Is Broken
+% (1)
+% see deriv.m
+x = linspace(-5, 5, 1e3);
+sinc_x = sinc(x);
+deriv_sinc = deriv(sinc_x, x);
+
+% see antideriv.m
+% approximates an integral of a given function using `cumtrapz`
+int_sinc = antideriv(sinc_x, x);
+
+% (2)
+% see switchsign.m
+
+% (3)
+[extrema_sinc_x, extrema_x] = extrema(sinc_x, x);
+[inflect_sinc_x, inflect_x] = inflections(sinc_x, x);
+
+% (4)
+figure;
+plot(x, sinc_x, x, int_sinc, x, deriv_sinc, ...
+ extrema_x, extrema_sinc_x, 'r*', ...
+ inflect_x, inflect_sinc_x, 'bo');
+ylim([-2 2.45]);
+xlim([-5, 5]);
+ylabel("y-axis");
+xlabel("x-axis");
+grid on;
+title( {'y = sinc(x) plotted over 1000 evenly spaced pts between x \epsilon [-5,5]', ...
+ 'Plotted alongside its first derivative and antiderivative.'});
+legend( 'y = sinc(x)',"$\int [sinc(x)] dx$",'$\frac{d}{dx} [sinc(x)]$', ...
+ 'Points of Extrema', 'Inflection Points', ...
+ "Interpreter","latex");
+
+%% Objectification
+% (1)
+% Matlab Classic Noteworthy Logo (TM) (C) (Not-for-individual-sale)
+L = 160*membrane(1,100);
+
+f = figure;
+ax = axes;
+
+s = surface(L);
+s.EdgeColor = 'none';
+view(3)
+
+ax.XLim = [1 201];
+ax.YLim = [1 201];
+ax.ZLim = [-53.4 160];
+
+ax.CameraPosition = [-145.5 -229.7 283.6];
+ax.CameraTarget = [77.4 60.2 63.9];
+ax.CameraUpVector = [0 0 1];
+ax.CameraViewAngle = 36.7;
+
+ax.Position = [0 0 1 1];
+ax.DataAspectRatio = [1 1 .9];
+
+l1 = light;
+l1.Position = [160 400 80];
+l1.Style = 'local';
+l1.Color = [0 0.8 0.8];
+
+l2 = light; % l2 = l1;
+l2.Position = [.5 -1 .4];
+l2.Color = [0.8 0.8 0];
+
+s.FaceColor = [0.9 0.2 0.2];
+
+s.FaceLighting = 'gouraud';
+s.AmbientStrength = 0.3;
+s.DiffuseStrength = 0.6;
+s.BackFaceLighting = 'lit';
+
+s.SpecularStrength = 1;
+s.SpecularColorReflectance = 1;
+s.SpecularExponent = 7;
+
+axis off
+f.Color = 'black';
+
+% (2)
+% Copyobj is not cooperating. Time to be lazy!
+% Matlab Classic Noteworthy Logo (TM) (C) (Not-for-individual-sale)
+
+f_dupe = figure;
+ax_dupe = axes;
+
+s_dupe = surface(L);
+s_dupe.EdgeColor = 'none';
+view(3)
+
+ax_dupe.XLim = [1 201];
+ax_dupe.YLim = [1 201];
+ax_dupe.ZLim = [-53.4 160];
+
+ax_dupe.CameraPosition = [-145.5 -229.7 283.6];
+ax_dupe.CameraTarget = [77.4 60.2 63.9];
+ax_dupe.CameraUpVector = [0 0 1];
+ax_dupe.CameraViewAngle = 36.7;
+
+ax_dupe.Position = [0 0 1 1];
+ax_dupe.DataAspectRatio = [1 1 .9];
+
+l3 = light;
+l3.Position = [160 400 80];
+l3.Style = 'local';
+l3.Color = [0 0.8 0.8];
+
+l4 = light; % l2 = l1;
+l4.Position = [.5 -1 .4];
+l4.Color = [0.8 0.8 0];
+
+s_dupe.FaceColor = [0.9 0.2 0.2];
+
+s_dupe.FaceLighting = 'gouraud';
+s_dupe.AmbientStrength = 0.3;
+s_dupe.DiffuseStrength = 0.6;
+s_dupe.BackFaceLighting = 'lit';
+
+s_dupe.SpecularStrength = 1;
+s_dupe.SpecularColorReflectance = 1;
+s_dupe.SpecularExponent = 7;
+
+axis off
+f_dupe.Color = 'black';
+
+% James's Changes
+% Change one - light at the origin!
+l4.Position = [0 0 0];
+l4.Color = [1 1 1];
+% This made the logo significantly more purple!
+
+% Change two - colormap
+colormap winter;
+% I didnt really notice a change with this one.
+
+% Change three - function
+s_dupe.FaceLighting = 'flat';
+% I changed the surface! It has different face lighting. Thats changing
+% the function, right?
+% It helped emphasize the purple-maroon which possesses the shape at the moment
+
+% Change four - backround
+f_dupe.Color = 'red';
+% Hard to miss - the entire background is now red!
+
+% (3)
+figure;
+
+% Matlab <TM>
+ax_former = subplot(2,1,1);
+copyobj(ax.Children, ax_former);
+title('Matlab <TM>');
+
+ax_former.XLim = [1 201];
+ax_former.YLim = [1 201];
+ax_former.ZLim = [-53.4 160];
+ax_former.XTick = [];
+ax_former.YTick = [];
+ax_former.ZTick = [];
+
+ax_former.CameraPosition = [-145.5 -229.7 283.6];
+ax_former.CameraTarget = [77.4 60.2 63.9];
+ax_former.CameraUpVector = [0 0 1];
+ax_former.CameraViewAngle = 36.7;
+
+l5 = light;
+l5.Position = [160 400 80];
+l5.Style = 'local';
+l5.Color = [0 0.8 0.8];
+
+l6 = light; % l2 = l1;
+l6.Position = [.5 -1 .4];
+l6.Color = [0.8 0.8 0];
+
+% James Plot
+ax_latter = subplot(2,1,2);
+copyobj(ax_dupe.Children, ax_latter);
+title('Conspicuous Copycat');
+
+ax_latter.XLim = [1 201];
+ax_latter.YLim = [1 201];
+ax_latter.ZLim = [-53.4 160];
+ax_latter.XTick = [];
+ax_latter.YTick = [];
+ax_latter.ZTick = [];
+
+
+ax_latter.CameraPosition = [-145.5 -229.7 283.6];
+ax_latter.CameraTarget = [77.4 60.2 63.9];
+ax_latter.CameraUpVector = [0 0 1];
+ax_latter.CameraViewAngle = 36.7;
+
+l7 = light;
+l7.Position = [160 400 80];
+l7.Style = 'local';
+l7.Color = [0 0.8 0.8];
+
+l8 = light; % l2 = l1;
+l8.Position = [.5 -1 .4];
+l8.Color = [0.8 0.8 0];
+
+% James's Changes
+% Change one - light at the origin!
+l8.Position = [0 0 0];
+l8.Color = [1 1 1];
+% This made the logo significantly more purple!
+
+% (4)
+get(f)
+f.PaperOrientation = 'landscape';
+f.PaperType = 'a4';
+f.PaperUnits = 'centimeters';
+% My figure now is configured to print in Europe! And its sideways.
diff --git a/hw4/antideriv.m b/hw4/antideriv.m
new file mode 100644
index 0000000..2f5ccdd
--- /dev/null
+++ b/hw4/antideriv.m
@@ -0,0 +1,3 @@
+function int_Y_dx = antideriv(y, x)
+ int_Y_dx = ( cumtrapz(y) .* (x(2) - x(1)) );
+end
diff --git a/hw4/deriv.m b/hw4/deriv.m
new file mode 100644
index 0000000..709929b
--- /dev/null
+++ b/hw4/deriv.m
@@ -0,0 +1,4 @@
+function dy = deriv(y, x)
+ dy = diff(y) ./ (x(2) - x(1));
+ dy = dy([1 1:end]);
+end
diff --git a/hw4/extrema.m b/hw4/extrema.m
new file mode 100644
index 0000000..9f68056
--- /dev/null
+++ b/hw4/extrema.m
@@ -0,0 +1,5 @@
+function [y_ext, x_ext] = extrema(y, x)
+ y_switch = switchsign( deriv(y, x) );
+ y_ext = nonzeros(y_switch .* y);
+ x_ext = nonzeros(y_switch .* x);
+end
diff --git a/hw4/feedback/Ryan_HW4_022124_feedback.m b/hw4/feedback/Ryan_HW4_022124_feedback.m
new file mode 100644
index 0000000..0fc9207
--- /dev/null
+++ b/hw4/feedback/Ryan_HW4_022124_feedback.m
@@ -0,0 +1,231 @@
+%%%% ECE-210-B HW 4 - Functions and Objects
+% James Ryan, 02/21/24
+
+% preamble
+close all; clear; clc;
+
+%% My Sinc Is Broken
+% (1)
+% see deriv.m
+x = linspace(-5, 5, 1e3);
+sinc_x = sinc(x);
+deriv_sinc = deriv(sinc_x, x);
+
+% see antideriv.m
+% approximates an integral of a given function using `cumtrapz`
+int_sinc = antideriv(sinc_x, x);
+
+% (2)
+% see switchsign.m
+
+% (3)
+[extrema_sinc_x, extrema_x] = extrema(sinc_x, x);
+[inflect_sinc_x, inflect_x] = inflections(sinc_x, x);
+
+% (4)
+figure;
+plot(x, sinc_x, x, int_sinc, x, deriv_sinc, ...
+ extrema_x, extrema_sinc_x, 'r*', ...
+ inflect_x, inflect_sinc_x, 'bo');
+ylim([-2 2.45]);
+xlim([-5, 5]);
+ylabel("y-axis");
+xlabel("x-axis");
+grid on;
+title( {'y = sinc(x) plotted over 1000 evenly spaced pts between x \epsilon [-5,5]', ...
+ 'Plotted alongside its first derivative and antiderivative.'});
+legend( 'y = sinc(x)',"$\int [sinc(x)] dx$",'$\frac{d}{dx} [sinc(x)]$', ...
+ 'Points of Extrema', 'Inflection Points', ...
+ "Interpreter","latex");
+ % <+.02> this is a very pretty legend.
+
+%% Objectification
+% (1)
+% Matlab Classic Noteworthy Logo (TM) (C) (Not-for-individual-sale)
+L = 160*membrane(1,100);
+
+f = figure;
+ax = axes;
+
+s = surface(L);
+s.EdgeColor = 'none';
+view(3)
+
+ax.XLim = [1 201];
+ax.YLim = [1 201];
+ax.ZLim = [-53.4 160];
+
+ax.CameraPosition = [-145.5 -229.7 283.6];
+ax.CameraTarget = [77.4 60.2 63.9];
+ax.CameraUpVector = [0 0 1];
+ax.CameraViewAngle = 36.7;
+
+ax.Position = [0 0 1 1];
+ax.DataAspectRatio = [1 1 .9];
+
+l1 = light;
+l1.Position = [160 400 80];
+l1.Style = 'local';
+l1.Color = [0 0.8 0.8];
+
+l2 = light; % l2 = l1;
+l2.Position = [.5 -1 .4];
+l2.Color = [0.8 0.8 0];
+
+s.FaceColor = [0.9 0.2 0.2];
+
+s.FaceLighting = 'gouraud';
+s.AmbientStrength = 0.3;
+s.DiffuseStrength = 0.6;
+s.BackFaceLighting = 'lit';
+
+s.SpecularStrength = 1;
+s.SpecularColorReflectance = 1;
+s.SpecularExponent = 7;
+
+axis off
+f.Color = 'black';
+
+% (2)
+% Copyobj is not cooperating. Time to be lazy!
+% Matlab FOR MS-DOS! Noteworthy Logo (TM) (C) (Not-for-individual-sale)
+disk_operating_system = 2;
+L_disk_operating_system = disk_operating_system .* L;
+
+
+f_dupe = figure;
+ax_dupe = axes;
+
+s_dupe = surface(L_disk_operating_system);
+s_dupe.EdgeColor = 'none';
+view(3)
+
+ax_dupe.XLim = disk_operating_system .* [1 201];
+ax_dupe.YLim = disk_operating_system .* [1 201];
+ax_dupe.ZLim = disk_operating_system .* [-53.4 160];
+
+ax_dupe.CameraPosition = disk_operating_system .* [-145.5 -229.7 283.6];
+ax_dupe.CameraTarget = disk_operating_system .* [77.4 60.2 63.9];
+ax_dupe.CameraUpVector = [0 0 1];
+ax_dupe.CameraViewAngle = 36.7;
+
+ax_dupe.Position = [0 0 1 1];
+ax_dupe.DataAspectRatio = [1 1 .9];
+
+l3 = light;
+l3.Position = [160 400 80];
+l3.Style = 'local';
+l3.Color = [0 0.8 0.8];
+
+l4 = light; % l2 = l1;
+l4.Position = [.5 -1 .4];
+l4.Color = [0.8 0.8 0];
+
+s_dupe.FaceColor = [0.9 0.2 0.2];
+
+s_dupe.FaceLighting = 'gouraud';
+s_dupe.AmbientStrength = 0.3;
+s_dupe.DiffuseStrength = 0.6;
+s_dupe.BackFaceLighting = 'lit';
+
+s_dupe.SpecularStrength = 1;
+s_dupe.SpecularColorReflectance = 1;
+s_dupe.SpecularExponent = 7;
+
+axis off
+f_dupe.Color = 'black';
+
+% James's Changes
+% Change one - light at the origin!
+l4.Position = [0 0 0];
+l4.Color = [1 1 1];
+% This made the logo significantly more purple!
+
+% Change two - colormap
+colormap winter;
+% I didnt really notice a change with this one.
+
+% Change three - function
+s_dupe.FaceLighting = 'flat';
+% I changed the surface! It has different face lighting. Thats changing
+% the function, right?
+% It helped emphasize the purple-maroon which possesses the shape at the moment
+
+% Change four - backround
+f_dupe.Color = 'red';
+% Hard to miss - the entire background is now red!
+
+% (3)
+f_both = figure;
+
+% Matlab <TM>
+ax_former = subplot(1,2,1);
+copyobj(ax.Children, ax_former);
+title('Matlab <TM>');
+
+ax_former.XLim = [1 201];
+ax_former.YLim = [1 201];
+ax_former.ZLim = [-53.4 160];
+ax_former.XTick = [];
+ax_former.YTick = [];
+ax_former.ZTick = [];
+
+ax_former.CameraPosition = [-145.5 -229.7 283.6];
+ax_former.CameraTarget = [77.4 60.2 63.9];
+ax_former.CameraUpVector = [0 0 1];
+ax_former.CameraViewAngle = 36.7;
+
+l5 = light;
+l5.Position = [160 400 80];
+l5.Style = 'local';
+l5.Color = [0 0.8 0.8];
+
+l6 = light; % l2 = l1;
+l6.Position = [.5 -1 .4];
+l6.Color = [0.8 0.8 0];
+
+% James Plot Matlab for DOS
+ax_latter = subplot(1,2,2);
+copyobj(ax_dupe.Children, ax_latter);
+title('Conspicuous Copycat');
+ % <+.02> MATLAB for DOS lol
+
+% disk operating system, or DOS, is 2 for short :)
+
+ax_latter.XLim = disk_operating_system .* [1 201];
+ax_latter.YLim = disk_operating_system .* [1 201];
+ax_latter.ZLim = disk_operating_system .* [-53.4 160];
+ax_latter.XTick = [];
+ax_latter.YTick = [];
+ax_latter.ZTick = [];
+
+
+ax_latter.CameraPosition = disk_operating_system .* [-145.5 -229.7 283.6];
+ax_latter.CameraTarget = disk_operating_system .* [77.4 60.2 63.9];
+ax_latter.CameraUpVector = [0 0 1];
+ax_latter.CameraViewAngle = 36.7;
+
+l7 = light;
+l7.Position = [160 400 80];
+l7.Style = 'local';
+l7.Color = [0 0.8 0.8];
+
+l8 = light; % l2 = l1;
+l8.Position = [.5 -1 .4];
+l8.Color = [0.8 0.8 0];
+
+f_both.Color = 'green';
+
+% James's Changes
+% Change one - light at the origin!
+l8.Position = [0 0 0];
+l8.Color = [1 1 1];
+% This made the logo significantly more purple!
+
+% (4)
+%get(f)
+f.PaperOrientation = 'landscape';
+f.PaperType = 'a4';
+f.PaperUnits = 'centimeters';
+% My figure now is configured to print in Europe! And its sideways.
+ % <+.01> lmao, incredible
diff --git a/hw4/feedback/antideriv.m b/hw4/feedback/antideriv.m
new file mode 100644
index 0000000..2f5ccdd
--- /dev/null
+++ b/hw4/feedback/antideriv.m
@@ -0,0 +1,3 @@
+function int_Y_dx = antideriv(y, x)
+ int_Y_dx = ( cumtrapz(y) .* (x(2) - x(1)) );
+end
diff --git a/hw4/feedback/deriv.m b/hw4/feedback/deriv.m
new file mode 100644
index 0000000..709929b
--- /dev/null
+++ b/hw4/feedback/deriv.m
@@ -0,0 +1,4 @@
+function dy = deriv(y, x)
+ dy = diff(y) ./ (x(2) - x(1));
+ dy = dy([1 1:end]);
+end
diff --git a/hw4/feedback/extrema.m b/hw4/feedback/extrema.m
new file mode 100644
index 0000000..9f68056
--- /dev/null
+++ b/hw4/feedback/extrema.m
@@ -0,0 +1,5 @@
+function [y_ext, x_ext] = extrema(y, x)
+ y_switch = switchsign( deriv(y, x) );
+ y_ext = nonzeros(y_switch .* y);
+ x_ext = nonzeros(y_switch .* x);
+end
diff --git a/hw4/feedback/inflections.m b/hw4/feedback/inflections.m
new file mode 100644
index 0000000..ad23a8b
--- /dev/null
+++ b/hw4/feedback/inflections.m
@@ -0,0 +1,5 @@
+function [y_inf, x_inf] = inflections(y, x)
+ y_switch = switchsign( deriv(deriv(y, x), x) );
+ y_inf = nonzeros(y_switch .* y);
+ x_inf = nonzeros(y_switch .* x);
+end
diff --git a/hw4/feedback/switchsign.m b/hw4/feedback/switchsign.m
new file mode 100644
index 0000000..16677d6
--- /dev/null
+++ b/hw4/feedback/switchsign.m
@@ -0,0 +1,17 @@
+% cursed
+ % lmao
+% basically - I take pairs of adjacent values on the vector `x` and add them
+% together. These sums are then checked against both members of the pair,
+% and if either member is *larger* than the sum, that indicates a sign change
+% since one of the values acted as a subtractor.
+% "Glue" is added in the comparison stage, to skew ONLY on cases where
+% we are comparing zero to zero (origin case). Its really a rounding error otherwise. lol
+% The front is padded w/ a zero.
+function ret = switchsign(x)
+ sum_check = x(1:end-1) + x(2:end);
+ ret = [0, ( abs(sum_check) < abs(x(1:end-1)+1e-7) ) | ( abs(sum_check) < abs(x(2:end)+1e-7) ) ];
+end
+ % <-.01> works here (this is insane), but technically has an
+ % error: calling `switchsign([-1 2 1 0 1 -3])` (the example i
+ % gave) returns `[0 1 0 1 1 1]` rather than `[0 1 0 0 0 1]`.
+ % not that that really matters. sorry...
diff --git a/hw4/inflections.m b/hw4/inflections.m
new file mode 100644
index 0000000..ad23a8b
--- /dev/null
+++ b/hw4/inflections.m
@@ -0,0 +1,5 @@
+function [y_inf, x_inf] = inflections(y, x)
+ y_switch = switchsign( deriv(deriv(y, x), x) );
+ y_inf = nonzeros(y_switch .* y);
+ x_inf = nonzeros(y_switch .* x);
+end
diff --git a/hw4/switchsign.m b/hw4/switchsign.m
new file mode 100644
index 0000000..c8c02ac
--- /dev/null
+++ b/hw4/switchsign.m
@@ -0,0 +1,12 @@
+% cursed
+% basically - I take pairs of adjacent values on the vector `x` and add them
+% together. These sums are then checked against both members of the pair,
+% and if either member is *larger* than the sum, that indicates a sign change
+% since one of the values acted as a subtractor.
+% "Glue" is added in the comparison stage, to skew ONLY on cases where
+% we are comparing zero to zero (origin case). Its really a rounding error otherwise. lol
+% The front is padded w/ a zero.
+function ret = switchsign(x)
+ sum_check = x(1:end-1) + x(2:end);
+ ret = [0, ( abs(sum_check) < abs(x(1:end-1)+1e-7) ) | ( abs(sum_check) < abs(x(2:end)+1e-7) ) ];
+end
diff --git a/hw5/Ryan_HW5_ECE210.m b/hw5/Ryan_HW5_ECE210.m
new file mode 100644
index 0000000..bd512a6
--- /dev/null
+++ b/hw5/Ryan_HW5_ECE210.m
@@ -0,0 +1,121 @@
+%%%% ECE-210-B HW5 - Advanced Manipulations - James Ryan
+% James Ryan, 03/18/24
+
+% preamble
+close all; clc; clear;
+
+%% On the Air
+% 1
+R = [-9:9] + fliplr([-9:9]).'.*j;
+
+% 2
+n = [1:50];
+t = [linspace(0,1,1e5)].';
+sin_int = sum(trapz(sin(t.^n)) .* (t(2) - t(1)), 'all');
+
+% 3
+% a
+[theta, phi] = meshgrid(...
+ linspace(0, 2*pi, 5), ...
+ linspace(0, pi, 5));
+% b
+x = sin(phi) .* cos(theta);
+y = sin(phi) .* sin(theta);
+z = cos(phi);
+
+% c
+figure;
+surf(x,y,z);
+pbaspect([1 1 1]);
+colormap winter;
+
+title("The best sphere");
+xlabel("over axis")
+ylabel("yonder axis")
+zlabel("up axis")
+
+%% Under the Sea
+% 1
+interest = sin(linspace(0,5,100).*linspace(-5,0,100).');
+half_extract = ( abs(interest - .5) < (5/1000) ) .* interest; % < min quantized val
+indices = find(half_extract);
+
+% 2
+% find area on xy where both intersect
+% volume of z2 - volume of z1
+% incomplete for now
+x = linspace(0, 10, 1000);
+intersect = (1/4).*sqrt(x.^2 + (x.').^2) == exp(-(1-x.*x.').^2);
+volume = (...
+ ( cumtrapz( (1/4).*sqrt(x.^2 + (x.').^2) .* (x(2) - x(1)) )) - ...
+ ( cumtrapz( exp(-(1-x.*x.').^2 ) .* (x(2) - x(1)) ))...
+ ) .* intersect;
+
+%% I Need a Vacation
+% 1
+A_vals = sqrt(...
+ ([1:256] - 99).^2 + ...
+ ([1:256].' - 99).^2);
+A = A_vals < 29;
+
+% What's happening here is that our A indices are taken, shifted "down" 99
+% values, and then a norm is taken on them. If its final norm is < 29, its true!
+% otherwise, they're false.
+% Theres a circle localized around an origin of (i,j) = (99,99), and extends
+% out. This makes sense! relative to everywhere else, this is our lowest value,
+% and it creates the smallest norm we see (zer0).
+% This holds experimentally! If we change 29 to 1, we just get a single teeny
+% pixel at (99,99), surrounded by a false sea. So, the equality we have acts
+% to give our circle some radius.
+figure;
+imshow(A);
+
+% 2
+B_vals = sqrt(...
+ ([1:256] - 62).^2 + ...
+ ([1:256].' - 62).^2);
+B = B_vals < 58;
+
+% Close to the previous idea, however we shift down only by 68 units, so our
+% origin is (68, 68).
+% We have a larger radius, given we doubled our tolerance for whats true.
+
+figure;
+imshow(B);
+
+% 3
+C_vals = [1:256] - 4.*sin(([1:256].'./10));
+C = C_vals > 200;
+
+% We get a vertical ripple!! The ripple runs along our y axis and has a period
+% of 10 and a peak-to-trough distance of 8. This acts as a wavefront from our
+% true shoreline and our false sea.
+figure;
+imshow(C);
+
+% 4
+S_vals = rand(256, 256, 3);
+S = permute(permute(S_vals,[3 1 2]) .* [0 0 1].', [2 3 1]);
+
+% Interpreting this as a 256 x 256 image with 3 color channels, it looks
+% like we applied a filter to kill R and G! We're left with the blue channel
+figure;
+imshow(S);
+
+% 5
+M = A & ~B;
+
+% We get a crescent moon! The portion of B that cast over A was negated and
+% compared, which resulted in that poriton being removed.
+% We're left with the parts of A that was free from B's grasp
+figure;
+imshow(M);
+
+% 6
+Z = (C .* S) + M;
+
+% Its a scene of a beach! Our moon is casting its light onto the dark beach,
+% as the water casts itself onto the shore.
+% reminds me of good times :D
+figure;
+imshow(Z);
diff --git a/hw5/feedback/JamesRyan_HW5_ECE210_feedback.m b/hw5/feedback/JamesRyan_HW5_ECE210_feedback.m
new file mode 100644
index 0000000..676e3cd
--- /dev/null
+++ b/hw5/feedback/JamesRyan_HW5_ECE210_feedback.m
@@ -0,0 +1,150 @@
+%%%% ECE-210-B HW5 - Advanced Manipulations - James Ryan
+% James Ryan, 03/18/24
+
+% preamble
+close all; clc; clear;
+
+%% On the Air
+% 1
+R = [-9:9] + fliplr([-9:9]).'.*j;
+
+% 2
+n = [1:50];
+t = [linspace(0,1,1e5)].';
+sin_int = sum(trapz(sin(t.^n)) .* (t(2) - t(1)), 'all');
+ % <-.02> erm... the integral calculation seems to be correct
+ % (though `trapz` has an independent variable argument), but
+ % why the `sum` at the end? should have said this was supposed
+ % to be 50 different integral estimates in parallel. :p
+
+% 3
+% a
+[theta, phi] = meshgrid(...
+ linspace(0, 2*pi, 5), ...
+ linspace(0, pi, 5));
+% b
+x = sin(phi) .* cos(theta);
+y = sin(phi) .* sin(theta);
+z = cos(phi);
+
+% c
+figure;
+surf(x,y,z);
+pbaspect([1 1 1]);
+colormap winter;
+
+title("The best sphere");
+xlabel("over axis")
+ylabel("yonder axis")
+zlabel("up axis")
+
+%% Under the Sea
+% 1
+interest = sin(linspace(0,5,100).*linspace(-5,0,100).');
+half_extract = ( abs(interest - .5) < (5/1000) ) .* interest; % < min quantized val
+indices = find(half_extract);
+ % <-.06> two things:
+ % 1. the minimum quantized value is pretty uncertain after
+ % you pass the input through the sine function -- in fact,
+ % the nearest points are (i think) significantly closer than
+ % this. you'll have to use `min` to find them.
+ % 2. i did say something about 2D indices, but that's less
+ % important -- linear indices often work just as well.
+
+% 2
+X = linspace(-5,5,100);
+[x, y] = meshgrid(X, X);
+intersect = (1/4).*sqrt(x.^2 + y.^2) < exp(-(1-x.*y).^2);
+% integral(end) = volume
+integral = cumtrapz(X, cumtrapz(X, exp(-(1-x.*y).^2).*intersect, 2));
+ % <-.02> not, alas, the volume contained between the surfaces,
+ % but just the volume under the first surface and above the
+ % xy-plane. i think that's just a small oversight, though.
+
+
+figure;
+surf(x, y, exp(-(1-x.*y).^2).*intersect);
+xlabel("X Axis");
+ylabel("Y Axis");
+zlabel("Z Axis");
+title('Volume representing $\frac{1}{4}\sqrt{x^2 + y^2} < e^{-(1-x * y)^2}$',...
+ 'interpreter', 'latex');
+pbaspect([4 4 3])
+ % <+.02> volume is consistent with the above, so i'll give a
+ % few points.
+
+
+%% I Need a Vacation
+% 1
+A_vals = sqrt(...
+ ([1:256] - 99).^2 + ...
+ ([1:256].' - 99).^2);
+A = A_vals < 29;
+
+% What's happening here is that our A indices are taken, shifted "down" 99
+% values, and then a norm is taken on them. If its final norm is < 29, its true!
+% otherwise, they're false.
+% Theres a circle localized around an origin of (i,j) = (99,99), and extends
+% out. This makes sense! relative to everywhere else, this is our lowest value,
+% and it creates the smallest norm we see (zer0).
+% This holds experimentally! If we change 29 to 1, we just get a single teeny
+% pixel at (99,99), surrounded by a false sea. So, the equality we have acts
+% to give our circle some radius.
+figure;
+imshow(A);
+
+% 2
+B_vals = sqrt(...
+ ([1:256] - 62).^2 + ...
+ ([1:256].' - 62).^2);
+B = B_vals < 58;
+
+% Close to the previous idea, however we shift down only by 68 units, so our
+% origin is (68, 68).
+% We have a larger radius, given we doubled our tolerance for whats true.
+
+figure;
+imshow(B);
+
+% 3
+C_vals = [1:256] - 4.*sin(([1:256].'./10));
+C = C_vals > 200;
+ % <-.02> traditionally, the i index is the first index in the
+ % MATLAB sense -- that is, the index which increments down each
+ % column, not across each row. in short, (x, y) ←→ (j, i),
+ % whereas you effectively have y equivalent to j.
+
+% We get a vertical ripple!! The ripple runs along our y axis and has a period
+% of 10 and a peak-to-trough distance of 8. This acts as a wavefront from our
+% true shoreline and our false sea.
+figure;
+imshow(C);
+
+% 4
+S_vals = rand(256, 256, 3);
+S = permute(permute(S_vals,[3 1 2]) .* [0 0 1].', [2 3 1]);
+
+% Interpreting this as a 256 x 256 image with 3 color channels, it looks
+% like we applied a filter to kill R and G! We're left with the blue channel
+figure;
+imshow(S);
+
+% 5
+M = A & ~B;
+
+% We get a crescent moon! The portion of B that cast over A was negated and
+% compared, which resulted in that poriton being removed.
+% We're left with the parts of A that was free from B's grasp
+figure;
+imshow(M);
+
+% 6
+Z = (C .* S) + M;
+
+% Its a scene of a beach! Our moon is casting its light onto the dark beach,
+% as the water casts itself onto the shore.
+% reminds me of good times :D
+figure;
+imshow(Z);
+ % ahhhh, that it does me. figured y'all could use something
+ % calming.
diff --git a/hw6/Ryan_HW6_ECE210.m b/hw6/Ryan_HW6_ECE210.m
new file mode 100644
index 0000000..a8fe128
--- /dev/null
+++ b/hw6/Ryan_HW6_ECE210.m
@@ -0,0 +1,136 @@
+%%%% ECE-210-B HW6 - Filters - James Ryan
+% James Ryan, 03/27/24
+
+% preamble
+close all; clc; clear;
+
+%% Generate white noise
+Fs = 44100; % Hz = s^-1
+N = Fs*2; % 2s * 44100 samples/sec = 88200 samples of noise
+F = linspace(-Fs/2, Fs/2, N); % range of noise
+
+% noise generation
+% good way
+noise = randn(1, N); % intensity of noise
+
+% funny way - requires `fortune` as a dependency
+%noise = fortune_favors_the_fontaine(Fs, 2);
+
+% thank you!
+shifted_fft_mag = @(sig, len) fftshift(abs(fft(sig, len))) / len;
+
+
+%% Butterworth Filter
+Hbutter = butterworth;
+fvm_butter = fvtool(Hbutter, 'magnitude');
+axm_butter = fvm_butter.Children(5);
+axm_butter.YLim = [-60 5];
+axm_butter.Title.String = 'Butterworth Bandpass Filter Magnitude Plot';
+
+fvp_butter = fvtool(Hbutter, 'phase');
+axp_butter = fvp_butter.Children(5);
+axp_butter.Title.String = 'Butterworth Bandpass Filter Phase Plot';
+
+butter_noise = Hbutter.filter(noise);
+
+fftm_butter = shifted_fft_mag(butter_noise, N);
+
+figure;
+plot(F, fftm_butter);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Butterworth Bandpass Filter");
+
+% to save you, i played it back in the matlab terminal:
+%soundsc(noise)
+%soundsc(fftm_butter)
+% to me, its just a mostly mute audio track minus two bursts of sound.
+% that must be the negative and positive passband regions
+% so the filter took the noise, and just took the regions outside of
+% the band and killed them.
+
+%% Elliptic Filter
+Hellip = elliptic;
+fvm_ellip = fvtool(Hellip, 'magnitude');
+axm_ellip = fvm_ellip.Children(5);
+axm_ellip.YLim = [-60 5];
+axm_ellip.Title.String = 'Elliptical Bandstop Filter Magnitude Plot';
+
+fvp_ellip = fvtool(Hellip, 'phase');
+axp_ellip = fvp_ellip.Children(5);
+axp_ellip.Title.String = 'Elliptical Bandstop Filter Phase Plot';
+
+ellip_noise = Hellip.filter(noise);
+
+fftm_ellip = shifted_fft_mag(ellip_noise, N);
+
+figure;
+plot(F, fftm_ellip);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Elliptical Bandstop Filter");
+
+%soundsc(fftm_ellip)
+% I can hear this one.. come back? like very faintly
+% I guess it goes to show how butterworth doesnt have that elliptic
+% stopband portion, and everything decays to -inf dB. Interesting
+% difference!
+% same pattern as the past passband
+
+%% Chebyshev Type 1 Filter
+Hcheb1 = chebyshev1;
+fvm_cheb1 = fvtool(Hcheb1, 'magnitude');
+axm_cheb1 = fvm_cheb1.Children(5);
+axm_cheb1.YLim = [-60 5];
+axm_cheb1.Title.String = 'Chebyshev Type I Lowpass Filter Magnitude Plot';
+
+fvp_cheb1 = fvtool(Hcheb1, 'phase');
+axp_cheb1 = fvp_cheb1.Children(5);
+axp_cheb1.Title.String = 'Chebyshev Type I Lowpass Filter Phase Plot';
+
+cheb1_noise = Hcheb1.filter(noise);
+
+fftm_cheb1 = shifted_fft_mag(cheb1_noise, N);
+
+figure;
+plot(F, fftm_cheb1);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Chebyshev I Lowpass Filter");
+
+%soundsc(fftm_cheb1)
+% It sounds choppy, almost, where it wibbles in and out for the starting portion
+% where its present. Definitely not the choice for audio filters!
+% Unless you `want` that distortion in like music production, in which case, go for it
+% Sort of sounds like a car engine firing
+
+%% Chebyshev Type 2 Filter
+Hcheb2 = chebyshev2;
+fvm_cheb2 = fvtool(Hcheb2, 'magnitude');
+axm_cheb2 = fvm_cheb2.Children(5);
+axm_cheb2.YLim = [-60 5];
+axm_cheb2.Title.String = 'Chebyshev Type I Highpass Filter Magnitude Plot';
+
+fvp_cheb2 = fvtool(Hcheb2, 'phase');
+axp_cheb2 = fvp_cheb2.Children(5);
+axp_cheb2.Title.String = 'Chebyshev Type II Highpass Filter Phase Plot';
+
+cheb2_noise = Hcheb2.filter(noise);
+
+fftm_cheb2 = shifted_fft_mag(cheb2_noise, N);
+
+figure;
+plot(F, fftm_cheb2);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Chebyshev II Highpass Filter");
+
+%soundsc(fftm_cheb2)
+% Its passband is sorta similar to the butterworth passband, but its die-out
+% has an audible drop off to me. maybe i missed that in others with elliptic
+% stop bands? No just played back fftm_ellip, and that dropped immediately. hmm
+% placebo, perhaps
diff --git a/hw6/butterworth.m b/hw6/butterworth.m
new file mode 100644
index 0000000..5934d1b
--- /dev/null
+++ b/hw6/butterworth.m
@@ -0,0 +1,27 @@
+function Hd = butterworth
+%BUTTERWORTH Returns a discrete-time filter object.
+
+% MATLAB Code
+% Generated by MATLAB(R) 23.2 and Signal Processing Toolbox 23.2.
+% Generated on: 27-Mar-2024 18:56:52
+
+% Butterworth Bandpass filter designed using FDESIGN.BANDPASS.
+
+% All frequency values are in Hz.
+Fs = 44100; % Sampling Frequency
+
+Fstop1 = 6300; % First Stopband Frequency
+Fpass1 = 7350; % First Passband Frequency
+Fpass2 = 14700; % Second Passband Frequency
+Fstop2 = 17640; % Second Stopband Frequency
+Astop1 = 50; % First Stopband Attenuation (dB)
+Apass = 1; % Passband Ripple (dB)
+Astop2 = 50; % Second Stopband Attenuation (dB)
+match = 'stopband'; % Band to match exactly
+
+% Construct an FDESIGN object and call its BUTTER method.
+h = fdesign.bandpass(Fstop1, Fpass1, Fpass2, Fstop2, Astop1, Apass, ...
+ Astop2, Fs);
+Hd = design(h, 'butter', 'MatchExactly', match);
+
+% [EOF]
diff --git a/hw6/chebyshev1.m b/hw6/chebyshev1.m
new file mode 100644
index 0000000..c3d2d87
--- /dev/null
+++ b/hw6/chebyshev1.m
@@ -0,0 +1,12 @@
+function Hcheb = chebyshev1
+ Fs = 44100;
+
+ Fpass = Fs/9;
+ Fstop = Fs/8;
+
+ Apass = 5;
+ Astop = 40;
+
+ specs = fdesign.lowpass(Fpass, Fstop, Apass, Astop, Fs);
+ Hcheb = design(specs,"cheby1",MatchExactly="passband");
+end
diff --git a/hw6/chebyshev2.m b/hw6/chebyshev2.m
new file mode 100644
index 0000000..b45b2a7
--- /dev/null
+++ b/hw6/chebyshev2.m
@@ -0,0 +1,12 @@
+function Hcheb = chebyshev2
+ Fs = 44100;
+
+ Fpass = Fs/3;
+ Fstop = Fs/4;
+
+ Apass = 5;
+ Astop = 40;
+
+ specs = fdesign.highpass(Fstop, Fpass, Astop, Apass, Fs);
+ Hcheb = design(specs,"cheby2",MatchExactly="passband");
+end
diff --git a/hw6/elliptic.m b/hw6/elliptic.m
new file mode 100644
index 0000000..bbc85b4
--- /dev/null
+++ b/hw6/elliptic.m
@@ -0,0 +1,27 @@
+function Hd = elliptic
+%ELLIPTIC Returns a discrete-time filter object.
+
+% MATLAB Code
+% Generated by MATLAB(R) 23.2 and Signal Processing Toolbox 23.2.
+% Generated on: 27-Mar-2024 18:58:12
+
+% Elliptic Bandstop filter designed using FDESIGN.BANDSTOP.
+
+% All frequency values are in Hz.
+Fs = 44100; % Sampling Frequency
+
+Fpass1 = 6300; % First Passband Frequency
+Fstop1 = 7350; % First Stopband Frequency
+Fstop2 = 14700; % Second Stopband Frequency
+Fpass2 = 17640; % Second Passband Frequency
+Apass1 = 1; % First Passband Ripple (dB)
+Astop = 50; % Stopband Attenuation (dB)
+Apass2 = 1; % Second Passband Ripple (dB)
+match = 'both'; % Band to match exactly
+
+% Construct an FDESIGN object and call its ELLIP method.
+h = fdesign.bandstop(Fpass1, Fstop1, Fstop2, Fpass2, Apass1, Astop, ...
+ Apass2, Fs);
+Hd = design(h, 'ellip', 'MatchExactly', match);
+
+% [EOF]
diff --git a/hw6/feedback/Ryan_HW6_ECE210.m b/hw6/feedback/Ryan_HW6_ECE210.m
new file mode 100644
index 0000000..7985bfb
--- /dev/null
+++ b/hw6/feedback/Ryan_HW6_ECE210.m
@@ -0,0 +1,138 @@
+%%%% ECE-210-B HW6 - Filters - James Ryan
+% James Ryan, 03/27/24
+
+% preamble
+close all; clc; clear;
+
+%% Generate white noise
+Fs = 44100; % Hz = s^-1
+N = Fs*2; % 2s * 44100 samples/sec = 88200 samples of noise
+F = linspace(-Fs/2, Fs/2, N); % range of noise
+
+% noise generation
+% good way
+noise = randn(1, N); % intensity of noise
+
+% funny way - requires `fortune` as a dependency
+%noise = fortune_favors_the_fontaine(Fs, 2);
+
+% thank you!
+ % you're welcome!
+shifted_fft_mag = @(sig, len) fftshift(abs(fft(sig, len))) / len;
+
+
+%% Butterworth Filter
+ % had to change `5` to `end`, but otherwise perfect.
+Hbutter = butterworth;
+fvm_butter = fvtool(Hbutter, 'magnitude');
+axm_butter = fvm_butter.Children(end);
+axm_butter.YLim = [-60 5];
+axm_butter.Title.String = 'Butterworth Bandpass Filter Magnitude Plot';
+
+fvp_butter = fvtool(Hbutter, 'phase');
+axp_butter = fvp_butter.Children(end);
+axp_butter.Title.String = 'Butterworth Bandpass Filter Phase Plot';
+
+butter_noise = Hbutter.filter(noise);
+
+fftm_butter = shifted_fft_mag(butter_noise, N);
+
+figure;
+plot(F, fftm_butter);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Butterworth Bandpass Filter");
+
+% to save you, i played it back in the matlab terminal:
+%soundsc(noise)
+%soundsc(fftm_butter)
+% to me, its just a mostly mute audio track minus two bursts of sound.
+% that must be the negative and positive passband regions
+% so the filter took the noise, and just took the regions outside of
+% the band and killed them.
+
+%% Elliptic Filter
+Hellip = elliptic;
+fvm_ellip = fvtool(Hellip, 'magnitude');
+axm_ellip = fvm_ellip.Children(end);
+axm_ellip.YLim = [-60 5];
+axm_ellip.Title.String = 'Elliptical Bandstop Filter Magnitude Plot';
+
+fvp_ellip = fvtool(Hellip, 'phase');
+axp_ellip = fvp_ellip.Children(end);
+axp_ellip.Title.String = 'Elliptical Bandstop Filter Phase Plot';
+
+ellip_noise = Hellip.filter(noise);
+
+fftm_ellip = shifted_fft_mag(ellip_noise, N);
+
+figure;
+plot(F, fftm_ellip);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Elliptical Bandstop Filter");
+
+%soundsc(fftm_ellip)
+% I can hear this one.. come back? like very faintly
+% I guess it goes to show how butterworth doesnt have that elliptic
+% stopband portion, and everything decays to -inf dB. Interesting
+% difference!
+% same pattern as the past passband
+
+%% Chebyshev Type 1 Filter
+Hcheb1 = chebyshev1;
+fvm_cheb1 = fvtool(Hcheb1, 'magnitude');
+axm_cheb1 = fvm_cheb1.Children(end);
+axm_cheb1.YLim = [-60 5];
+axm_cheb1.Title.String = 'Chebyshev Type I Lowpass Filter Magnitude Plot';
+
+fvp_cheb1 = fvtool(Hcheb1, 'phase');
+axp_cheb1 = fvp_cheb1.Children(end);
+axp_cheb1.Title.String = 'Chebyshev Type I Lowpass Filter Phase Plot';
+
+cheb1_noise = Hcheb1.filter(noise);
+
+fftm_cheb1 = shifted_fft_mag(cheb1_noise, N);
+
+figure;
+plot(F, fftm_cheb1);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Chebyshev I Lowpass Filter");
+
+%soundsc(fftm_cheb1)
+% It sounds choppy, almost, where it wibbles in and out for the starting portion
+% where its present. Definitely not the choice for audio filters!
+% Unless you `want` that distortion in like music production, in which case, go for it
+% Sort of sounds like a car engine firing
+
+%% Chebyshev Type 2 Filter
+Hcheb2 = chebyshev2;
+fvm_cheb2 = fvtool(Hcheb2, 'magnitude');
+axm_cheb2 = fvm_cheb2.Children(end);
+axm_cheb2.YLim = [-60 5];
+axm_cheb2.Title.String = 'Chebyshev Type I Highpass Filter Magnitude Plot';
+
+fvp_cheb2 = fvtool(Hcheb2, 'phase');
+axp_cheb2 = fvp_cheb2.Children(end);
+axp_cheb2.Title.String = 'Chebyshev Type II Highpass Filter Phase Plot';
+
+cheb2_noise = Hcheb2.filter(noise);
+
+fftm_cheb2 = shifted_fft_mag(cheb2_noise, N);
+
+figure;
+plot(F, fftm_cheb2);
+xlim([-Fs/2 Fs/2]);
+xlabel("Frequency (Hz)");
+ylabel("Magnitude (dB)");
+title("FFT of noisy signal passed through Chebyshev II Highpass Filter");
+
+%soundsc(fftm_cheb2)
+% Its passband is sorta similar to the butterworth passband, but its die-out
+% has an audible drop off to me. maybe i missed that in others with elliptic
+% stop bands? No just played back fftm_ellip, and that dropped immediately. hmm
+% placebo, perhaps
diff --git a/hw6/feedback/butterworth.m b/hw6/feedback/butterworth.m
new file mode 100644
index 0000000..5934d1b
--- /dev/null
+++ b/hw6/feedback/butterworth.m
@@ -0,0 +1,27 @@
+function Hd = butterworth
+%BUTTERWORTH Returns a discrete-time filter object.
+
+% MATLAB Code
+% Generated by MATLAB(R) 23.2 and Signal Processing Toolbox 23.2.
+% Generated on: 27-Mar-2024 18:56:52
+
+% Butterworth Bandpass filter designed using FDESIGN.BANDPASS.
+
+% All frequency values are in Hz.
+Fs = 44100; % Sampling Frequency
+
+Fstop1 = 6300; % First Stopband Frequency
+Fpass1 = 7350; % First Passband Frequency
+Fpass2 = 14700; % Second Passband Frequency
+Fstop2 = 17640; % Second Stopband Frequency
+Astop1 = 50; % First Stopband Attenuation (dB)
+Apass = 1; % Passband Ripple (dB)
+Astop2 = 50; % Second Stopband Attenuation (dB)
+match = 'stopband'; % Band to match exactly
+
+% Construct an FDESIGN object and call its BUTTER method.
+h = fdesign.bandpass(Fstop1, Fpass1, Fpass2, Fstop2, Astop1, Apass, ...
+ Astop2, Fs);
+Hd = design(h, 'butter', 'MatchExactly', match);
+
+% [EOF]
diff --git a/hw6/feedback/chebyshev1.m b/hw6/feedback/chebyshev1.m
new file mode 100644
index 0000000..c3d2d87
--- /dev/null
+++ b/hw6/feedback/chebyshev1.m
@@ -0,0 +1,12 @@
+function Hcheb = chebyshev1
+ Fs = 44100;
+
+ Fpass = Fs/9;
+ Fstop = Fs/8;
+
+ Apass = 5;
+ Astop = 40;
+
+ specs = fdesign.lowpass(Fpass, Fstop, Apass, Astop, Fs);
+ Hcheb = design(specs,"cheby1",MatchExactly="passband");
+end
diff --git a/hw6/feedback/chebyshev2.m b/hw6/feedback/chebyshev2.m
new file mode 100644
index 0000000..b45b2a7
--- /dev/null
+++ b/hw6/feedback/chebyshev2.m
@@ -0,0 +1,12 @@
+function Hcheb = chebyshev2
+ Fs = 44100;
+
+ Fpass = Fs/3;
+ Fstop = Fs/4;
+
+ Apass = 5;
+ Astop = 40;
+
+ specs = fdesign.highpass(Fstop, Fpass, Astop, Apass, Fs);
+ Hcheb = design(specs,"cheby2",MatchExactly="passband");
+end
diff --git a/hw6/feedback/elliptic.m b/hw6/feedback/elliptic.m
new file mode 100644
index 0000000..bbc85b4
--- /dev/null
+++ b/hw6/feedback/elliptic.m
@@ -0,0 +1,27 @@
+function Hd = elliptic
+%ELLIPTIC Returns a discrete-time filter object.
+
+% MATLAB Code
+% Generated by MATLAB(R) 23.2 and Signal Processing Toolbox 23.2.
+% Generated on: 27-Mar-2024 18:58:12
+
+% Elliptic Bandstop filter designed using FDESIGN.BANDSTOP.
+
+% All frequency values are in Hz.
+Fs = 44100; % Sampling Frequency
+
+Fpass1 = 6300; % First Passband Frequency
+Fstop1 = 7350; % First Stopband Frequency
+Fstop2 = 14700; % Second Stopband Frequency
+Fpass2 = 17640; % Second Passband Frequency
+Apass1 = 1; % First Passband Ripple (dB)
+Astop = 50; % Stopband Attenuation (dB)
+Apass2 = 1; % Second Passband Ripple (dB)
+match = 'both'; % Band to match exactly
+
+% Construct an FDESIGN object and call its ELLIP method.
+h = fdesign.bandstop(Fpass1, Fstop1, Fstop2, Fpass2, Apass1, Astop, ...
+ Apass2, Fs);
+Hd = design(h, 'ellip', 'MatchExactly', match);
+
+% [EOF]
diff --git a/hw6/feedback/fortune_favors_the_fontaine.m b/hw6/feedback/fortune_favors_the_fontaine.m
new file mode 100644
index 0000000..bec843e
--- /dev/null
+++ b/hw6/feedback/fortune_favors_the_fontaine.m
@@ -0,0 +1,12 @@
+function noise = fortune_favors_the_fontaine(Fs, duration)
+ % make some noise
+ output = "";
+ while strlength(output) < Fs*duration
+ [ret dirty_snippet] = unix('fortune');
+ % clean
+ snippet = regexprep(dirty_snippet, '[^A-Za-z]+', '');
+ output = strcat(output, snippet);
+ end
+ noise = normalize(letters2numbers(convertStringsToChars(output)));
+end
+ % <+.03> what the fuck
diff --git a/hw6/feedback/letters2numbers.m b/hw6/feedback/letters2numbers.m
new file mode 100644
index 0000000..67ace84
--- /dev/null
+++ b/hw6/feedback/letters2numbers.m
@@ -0,0 +1,4 @@
+function num = letters2numbers(word)
+ asc = double( upper(word) ); % http://www.asciitable.com/
+ num = 26 + double('A') - asc; % simple arithmetic
+end
diff --git a/hw6/fortune_favors_the_fontaine.m b/hw6/fortune_favors_the_fontaine.m
new file mode 100644
index 0000000..e15f8ac
--- /dev/null
+++ b/hw6/fortune_favors_the_fontaine.m
@@ -0,0 +1,11 @@
+function noise = fortune_favors_the_fontaine(Fs, duration)
+ % make some noise
+ output = "";
+ while strlength(output) < Fs*duration
+ [ret dirty_snippet] = unix('fortune');
+ % clean
+ snippet = regexprep(dirty_snippet, '[^A-Za-z]+', '');
+ output = strcat(output, snippet);
+ end
+ noise = normalize(letters2numbers(convertStringsToChars(output)));
+end
diff --git a/hw6/letters2numbers.m b/hw6/letters2numbers.m
new file mode 100644
index 0000000..67ace84
--- /dev/null
+++ b/hw6/letters2numbers.m
@@ -0,0 +1,4 @@
+function num = letters2numbers(word)
+ asc = double( upper(word) ); % http://www.asciitable.com/
+ num = 26 + double('A') - asc; % simple arithmetic
+end
diff --git a/lessons/lesson01/matlab_intro.m b/lessons/lesson01/matlab_intro.m
new file mode 100644
index 0000000..225dfae
--- /dev/null
+++ b/lessons/lesson01/matlab_intro.m
@@ -0,0 +1,180 @@
+%% Lesson 1
+
+%% Objective
+% After this class, you should be able to:
+%%
+%
+% * Know why you need MATLAB
+% * Manuever around the MATLAB interface
+% * Understand arithmetic and basic functions in MATLAB
+% * Know how to make scalar, vector and matrix variables in MATLAB
+% * Know how to perform matrix operations in MATLAB
+%
+%% MATLAB overview
+% MATLAB (short for MATrix LABoratory) is a commonly used interactive
+% software amongst engineers. As the name suggests, MATLAB organizes its
+% data as matrices and is specially designed for matrix multiplication. In
+% addition, it has a plethora of plugins and functions that engineers can
+% use, such as machine learning, financial analysis, filter design etc.
+%%
+% In Cooper, MATLAB is widely used in electrical and computer engineering
+% classes (signals, comm theory, machine learning, etc.), and is more broadly
+% used for these purposes and others (physics simulations, controls design,
+% etc.) across many engineering and scientific disciplines.
+%
+%% MATLAB Environment
+%%% Command window
+% The command window is sort of the equivalent of a terminal in Linux, or
+% Cygwin in Windows. When you type a command into the command window, an
+% operation performs. You can type a MATLAB command, such as 5+10, and the
+% answer would be printed out. If a variable is not assigned to the
+% command, the result would be stored in the variable ans automatically. If
+% a semicolon is added at the end of the line, the result would be
+% suppressed. You can clear the command window by typing clc. Moreover, you
+% can also type command line commands in the command window, such as ls,
+% pwd etc.
+
+%%% Command history
+% When you are playing around with different functions in MATLAB, you might
+% want to trace back what functions you played with. At that time, you can
+% press the up arrow, which would show you your command history.
+
+%%% Workspace
+% The workspace is where all the variables are stored. Each variable is
+% displayed as a name value pair in the workspace. If the variable is a
+% scalar, then the actual value would be shown. If it is a vector or matrix
+% , then depending on the size of the vector / matrix, it would either be
+% shown as its value or simply the size of the vector / array and its type.
+% You can double click on the variables to investigate its actual value in
+% a spreadsheet.
+
+%%% Current Folder
+% The current folder shows you where you are located at in MATLAB. If you
+% execute the command pwd on the command window, it should show you the
+% location of the current folder. You might find a time where you need to
+% add a folder and link it to your current folder location. At that time,
+% you can right click and select "Add to Path". To change current folder,
+% you can execute the cd command on your command window
+
+%%% Editor
+% The editor is where you can write a script and execute it. All MATLAB
+% scripts are saved as .m files. To execute a script, press the play button
+% on top in EDITOR tab. When you are executing a script, you can use the
+% semicolon to suppress the output of each line. To display a certain
+% variable at an arbitrary location in your script, you can use disp()
+% function.
+
+%% Arithmetic and Basic functions
+
+%% Basic Operations
+5+10; % Addition
+ans; % Prints out previous answer
+25-7; % Subtraction
+24*86; % Multiplication
+123.456*78.90; % Multiplication
+145/123; % Division
+2^5; % Exponential
+log10(1000); % Logarithm base 10
+log(exp(5)); % Natural logarithm
+sqrt(625); % Square root
+sin(pi); % sine function
+asin(0); % arc sine function
+1e5; % e5 multiplies 1 by 10^5
+1e-2; % e-2 multiplies 1 by 10^-2
+
+%% Complex Numbers
+2+1i; % equivalently, 2+i
+2+1j; % equivalently, 2+j
+(2+2i)*(3+4j);
+
+%% Special Numbers
+pi;
+exp(2*pi*j);
+inf;
+
+%% Complex number operations
+conj(2+i); % complex conjugate
+real(2+i); % real part
+imag(2+i); % imaginary part
+abs(2+i); % magnitude/absolute value
+angle(2+i); % angle or phase
+
+%% Variables
+% In matlab, there are 3 (main) different kinds of variables
+%%
+% * Scalar - A scalar appears as 1-by-1 and it is a single real or complex
+% number
+% * Vector - A vector is 1-by-n or n-by-1, and appears in MATLAB as a row or
+% column of complex numbers
+% * Matrix - A matrix is m-by-n, and appears in MATLAB as, essentially, a
+% matrix. A matrix is a 2-D array
+% If you want to see what variables you've declared, either look in the
+% Workspace section of the MATLAB window, or type:
+who;
+whos;
+
+%% Scalar Variables
+a = 5;
+b = 10;
+c = a+b;
+z1 = 2+j;
+z2 = 3+4j;
+z = z1*z2;
+
+%% Vector Variables
+x = [1 2+3j 2.718 pi cos(pi)]; % row vector
+x = [1, 2+3j, 2.718, pi, cos(pi)]; % same thing with commas
+xT = transpose(x); % now you created the column vector
+xT = x.'; % regular tranpose
+xT = x'; % complex tranpose
+y = [1 ; 2.5 ; 3.2 ; 4*pi; cos(pi)]; % column vector
+xlen = length(x); % length of row/col vector
+ylen = length(y); % same value as length(x)!
+
+%% BE CAREFUL!
+% The following two vectors produces vectors of different sizes, the reason
+% being linspace(x1, x2, n) creates n evenly spaced points between x1 and
+% x2 , with the value of interval (x2-x1)/(n+1), while the colon operator
+% (used in the form of x1:i:x2) creates an array with [x1, x1+i, x1+2i...,
+% x1+mi], where m = (x2-x1)/i. Hence when creating a vector with the colon
+% operator or linspace, make sure you know when to use it. In conclusion,
+% linspace works with number of points, whereas the colon operator works
+% with increments.
+
+%%
+v1 = linspace(-5,5,10);
+v2 = -5:1:5;
+
+%% Matrix Variables
+A = [1 2 3; 4 5 6; 7 8 9]; % basic construction of matrix
+B = repmat(A,2,1); % you concatenated A one above the other
+C = [A; A]; % same as above
+C1 = transpose(C); % now you transposed C!
+C2 = C.'; % still transposed! if it is only C' then it is
+ % conjugate transpose
+size(C); % Confirm that they are tranposes of each other
+size(C1);
+size(C1,1); % You get the dimension you want!
+eye(3); % Create identity matrix
+speye(30000000); % Create sparse identity matrix
+D = ones(50,60); % D is 50-by-60 ones
+E = zeros(40); % E is 40-by-40 zeros
+
+%% Matrix Operations
+B+C; % addition
+B-C; % subtraction
+4*B + C/5; % multiplication and division with a constant
+A+ones(size(A)); % elementwise addition with a constant
+B*C'; % matrix multiplication
+B.*C; % elementwise multiplication
+B.^3; % elementwise exponentiation! note: do not use B^3
+2*(eye(3))^3; % only possible with square matrices
+
+%% Documentation
+% If you don't know how to use a function, look it up using one of the
+% following commands. help opens a textual documentation in the
+% command window (just like Linux's man command), while doc will open a
+% new window with graphical documentation just like their website. The
+% MATLAB documentation website is also a great resource!
+help clc;
+doc size;
diff --git a/lessons/lesson02/numerical_calculus.m b/lessons/lesson02/numerical_calculus.m
new file mode 100644
index 0000000..afd9dc4
--- /dev/null
+++ b/lessons/lesson02/numerical_calculus.m
@@ -0,0 +1,75 @@
+%% Lesson 2c: Numerical estimation of integrals and derivatives
+% Numerical integration and differentiation are a staple of numerical
+% computing. We will now see how easy these are in MATLAB!
+clc; clear; close all;
+
+%% The diff and cumsum functions
+% Note the lengths of z, zdiff, and zcumsum! (Fencepost problem)
+z = [0 5 -2 3 4];
+zdiff = diff(z);
+zcumsum = cumsum(z);
+
+%% Setup: A simple function
+% Let's start with a simple example: y = x.^2. The domain is N points
+% linearly sampled from lo to hi.
+lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');
+
+%% Numerical (Approximate) Derivatives
+% We can calculate a difference quotient between each pair of (x,y) points
+% using the diff() function.
+dydx = diff(y)./diff(x); % difference quotient
+
+figure();
+plot(x(1:end-1), dydx);
+
+%% Numerical (Approximate) Integrals
+% Now suppose we want to approximate the cumulative integral (Riemann sum)
+% of a function.
+xdiff = diff(x);
+dx = xdiff(1); % spacing between points
+dx = (hi-lo) / (N-1); % alternative to the above (note: N-1)
+
+Y = cumsum(y) * dx; % Riemann sum
+
+figure();
+plot(x, Y);
+
+%% Error metrics: Check how close we are!
+% dydx should be derivative of x.^2 = 2*x
+dydx_actual = 2 * x;
+
+% Y should be \int_{-2}^{x}{t.^2 dt}
+Y_actual = (x .^3 - (-2)^3) / 3;
+
+% Slightly more accurate -- can you figure out why?
+% Y_actual = (x .^3 - (-2-dx)^3) / 3;
+
+% Error metric: MSE (mean square error) or RMSE (root mean square error)
+% Try changing N and see how the error changes. Try this with both the
+% integral and derivative.
+
+% estimated = dydx;
+% actual = dydx_actual(1:end-1);
+estimated = Y;
+actual = Y_actual;
+mse = mean((estimated - actual) .^ 2);
+rmse = rms(estimated - actual);
+
+%% Fundamental Theorem of Calculus
+% Now, use the approximate derivative to get the original function, y back
+% as yhat and plot it. You may need to use/create another variable for
+% the x axis when plotting.
+figure();
+yhat = diff(Y)./diff(x); % differentiate Y
+plot(x(1:end-1), yhat);
+
+figure();
+yhat2 = cumsum(dydx) * dx; % integrate dydx
+plot(x(1:end-1), yhat2);
\ No newline at end of file
diff --git a/lessons/lesson02/octave-log.txt b/lessons/lesson02/octave-log.txt
new file mode 100644
index 0000000..0cb7af8
--- /dev/null
+++ b/lessons/lesson02/octave-log.txt
@@ -0,0 +1,1378 @@
+[cat@lazarus:~/classes/ece210-materials/2024-van-west/lessons/lesson02]
+$ octave
+GNU Octave, version 7.3.0
+octave:2> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:9> close all
+octave:10>
+octave:10>
+octave:10>
+octave:10>
+octave:10>
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');^[[201~octave:10>
+octave:10>
+octave:10>
+octave:10>
+octave:10> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:17> close all
+octave:18>
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');^[[201~octave:18>
+octave:18>
+octave:18>
+octave:18> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:25> close all
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26>
+octave:26> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:33>
+octave:33>
+octave:33>
+octave:33>
+octave:33> x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+octave:40> close all
+octave:41>
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+octave:53> x = 1:5:10
+x =
+
+ 1 6
+
+octave:54> x = 1:10
+x =
+
+ 1 2 3 4 5 6 7 8 9 10
+
+octave:55> x(1)
+ans = 1
+octave:56> x(1:3)
+ans =
+
+ 1 2 3
+
+octave:57> x(2:7)
+ans =
+
+ 2 3 4 5 6 7
+
+octave:58> x = 0:9
+x =
+
+ 0 1 2 3 4 5 6 7 8 9
+
+octave:59> x = linspace(0, 1, 10)
+x =
+
+ Columns 1 through 8:
+
+ 0 0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778
+
+ Columns 9 and 10:
+
+ 0.8889 1.0000
+
+octave:60> x = linspace(0, 1, 10);
+octave:61> x = linspace(0, 1, 10).';
+octave:62> x
+x =
+
+ 0
+ 0.1111
+ 0.2222
+ 0.3333
+ 0.4444
+ 0.5556
+ 0.6667
+ 0.7778
+ 0.8889
+ 1.0000
+
+octave:63> x(2:6)
+ans =
+
+ 0.1111
+ 0.2222
+ 0.3333
+ 0.4444
+ 0.5556
+
+octave:64> x([1 3 5 7])
+ans =
+
+ 0
+ 0.2222
+ 0.4444
+ 0.6667
+
+octave:65> x(:)
+ans =
+
+ 0
+ 0.1111
+ 0.2222
+ 0.3333
+ 0.4444
+ 0.5556
+ 0.6667
+ 0.7778
+ 0.8889
+ 1.0000
+
+octave:66> x(end)
+ans = 1
+octave:67> x(end:-1:3)
+ans =
+
+ 1.0000
+ 0.8889
+ 0.7778
+ 0.6667
+ 0.5556
+ 0.4444
+ 0.3333
+ 0.2222
+
+octave:68> x(end:-1:1)
+ans =
+
+ 1.0000
+ 0.8889
+ 0.7778
+ 0.6667
+ 0.5556
+ 0.4444
+ 0.3333
+ 0.2222
+ 0.1111
+ 0
+
+octave:69> x(1:2:end)
+ans =
+
+ 0
+ 0.2222
+ 0.4444
+ 0.6667
+ 0.8889
+
+octave:70> M = 1:100;
+octave:71> size(M)
+ans =
+
+ 1 100
+
+octave:72> help reshape
+'reshape' is a built-in function from the file libinterp/corefcn/data.cc
+
+ -- reshape (A, M, N, ...)
+ -- reshape (A, [M N ...])
+ -- reshape (A, ..., [], ...)
+ -- reshape (A, SIZE)
+ Return a matrix with the specified dimensions (M, N, ...) whose
+ elements are taken from the matrix A.
+
+ The elements of the matrix are accessed in column-major order (like
+ Fortran arrays are stored).
+
+ The following code demonstrates reshaping a 1x4 row vector into a
+ 2x2 square matrix.
+
+ reshape ([1, 2, 3, 4], 2, 2)
+ => 1 3
+ 2 4
+
+ Note that the total number of elements in the original matrix
+ ('prod (size (A))') must match the total number of elements in the
+ new matrix ('prod ([M N ...])').
+
+ A single dimension of the return matrix may be left unspecified and
+ Octave will determine its size automatically. An empty matrix ([])
+ is used to flag the unspecified dimension.
+
+ See also: resize, vec, postpad, cat, squeeze.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:73> N1 = reshape(M, round(sqrt(100)), [])
+N1 =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:74> N1
+N1 =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:75> N1(:, :)
+ans =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:76> N1(1:3, :)
+ans =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+
+octave:77> N1(1:3, 2:6)
+ans =
+
+ 11 21 31 41 51
+ 12 22 32 42 52
+ 13 23 33 43 53
+
+octave:78> N1(1:3, 1:3)
+ans =
+
+ 1 11 21
+ 2 12 22
+ 3 13 23
+
+octave:79> N1(end-3:end)
+ans =
+
+ 97 98 99 100
+octave:83>
+Display all 1790 possibilities? (y or n)
+abs
+accumarray
+__accumarray_max__
+__accumarray_min__
+__accumarray_sum__
+accumdim
+octave:84> x = 3
+x = 3
+octave:85> x = [1 2 3];
+octave:86> x = 0:15
+x =
+
+ Columns 1 through 15:
+
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
+
+ Column 16:
+
+ 15
+
+octave:87> y = x.^2
+y =
+
+ Columns 1 through 13:
+
+ 0 1 4 9 16 25 36 49 64 81 100 121 144
+
+ Columns 14 through 16:
+
+ 169 196 225
+
+octave:88> x
+x =
+
+ Columns 1 through 15:
+
+ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
+
+ Column 16:
+
+ 15
+
+octave:89> xt = x.'
+xt =
+
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+ 11
+ 12
+ 13
+ 14
+ 15
+
+octave:90> x = (1:9).'
+x =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:91> sin(x)
+ans =
+
+ 0.8415
+ 0.9093
+ 0.1411
+ -0.7568
+ -0.9589
+ -0.2794
+ 0.6570
+ 0.9894
+ 0.4121
+
+octave:92> rad2deg(pi/2)
+ans = 90
+octave:93> length(x)
+ans = 9
+octave:94> mean(x)
+ans = 5
+octave:95> x
+x =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:96> mean(sin(x))
+ans = 0.2172
+octave:97>
+^[[200~T = 1e-6; % Sampling period (s)
+t = 0:T:2e-3; % Time (domain/x-axis)
+f0 = 50; % Initial frequency (Hz)
+b = 10e6; % Chirp rate (Hz/s)
+A = 10; % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);^[[201~octave:97>
+octave:97>
+octave:97>
+octave:97>
+octave:97> T = 1e-6; % Sampling period (s)
+t = 0:T:2e-3; % Time (domain/x-axis)
+f0 = 50; % Initial frequency (Hz)
+b = 10e6; % Chirp rate (Hz/s)
+A = 10; % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);
+octave:105> close all
+octave:109> x = linspace(0, 1, 9)
+x =
+
+ Columns 1 through 8:
+
+ 0 0.1250 0.2500 0.3750 0.5000 0.6250 0.7500 0.8750
+
+ Column 9:
+
+ 1.0000
+
+octave:110> x = linspace(0, 1, 9).'
+x =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:111> x(1)
+ans = 0
+octave:112> x(9)
+ans = 1
+octave:113> x(10)
+error: x(10): out of bound 9 (dimensions are 9x1)
+octave:114> x(5)
+ans = 0.5000
+octave:115> x([1 2 3])
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+
+octave:116> x([1 3 2])
+ans =
+
+ 0
+ 0.2500
+ 0.1250
+
+octave:117> x(1:9)
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:118> x(1:end)
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:119> x(end)
+ans = 1
+octave:120> x(3:end)
+ans =
+
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:121> x(end:-1:1)
+ans =
+
+ 1.0000
+ 0.8750
+ 0.7500
+ 0.6250
+ 0.5000
+ 0.3750
+ 0.2500
+ 0.1250
+ 0
+
+octave:122> 9:-1:1
+ans =
+
+ 9 8 7 6 5 4 3 2 1
+
+octave:123> 9:-.5:1
+ans =
+
+ Columns 1 through 7:
+
+ 9.0000 8.5000 8.0000 7.5000 7.0000 6.5000 6.0000
+
+ Columns 8 through 14:
+
+ 5.5000 5.0000 4.5000 4.0000 3.5000 3.0000 2.5000
+
+ Columns 15 through 17:
+
+ 2.0000 1.5000 1.0000
+
+octave:124> x(:)
+ans =
+
+ 0
+ 0.1250
+ 0.2500
+ 0.3750
+ 0.5000
+ 0.6250
+ 0.7500
+ 0.8750
+ 1.0000
+
+octave:125> A = [1 2; 3 4]
+A =
+
+ 1 2
+ 3 4
+
+octave:126> A(:)
+ans =
+
+ 1
+ 3
+ 2
+ 4
+
+octave:127> z = 4
+z = 4
+octave:128> z(1)
+ans = 4
+octave:129> inc
+error: 'inc' undefined near line 1, column 1
+octave:130> a++
+error: in x++ or ++x, x must be defined first
+octave:131> ++x
+ans =
+
+ 1.0000
+ 1.1250
+ 1.2500
+ 1.3750
+ 1.5000
+ 1.6250
+ 1.7500
+ 1.8750
+ 2.0000
+
+octave:132> M = reshape(1:100, 10, 10)
+M =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:133> M(1, 1)
+ans = 1
+octave:134> M(10, 10)
+ans = 100
+octave:135> M(end, end)
+ans = 100
+octave:136> M(end, 1)
+ans = 10
+octave:137> M(1, end)
+ans = 91
+octave:138> M(1:3, :)
+ans =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+
+octave:139> M(1:3, 2:5)
+ans =
+
+ 11 21 31 41
+ 12 22 32 42
+ 13 23 33 43
+
+octave:140> M([1 3 5 7], 2:5)
+ans =
+
+ 11 21 31 41
+ 13 23 33 43
+ 15 25 35 45
+ 17 27 37 47
+
+octave:141> x
+x =
+
+ 1.0000
+ 1.1250
+ 1.2500
+ 1.3750
+ 1.5000
+ 1.6250
+ 1.7500
+ 1.8750
+ 2.0000
+
+octave:142> x(1:5)
+ans =
+
+ 1.0000
+ 1.1250
+ 1.2500
+ 1.3750
+ 1.5000
+
+octave:143> x(1:5) = 1:5
+x =
+
+ 1.0000
+ 2.0000
+ 3.0000
+ 4.0000
+ 5.0000
+ 1.6250
+ 1.7500
+ 1.8750
+ 2.0000
+
+octave:144> x(6:end) = x(6:end).^2
+x =
+
+ 1.0000
+ 2.0000
+ 3.0000
+ 4.0000
+ 5.0000
+ 2.6406
+ 3.0625
+ 3.5156
+ 4.0000
+
+octave:145> M
+M =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 23 33 43 53 63 73 83 93
+ 4 14 24 34 44 54 64 74 84 94
+ 5 15 25 35 45 55 65 75 85 95
+ 6 16 26 36 46 56 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:146> M(3:6, 3:6) = 0
+M =
+
+ 1 11 21 31 41 51 61 71 81 91
+ 2 12 22 32 42 52 62 72 82 92
+ 3 13 0 0 0 0 63 73 83 93
+ 4 14 0 0 0 0 64 74 84 94
+ 5 15 0 0 0 0 65 75 85 95
+ 6 16 0 0 0 0 66 76 86 96
+ 7 17 27 37 47 57 67 77 87 97
+ 8 18 28 38 48 58 68 78 88 98
+ 9 19 29 39 49 59 69 79 89 99
+ 10 20 30 40 50 60 70 80 90 100
+
+octave:147> x
+x =
+
+ 1.0000
+ 2.0000
+ 3.0000
+ 4.0000
+ 5.0000
+ 2.6406
+ 3.0625
+ 3.5156
+ 4.0000
+
+octave:148> x = 1:9.'
+x =
+
+ 1 2 3 4 5 6 7 8 9
+
+octave:149> x = (1:9).'
+x =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:150> reshape(x, 3, 3)
+ans =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:151> x = 1:9
+x =
+
+ 1 2 3 4 5 6 7 8 9
+
+octave:152> reshape(x, 3, 3)
+ans =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:153> x
+x =
+
+ 1 2 3 4 5 6 7 8 9
+
+octave:154> xm = reshape(x, 3, 3)
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:155> xm(1, 1)
+ans = 1
+octave:156> xm(1)
+ans = 1
+octave:157> xm(2)
+ans = 2
+octave:158> xm(3)
+ans = 3
+octave:159> xm(4)
+ans = 4
+octave:160> xm
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:161> reshape(xm, 9, 1)
+ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:162> mx
+error: 'mx' undefined near line 1, column 1
+octave:163> xm
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:164> reshape(xm, 9, 1)
+ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:165> reshape(xm, 9, [])
+ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+
+octave:166> reshape(xm, 3, [])
+ans =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:167> xm
+xm =
+
+ 1 4 7
+ 2 5 8
+ 3 6 9
+
+octave:168> xm(2, :) = []
+xm =
+
+ 1 4 7
+ 3 6 9
+
+octave:169> magic(4)
+ans =
+
+ 16 2 3 13
+ 5 11 10 8
+ 9 7 6 12
+ 4 14 15 1
+
+octave:170> sum(magic(4))
+ans =
+
+ 34 34 34 34
+
+octave:171> sum(magic(4), 2)
+ans =
+
+ 34
+ 34
+ 34
+ 34
+
+octave:172>
+^[[200~lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');^[[201~octave:172>
+octave:172>
+octave:172>
+octave:172>
+octave:172> lo = -2;
+hi = 2;
+N = 1e2;
+
+x = linspace(lo, hi, N);
+y = x.^2;
+plot(x, y);
+title('x^2');
+octave:179> close all
+octave:180> diff(x)
+ans =
+
+ Columns 1 through 7:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 8 through 14:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 15 through 21:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 22 through 28:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 29 through 35:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 36 through 42:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 43 through 49:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 50 through 56:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 57 through 63:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 64 through 70:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 71 through 77:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 78 through 84:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 85 through 91:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Columns 92 through 98:
+
+ 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404 0.040404
+
+ Column 99:
+
+ 0.040404
+
+octave:181> diff(y)
+ans =
+
+ Columns 1 through 8:
+
+ -0.1600 -0.1567 -0.1535 -0.1502 -0.1469 -0.1437 -0.1404 -0.1371
+
+ Columns 9 through 16:
+
+ -0.1339 -0.1306 -0.1273 -0.1241 -0.1208 -0.1175 -0.1143 -0.1110
+
+ Columns 17 through 24:
+
+ -0.1077 -0.1045 -0.1012 -0.0979 -0.0947 -0.0914 -0.0882 -0.0849
+
+ Columns 25 through 32:
+
+ -0.0816 -0.0784 -0.0751 -0.0718 -0.0686 -0.0653 -0.0620 -0.0588
+
+ Columns 33 through 40:
+
+ -0.0555 -0.0522 -0.0490 -0.0457 -0.0424 -0.0392 -0.0359 -0.0326
+
+ Columns 41 through 48:
+
+ -0.0294 -0.0261 -0.0229 -0.0196 -0.0163 -0.0131 -0.0098 -0.0065
+
+ Columns 49 through 56:
+
+ -0.0033 0 0.0033 0.0065 0.0098 0.0131 0.0163 0.0196
+
+ Columns 57 through 64:
+
+ 0.0229 0.0261 0.0294 0.0326 0.0359 0.0392 0.0424 0.0457
+
+ Columns 65 through 72:
+
+ 0.0490 0.0522 0.0555 0.0588 0.0620 0.0653 0.0686 0.0718
+
+ Columns 73 through 80:
+
+ 0.0751 0.0784 0.0816 0.0849 0.0882 0.0914 0.0947 0.0979
+
+ Columns 81 through 88:
+
+ 0.1012 0.1045 0.1077 0.1110 0.1143 0.1175 0.1208 0.1241
+
+ Columns 89 through 96:
+
+ 0.1273 0.1306 0.1339 0.1371 0.1404 0.1437 0.1469 0.1502
+
+ Columns 97 through 99:
+
+ 0.1535 0.1567 0.1600
+
+octave:182> diff(y)./diff(x)
+ans =
+
+ Columns 1 through 8:
+
+ -3.9596 -3.8788 -3.7980 -3.7172 -3.6364 -3.5556 -3.4747 -3.3939
+
+ Columns 9 through 16:
+
+ -3.3131 -3.2323 -3.1515 -3.0707 -2.9899 -2.9091 -2.8283 -2.7475
+
+ Columns 17 through 24:
+
+ -2.6667 -2.5859 -2.5051 -2.4242 -2.3434 -2.2626 -2.1818 -2.1010
+
+ Columns 25 through 32:
+
+ -2.0202 -1.9394 -1.8586 -1.7778 -1.6970 -1.6162 -1.5354 -1.4545
+
+ Columns 33 through 40:
+
+ -1.3737 -1.2929 -1.2121 -1.1313 -1.0505 -0.9697 -0.8889 -0.8081
+
+ Columns 41 through 48:
+
+ -0.7273 -0.6465 -0.5657 -0.4848 -0.4040 -0.3232 -0.2424 -0.1616
+
+ Columns 49 through 56:
+
+ -0.0808 0 0.0808 0.1616 0.2424 0.3232 0.4040 0.4848
+
+ Columns 57 through 64:
+
+ 0.5657 0.6465 0.7273 0.8081 0.8889 0.9697 1.0505 1.1313
+
+ Columns 65 through 72:
+
+ 1.2121 1.2929 1.3737 1.4545 1.5354 1.6162 1.6970 1.7778
+
+ Columns 73 through 80:
+
+ 1.8586 1.9394 2.0202 2.1010 2.1818 2.2626 2.3434 2.4242
+
+ Columns 81 through 88:
+
+ 2.5051 2.5859 2.6667 2.7475 2.8283 2.9091 2.9899 3.0707
+
+ Columns 89 through 96:
+
+ 3.1515 3.2323 3.3131 3.3939 3.4747 3.5556 3.6364 3.7172
+
+ Columns 97 through 99:
+
+ 3.7980 3.8788 3.9596
+
+octave:183>
+^[[200~figure();
+plot(x(1:end-1), dydx);^[[201~octave:183>
+octave:183>
+octave:183>
+octave:183>
+octave:183> figure();
+plot(x(1:end-1), dydx);
+error: 'dydx' undefined near line 1, column 18
+octave:201> xdiff = diff(x);
+octave:202> dx = xdiff(1);
+octave:203> dx
+dx = 0.040404
+octave:204> int_of_y = sum(y)*dx
+int_of_y = 5.4960
+octave:205> Y = cumsum(y)*dx
+Y =
+
+ Columns 1 through 8:
+
+ 0.1616 0.3168 0.4656 0.6082 0.7448 0.8754 1.0002 1.1193
+
+ Columns 9 through 16:
+
+ 1.2329 1.3411 1.4440 1.5418 1.6345 1.7224 1.8055 1.8841
+
+ Columns 17 through 24:
+
+ 1.9581 2.0277 2.0932 2.1546 2.2120 2.2655 2.3154 2.3617
+
+ Columns 25 through 32:
+
+ 2.4046 2.4442 2.4806 2.5140 2.5445 2.5722 2.5973 2.6199
+
+ Columns 33 through 40:
+
+ 2.6401 2.6581 2.6739 2.6878 2.6998 2.7101 2.7188 2.7261
+
+ Columns 41 through 48:
+
+ 2.7320 2.7368 2.7405 2.7433 2.7453 2.7466 2.7474 2.7479
+
+ Columns 49 through 56:
+
+ 2.7480 2.7480 2.7480 2.7482 2.7486 2.7494 2.7507 2.7527
+
+ Columns 57 through 64:
+
+ 2.7555 2.7592 2.7640 2.7700 2.7772 2.7859 2.7963 2.8083
+
+ Columns 65 through 72:
+
+ 2.8221 2.8380 2.8559 2.8761 2.8987 2.9238 2.9515 2.9820
+
+ Columns 73 through 80:
+
+ 3.0154 3.0518 3.0914 3.1343 3.1806 3.2305 3.2841 3.3415
+
+ Columns 81 through 88:
+
+ 3.4028 3.4683 3.5380 3.6120 3.6905 3.7736 3.8615 3.9542
+
+ Columns 89 through 96:
+
+ 4.0520 4.1549 4.2631 4.3767 4.4959 4.6207 4.7513 4.8878
+
+ Columns 97 through 100:
+
+ 5.0304 5.1793 5.3344 5.4960
+
+octave:206> figure
+octave:207> hold on
+octave:208> plot(x, y)
+octave:209> plot(x, Y)
+octave:210> close all
+octave:211> help cumtrapz
+'cumtrapz' is a function from the file /usr/share/octave/7.3.0/m/general/cumtr
+apz.m
+
+ -- Q = cumtrapz (Y)
+ -- Q = cumtrapz (X, Y)
+ -- Q = cumtrapz (..., DIM)
+ Cumulative numerical integration of points Y using the trapezoidal
+ method.
+
+ 'cumtrapz (Y)' computes the cumulative integral of Y along the
+ first non-singleton dimension. Where 'trapz' reports only the
+ overall integral sum, 'cumtrapz' reports the current partial sum
+ value at each point of Y.
+
+ When the argument X is omitted an equally spaced X vector with unit
+ spacing (1) is assumed. 'cumtrapz (X, Y)' evaluates the integral
+ with respect to the spacing in X and the values in Y. This is
+ useful if the points in Y have been sampled unevenly.
+
+ If the optional DIM argument is given, operate along this
+ dimension.
+
+ Application Note: If X is not specified then unit spacing will be
+ used. To scale the integral to the correct value you must multiply
+ by the actual spacing value (deltaX).
+
+ See also: trapz, cumsum.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:212> help trapz=
+error: help: 'trapz=' not found
+octave:213> help trapz
+'trapz' is a function from the file /usr/share/octave/7.3.0/m/general/trapz.m
+
+ -- Q = trapz (Y)
+ -- Q = trapz (X, Y)
+ -- Q = trapz (..., DIM)
+
+ Numerically evaluate the integral of points Y using the trapezoidal
+ method.
+
+ 'trapz (Y)' computes the integral of Y along the first
+ non-singleton dimension. When the argument X is omitted an equally
+ spaced X vector with unit spacing (1) is assumed. 'trapz (X, Y)'
+ evaluates the integral with respect to the spacing in X and the
+ values in Y. This is useful if the points in Y have been sampled
+ unevenly.
+
+ If the optional DIM argument is given, operate along this
+ dimension.
+
+ Application Note: If X is not specified then unit spacing will be
+ used. To scale the integral to the correct value you must multiply
+ by the actual spacing value (deltaX). As an example, the integral
+ of x^3 over the range [0, 1] is x^4/4 or 0.25. The following code
+ uses 'trapz' to calculate the integral in three different ways.
+
+ x = 0:0.1:1;
+ y = x.^3;
+ ## No scaling
+ q = trapz (y)
+ => q = 2.5250
+ ## Approximation to integral by scaling
+ q * 0.1
+ => 0.25250
+ ## Same result by specifying X
+ trapz (x, y)
+ => 0.25250
+
+ See also: cumtrapz.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:214> trapz(x, y)
+ans = 5.3344
+octave:215> cmutrapz(x, y)
+error: 'cmutrapz' undefined near line 1, column 1
+octave:216> cumtrapz(x, y)
+ans =
+
+ Columns 1 through 8:
+
+ 0 0.1584 0.3104 0.4561 0.5957 0.7293 0.8570 0.9789
+
+ Columns 9 through 16:
+
+ 1.0953 1.2062 1.3118 1.4121 1.5074 1.5977 1.6832 1.7640
+
+ Columns 17 through 24:
+
+ 1.8403 1.9121 1.9797 2.0431 2.1024 2.1579 2.2097 2.2578
+
+ Columns 25 through 32:
+
+ 2.3024 2.3436 2.3816 2.4165 2.4485 2.4776 2.5040 2.5278
+
+ Columns 33 through 40:
+
+ 2.5492 2.5683 2.5852 2.6000 2.6130 2.6241 2.6336 2.6416
+
+ Columns 41 through 48:
+
+ 2.6483 2.6536 2.6579 2.6611 2.6635 2.6652 2.6662 2.6668
+
+ Columns 49 through 56:
+
+ 2.6671 2.6672 2.6672 2.6673 2.6676 2.6682 2.6693 2.6709
+
+ Columns 57 through 64:
+
+ 2.6733 2.6766 2.6808 2.6862 2.6928 2.7008 2.7103 2.7215
+
+ Columns 65 through 72:
+
+ 2.7344 2.7493 2.7662 2.7852 2.8066 2.8305 2.8569 2.8860
+
+ Columns 73 through 80:
+
+ 2.9179 2.9528 2.9908 3.0321 3.0767 3.1248 3.1765 3.2320
+
+ Columns 81 through 88:
+
+ 3.2914 3.3548 3.4223 3.4942 3.5704 3.6512 3.7367 3.8271
+
+ Columns 89 through 96:
+
+ 3.9223 4.0227 4.1282 4.2391 4.3555 4.4774 4.6052 4.7387
+
+ Columns 97 through 100:
+
+ 4.8783 5.0241 5.1760 5.3344
+
+octave:217> Y = cumtrapz(x, y)
+Y =
+
+ Columns 1 through 8:
+
+ 0 0.1584 0.3104 0.4561 0.5957 0.7293 0.8570 0.9789
+
+ Columns 9 through 16:
+
+ 1.0953 1.2062 1.3118 1.4121 1.5074 1.5977 1.6832 1.7640
+
+ Columns 17 through 24:
+
+ 1.8403 1.9121 1.9797 2.0431 2.1024 2.1579 2.2097 2.2578
+
+ Columns 25 through 32:
+
+ 2.3024 2.3436 2.3816 2.4165 2.4485 2.4776 2.5040 2.5278
+
+ Columns 33 through 40:
+
+ 2.5492 2.5683 2.5852 2.6000 2.6130 2.6241 2.6336 2.6416
+
+ Columns 41 through 48:
+
+ 2.6483 2.6536 2.6579 2.6611 2.6635 2.6652 2.6662 2.6668
+
+ Columns 49 through 56:
+
+ 2.6671 2.6672 2.6672 2.6673 2.6676 2.6682 2.6693 2.6709
+
+ Columns 57 through 64:
+
+ 2.6733 2.6766 2.6808 2.6862 2.6928 2.7008 2.7103 2.7215
+
+ Columns 65 through 72:
+
+ 2.7344 2.7493 2.7662 2.7852 2.8066 2.8305 2.8569 2.8860
+
+ Columns 73 through 80:
+
+ 2.9179 2.9528 2.9908 3.0321 3.0767 3.1248 3.1765 3.2320
+
+ Columns 81 through 88:
+
+ 3.2914 3.3548 3.4223 3.4942 3.5704 3.6512 3.7367 3.8271
+
+ Columns 89 through 96:
+
+ 3.9223 4.0227 4.1282 4.2391 4.3555 4.4774 4.6052 4.7387
+
+ Columns 97 through 100:
+
+ 4.8783 5.0241 5.1760 5.3344
+
+octave:218>
diff --git a/lessons/lesson02/vectorized_operations.m b/lessons/lesson02/vectorized_operations.m
new file mode 100644
index 0000000..d538411
--- /dev/null
+++ b/lessons/lesson02/vectorized_operations.m
@@ -0,0 +1,124 @@
+%% Lesson 2a: More vector and matrix operations
+%
+% Objectives:
+% * Understand how to perform vector operations in MATLAB
+% * Understand arithmetic and basic functions in MATLAB
+
+%% Vector operations
+% In lesson 1, we saw how to create a vector with the colon operator and
+% linspace. Now let's perform some operations on them!
+%
+% There are two common classes of operations that you can perform on vectors:
+% element-wise operations (which produce another vector) and aggregate
+% operations (which produce a scalar value). There are also many functions that
+% don't fall under these categories, but these cover many of the common
+% functions.
+
+%% Element-wise operations
+% Many operations that work on scalars (which are really degenerate matrices)
+% also work element-wise on vectors (or matrices).
+x = 0:0.01:2*pi; % Create a linearly-spaced vector
+y = sin(x); % sin() works element-wise on vectors!
+y = abs(x); % same with abs()!
+y = x .^ 4; % element-wise power
+y = power(x, 4); % same as above
+
+plot(x, y); % Plot y vs. x (line graph)
+title('y vs. x');
+
+%% Aggregate operations
+% Another common class of operations produce a single output or statistic about
+% a vector (or matrix).
+length(x); % number of elements in x
+sum(x); % sum of the elements of x
+mean(x); % average of the elements of x
+min(x); % minimum element of x
+diff(x); % difference between adjacent elements of x
+
+%% Exercise 1 : Vector operations
+T = 1e-6; % Sampling period (s)
+t = 0:T:2e-3; % Time (domain/x-axis)
+f0 = 50; % Initial frequency (Hz)
+b = 10e6; % Chirp rate (Hz/s)
+A = 10; % Amplitude
+y1 = A * cos(2*pi*f0*t + pi*b*t.^2);
+
+figure;
+plot(t,y1);
+
+%% Exercise: Numerical calculus
+% See numerical_calculus.m.
+
+%% Basic indexing in MATLAB
+% The process of extracting values from a vector (or matrix) is called
+% "indexing." In MATLAB, indices start at 1, rather than 0 in most languages
+% (in which it is more of an "offset" than a cardinal index).
+
+%% Exercise 2 : Basic indexing
+% The syntax for indexing is "x(indices)", where x is the variable to index,
+% and indices is a scalar or a vector of indices. There are many variations on
+% this. Note that indices can be any vector
+x(1); % first element of x
+x(1:3); % elements 1, 2 and 3 (inclusive!)
+x(1:length(x)); % all elements in x
+x(1:end); % same as above
+x(:); % same as above
+x(end); % last element of x
+x(3:end); % all elements from 3 onwards
+x([1,3,5]); % elements 1, 3, and 5 from x
+
+x(1:2:end); % all odd-indexed elements of x
+ind = 1:2:length(x);
+x(ind); % same as the previous example
+
+%% Exercises to improve your understanding
+% Take some time to go through these on your own.
+x([1,2,3]); % Will these produce the same result?
+x([3,2,1]);
+
+x2 = 1:5;
+x2(6); % What will this produce?
+x2(0); % What will this produce?
+x2(1:1.5:4); % What will this produce?
+ind = 1:1.5:4;
+x2(ind); % What will this produce?
+
+z = 4;
+z(1); % What will this produce?
+
+%% Matrix operations
+% Matrices is closely related to vectors, and we have also explored some matrix
+% operations last class. This class, we are going to explore functions that are
+% very useful but are hard to grasp for beginners, namely reshape, meshgrid,
+% row-wise and column-wise operations.
+
+%% Reshape
+% Change a matrix from one shape to another. The new shape has to have the same
+% number of elements as the original shape.
+%
+% When you are reshaping an array / matrix, the first dimension is filled
+% first, and then the second dimension, so on and so forth. I.e., elements
+% start filling down columns, then rows, etc.
+M = 1:100;
+N1 = reshape(M,2,2,[]); % It would create a 2*2*25 matrix
+N2 = reshape(M,[2,2,25]); % Same as N1
+N2(:,:,25); % Gives you 97,98,99,100
+N2(:,1,25); % Gives you 97 and 98
+
+%% Row-wise / Column-wise operations
+% Vector operations can also be performed on matrices. We can perform a vector
+% operation on each row/column of a matrix, or on a particular row/column by
+% indexing.
+H = magic(4); % create the magical matrix H
+sum(H,1); % column wise sum, note that this is a row vector(default)
+fliplr(H); % flip H from left to right
+flipud(H); % flip H upside down
+H(1,:) = fliplr(H(1,:)); % flip only ONE row left to right
+H(1,:) = []; % delete the first row
+
+%% Exercise 7 : Matrix Operations
+H2 = randi(20,4,5); % random 4x5 matrix with integers from 1 to 20
+sum(H2(:,2));
+mean(H2(3,:));
+C = reshape(H2,2,2,5);
+C(2,:,:) = [];
diff --git a/lessons/lesson03/matlab_logo.m b/lessons/lesson03/matlab_logo.m
new file mode 100644
index 0000000..691d4ea
--- /dev/null
+++ b/lessons/lesson03/matlab_logo.m
@@ -0,0 +1,57 @@
+%% Creating the MATLAB logo
+% from https://www.mathworks.com/help/matlab/visualize/creating-the-matlab-logo.html
+% Copyright (C) 2014 Mathworks Inc.
+close all; clear; clc;
+
+%% Create the surface
+L = 160*membrane(1,100);
+
+%% Create the figure and axes
+f = figure;
+ax = axes;
+
+s = surface(L);
+s.EdgeColor = 'none';
+view(3)
+
+%% Adjust axis limits
+ax.XLim = [1 201];
+ax.YLim = [1 201];
+ax.ZLim = [-53.4 160];
+
+%% Adjust the camera position
+ax.CameraPosition = [-145.5 -229.7 283.6];
+ax.CameraTarget = [77.4 60.2 63.9];
+ax.CameraUpVector = [0 0 1];
+ax.CameraViewAngle = 36.7;
+
+%% Adjust the position of the x, y, z axes themselves
+ax.Position = [0 0 1 1];
+ax.DataAspectRatio = [1 1 .9];
+
+%% Add some light
+l1 = light;
+l1.Position = [160 400 80];
+l1.Style = 'local';
+l1.Color = [0 0.8 0.8];
+
+l2 = light;
+l2.Position = [.5 -1 .4];
+l2.Color = [0.8 0.8 0];
+
+%% Change the surface color
+s.FaceColor = [0.9 0.2 0.2];
+
+%% Adjust lighting algorithm
+s.FaceLighting = 'gouraud';
+s.AmbientStrength = 0.3;
+s.DiffuseStrength = 0.6;
+s.BackFaceLighting = 'lit';
+
+s.SpecularStrength = 1;
+s.SpecularColorReflectance = 1;
+s.SpecularExponent = 7;
+
+%% Remove background
+axis off
+f.Color = 'black';
diff --git a/lessons/lesson03/octave-log.txt b/lessons/lesson03/octave-log.txt
new file mode 100644
index 0000000..c6c453f
--- /dev/null
+++ b/lessons/lesson03/octave-log.txt
@@ -0,0 +1,711 @@
+octave:2>
+octave:2>
+octave:2>
+octave:16>
+
+y2 = x.^2;^[[201~octave:16>
+octave:16>
+octave:16>
+octave:16>
+octave:16> x = -10:0.1:10;
+y = x.^3;
+y2 = x.^2;
+octave:19> size(y)
+ans =
+
+ 1 201
+
+octave:20> size(x)
+ans =
+
+ 1 201
+
+octave:21> plot(x, y)
+octave:22> plot(x, y2)
+octave:23> hold on
+octave:24> plot(x, y)
+octave:25> figure
+octave:26> plot(x, y)
+octave:27> close all
+octave:28> figure
+octave:29> plot(x, y)
+octave:30> xlabel('x axis')
+octave:31> ylabel('y axis')
+octave:32> ylabel('y axis! yay!')
+octave:33> ylabel('y axis! yay! more letters')
+octave:34> title('a plot of a cubic')
+octave:35> grid on
+octave:36> legend('x^3')
+octave:37> hold on
+octave:38> close all
+octave:39>
+^[[201~octave:39>
+octave:39>
+octave:39>
+octave:39>
+octave:39> hold on; % plotting more than 1 plot on 1 figure rather than overwriting
+
+
+plot(x, y, 'DisplayName', 'x^3');
+plot(x, y2, 'DisplayName', 'x^2');
+
+hold off;
+
+xlabel 'x axis';
+ylabel 'y axis';
+title 'Example 1';
+xlim([-10 10]);
+ylim([-10 10]);
+% axis([-10 10 -10 10]);
+grid on;
+legend show; % 'DisplayName does thisi
+octave:50> close all
+octave:51> help axis
+'axis' is a function from the file /usr/share/octave/7.3.0/m/plot/appearance/axis.m
+
+ -- axis ()
+ -- axis ([X_LO X_HI])
+ -- axis ([X_LO X_HI Y_LO Y_HI])
+ -- axis ([X_LO X_HI Y_LO Y_HI Z_LO Z_HI])
+ -- axis ([X_LO X_HI Y_LO Y_HI Z_LO Z_HI C_LO C_HI])
+ -- axis (OPTION)
+ -- axis (OPTION1, OPTION2, ...)
+ -- axis (HAX, ...)
+ -- LIMITS = axis ()
+ Set axis limits and appearance.
+
+ The argument LIMITS should be a 2-, 4-, 6-, or 8-element vector.
+ The first and second elements specify the lower and upper limits
+ for the x-axis. The third and fourth specify the limits for the
+ y-axis, the fifth and sixth specify the limits for the z-axis, and
+ the seventh and eighth specify the limits for the color axis. The
+ special values '-Inf' and 'Inf' may be used to indicate that the
+ limit should be automatically computed based on the data in the
+ axes.
+
+ Without any arguments, 'axis' turns autoscaling on.
+
+ With one output argument, 'LIMITS = axis' returns the current axis
+ limits.
+
+ The vector argument specifying limits is optional, and additional
+ string arguments may be used to specify various axis properties.
+
+ The following options control the aspect ratio of the axes.
+
+ "equal"
+ Force x-axis unit distance to equal y-axis (and z-axis) unit
+ distance.
+
+ "square"
+ Force a square axis aspect ratio.
+
+ "vis3d"
+ Set aspect ratio modes ("DataAspectRatio",
+ "PlotBoxAspectRatio") to "manual" for rotation without
+ stretching.
+
+ "normal"
+ "fill"
+ Restore default automatically computed aspect ratios.
+
+ The following options control the way axis limits are interpreted.
+
+ "auto"
+ "auto[xyz]"
+ "auto [xyz]"
+ Set nice auto-computed limits around the data for all axes, or
+ only the specified axes.
+
+ "manual"
+ Fix the current axes limits.
+
+ "tight"
+ Fix axes to the limits of the data.
+
+ "image"
+ Equivalent to "tight" and "equal".
+
+ The following options affect the appearance of tick marks.
+
+ "tic"
+ "tic[xyz]"
+ "tic [xyz]"
+ Turn tick marks on for all axes, or turn them on for the
+ specified axes and off for the remainder.
+
+ "label"
+ "label[xyz]"
+ "label [xyz]"
+ Turn tick labels on for all axes, or turn them on for the
+ specified axes and off for the remainder.
+
+ "nolabel"
+ Turn tick labels off for all axes.
+
+ Note: If there are no tick marks for an axes then there can be no
+ labels.
+
+ The following options affect the direction of increasing values on
+ the axes.
+
+ "xy"
+ Default y-axis, larger values are near the top.
+
+ "ij"
+ Reverse y-axis, smaller values are near the top.
+
+ The following options affects the visibility of the axes.
+
+ "on"
+ Make the axes visible.
+
+ "off"
+ Hide the axes.
+
+ If the first argument HAX is an axes handle, then operate on this
+ axes rather than the current axes returned by 'gca'.
+
+ Example 1: set X/Y limits and force a square aspect ratio
+
+ axis ([1, 2, 3, 4], "square");
+
+ Example 2: enable tick marks on all axes, enable tick mark labels
+ only on the y-axis
+
+ axis ("tic", "labely");
+
+ See also: xlim, ylim, zlim, caxis, daspect, pbaspect, box, grid.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:52>
+
+d2 = cos(t);^[[201~octave:52>
+octave:52>
+octave:52>
+octave:52>
+octave:52> t = 0:.1:10;
+d1 = sin(t);
+d2 = cos(t);
+octave:55> figure
+octave:56>
+^[[200~hold on;
+plot(t, d1);
+plot(t, d2);
+hold off;^[[201~octave:56>
+octave:56>
+octave:56>
+octave:56>
+octave:56> hold on;
+plot(t, d1);
+plot(t, d2);
+hold off;
+octave:60>
+^[[200~title 'Trig Functions';^[[201~octave:60>
+octave:60>
+octave:60>
+octave:60>
+octave:60> title 'Trig Functions';
+octave:61>
+^[[200~xlabel 'time (\mu)s';^[[201~octave:61>
+octave:61>
+octave:61>
+octave:61>
+octave:61> xlabel('time (\mu)s');
+octave:62> ylabel('voltage')
+octave:63> legend('sin', 'cos')
+octave:64> close all
+octave:65> figure
+octave:66>
+octave:66>
+octave:66>
+octave:66>
+octave:66>
+octave:66> plot(t, d1, 'b-.', t, d2, 'rp');
+octave:67>
+
+^[[200~title 'Trig Functions';
+xlabel 'time ($\mu$s)' Interpreter latex
+ylabel voltage;
+legend('sin', 'cos');
+xticks(0:pi/2:10);
+xticklabels({'0', '\pi/2', '\pi', '3\pi/2', '2\pi', '5\pi/2', '3\pi'});^[[201~octave:6
+7>
+octave:67>
+octave:67>
+octave:67>
+octave:67> title 'Trig Functions';
+xlabel 'time ($\mu$s)' Interpreter latex
+ylabel voltage;
+legend('sin', 'cos');
+xticks(0:pi/2:10);
+xticklabels({'0', '\pi/2', '\pi', '3\pi/2', '2\pi', '5\pi/2', '3\pi'});
+sh: 1: dvipng: not found
+warning: latex_renderer: a run-time test failed and the 'latex' interpreter has been d
+isabled.
+warning: called from
+ __axis_label__ at line 36 column 6
+ xlabel at line 59 column 8
+
+octave:73> title 'Trig Functions';
+xlabel 'time ($\mu$s)' Interpreter latex
+ylabel voltage;
+legend('sin', 'cos');
+xticks(0:pi/2:10);
+xticklabels({'0', '\pi/2', '\pi', '3\pi/2',
+octave:73>
+^[[200~xticks(0:pi/2:10);^[[201~octave:73>
+octave:73>
+octave:73>
+octave:73>
+octave:73> xticks(0:pi/2:10);
+octave:74>
+^[[200~xticklabels({'0', '\pi/2', '\pi', '3\pi/2', '2\pi', '5\pi/2', '3\pi'});^[[201~o
+ctave:74>
+octave:74>
+octave:74>
+octave:74>
+octave:74> xticklabels({'0', '\pi/2', '\pi', '3\pi/2', '2\pi', '5\pi/2', '3\pi'});
+octave:75> close all
+octave:76> figure
+octave:77> subplot(2, 1, 1)
+octave:78>
+^[[200~octave:75> close all^[[201~octave:78>
+octave:78>
+octave:78>
+octave:78>
+octave:78> octave:75> close all
+octave:78> figure
+octave:79> subplot(2, 1, 1)
+octave:80> plot(t, d1)
+octave:81> hold on
+octave:82> plot(t, d2)
+octave:83> title('an ordinary plot')
+octave:84> subplot(2, 1, 2)
+octave:85>
+octave:85>
+octave:85>
+octave:85>
+octave:85>
+octave:85> plot(t, d1, 'b-.', t, d2, 'rp');
+title 'Customized plot';
+octave:87> close all
+octave:88> figure
+octave:89> stem(t, d1)
+octave:90> hold on
+octave:91> scatter(t, d2)
+octave:92> close all
+octave:93>
+^[[200~t = linspace(0,10*pi);^[[201~octave:93>
+octave:93>
+octave:93>
+octave:93>
+octave:93> t = linspace(0,10*pi);
+octave:94> figure
+octave:95> plot3(sin(t), cos(t), t)
+octave:96> zlabel('t')
+octave:97> zlabel('tttttttttttttttttttttttttttttttttttt')
+octave:98> title('a helix! in space!')
+octave:99> text(0, 0, 0, 'origin')
+octave:100> close all
+octave:101>
+^[[200~a1 = -2:0.25:2;
+b1 = a1;
+[A1, B1] = meshgrid(a1);
+F = A1.*exp(-A1.^2-B1.^2);^[[201~octave:101>
+octave:101>
+octave:101>
+octave:101>
+octave:101> a1 = -2:0.25:2;
+b1 = a1;
+[A1, B1] = meshgrid(a1);
+F = A1.*exp(-A1.^2-B1.^2);
+octave:105> A1
+A1 =
+
+ Columns 1 through 9:
+
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+ -2.0000 -1.7500 -1.5000 -1.2500 -1.0000 -0.7500 -0.5000 -0.2500 0
+
+ Columns 10 through 17:
+
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+ 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 1.7500 2.0000
+
+octave:106> B1
+B1 =
+
+ Columns 1 through 9:
+
+ -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000
+ -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500
+ -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000
+ -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500
+ -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000
+ -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500
+ -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000
+ -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500
+ 0 0 0 0 0 0 0 0 0
+ 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500
+ 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000
+ 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500
+ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
+ 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500
+ 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000
+ 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500
+ 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
+
+ Columns 10 through 17:
+
+ -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000
+ -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500 -1.7500
+ -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000 -1.5000
+ -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500 -1.2500
+ -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000
+ -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500 -0.7500
+ -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000 -0.5000
+ -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500 -0.2500
+ 0 0 0 0 0 0 0 0
+ 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500
+ 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000
+ 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500
+ 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
+ 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500 1.2500
+ 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000
+ 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500 1.7500
+ 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
+
+octave:107> F
+F =
+
+ Columns 1 through 9:
+
+ -0.0007 -0.0015 -0.0029 -0.0048 -0.0067 -0.0078 -0.0071 -0.0043 0
+ -0.0017 -0.0038 -0.0074 -0.0123 -0.0172 -0.0200 -0.0182 -0.0110 0
+ -0.0039 -0.0086 -0.0167 -0.0276 -0.0388 -0.0450 -0.0410 -0.0248 0
+ -0.0077 -0.0172 -0.0331 -0.0549 -0.0771 -0.0896 -0.0816 -0.0492 0
+ -0.0135 -0.0301 -0.0582 -0.0964 -0.1353 -0.1572 -0.1433 -0.0864 0
+ -0.0209 -0.0466 -0.0901 -0.1493 -0.2096 -0.2435 -0.2219 -0.1338 0
+ -0.0285 -0.0637 -0.1231 -0.2041 -0.2865 -0.3328 -0.3033 -0.1829 0
+ -0.0344 -0.0769 -0.1485 -0.2461 -0.3456 -0.4014 -0.3658 -0.2206 0
+ -0.0366 -0.0818 -0.1581 -0.2620 -0.3679 -0.4273 -0.3894 -0.2349 0
+ -0.0344 -0.0769 -0.1485 -0.2461 -0.3456 -0.4014 -0.3658 -0.2206 0
+ -0.0285 -0.0637 -0.1231 -0.2041 -0.2865 -0.3328 -0.3033 -0.1829 0
+ -0.0209 -0.0466 -0.0901 -0.1493 -0.2096 -0.2435 -0.2219 -0.1338 0
+ -0.0135 -0.0301 -0.0582 -0.0964 -0.1353 -0.1572 -0.1433 -0.0864 0
+ -0.0077 -0.0172 -0.0331 -0.0549 -0.0771 -0.0896 -0.0816 -0.0492 0
+ -0.0039 -0.0086 -0.0167 -0.0276 -0.0388 -0.0450 -0.0410 -0.0248 0
+ -0.0017 -0.0038 -0.0074 -0.0123 -0.0172 -0.0200 -0.0182 -0.0110 0
+ -0.0007 -0.0015 -0.0029 -0.0048 -0.0067 -0.0078 -0.0071 -0.0043 0
+
+ Columns 10 through 17:
+
+ 0.0043 0.0071 0.0078 0.0067 0.0048 0.0029 0.0015 0.0007
+ 0.0110 0.0182 0.0200 0.0172 0.0123 0.0074 0.0038 0.0017
+ 0.0248 0.0410 0.0450 0.0388 0.0276 0.0167 0.0086 0.0039
+ 0.0492 0.0816 0.0896 0.0771 0.0549 0.0331 0.0172 0.0077
+ 0.0864 0.1433 0.1572 0.1353 0.0964 0.0582 0.0301 0.0135
+ 0.1338 0.2219 0.2435 0.2096 0.1493 0.0901 0.0466 0.0209
+ 0.1829 0.3033 0.3328 0.2865 0.2041 0.1231 0.0637 0.0285
+ 0.2206 0.3658 0.4014 0.3456 0.2461 0.1485 0.0769 0.0344
+ 0.2349 0.3894 0.4273 0.3679 0.2620 0.1581 0.0818 0.0366
+ 0.2206 0.3658 0.4014 0.3456 0.2461 0.1485 0.0769 0.0344
+ 0.1829 0.3033 0.3328 0.2865 0.2041 0.1231 0.0637 0.0285
+ 0.1338 0.2219 0.2435 0.2096 0.1493 0.0901 0.0466 0.0209
+ 0.0864 0.1433 0.1572 0.1353 0.0964 0.0582 0.0301 0.0135
+ 0.0492 0.0816 0.0896 0.0771 0.0549 0.0331 0.0172 0.0077
+ 0.0248 0.0410 0.0450 0.0388 0.0276 0.0167 0.0086 0.0039
+ 0.0110 0.0182 0.0200 0.0172 0.0123 0.0074 0.0038 0.0017
+ 0.0043 0.0071 0.0078 0.0067 0.0048 0.0029 0.0015 0.0007
+
+octave:108> figure
+octave:109> surf(A1, B1, F)
+octave:110> figure
+octave:111> mesh(A1, B1, F)
+octave:112> close all
+octave:113> help quiver
+'quiver' is a function from the file /usr/share/octave/7.3.0/m/plot/draw/quiver.m
+
+ -- quiver (U, V)
+ -- quiver (X, Y, U, V)
+ -- quiver (..., S)
+ -- quiver (..., STYLE)
+ -- quiver (..., "filled")
+ -- quiver (HAX, ...)
+ -- H = quiver (...)
+
+ Plot a 2-D vector field with arrows.
+
+ Plot the (U, V) components of a vector field at the grid points
+ defined by (X, Y). If the grid is uniform then X and Y can be
+ specified as vectors and 'meshgrid' is used to create the 2-D grid.
+
+ If X and Y are not given they are assumed to be '(1:M, 1:N)' where
+ '[M, N] = size (U)'.
+
+ The optional input S is a scalar defining a scaling factor to use
+ for the arrows of the field relative to the mesh spacing. A value
+ of 1.0 will result in the longest vector exactly filling one grid
+ square. A value of 0 disables all scaling. The default value is
+ 0.9.
+
+ The style to use for the plot can be defined with a line style
+ STYLE of the same format as the 'plot' command. If a marker is
+ specified then the markers are drawn at the origin of the vectors
+ (which are the grid points defined by X and Y). When a marker is
+ specified, the arrowhead is not drawn. If the argument "filled" is
+ given then the markers are filled.
+
+ If the first argument HAX is an axes handle, then plot into this
+ axes, rather than the current axes returned by 'gca'.
+
+ The optional return value H is a graphics handle to a quiver
+ object. A quiver object regroups the components of the quiver plot
+ (body, arrow, and marker), and allows them to be changed together.
+
+ Example:
+
+ [x, y] = meshgrid (1:2:20);
+ h = quiver (x, y, sin (2*pi*x/10), sin (2*pi*y/10));
+ set (h, "maxheadsize", 0.33);
+
+ See also: quiver3, compass, feather, plot.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:114> help quiver3
+'quiver3' is a function from the file /usr/share/octave/7.3.0/m/plot/draw/quiver3.m
+
+ -- quiver3 (X, Y, Z, U, V, W)
+ -- quiver3 (Z, U, V, W)
+ -- quiver3 (..., S)
+ -- quiver3 (..., STYLE)
+ -- quiver3 (..., "filled")
+ -- quiver3 (HAX, ...)
+ -- H = quiver3 (...)
+
+ Plot a 3-D vector field with arrows.
+
+ Plot the (U, V, W) components of a vector field at the grid points
+ defined by (X, Y, Z). If the grid is uniform then X, Y, and Z can
+ be specified as vectors and 'meshgrid' is used to create the 3-D
+ grid.
+
+ If X and Y are not given they are assumed to be '(1:M, 1:N)' where
+ '[M, N] = size (U)'.
+
+ The optional input S is a scalar defining a scaling factor to use
+ for the arrows of the field relative to the mesh spacing. A value
+ of 1.0 will result in the longest vector exactly filling one grid
+ cube. A value of 0 disables all scaling. The default value is
+ 0.9.
+
+ The style to use for the plot can be defined with a line style
+ STYLE of the same format as the 'plot' command. If a marker is
+ specified then the markers are drawn at the origin of the vectors
+ (which are the grid points defined by X, Y, Z). When a marker is
+ specified, the arrowhead is not drawn. If the argument "filled" is
+ given then the markers are filled.
+
+ If the first argument HAX is an axes handle, then plot into this
+ axes, rather than the current axes returned by 'gca'.
+
+ The optional return value H is a graphics handle to a quiver
+ object. A quiver object regroups the components of the quiver plot
+ (body, arrow, and marker), and allows them to be changed together.
+
+ [x, y, z] = peaks (25);
+ surf (x, y, z);
+ hold on;
+ [u, v, w] = surfnorm (x, y, z / 10);
+ h = quiver3 (x, y, z, u, v, w);
+ set (h, "maxheadsize", 0.33);
+
+ See also: quiver, compass, feather, plot.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:115> help feather
+'feather' is a function from the file /usr/share/octave/7.3.0/m/plot/draw/feather.m
+
+ -- feather (U, V)
+ -- feather (Z)
+ -- feather (..., STYLE)
+ -- feather (HAX, ...)
+ -- H = feather (...)
+
+ Plot the '(U, V)' components of a vector field emanating from
+ equidistant points on the x-axis.
+
+ If a single complex argument Z is given, then 'U = real (Z)' and 'V
+ = imag (Z)'.
+
+ The style to use for the plot can be defined with a line style
+ STYLE of the same format as the 'plot' command.
+
+ If the first argument HAX is an axes handle, then plot into this
+ axes, rather than the current axes returned by 'gca'.
+
+ The optional return value H is a vector of graphics handles to the
+ line objects representing the drawn vectors.
+
+ phi = [0 : 15 : 360] * pi/180;
+ feather (sin (phi), cos (phi));
+
+ See also: plot, quiver, compass.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:116> figure
+octave:117> imshow([1 0; 0 1])
+octave:118> imshow(1:255)
+octave:119> imshow([1 .5; .5 1])
+octave:120> whos
+Variables visible from the current scope:
+
+variables in scope: top scope
+
+ Attr Name Size Bytes Class
+octave:124> whos
+octave:125> x = [1 0; 0 1]
+x =
+
+ 1 0
+ 0 1
+
+octave:126> whose
+error: 'whose' undefined near line 1, column 1
+octave:127> whos
+Variables visible from the current scope:
+
+variables in scope: top scope
+
+ Attr Name Size Bytes Class
+ ==== ==== ==== ===== =====
+ x 2x2 32 double
+
+Total is 4 elements using 32 bytes
+
+octave:128> imshow(x)
+octave:129> imshow(int(x))
+error: 'int' undefined near line 1, column 8
+octave:130> int
+int16 int64 integral2 interp2 interpn intmin
+int2str int8 integral3 interp3 intersect
+int32 integral interp1 interpft intmax
+octave:130> imshow(int8(x))
+error: imshow: invalid data type for image
+error: called from
+ imshow at line 199 column 9
+octave:131> close all
+octave:132> image = reshape(linspace(0, 1, 12), 2, 2, 3)
+image =
+
+ans(:,:,1) =
+
+ 0 0.1818
+ 0.0909 0.2727
+
+ans(:,:,2) =
+
+ 0.3636 0.5455
+ 0.4545 0.6364
+
+ans(:,:,3) =
+
+ 0.7273 0.9091
+ 0.8182 1.0000
+
+octave:133> imshow(image)
+octave:134> image(:, :, 3) = 0
+image =
+
+ans(:,:,1) =
+
+ 0 0.1818
+ 0.0909 0.2727
+
+ans(:,:,2) =
+
+ 0.3636 0.5455
+ 0.4545 0.6364
+
+ans(:,:,3) =
+
+ 0 0
+ 0 0
+
+octave:135> imshow(image)
+octave:136> image(:, :, 2) = 0
+image =
+
+ans(:,:,1) =
+
+ 0 0.1818
+ 0.0909 0.2727
+
+ans(:,:,2) =
+
+ 0 0
+ 0 0
+
+ans(:,:,3) =
+
+ 0 0
+ 0 0
+
+octave:137> imshow(image)
+octave:138> close all
+octave:139>
diff --git a/lessons/lesson03/plotting.m b/lessons/lesson03/plotting.m
new file mode 100644
index 0000000..1b32273
--- /dev/null
+++ b/lessons/lesson03/plotting.m
@@ -0,0 +1,152 @@
+%% Lesson 5a: Plotting
+%
+% We are going to go through several plotting schemes, and explore how you
+% can customize plotting. We would go through 2D plotting, surface
+% plotting, subplot, stem plot and 3D plotting
+%
+% In this file the `command syntax` style of functions will be used when
+% possible, just to get you familiar with the style.
+clear; clc; close all;
+
+%% 2D plotting: line graphs
+% In general, always annotate your plots appropriately! Use a title,
+% axis labels, legends, etc. as necessary. Set appropriate bounds,
+% appropriate scaling (e.g., linear vs. logarithmic), and the correct
+% type of plot. We'll start simple with line plots.
+x = -10:0.1:10;
+y = x.^3;
+y2 = x.^2;
+
+hold on; % plotting more than 1 plot on 1 figure rather than overwriting
+
+plot(x, y, 'DisplayName', 'x^3');
+plot(x, y2, 'DisplayName', 'x^2');
+
+hold off;
+
+xlabel 'x axis';
+ylabel 'y axis';
+title 'Example 1';
+xlim([-10 10]);
+ylim([-10 10]);
+% axis([-10 10 -10 10]);
+grid on;
+legend show; % 'DisplayName does this
+
+%% Example 2: Plotting sine and cosine
+t = 0:.1:10;
+d1 = sin(t);
+d2 = cos(t);
+
+figure;
+
+% plot(t, [d1.' d2.']);
+% plot(t, d1, t, d2);
+
+hold on;
+plot(t, d1);
+plot(t, d2);
+hold off;
+
+title 'Trig Functions';
+
+% We can use some LaTeX-like symbols like \mu, \beta, \pi, \leq, \infty.
+% For full LaTeX support use the `Interpreter: latex` option
+xlabel 'time (\mu)s';
+
+ylabel voltage;
+legend sin cos;
+
+% Save to file; gcf() is "get current figure"
+exportgraphics(gcf(), "sample_plot.png");
+
+%% More plotting options
+% Legends, axis ticks (and labels), LaTeX interpreter
+figure;
+
+% Options for changing line pattern and color.
+% Don't need `hold on`/`hold off` if multiple lines plotted with a single
+% `plot` function.
+plot(t, d1, 'b-.', t, d2, 'rp');
+
+title 'Trig Functions';
+xlabel 'time ($\mu$s)' Interpreter latex
+ylabel voltage;
+legend('sin', 'cos');
+xticks(0:pi/2:10);
+xticklabels({'0', '\pi/2', '\pi', '3\pi/2', '2\pi', '5\pi/2', '3\pi'});
+
+% Many other options availible for plotting. Check the documentation or
+% search online for options.
+
+%% Subplots
+% Subplots exist for stylistic purposes. Let's say you have a signal and
+% you want to plot the magnitude and phase of the signal itself. It would
+% make more sense if the magnitude and phase plots exist in the same
+% figure. There are several examples fo subplot below to explain how it
+% works. Note that linear indexing of plots is different from normal linear
+% indexing.
+
+figure;
+subplot(2,2,1); % subplot(# of rows, # of columns, index)
+plot(t,d1);
+hold on;
+plot(t, d2);
+title 'Normal plot';
+
+subplot(2, 2, 2); % index runs down rows, not columns!
+plot(t, d1, 'b-.', t, d2, 'rp');
+title 'Customized plot';
+
+%% Stem plots
+% stem plots are particularly useful when you are representing digital
+% signals, hence it is good (and necessary) to learn them too!
+% subplot(2,2,[3 4]) % takes up two slots
+subplot(2, 1, 2);
+hold on;
+stem(t, d1);
+stem(t, d2);
+hold off;
+title 'Stem plots';
+
+sgtitle Subplots;
+
+%% Tiling -- like subplots but newer
+figure;
+tiledlayout(2, 2);
+
+nexttile;
+plot(t, d1);
+
+nexttile;
+plot(t, d2);
+
+%% A 3-D parametric function
+% A helix curve
+t = linspace(0,10*pi);
+
+figure;
+plot3(sin(t), cos(t), t);
+xlabel sin(t);
+ylabel cos(t);
+zlabel t;
+text(0, 0, 0, 'origin');
+grid on;
+title Helix;
+
+%% Surface plot
+% A shaded look for 2-D functions
+%
+% $f(x) = x\exp -(x^2+y^2)$
+a1 = -2:0.25:2;
+b1 = a1;
+[A1, B1] = meshgrid(a1);
+F = A1.*exp(-A1.^2-B1.^2);
+
+figure;
+surf(A1,B1,F);
+
+%% Mesh plot
+% A wireframe look for 2-D functions
+figure;
+mesh(A1,B1,F);
diff --git a/lessons/lesson04/BasicClass.m b/lessons/lesson04/BasicClass.m
new file mode 100644
index 0000000..2cc8ad4
--- /dev/null
+++ b/lessons/lesson04/BasicClass.m
@@ -0,0 +1,24 @@
+classdef BasicClass
+ % BasicClass is a simple sample class definition
+ properties (Access = private)
+ vals % vector
+ end
+
+ methods
+ function obj = BasicClass(x)
+ % Construct an instance of this class
+ % Sets x to the object's vals property
+ obj.vals = x;
+ end
+
+ function z = getVals(obj)
+ z = obj.vals;
+ end
+
+ function z = findClosest(obj,n)
+ % Find closest entry in obj.vals to n
+ [~, ind] = min(abs(obj.vals - n));
+ z = obj.vals(ind);
+ end
+ end
+end
\ No newline at end of file
diff --git a/lessons/lesson04/controlflow.m b/lessons/lesson04/controlflow.m
new file mode 100644
index 0000000..4091576
--- /dev/null
+++ b/lessons/lesson04/controlflow.m
@@ -0,0 +1,57 @@
+%% Lesson 4b: Control Sequences
+%
+% Since it likes to play at being a general purpose programming language,
+% MATLAB has control flow constructs (if, loops, etc.). Sometimes not
+% everything is achievable with a linear set of instructions and vectorization,
+% such as in particularly complex situations.
+
+clear; clc; close all;
+
+%% For loops (a review)
+total=0;
+for i=1:10
+ total = total + i;
+end
+total
+
+%% If...Else Statements
+gpa = 2.3;
+if gpa < 2
+ fprintf('You are in DANGER!!!\n')
+elseif gpa >= 2 && gpa < 3.5
+ fprintf('You are safe.\n')
+elseif gpa >= 3.5 && gpa < 3.7
+ fprintf('Cum Laude\n')
+elseif gpa >= 3.7 && gpa < 3.8
+ fprintf('Magna Cum Laude\n')
+elseif gpa >= 3.8 && gpa < 4
+ fprintf('Summa Cum Laude\n')
+else
+ fprintf("invalid gpa\n")
+end
+
+%% While loop
+% Iterate while a condition is true, as opposed to iterating over a vector.
+total = 0;
+i = 0;
+while i <= 10
+ total = total + i;
+ i = i+1;
+end
+total
+
+%% Try/catch blocks
+% Try and catch allow for error handling, which can be useful when using
+% user-supplied input that may be invalid. See the documentation of any
+% function to see what errors it may throw.
+%
+% It is also a good idea for you to validate input if writing a function
+% intended to be used by other people. This is very common in all MATLAB
+% builtin/toolbox functions.
+try
+ a = [1 2; 3 4];
+ b = [1 2 3 4];
+ c = a*b % can't do this! not the right dimensions!
+catch err
+ fprintf('Hey! Not allowed!\n');
+end
diff --git a/lessons/lesson04/datatypes.m b/lessons/lesson04/datatypes.m
new file mode 100644
index 0000000..5a95701
--- /dev/null
+++ b/lessons/lesson04/datatypes.m
@@ -0,0 +1,162 @@
+%% Lesson 5b: Datatypes
+% There are several basic data types in MATLAB:
+%
+% - single, double (floating pt. 32 and 64 bit respectively)
+% - int8, int16, int32, int64 (different size integers)
+% - uint8, uint16, uint32, uint64 (different size unsigned intergers)
+% - logicals
+% - char arrays, strings
+% - cell arrays
+clc; clear; close all;
+
+%% What type is this?
+% You can get the data type of a variable using the class function. Most
+% of the time, values default to fp64, which is often good for
+% scientific computing with high accuracy.
+a = 10;
+class(a);
+
+b = int8(a);
+class(b);
+
+%% Data Type Sizes
+% Different data types take up different amounts of space in your memory
+% and hard drive. Let's take a look at some standard sizes in MATLAB.
+A = randn(1, 'double');
+B = randn(1, 'single');
+C = true(1);
+D = 'D';
+
+% lists info of the variables in the current workspace
+whos;
+
+% If your data is getting too large, it can help to cast to smaller types.
+% `single` is a smaller fp type, and uint8/uint16 are smaller
+
+%% Different Interpretations
+% Be careful with what data types you feed into built in functions.
+% MATLAB will have different responses to different types.
+
+% imshow with ints
+imshow(uint8(rand(128, 128, 3)));
+
+%%
+% imshow with doubles
+imshow(double(rand(128, 128, 3)));
+
+%% Overflow and Underflow
+% With floating point, we are trying to represent real numbers. Obviously
+% there must be some spacing between representable numbers using a
+% fixed-size representation. Let's take a look.
+%
+% eps() shows the error between a number and its true value given IEEE
+% floating-point.
+L = logspace(-400, 400, 4096);
+loglog(L, eps(L), single(L), eps(single(L)));
+
+% As the plot shows, doubles have a much larger range as well as higher
+% precision at each point. Let's see how this applies in practice.
+
+%% More examples examples of eps
+eps(1);
+(1 + 0.5001*eps(1)) - 1;
+(1 + 0.4999*eps(1)) - 1;
+
+eps(1e16);
+1 + 1e16 == 1e16;
+
+%% More examples of overflow/underflow
+single(10^50);
+single(10^-50);
+
+uint8(256);
+int8(-129);
+
+%% Cell Arrays
+% Matrices can only store values of the same type (homogenous data).
+% Cell arrays can store heterogeneous (but still rectangular) data.
+% However, they are relatively slow and should be used sparingly.
+courses = {
+ '', 'Modern Physics', 'Signals and Systems', ...
+ 'Complex Analysis', 'Drawing and Sketching for Engineers'; ...
+
+ 'Grades', 70 + 3*randn(132,1), 80 + 3*randn(41,1), ...
+ 40 + 3*randn(20,1), 100*ones(29,1); ...
+
+ 'Teachers', {'Debroy'; 'Yecko'}, {'Fontaine'}, {'Smyth'}, {'Dell'}
+};
+
+%% Difference between {} and ()
+% As we've seen before, () performs matrix indexing. A cell array behaves
+% as a matrix of cell-arrays; indexing it using () will also produce a cell
+% array.
+%
+% {} performs matrix indexing, and also extracts the values of the cell(s).
+% Usually, this is what you want when immediately indexing a cell array.
+%
+% Confusingly (or not?) you can double-index cell arrays using `{}`, but
+% you cannot double-index matrices using `()`.
+courses(1,2);
+courses{1,2};
+courses{2,2}(3:5,1);
+courses(3,2);
+courses{3,2};
+courses{3,2}{1}(1);
+
+%% Objects and Classes
+% MATLAB supports OOP programming. We saw a basic class definition last
+% class. Classes are also used in all of MATLAB's toolboxes (e.g., ML,
+% signal processing).
+x = randn(10000, 1);
+h = histogram(x); % h is a histogram class
+
+properties(h);
+methods(h);
+
+% Check out the documentation for more info on the histogram class and
+% other classes.
+
+%% Structs and Objects
+% MATLAB also supports `struct`s. Structs are like objects in that they
+% can combine multiple values under the same variable (composite data),
+% and use the same dot-indexing syntax. However, they are not defined by
+% and constructed using a class, but rather using the `struct` constructor.
+
+sfield1 = 'a'; svalue1 = 5;
+sfield2 = 'b'; svalue2 = [3 6];
+sfield3 = 'c'; svalue3 = "Hello World";
+sfield4 = 'd'; svalue4 = {'cell array'};
+
+s = struct( ...
+ sfield1, svalue1, ...
+ sfield2, svalue2, ...
+ sfield3, svalue3, ...
+ sfield4, svalue4 ...
+);
+
+s.a;
+s.b(2);
+
+%% Struct with cell array fields
+% Cell arrays as struct fields may be used to generate an array of structs.
+tfield1 = 'f1'; tvalue1 = zeros(1,10);
+tfield2 = 'f2'; tvalue2 = {'a', 'b'};
+tfield3 = 'f3'; tvalue3 = {pi, pi.^2};
+tfield4 = 'f4'; tvalue4 = {'fourth'};
+t = struct( ...
+ tfield1, tvalue1, ...
+ tfield2, tvalue2, ...
+ tfield3, tvalue3, ...
+ tfield4, tvalue4 ...
+);
+
+t(1);
+t(2);
+
+%% Adding fields to a struct
+% Struct fields are not fixed; you can add fields to it after it is
+% constructed using dot-indexing.
+u = struct;
+u.a = [1 3 5 7];
+u.b = ["Hello","World!"];
+u;
diff --git a/lessons/lesson04/distance.m b/lessons/lesson04/distance.m
new file mode 100644
index 0000000..9cd425a
--- /dev/null
+++ b/lessons/lesson04/distance.m
@@ -0,0 +1,19 @@
+% In a file like this one (which only defines normal functions, and in which
+% the first function is hononymous with the filename), the first function is a
+% public function, and all other functions are local helper functions.
+%
+% Note that you DO NOT want to put clc; clear; close all; at the top of a file
+% like this; it only makes sense to do this at the top of script files when you
+% want to clean your workspace!
+
+% Calculate Euclidean distance
+% A public function, callable by other files and from the Command Window if
+% this is in the MATLAB path.
+function d = distance(x, y)
+ d = sqrt(sq(x) + sq(y));
+end
+
+% A local helper function. Only accessible within this file.
+function x = sq(x)
+ x = x * x;
+end
diff --git a/lessons/lesson04/for_loops.m b/lessons/lesson04/for_loops.m
new file mode 100644
index 0000000..2fe57b5
--- /dev/null
+++ b/lessons/lesson04/for_loops.m
@@ -0,0 +1,34 @@
+%% Lesson 2b: Control Sequence - For loops
+% Similar to other languages, MATLAB have if, while and for control
+% sequences. For loops are one of the commonly used control sequences in
+% MATLAB. We will continue the discussion of if and while in the next
+% lesson.
+
+clear;
+n = 1e7;
+% D = zeros(1,n); % This is called pre-allocation
+ % try uncommenting this line to see the
+ % difference. Memory allocations are slow!
+
+% Calculate fib(n). Hard to do vectorized unless we know Binet's formula.
+% (Note that this will quickly overflow, but that's not our objective
+% here.)
+D(1) = 1;
+D(2) = 2;
+tic % start timer
+for i = 3:n
+ D(i) = D(i-1) + D(i-2);
+end
+toc % print elapsed time since last tic
+
+%% Be careful!
+% For loops are considered the more inefficient type of operation in
+% MATLAB. Performing operations on vectors is faster because your hardware
+% can optimize vectorized instructions, and because the "for" construct
+% exists in MATLAB's runtime, which is essentially a scripting language and
+% thus is slow.
+%
+% However, things are changing, and things like parfor (parallel-for) and
+% advanced JIT (just-in-time) compilation are improving the performance of
+% loops. But vectorized operations are almost always faster and should
+% be used where possible.
\ No newline at end of file
diff --git a/lessons/lesson04/funnnnnn.m b/lessons/lesson04/funnnnnn.m
new file mode 100644
index 0000000..7def6a2
--- /dev/null
+++ b/lessons/lesson04/funnnnnn.m
@@ -0,0 +1,177 @@
+%% Lesson 4a: Functions
+%
+% Objectives:
+% - Explore MATLAB's function-calling syntax and semantics.
+% - Explore anonymous functions.
+% - Explore local functions.
+% - Explore regular functions.
+clear; clc; close all;
+
+%% Functions
+% Functions in any programming language are the means of abstraction. In
+% functional programming, they are a fundamental structure with the power to
+% build any complex form. In imperative and scientific computing, functions are
+% useful to prevent code duplication and improve maintainability.
+%
+% There are a few varieties of MATLAB, each of which has very different syntax
+% and semantics, so buckle up!
+%
+% (Insider information from MathWorks: MATLAB is working on improving the
+% support/accesibility of functional programming! So there may be some
+% improvements in the near future!)
+%
+% Note: this lecture was heavily modified by a functional-programming fan!
+
+%% Using functions
+% All functions are called (invoked) in the same manner: by using the
+% function's name followed by the list of parameters with which to invoke the
+% function.
+z = zeros(1, 3);
+mean([1, 2, 3, 4, 5]);
+
+% Functions may have optional parameters, and they may behave differently when
+% called with different inputs (polymorphism) or when different numbers of
+% outputs are expected. They may also return zero, one, or multiple values.
+size1 = size(z);
+size2 = size(z, 2); % optional second parameter
+
+hist(1:10); % displays a historgram
+a = hist(1:10); % stores a histogram into the variable a, and does
+ % not display a histogram
+
+[r1, r2] = HelloWorld("Jon", "Lam"); % Function returning multiple
+ % values (HelloWorld defined later in
+ % file)
+
+% There is a second way that you've seen to call functions, called "command
+% syntax." These are translatable to the regular function syntax. Any textual
+% arguments in this syntax will be interpreted as strings, which is similar to
+% UNIX sh commands arguments (hence "command" syntax).
+clear a b; % clear("a", "b")
+close all; % close("all")
+mkdir test; % mkdir("test")
+rmdir test; % rmdir("test")
+
+% Note that if a function takes no arguments, the parentheses may also be
+% omitted. (This is not true for anonymous functions, as we will see.)
+clc; % clc()
+figure; % figure()
+
+%% Anonymous functions
+% Anonymous functions are defined with the following syntax:
+%
+% @(arg1, arg2, ...) fun_body
+%
+% These allow us to conveniently define functions inline, and perform some
+% functional programming capabilities. In other programming languages, they may
+% be known as "arrow functions" or "lambda functions."
+cube = @(x) x.^3;
+parabaloid = @(x, y) x.^2 + y.^2;
+
+% Call these functions like any other function.
+cube(3);
+parabaloid(3, 4);
+
+% As the name suggests, anonymous functions don't have to be assigned names; by
+% themselves, they are "function values" or "function expressions" and can be
+% handled like any other expression. (I.e., "first-class" functions). Usually,
+% we assign it to a variable to name it, just as we do for most useful
+% expressions.
+@(x) x.^3;
+cube_fnval = ans;
+
+% Anonymous functions can capture variables from the surrounding scope. They
+% may exist anywhere an expression may exist (e.g., within the body of another
+% function).
+outside_variable = 3;
+capture_fn = @() outside_variable;
+outside_variable = 4;
+
+capture_fn; % What will this return?
+capture_fn(); % What will this return?
+
+% Lexical scoping and closures apply! This allows for some degree of functional
+% programming (e.g., function currying).
+curried_sum = @(x) @(y) x + y;
+add_two = curried_sum(2);
+add_two(5); % The binding x <- 2 is captured in
+ % add_two's lexical closure
+
+% add_two(2)(5) % Can't do this due to MATLAB's syntax,
+ % just like how you can't double-index
+ % an expression
+
+%% Example: Higher-order functions
+% An example of passing functions around like any other variable. (Functions
+% that accept other functions as parameters or return a function are called
+% "higher-order functions.")
+%
+% In this example, compose is the function composition operator. compose(f,g)
+% = (f o g). It is a HOF because it takes two functions as parameters, and it
+% returns a function.
+compose = @(f, g) @(x) f(g(x));
+add_three = @(x) x + 3;
+times_four = @(x) x * 4;
+x_times_four_plus_three = compose(add_three, times_four);
+
+x = 0:10;
+y = x_times_four_plus_three(x);
+plot(x, y);
+grid on;
+
+%% Normal Functions
+% "Normal functions" (so-called because they were the first type of function in
+% MATLAB) are defined using the syntax shown below, and are invoked in the same
+% way as anonymous functions.
+%
+% Note that normal functions cannot be passed around as variables, and this
+% makes a functional-programming style impossible using normal functions.
+%
+% Unlike anonymous functions, normal functions may include multiple statements,
+% may return multiple values, are invoked if appearing in an expression without
+% parentheses (unless you use a function handle), and do not capture variables
+% from the surrounding scope/workspace.
+%
+% There are two types of normal functions: "public" normal functions, which
+% must be placed at the beginning of a *.m file with the same name as the
+% function, or local/private normal functions, which may appear at the end of a
+% *.m file (after anything that is not a function). Both types of normal
+% functions must appear after any statements in the file; in other words, a
+% file defining a non-local function can only contain a list of function
+% definitions, the first of is public and the rest of which are private. As the
+% name suggests, local functions are only in scope in the file in which they
+% are defined, whereas public functions may be accessible if they are on the
+% MATLAB path.
+%
+% You don't need to understand all this right now; this brief explanation and
+% the MATLAB documentation can serve as a reference for your future MATLAB
+% endeavors.
+
+%% The MATLAB Path
+% In order to call a public function defined in another file, the function file
+% must be in the current folder or on the MATLAB path.
+matlabpath; % Print out MATLAB path
+
+%% Invoking regular functions
+% Note that this section is before the local function definition because the
+% local function must be defined after all statements in the script.
+%
+% eye() is a builtin function. Here we call it with no arguments, so the first
+% argument's value defaults to 1. distance() is a public function defined in
+% the file distance.m HelloWorld() is a local function defined below
+eye;
+distance(3, 4);
+HelloWorld("Jon", "Lam"); % Invoking HelloWorld and ignoring outputs
+
+%% Defining a local function
+function [res1, res2] = HelloWorld(name1, name2)
+ fprintf("Hello, world! %s %s\n", name1, name2);
+ res1 = 3;
+ res2 = [5 6];
+end
+
+%% Other fun stuff:
+% Function handles
+% `feval` on function handles and strings
+% Double-indexing using functions
+% Viewing source code using `edit`
diff --git a/lessons/lesson04/octave-log.txt b/lessons/lesson04/octave-log.txt
new file mode 100644
index 0000000..0fb44bd
--- /dev/null
+++ b/lessons/lesson04/octave-log.txt
@@ -0,0 +1,572 @@
+octave:2> z = zeros(3)
+z =
+
+ 0 0 0
+ 0 0 0
+ 0 0 0
+
+octave:3> mean(1:5)
+ans = 3
+octave:4> zeros(3)
+ans =
+
+ 0 0 0
+ 0 0 0
+ 0 0 0
+
+octave:5> zeros(3, 1)
+ans =
+
+ 0
+ 0
+ 0
+
+octave:6> help sum
+'sum' is a built-in function from the file libinterp/corefcn/data.cc
+
+ -- sum (X)
+ -- sum (X, DIM)
+ -- sum (..., "native")
+ -- sum (..., "double")
+ -- sum (..., "extra")
+ Sum of elements along dimension DIM.
+
+ If DIM is omitted, it defaults to the first non-singleton
+ dimension.
+
+ The optional "type" input determines the class of the variable used
+ for calculations. By default, operations on floating point inputs
+ (double or single) are performed in their native data type, while
+ operations on integer, logical, and character data types are
+ performed using doubles. If the argument "native" is given, then
+ the operation is performed in the same type as the original
+ argument.
+
+ For example:
+
+ sum ([true, true])
+ => 2
+ sum ([true, true], "native")
+ => true
+
+ If "double" is given the sum is performed in double precision even
+ for single precision inputs.
+
+ For double precision inputs, the "extra" option will use a more
+ accurate algorithm than straightforward summation. For single
+ precision inputs, "extra" is the same as "double". For all other
+ data type "extra" has no effect.
+
+ See also: cumsum, sumsq, prod.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:7> sum(eye(3))
+ans =
+
+ 1 1 1
+
+octave:8> eye(3)
+ans =
+
+Diagonal Matrix
+
+ 1 0 0
+ 0 1 0
+ 0 0 1
+
+octave:9> sum(eye(3))
+ans =
+
+ 1 1 1
+
+octave:10> sum(eye(3), 2)
+ans =
+
+ 1
+ 1
+ 1
+
+octave:11> sum(eye(3), 3)
+ans =
+
+ 1 0 0
+ 0 1 0
+ 0 0 1
+
+octave:12> hold on
+octave:13> close all
+octave:14> close("all")
+octave:15> hold("on")
+octave:16> close("all")
+octave:17> figure
+octave:18> figure()
+octave:19> close("all")
+octave:20> hist(1:10)
+octave:21> close all
+octave:22> a = hist(1:10)
+a =
+
+ 1 1 1 1 1 1 1 1 1 1
+
+octave:23> help find
+'find' is a built-in function from the file libinterp/corefcn/find.cc
+
+ -- IDX = find (X)
+ -- IDX = find (X, N)
+ -- IDX = find (X, N, DIRECTION)
+ -- [i, j] = find (...)
+ -- [i, j, v] = find (...)
+ Return a vector of indices of nonzero elements of a matrix, as a
+ row if X is a row vector or as a column otherwise.
+
+ To obtain a single index for each matrix element, Octave pretends
+ that the columns of a matrix form one long vector (like Fortran
+ arrays are stored). For example:
+
+ find (eye (2))
+ => [ 1; 4 ]
+
+ If two inputs are given, N indicates the maximum number of elements
+ to find from the beginning of the matrix or vector.
+
+ If three inputs are given, DIRECTION should be one of "first" or
+ "last", requesting only the first or last N indices, respectively.
+ However, the indices are always returned in ascending order.
+
+ If two outputs are requested, 'find' returns the row and column
+ indices of nonzero elements of a matrix. For example:
+
+ [i, j] = find (2 * eye (2))
+ => i = [ 1; 2 ]
+ => j = [ 1; 2 ]
+
+ If three outputs are requested, 'find' also returns a vector
+ containing the nonzero values. For example:
+
+ [i, j, v] = find (3 * eye (2))
+ => i = [ 1; 2 ]
+ => j = [ 1; 2 ]
+ => v = [ 3; 3 ]
+
+ If X is a multi-dimensional array of size m x n x p x ..., J
+ contains the column locations as if X was flattened into a
+ two-dimensional matrix of size m x (n + p + ...).
+
+ Note that this function is particularly useful for sparse matrices,
+ as it extracts the nonzero elements as vectors, which can then be
+ used to create the original matrix. For example:
+
+ sz = size (a);
+ [i, j, v] = find (a);
+ b = sparse (i, j, v, sz(1), sz(2));
+
+ See also: nonzeros.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:24> cube = @(x) x.^3;
+octave:25> cube
+cube =
+
+@(x) x .^ 3
+
+octave:26> cube(3)
+ans = 27
+octave:27> paraboloid = @(x, y) x.^2 + y.^2;
+octave:28> paraboloid(3, 4)
+ans = 25
+octave:29> @(x) x.^3
+ans =
+octave:35> a = 9
+a = 9
+octave:36> capture = @() a;
+octave:37> capture
+capture =
+
+@() a
+
+octave:38> capture()
+ans = 9
+octave:39> capture = @() 9
+capture =
+
+@() 9
+
+octave:40> capture()
+ans = 9
+octave:41> capture = @() a;
+octave:42> capture()
+ans = 9
+octave:43> a = 3
+a = 3
+octave:44> capture()
+ans = 9
+octave:45> n = 4
+n = 4
+octave:46> n = 2
+n = 2
+octave:47> pown = @(x) x.^n
+pown =
+
+@(x) x .^ n
+
+octave:48> pown(3)
+ans = 9
+octave:49> n = 4
+n = 4
+octave:50> pown(3)
+ans = 9
+octave:51>
+^[[200~compose = @(f, g) @(x) f(g(x));^[[201~octave:51>
+octave:51>
+octave:51>
+octave:51>
+octave:51> compose = @(f, g) @(x) f(g(x));
+octave:52>
+octave:52>
+octave:52>
+octave:52>
+octave:52>
+octave:52> add_three = @(x) x + 3;
+times_four = @(x) x * 4;
+x_times_four_plus_three = compose(add_three, times_four);
+octave:55> x_times_four_plus_three
+x_times_four_plus_three =
+
+@(x) f (g (x))
+
+octave:56> function y = times_four_plus_three(x)
+> x2 = x.*4;
+> y = x2 + 3;
+> end
+octave:57> times_four_plus_three(1)
+ans = 7
+octave:58> answer = times_four_plus_three(1)
+answer = 7
+octave:59> function [y, z] = mystery_math(p, q)
+> y = p + q;
+> z = p - q;
+> disp('did the math!');
+> end
+octave:60> mystery_math(3, 4)
+did the math!
+ans = 7
+octave:61> [val1, val2] = mystery_math(3, 4)
+did the math!
+val1 = 7
+val2 = -1
+octave:62>
+
+[cat@lazarus:~/classes/ece210-materials/2024-van-west/lessons/lesson04]
+$ ls
+BasicClass.m datatypes.m for_loops.m lecture.txt
+controlflow.m distance.m funnnnnn.m
+[cat@lazarus:~/classes/ece210-materials/2024-van-west/lessons/lesson04]
+$ cat distance.m
+% In a file like this one (which only defines normal functions, and in which
+% the first function is hononymous with the filename), the first function is a
+% public function, and all other functions are local helper functions.
+%
+% Note that you DO NOT want to put clc; clear; close all; at the top of a file
+% like this; it only makes sense to do this at the top of script files when you
+% want to clean your workspace!
+
+% Calculate Euclidean distance
+% A public function, callable by other files and from the Command Window if
+% this is in the MATLAB path.
+function d = distance(x, y)
+ d = sqrt(sq(x) + sq(y));
+end
+
+% A local helper function. Only accessible within this file.
+function x = sq(x)
+ x = x * x;
+end
+[cat@lazarus:~/classes/ece210-materials/2024-van-west/lessons/lesson04]
+$ octave
+GNU Octave, version 7.3.0
+Copyright (C) 1993-2022 The Octave Project Developers.
+This is free software; see the source code for copying conditions.
+There is ABSOLUTELY NO WARRANTY; not even for MERCHANTABILITY or
+FITNESS FOR A PARTICULAR PURPOSE. For details, type 'warranty'.
+
+Octave was configured for "x86_64-pc-linux-gnu".
+
+Additional information about Octave is available at https://www.octave.org.
+
+Please contribute if you find this software useful.
+For more information, visit https://www.octave.org/get-involved.html
+
+Read https://www.octave.org/bugs.html to learn how to submit bug reports.
+For information about changes from previous versions, type 'news'.
+
+octave:1> who
+octave:2> distance(3, 4)
+ans = 5
+octave:3> path
+
+Octave's search path contains the following directories:
+
+.
+/usr/lib/x86_64-linux-gnu/octave/7.3.0/site/oct/x86_64-pc-linux-gnu
+/usr/lib/x86_64-linux-gnu/octave/site/oct/api-v57/x86_64-pc-linux-gnu
+/usr/lib/x86_64-linux-gnu/octave/site/oct/x86_64-pc-linux-gnu
+/usr/share/octave/7.3.0/site/m
+/usr/share/octave/site/api-v57/m
+/usr/share/octave/site/m
+/usr/share/octave/site/m/startup
+/usr/lib/x86_64-linux-gnu/octave/7.3.0/oct/x86_64-pc-linux-gnu
+/usr/share/octave/7.3.0/m
+/usr/share/octave/7.3.0/m/audio
+/usr/share/octave/7.3.0/m/deprecated
+/usr/share/octave/7.3.0/m/elfun
+/usr/share/octave/7.3.0/m/general
+/usr/share/octave/7.3.0/m/geometry
+/usr/share/octave/7.3.0/m/gui
+/usr/share/octave/7.3.0/m/help
+/usr/share/octave/7.3.0/m/image
+/usr/share/octave/7.3.0/m/io
+/usr/share/octave/7.3.0/m/java
+/usr/share/octave/7.3.0/m/legacy
+/usr/share/octave/7.3.0/m/linear-algebra
+/usr/share/octave/7.3.0/m/miscellaneous
+/usr/share/octave/7.3.0/m/ode
+/usr/share/octave/7.3.0/m/optimization
+/usr/share/octave/7.3.0/m/path
+/usr/share/octave/7.3.0/m/pkg
+/usr/share/octave/7.3.0/m/plot
+/usr/share/octave/7.3.0/m/plot/appearance
+/usr/share/octave/7.3.0/m/plot/draw
+/usr/share/octave/7.3.0/m/plot/util
+/usr/share/octave/7.3.0/m/polynomial
+/usr/share/octave/7.3.0/m/prefs
+/usr/share/octave/7.3.0/m/profiler
+/usr/share/octave/7.3.0/m/set
+/usr/share/octave/7.3.0/m/signal
+/usr/share/octave/7.3.0/m/sparse
+/usr/share/octave/7.3.0/m/specfun
+/usr/share/octave/7.3.0/m/special-matrix
+/usr/share/octave/7.3.0/m/startup
+/usr/share/octave/7.3.0/m/statistics
+/usr/share/octave/7.3.0/m/strings
+/usr/share/octave/7.3.0/m/testfun
+/usr/share/octave/7.3.0/m/time
+/usr/share/octave/7.3.0/m/web
+/usr/share/octave/7.3.0/data
+
+octave:4> power_to_dB = @(p) 10*log10(p);
+octave:5> dB_to_power = @(db) 10.^(db./10);
+octave:6> power_to_dB(10)
+ans = 10
+octave:7> power_to_dB(1000)
+ans = 30
+octave:8> power_to_dB(100)
+ans = 20
+octave:9> dB_to_power(10)
+ans = 10
+octave:10> dB_to_power(-10)
+ans = 0.1000
+octave:11> for v = [1 2 5 7]
+> v
+> end
+v = 1
+v = 2
+v = 5
+v = 7
+octave:12> for i = 1:10
+> i + 3
+> end
+ans = 4
+ans = 5
+ans = 6
+ans = 7
+ans = 8
+ans = 9
+ans = 10
+ans = 11
+ans = 12
+ans = 13
+octave:13> total = 0
+total = 0
+octave:14> for i = 1:10
+> total = total + i
+> end
+total = 1
+total = 3
+total = 6
+total = 10
+total = 15
+total = 21
+total = 28
+total = 36
+total = 45
+total = 55
+octave:15> sum(1:10)
+ans = 55
+octave:16> 1 < 2
+ans = 1
+octave:17> who
+Variables visible from the current scope:
+
+ans dB_to_power i power_to_dB total v
+
+octave:18> whos
+Variables visible from the current scope:
+
+variables in scope: top scope
+
+ Attr Name Size Bytes Class
+ ==== ==== ==== ===== =====
+ ans 1x1 1 logical
+ dB_to_power 1x1 0 function_handle
+ i 1x1 8 double
+ power_to_dB 1x1 0 function_handle
+ total 1x1 8 double
+ v 1x1 8 double
+
+Total is 6 elements using 25 bytes
+
+octave:19> if 1 < 2
+> disp('math still works today');
+> else
+> disp('oh no');
+> end
+math still works today
+octave:20> i = 0
+i = 0
+octave:21> while i < 10
+> i = i + 1
+> end
+i = 1
+i = 2
+i = 3
+i = 4
+i = 5
+i = 6
+i = 7
+i = 8
+i = 9
+i = 10
+octave:22> try
+> a = [1 2; 3 4];
+> b = [1 2 3 4];
+> c = a*b;
+> catch err
+> disp('yo! don't do that!')
+error: parse error:
+
+ syntax error
+
+>>> disp('yo! don't do that!')
+ ^
+octave:22> try
+> a = [1 2; 3 4];
+> b = [1 2 3 4];
+> c = a*b;
+> catch err
+> disp('yo! do not do that!')
+> end
+yo! do not do that!
+octave:23> a
+a =
+
+ 1 2
+ 3 4
+
+octave:24> b
+b =
+
+ 1 2 3 4
+
+octave:25> a*b
+error: operator *: nonconformant arguments (op1 is 2x2, op2 is 1x4)
+octave:26> clear
+octave:27> whos
+octave:28> a = 10
+a = 10
+octave:29> class(a)
+ans = double
+octave:30> b = int8(a)
+b = 10
+octave:31> class(b)
+ans = int8
+octave:32> b + .5
+ans = 11
+octave:33> whos
+Variables visible from the current scope:
+
+variables in scope: top scope
+
+ Attr Name Size Bytes Class
+ ==== ==== ==== ===== =====
+ a 1x1 8 double
+ ans 1x1 1 int8
+ b 1x1 1 int8
+
+Total is 3 elements using 10 bytes
+
+octave:34>
+octave:34>
+octave:34>
+^[[200~imshow(uint8(rand(128, 128, 3)));^[[201~octave:34>
+octave:34>
+octave:34> imshow(uint8(rand(128, 128, 3)));
+octave:35> close all
+octave:36>
+^[[200~imshow(double(rand(128, 128, 3)));^[[201~octave:36>
+octave:36>
+octave:36>
+octave:36>
+octave:36> imshow(double(rand(128, 128, 3)));
+octave:37> close all
+octave:38> imshow(uint8(rand(128, 128, 3)));
+octave:39> close all
+octave:40> imshow(uint8(rand(128, 128, 3)));
+octave:40> eps
+ans = 2.2204e-16
+octave:41> eps(1)
+ans = 2.2204e-16
+octave:42> eps(.001)
+ans = 2.1684e-19
+octave:43> eps(10000)
+ans = 1.8190e-12
+octave:44> eps(1e100)
+ans = 1.9427e+84
+octave:45> 10^(10^10)
+ans = Inf
+octave:46> a = 5
+a = 5
+octave:47> b = "hello!"
+b = hello!
+octave:48> s = struct('a', 5, 'b', "hello!")
+s =
+
+ scalar structure containing the fields:
+
+ a = 5
+ b = hello!
+
+octave:49> s
+s =
+
+ scalar structure containing the fields:
+
+ a = 5
+ b = hello!
+
+octave:50> s.a
+ans = 5
+octave:51> s.b
+ans = hello!
+octave:52>
diff --git a/lessons/lesson05/indexing.m b/lessons/lesson05/indexing.m
new file mode 100644
index 0000000..a1a2da6
--- /dev/null
+++ b/lessons/lesson05/indexing.m
@@ -0,0 +1,127 @@
+%% Lesson 3b: Advanced Indexing
+%
+% Objectives:
+% - Review basic vector indexing
+% - Introduce multidimensional indexing and linear indexing
+% - Introduce logical indexing
+clear; clc; close all;
+
+%% Review of basic vector indexing
+% Indexed expressions can appear on the right-hand side of an assignment or the
+% left-hand side (rvalues and lvalues in C).
+x = [2 4 8 16 32 64 128];
+x(3);
+x(4:6);
+x(:);
+x(end);
+x(2) = -4;
+
+%% Vectors indexing vectors
+% You can use a vector to index another vector. This can be useful if there is
+% a specific index pattern you want from a vector.
+
+x([1 3 5 7]); % Returns a vector with the 1st, 3rd, 5th, and 7th
+ % entries of x
+x(1:2:7); % Same thing uses the colon operator to create the
+ % index vector
+x([2:4 3:6]);
+x([1:4]) = 7; % Changes the values from the 1st and 4th entries
+ % of x to 7
+
+%% Indexing matrices with 2 subscripts
+% You can index a matrix using two indices, separated by a comma.
+A = [16 2 3 13; 5 11 10 8; 9 7 6 12; 4 14 15 1];
+
+A(1,1); % Returns entry in first row and first column
+A(1:3,1:2); % Returns the entries in the first three rows AND
+ % first two columns
+A(end,:);
+
+% What if I want to index the (2,1), (3,2), (4,4) entries?
+
+A([2 3 4],[1 2 4]); % Will this work?
+
+%% Linear Indexing
+% Instead you should index linearly! Linear indexing only uses one
+% subscript
+
+A(:); % What does this return and in what form and order?
+
+reshape(A, [], 1); % What about this?
+
+A(14); % What value does this return?
+
+
+%% Non-rectangular indices
+% Now that we know about linear indexing, what values does this return?
+A([2 7 16]);
+
+% Luckily enough, MATLAB has a function that would calculate the linear
+% index for you!
+ind = sub2ind(size(A),[2 3 4],[1 2 4]);
+A(ind);
+
+% Note that sub2ind follows a common naming convention in MATLAB for converting
+% one type to another. For example, there is also ind2sub, str2int, int2str,
+% zp2tf, tf2zp, etc.
+
+%% Logical Indexing
+% Logical indexing allows us to use boolean logic to build more complex
+% indices. This is based off of boolean logic: we have two boolean values TRUE
+% (represented by 1) and FALSE (represented by 0), and the boolean operators
+% AND, OR, and NOT.
+%
+% Logical scalar and matrix literals can be constructed from double and matrix
+% literals using the logical() function, or they may be generated from other
+% operations. (Note that logical and double values have different data types.)
+
+B = eye(4); % Regular identity matrix
+C = logical([1 1 1 1; 1 0 0 0; 1 0 0 0; 1 0 0 0]); % Logical entries
+islogical(B);
+
+islogical(C);
+
+%% Some logical operators
+B = logical(B); % Converts B into a logical array
+
+B & C; % and(B,C) is equivalent to B&C
+B | C; % or(B,C) is equivalent to B|C
+~(B & C); % not(and(B,C)) is equivalent to ~(B&C)
+not(B & C); % you can use both representations intermittenly
+
+%% Functions that generate logical values
+% The following functions are analogous to the ones() and zeros() functions
+% for double matrices.
+true(2,5); % creates a 2x5 logical matrix of trues(1s)
+false(2,5);
+
+%% Generating logical values from predicates
+% Relational (comparison) operators generate logical values. Relational
+% operators on matrices will perform the operation element-wise and return a
+% logical matrix.
+%
+% This is not anything really special, since we've seen element-wise operations
+% on matrices. The only change is that instead of functions that return
+% numbers, we have functions that return booleans (predicates).
+D = [1 2 3; 2 3 4; 3 4 5];
+E = [1 5 6; 9 2 8; 7 3 4];
+
+D == 3; % Returns true every place there is a 3.
+
+D < E; % Elementwise comparison
+D < 3; % Elementwise comparison against a scalar.
+ % (Note: this is broadcasting!)
+D < [7; 4; 1]; % Another example of broadcasting: comparison
+ % of each row against a scalar.
+
+%% Logical indexing
+% Logical indexing uses a logical value as the index. Elements in the indexed
+% matrix where the logical index is TRUE will be returned, and FALSE values
+% will be disregarded. The returned matrix will always be a column vector,
+% similar to indexing with (:).
+A(A > 12); % returns entries of A which are greater
+ % than 12 in a column vector
+
+% To return the linear indices of the TRUE values in a logical matrix.
+find(A > 12);
+A(find(A > 12)); % Gives the same results as above
diff --git a/lessons/lesson05/meshgrid_broadcasting.m b/lessons/lesson05/meshgrid_broadcasting.m
new file mode 100644
index 0000000..76bbcf0
--- /dev/null
+++ b/lessons/lesson05/meshgrid_broadcasting.m
@@ -0,0 +1,44 @@
+%% Lesson 3a: Meshes & Broadcasting
+
+%% Meshgrid
+% Meshgrid is quite hard to understand. Think of it as a way to replicate
+% arrays, like the following example:
+a = 1:3;
+b = 1:5;
+[A,B] = meshgrid(a,b);
+
+%% Using meshgrid for functions of multiple variables
+% You have created two arrays A and B, note that in A, the vector a is copied
+% row-wise, while the vector b is transposed and copied column-wise. This is
+% useful, because when you lay one above the other, you essentially create a
+% CARTESIAN PRODUCT, and it is useful when we need to plot a 3D graph.
+%
+% Here is a more complicated example to show you when meshgrid is useful
+a1 = -2:0.25:2;
+b1 = a1;
+[A1,B1] = meshgrid(a1);
+
+% Here we plot the surface of f(x) = x*exp^(x.^2+y.^2)
+F = A1.*exp(-A1.^2-B1.^2);
+surf(A1, B1, F);
+
+%% Broadcasting: an alternative to meshgrid
+% Broadcasting, like meshgrid, can be used as a way to write functions of
+% multiple variables, without generating the intermediate matrices A and B.
+% The way this works is that operations performed on a set of N orthogonal
+% vectors will automatically generate an N-dimensional result.
+%
+% See the following example, which recreates the above example. Note that b1 is
+% transposed.
+F2 = a1.*exp(-a1.^2-(b1.').^2);
+surf(A1, B1, F2);
+
+% Check that this matches the previous result.
+error = rms(F - F2, 'all');
+
+% Note: broadcasting doesn't generate the domains A1 and B1, so meshgrid() is
+% more useful when we need to supply the domain to a function like mesh() or
+% surf(). But when we only need the result, broadcasting is somewhat simpler.
+%
+% For homework assignments that require functions of multiple variables, use
+% whichever method is more intuitive to you.
diff --git a/lessons/lesson05/octave-log.txt b/lessons/lesson05/octave-log.txt
new file mode 100644
index 0000000..bc24213
--- /dev/null
+++ b/lessons/lesson05/octave-log.txt
@@ -0,0 +1,3902 @@
+octave:2> X, Y = meshgrid(linspace(-2, 2));
+error: 'X' undefined near line 1, column 1
+octave:3> [X, Y] = meshgrid(linspace(-2, 2));
+octave:4> size(X)
+ans =
+
+ 100 100
+
+octave:5> z = p - q;
+octave:5> size(Y)
+ans =
+
+ 100 100
+
+octave:6> Z = X.^2 + Y.^2;
+octave:7> figure
+octave:8> plot3(X, Y, Z)
+octave:9> surf(X, Y, Z)
+octave:10> close all
+octave:11> a = 1:3
+a =
+
+ 1 2 3
+
+octave:12> b = 1:5
+b =
+
+ 1 2 3 4 5
+
+octave:13> [A, B] = meshgrid(a, b);
+octave:14> A
+A =
+
+ 1 2 3
+ 1 2 3
+ 1 2 3
+ 1 2 3
+ 1 2 3
+
+octave:15> B
+B =
+
+ 1 1 1
+ 2 2 2
+ 3 3 3
+ 4 4 4
+ 5 5 5
+
+octave:16> [A, B] = meshgrid(-2:2, 1:4);
+octave:17> A
+A =
+
+ -2 -1 0 1 2
+ -2 -1 0 1 2
+ -2 -1 0 1 2
+ -2 -1 0 1 2
+
+octave:18> B
+B =
+
+ 1 1 1 1 1
+ 2 2 2 2 2
+ 3 3 3 3 3
+ 4 4 4 4 4
+
+octave:19> T = eye(3)
+T =
+
+Diagonal Matrix
+
+ 1 0 0
+ 0 1 0
+ 0 0 1
+
+octave:20> T + 1
+ans =
+
+ 2 1 1
+ 1 2 1
+ 1 1 2
+
+octave:21> T + [1 2 3]
+error: operator +: nonconformant arguments (op1 is 3x3, op2 is 1x3)
+octave:22> T .+ [1 2 3]
+warning: the '.+' operator was deprecated in version 7 and will not be allowed in a fu
+ture version of Octave; please use '+' instead
+error: operator +: nonconformant arguments (op1 is 3x3, op2 is 1x3)
+octave:23> T + 1
+ans =
+
+ 2 1 1
+ 1 2 1
+ 1 1 2
+
+octave:24> [1 2 3] + [1; 2; 3]
+ans =
+
+ 2 3 4
+ 3 4 5
+ 4 5 6
+
+octave:25> [1 2 3] + [1; 2; 3] + T
+ans =
+
+ 3 3 4
+ 3 5 5
+ 4 5 7
+
+octave:26> T + [1; 2; 3]
+error: operator +: nonconformant arguments (op1 is 3x3, op2 is 3x1)
+octave:27> T.*[1; 2; 3]
+ans =
+
+ 1 0 0
+ 0 2 0
+ 0 0 3
+
+octave:28> T*[1; 2; 3]
+ans =
+
+ 1
+ 2
+ 3
+
+octave:29> T.*[1; 2; 3]
+ans =
+
+ 1 0 0
+ 0 2 0
+ 0 0 3
+
+octave:30> [1; 2; 3]
+ans =
+
+ 1
+ 2
+ 3
+
+octave:31> [1; 2; 3].*ones(3)
+ans =
+
+ 1 1 1
+ 2 2 2
+ 3 3 3
+
+octave:32> t = linspace(0, 1, 100);
+octave:33> omega = 4
+omega = 4
+octave:34> signal = sin(omega.*t);
+octave:35> figure
+octave:36> plot(t, signal)
+octave:37> omega = 44
+omega = 44
+octave:38> signal = sin(omega.*t);
+octave:39> plot(t, signal)
+octave:40> noise_db = (0:5).';
+octave:41> noise_db
+noise_db =
+
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+
+octave:42> close all
+octave:43> noise_amp = 10.^(noise_db/20)
+noise_amp =
+
+ 1.0000
+ 1.1220
+ 1.2589
+ 1.4125
+ 1.5849
+ 1.7783
+
+octave:44> size(sig
+sighup_dumps_octave_core signal sigquit_dumps_octave_core
+sign signbit sigterm_dumps_octave_core
+octave:44> size(signal)
+ans =
+
+ 1 100
+
+octave:45> noise = randn(size(signal));
+octave:46> size(noise)
+ans =
+
+ 1 100
+
+octave:47> plot(t, noise)
+octave:48> close all
+octave:49> noisy_signal = signal + noise_amp.*noise;
+octave:50> size(noisy_signal)
+ans =
+
+ 6 100
+
+octave:51> figure
+octave:52> plot(t, noisy_signal(1, :))
+octave:53> plot(t, noisy_signal(2, :))
+octave:54> plot(t, noisy_signal(3, :))
+octave:55> plot(t, noisy_signal(4, :))
+octave:56> plot(t, noisy_signal)
+octave:57> noise_db = (-30:5:0)
+noise_db =
+
+ -30 -25 -20 -15 -10 -5 0
+
+(reverse-i-search)`noise_amp': noisy_signal = signal + noise_amp.*noise;
+octave:58> noise_amp = 10.^(noise_db/20)
+noise_amp =
+
+ 0.031623 0.056234 0.100000 0.177828 0.316228 0.562341 1.000000
+
+octave:59> noisy_signal = signal + noise_amp.*noise;
+error: product: nonconformant arguments (op1 is 1x7, op2 is 1x100)
+octave:60> noisy_signal = signal + (noise_amp.').*noise;
+octave:61> plot(t, noisy_signal)
+octave:62> close all
+octave:63> A
+A =
+
+ -2 -1 0 1 2
+ -2 -1 0 1 2
+ -2 -1 0 1 2
+ -2 -1 0 1 2
+
+octave:64> A(:, 2:3)
+ans =
+
+ -1 0
+ -1 0
+ -1 0
+ -1 0
+
+octave:65> A(:)
+ans =
+
+ -2
+ -2
+ -2
+ -2
+ -1
+ -1
+ -1
+ -1
+ 0
+ 0
+ 0
+ 0
+ 1
+ 1
+ 1
+ 1
+ 2
+ 2
+ 2
+ 2
+
+octave:66> A(1, 1)
+ans = -2
+octave:67> A(1, end)
+ans = 2
+octave:68> A([1 3], [2 4])
+ans =
+
+ -1 1
+ -1 1
+
+octave:69> A([1 3], linspace(1, 2, 3))
+error: A(_,1.5): subscripts must be either integers 1 to (2^63)-1 or logicals
+octave:70> A(2)
+ans = -2
+octave:71> A(5)
+ans = -1
+octave:72> boolean = logical(1)
+boolean = 1
+octave:73> whos
+Variables visible from the current scope:
+
+variables in scope: top scope
+
+ Attr Name Size Bytes Class
+ ==== ==== ==== ===== =====
+ A 4x5 160 double
+ B 4x5 160 double
+ T 3x3 24 double
+ X 100x100 80000 double
+ Y 100x100 80000 double
+ Z 100x100 80000 double
+ a 1x3 24 double
+ ans 1x1 8 double
+ b 1x5 24 double
+ boolean 1x1 1 logical
+ noise 1x100 800 double
+ noise_amp 1x7 56 double
+ noise_db 1x7 24 double
+ noisy_signal 7x100 5600 double
+ omega 1x1 8 double
+ signal 1x100 800 double
+ t 1x100 800 double
+
+Total is 31074 elements using 248489 bytes
+
+octave:74> M = eye(4)
+M =
+
+Diagonal Matrix
+
+ 1 0 0 0
+ 0 1 0 0
+ 0 0 1 0
+ 0 0 0 1
+
+octave:75> islogical(M)
+ans = 0
+octave:76> islogical(ans)
+ans = 1
+octave:77> logical(M)
+ans =
+
+ 1 0 0 0
+ 0 1 0 0
+ 0 0 1 0
+ 0 0 0 1
+
+octave:78> islogical(logical(M))
+ans = 1
+octave:79> B = logical(eye(4))
+B =
+
+ 1 0 0 0
+ 0 1 0 0
+ 0 0 1 0
+ 0 0 0 1
+
+octave:80> C = logical([1 1 1 1; 1 0 0 0; 1 0 0 0; 1 0 0 0])
+C =
+
+ 1 1 1 1
+ 1 0 0 0
+ 1 0 0 0
+ 1 0 0 0
+
+octave:81> double(C)
+ans =
+
+ 1 1 1 1
+ 1 0 0 0
+ 1 0 0 0
+ 1 0 0 0
+
+octave:82> C.*2
+ans =
+
+ 2 2 2 2
+ 2 0 0 0
+ 2 0 0 0
+ 2 0 0 0
+
+octave:83> B & C
+ans =
+
+ 1 0 0 0
+ 0 0 0 0
+ 0 0 0 0
+ 0 0 0 0
+
+octave:84> B | C
+ans =
+
+ 1 1 1 1
+ 1 1 0 0
+ 1 0 1 0
+ 1 0 0 1
+
+octave:85> ~B
+ans =
+
+ 0 1 1 1
+ 1 0 1 1
+ 1 1 0 1
+ 1 1 1 0
+
+octave:86> ~(B | C)
+ans =
+
+ 0 0 0 0
+ 0 0 1 1
+ 0 1 0 1
+ 0 1 1 0
+
+octave:87> xor(B, C)
+ans =
+
+ 0 1 1 1
+ 1 1 0 0
+ 1 0 1 0
+ 1 0 0 1
+
+octave:88> x = linspace(-2, 2, 10)
+x =
+
+ Columns 1 through 9:
+
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+
+ Column 10:
+
+ 2.0000
+
+octave:89> x = x.'
+x =
+
+ -2.0000
+ -1.5556
+ -1.1111
+ -0.6667
+ -0.2222
+ 0.2222
+ 0.6667
+ 1.1111
+ 1.5556
+ 2.0000
+
+octave:90> sum(x)
+ans = 6.6613e-16
+octave:91> sum(x(x > 0))
+ans = 5.5556
+octave:92> x > 0
+ans =
+
+ 0
+ 0
+ 0
+ 0
+ 0
+ 1
+ 1
+ 1
+ 1
+ 1
+
+octave:93> x(x > 0)
+ans =
+
+ 0.2222
+ 0.6667
+ 1.1111
+ 1.5556
+ 2.0000
+
+octave:94> sum(x(x > 0))
+ans = 5.5556
+octave:95> sum(x(x.^2 > 1))
+ans = 6.6613e-16
+octave:96> x(x.^2 > 1)
+ans =
+
+ -2.0000
+ -1.5556
+ -1.1111
+ 1.1111
+ 1.5556
+ 2.0000
+
+octave:97> x.^2
+ans =
+
+ 4.000000
+ 2.419753
+ 1.234568
+ 0.444444
+ 0.049383
+ 0.049383
+ 0.444444
+ 1.234568
+ 2.419753
+ 4.000000
+
+octave:98> x.^2 > 1
+ans =
+
+ 1
+ 1
+ 1
+ 0
+ 0
+ 0
+ 0
+ 1
+ 1
+ 1
+
+octave:99> X, Y = meshgrid(linspace(x))
+X =
+
+ Columns 1 through 7:
+
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+
+ Columns 8 through 14:
+
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
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+
+ Columns 15 through 21:
+
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+
+ Columns 22 through 28:
+
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+ Columns 50 through 56:
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+
+ Columns 57 through 63:
+
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+
+ Columns 64 through 70:
+
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+
+ Columns 71 through 77:
+
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+
+ Columns 78 through 84:
+
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+
+ Columns 85 through 91:
+
+ 1.393939 1.434343 1.474747 1.515152 1.555556 1.595960 1.636364
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+
+ Columns 92 through 98:
+
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+ 1.676768 1.717172 1.757576 1.797980 1.838384 1.878788 1.919192
+
+ Columns 99 and 100:
+
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+ 1.959596 2.000000
+
+error: Invalid call to linspace. Correct usage is:
+
+ -- linspace (START, END)
+ -- linspace (START, END, N)
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:100> X, Y = meshgrid(x)
+X =
+
+ Columns 1 through 7:
+
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
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+ -2.000000 -1.959596 -1.919192 -1.878788 -1.838384 -1.797980 -1.757576
+
+ Columns 8 through 14:
+
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+ -1.717172 -1.676768 -1.636364 -1.595960 -1.555556 -1.515152 -1.474747
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+
+ Columns 15 through 21:
+
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+ -1.434343 -1.393939 -1.353535 -1.313131 -1.272727 -1.232323 -1.191919
+
+ Columns 22 through 28:
+
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+ Columns 36 through 42:
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+ Columns 43 through 49:
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+
+ Columns 50 through 56:
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+
+ Columns 57 through 63:
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+
+ Columns 64 through 70:
+
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+
+ Columns 71 through 77:
+
+ 0.828283 0.868687 0.909091 0.949495 0.989899 1.030303 1.070707
+ 0.828283 0.868687 0.909091 0.949495 0.989899 1.030303 1.070707
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+
+ Columns 78 through 84:
+
+ 1.111111 1.151515 1.191919 1.232323 1.272727 1.313131 1.353535
+ 1.111111 1.151515 1.191919 1.232323 1.272727 1.313131 1.353535
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+ 1.111111 1.151515 1.191919 1.232323 1.272727 1.313131 1.353535
+
+ Columns 85 through 91:
+
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+
+ Columns 92 through 98:
+
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+
+ Columns 99 and 100:
+
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+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+ 1.959596 2.000000
+
+Y =
+
+ Columns 1 through 9:
+
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+
+ Column 10:
+
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+
+octave:101> size(x)
+ans =
+
+ 10 1
+
+octave:102> size(X)
+ans =
+
+ 100 100
+
+octave:103> size(Y)
+ans =
+
+ 10 10
+
+octave:104> [X, Y] = meshgrid(x)
+X =
+
+ Columns 1 through 9:
+
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+
+ Column 10:
+
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+
+Y =
+
+ Columns 1 through 9:
+
+ -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000
+ -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556
+ -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111
+ -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667
+ -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222
+ 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222
+ 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667
+ 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111
+ 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556
+ 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
+
+ Column 10:
+
+ -2.0000
+ -1.5556
+ -1.1111
+ -0.6667
+ -0.2222
+ 0.2222
+ 0.6667
+ 1.1111
+ 1.5556
+ 2.0000
+
+octave:105> size(X)
+ans =
+
+ 10 10
+
+octave:106> size(Y)
+ans =
+
+ 10 10
+
+octave:107> X
+X =
+
+ Columns 1 through 9:
+
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+ -2.0000 -1.5556 -1.1111 -0.6667 -0.2222 0.2222 0.6667 1.1111 1.5556
+
+ Column 10:
+
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+ 2.0000
+
+octave:108> Y
+Y =
+
+ Columns 1 through 9:
+
+ -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000 -2.0000
+ -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556 -1.5556
+ -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111 -1.1111
+ -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667 -0.6667
+ -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222 -0.2222
+ 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222 0.2222
+ 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667 0.6667
+ 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111 1.1111
+ 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556 1.5556
+ 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
+
+ Column 10:
+
+ -2.0000
+ -1.5556
+ -1.1111
+ -0.6667
+ -0.2222
+ 0.2222
+ 0.6667
+ 1.1111
+ 1.5556
+ 2.0000
+
+octave:109> X.^2 + Y.^2
+ans =
+
+ Columns 1 through 7:
+
+ 8.000000 6.419753 5.234568 4.444444 4.049383 4.049383 4.444444
+ 6.419753 4.839506 3.654321 2.864198 2.469136 2.469136 2.864198
+ 5.234568 3.654321 2.469136 1.679012 1.283951 1.283951 1.679012
+ 4.444444 2.864198 1.679012 0.888889 0.493827 0.493827 0.888889
+ 4.049383 2.469136 1.283951 0.493827 0.098765 0.098765 0.493827
+ 4.049383 2.469136 1.283951 0.493827 0.098765 0.098765 0.493827
+ 4.444444 2.864198 1.679012 0.888889 0.493827 0.493827 0.888889
+ 5.234568 3.654321 2.469136 1.679012 1.283951 1.283951 1.679012
+ 6.419753 4.839506 3.654321 2.864198 2.469136 2.469136 2.864198
+ 8.000000 6.419753 5.234568 4.444444 4.049383 4.049383 4.444444
+
+ Columns 8 through 10:
+
+ 5.234568 6.419753 8.000000
+ 3.654321 4.839506 6.419753
+ 2.469136 3.654321 5.234568
+ 1.679012 2.864198 4.444444
+ 1.283951 2.469136 4.049383
+ 1.283951 2.469136 4.049383
+ 1.679012 2.864198 4.444444
+ 2.469136 3.654321 5.234568
+ 3.654321 4.839506 6.419753
+ 5.234568 6.419753 8.000000
+
+octave:110> X.^2 + Y.^2 < 2^2
+ans =
+
+ 0 0 0 0 0 0 0 0 0 0
+ 0 0 1 1 1 1 1 1 0 0
+ 0 1 1 1 1 1 1 1 1 0
+ 0 1 1 1 1 1 1 1 1 0
+ 0 1 1 1 1 1 1 1 1 0
+ 0 1 1 1 1 1 1 1 1 0
+ 0 1 1 1 1 1 1 1 1 0
+ 0 1 1 1 1 1 1 1 1 0
+ 0 0 1 1 1 1 1 1 0 0
+ 0 0 0 0 0 0 0 0 0 0
+
+octave:111> X.^2 + Y.^2 < 1.5^2
+ans =
+
+ 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 1 1 1 1 0 0 0
+ 0 0 1 1 1 1 1 1 0 0
+ 0 0 1 1 1 1 1 1 0 0
+ 0 0 1 1 1 1 1 1 0 0
+ 0 0 1 1 1 1 1 1 0 0
+ 0 0 0 1 1 1 1 0 0 0
+ 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0
+
+octave:112> X(X.^2 + Y.^2 < 1.5^2)
+ans =
+
+ -1.1111
+ -1.1111
+ -1.1111
+ -1.1111
+ -0.6667
+ -0.6667
+ -0.6667
+ -0.6667
+ -0.6667
+ -0.6667
+ -0.2222
+ -0.2222
+ -0.2222
+ -0.2222
+ -0.2222
+ -0.2222
+ 0.2222
+ 0.2222
+ 0.2222
+ 0.2222
+ 0.2222
+ 0.2222
+ 0.6667
+ 0.6667
+ 0.6667
+ 0.6667
+ 0.6667
+ 0.6667
+ 1.1111
+ 1.1111
+ 1.1111
+ 1.1111
+
+octave:113> imshow(X.^2 + Y.^2 < 1.5^2)
+octave:114> close all
+octave:115> x = linspace(-2, 2, 1e4);
+octave:116> y = x.';
+octave:117> size(x.^2 + y.^2)
+ans =
+
+ 10000 10000
+
+octave:118> x = linspace(-2, 2, 1e3);
+octave:119> y = x.';
+octave:120> size(x.^2 + y.^2)
+ans =
+
+ 1000 1000
+
+octave:121> size(x.^2 + y.^2 < 1.5^2)
+ans =
+
+ 1000 1000
+
+octave:122> islogical(x.^2 + y.^2 < 1.5^2)
+ans = 1
+octave:123> imshow(x.^2 + y.^2 < 1.5^2)
+octave:124> close all
+octave:125> circle_1 = x.^2 + y.^2 < 1.5^2;
+octave:126> circle_2 = (x - .2).^2 + (y - .2).^2 < 1.5^2;
+octave:127> figure
+octave:128> imshow(circle_1)
+octave:129> imshow(circle_2)
+octave:130> circle_2 = (x - 1).^2 + (y - 1).^2 < 1.5^2;
+octave:131> imshow(circle_2)
+octave:132> imshow(circle_1 | circle_2)
+octave:133> imshow(circle_1 & circle_2)
+octave:134> imshow(xor(circle_1, circle_2))
+octave:135> close all
+octave:136>
diff --git a/lessons/lesson06/analog_freqs.m b/lessons/lesson06/analog_freqs.m
new file mode 100644
index 0000000..3348071
--- /dev/null
+++ b/lessons/lesson06/analog_freqs.m
@@ -0,0 +1,37 @@
+%% Lesson 6x: the S-plane
+% If you happen to have an analog filter represented as a ratio of polynomials,
+% MATLAB can analyze that too! `freqs()` will give you the analog frequency
+% response evaluated at frequencies in the given vector, interpreted in rad/s.
+
+%% a simple example
+% What sort of filter is this?
+a = [1 0.4 1];
+b = [0.2 0.3 1];
+
+w = logspace(-1, 1);
+H = freqs(b, a, w);
+
+Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+
+figure;
+subplot(2, 1, 1);
+semilogx(w, Hdb);
+xlabel("Frequency (rad/s)");
+ylabel("|H| (dB)");
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+semilogx(w, Hph);
+xlabel("Frequency (rad/s)");
+ylabel("Phase (deg)");
+title("Phase Response");
+
+%% some notes
+% `zp2tf` and its associates also work in the analog domain, as they deal
+% simply with ratios of polynomials. The interpretation of the results will be
+% different, though, as we're working with the *analog* plane rather than the
+% digital one. `splane.m` gives an example visualization of the s-plane.
+[z, p, k] = tf2zp(b, a);
+figure;
+splane(z, p);
diff --git a/lessons/lesson06/bw-cat.jpg b/lessons/lesson06/bw-cat.jpg
new file mode 100644
index 0000000..cc4ab21
Binary files /dev/null and b/lessons/lesson06/bw-cat.jpg differ
diff --git a/lessons/lesson06/fft_1d.m b/lessons/lesson06/fft_1d.m
new file mode 100644
index 0000000..7d07bdb
--- /dev/null
+++ b/lessons/lesson06/fft_1d.m
@@ -0,0 +1,84 @@
+%% Lesson 7a. The FFT
+% * Explore how to perform the FFT (DFT) on a 1-D signal
+% * Learn how to plot the FFT
+clear; clc; close all;
+
+%% Load a sample signal
+load handel; % builtin dataset that loads `y` and `Fs` to your workspace
+sound(y, Fs);
+T = 1/Fs;
+t = 0:T:(length(y)-1)*T;
+
+%% Loud noise warning!!!
+Fnoise = 2500;
+noise = 0.2*sin(2*pi*t*Fnoise).'; % additive "noise" with freq. 2.5kHz
+y = noise + y;
+% sound(y, Fs);
+
+%% Perform the FFT
+% Recall that the Fast Fourier Transform is a well-known and efficient
+% implementation of the DFT. The DFT is a Fourier transform on a
+% N-point fixed-length signal, and produces a N-point frequency spectrum.
+N = 2^15;
+S = fft(y, N); % N-point DFT (best to use power of 2)
+
+% Note the behavior of doing `fft(y, N)` (taken from the documentation):
+% * If X is a vector and the length of X is less than n, then X is padded
+% with trailing zeros to length n.
+% * If X is a vector and the length of X is greater than n, then X is
+% truncated to length n.
+% * If X is a matrix, then each column is treated as in the vector case.
+% * If X is a multidimensional array, then the first array dimension
+% whose size does not equal 1 is treated as in the vector case.
+%
+% This is the time when padding with zeros is acceptable (when performing
+% a FFT where N > length(y))!
+
+S = fftshift(abs(S)) / N;
+% fftshift is necessary since MATLAB returns from DFT the zeroth frequency
+% at the first index, then the positive frequencies, then the negative
+% frequencies when what you probably want is the zeroth frequency centered
+% between the negative and positive frequencies.
+
+% Get (analog) frequency domain. E.g., if you perform a 128-pt
+% FFT and sampled at 50kHz, you want to generate a 128-pt frequency range
+% from -25kHz to 25kHz.
+F = Fs .* (-N/2:N/2-1) / N;
+% Each frequency index represents a "bin" of frequencies between -Fs/2 and
+% Fs/2 (so index i represents i*Fs/N).
+
+figure;
+plot(F, S);
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
+
+%% Alternative ways to generate FFT plots:
+% Slightly more readable version of frequency domain.
+% Get the digital frequency
+wd = linspace(-pi, pi, N+1);
+wd = wd(1:end-1);
+F2 = wd * Fs / (2 * pi);
+
+% Equivalent formulation, without going through the digital domain.
+F3 = linspace(-Fs/2, Fs/2, N+1);
+F3 = F3(1:end-1);
+
+% Slightly more readable but less accurate version.
+F4 = linspace(-Fs/2, Fs/2, N);
+
+% Some arbitrariness in scaling factor. It's more important that
+% you use the same scaling when performing FFT as IFFT. See
+% https://www.mathworks.com/matlabcentral/answers/15770-scaling-the-fft-and-the-ifft#answer_21372
+% (use the formula Prof. Fontaine wants you to use.)
+S = fftshift(fft(y, N)) / Fs;
+
+% This S is complex; you can take either abs() or angle() of it to get
+% magnitude and phase responses, respectively
+figure;
+plot(F2, abs(S));
+% plot(F3, abs(S));
+% plot(F4, abs(S));
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
diff --git a/lessons/lesson06/fft_2d.m b/lessons/lesson06/fft_2d.m
new file mode 100644
index 0000000..7b8b07f
--- /dev/null
+++ b/lessons/lesson06/fft_2d.m
@@ -0,0 +1,74 @@
+%% Lesson 7b: The FFT -- for images!
+% * Explore how to perform and plot the FFT of a 2-D signal
+% * Implement a filter by masking in the frequency domain
+clc; clear; close all;
+
+%% Image Example
+% Low pass filtering
+% Here we look at images. Think frequencies spatially. High frequency
+% represents fast change in the image, low frequency represents slight
+% change. Also frequenies have two dimensions, vertial and horizontal.
+x = imread('bw-cat.jpg');
+imshow(x);
+
+%% 2-D FFT!
+F = fftshift(fft2(x));
+
+% Generate a mask (which will be a high-pass filter).
+% This mask will be 1 for low frequencies, and 0 for high frequencies.
+% Since the signal and its FFT are 2-D (an image), this will be a
+% white square centered in the FFT.
+lpf_mask = zeros(size(F));
+[H, W] = size(F);
+lpf_mask(floor(H/2-H/10):floor(H/2+H/10), ...
+ floor(W/2-W/10):floor(W/2+W/10)) = 1;
+
+% Show the 2-D FFT of the image
+figure;
+subplot(121);
+% Normalize FFT to range [0, 1]
+imshow(log(abs(F)) / max(log(abs(F(:)))));
+title 'Fourier transform';
+
+% Show the LPF mask
+subplot(122);
+imshow(lpf_mask);
+title 'LPF mask';
+
+%% Perform a LPF by only multiplying by frequency mask
+im_filtered_fft = lpf_mask .* F; % high frequencies removed
+f = ifft2(ifftshift(im_filtered_fft)); % ifft2 == ifft 2-D
+imshow(log(abs(im_filtered_fft)) / max(log(abs(im_filtered_fft(:)))));
+title 'Filtered FFT';
+
+%% Altered image vs. original image
+% Check out how the new image looks a bit blurry. That's because we removed
+% the high frequency band! All "edges" are less noticable.
+%
+% Alternatively, we can perform a high-pass filter. What would this do
+% to the image? (Hint: see next section.)
+figure;
+subplot(121);
+fnorm = abs(f) / max(abs(f(:)));
+imshow(fnorm);
+title 'High frequencies removed';
+
+subplot(122);
+xnorm = abs(double(x)) / max(double(x(:)));
+imshow(xnorm);
+title 'Original image';
+
+%% Basic Edge Detection
+% The difference between the old image and the new image: The edges!
+sharp_image = abs(fnorm - xnorm);
+sharp_image_norm = sharp_image / max(sharp_image(:));
+imshow(sharp_image_norm);
+
+%% Pinpoints these changes
+edge_image = zeros(size(sharp_image_norm));
+edge_image(sharp_image_norm > 3.5*std(sharp_image_norm(:))) = 1;
+imshow(edge_image);
+
+% Edge detection algorithms are one application of high-pass filters!
+% If you want to get started with these, look up Sobel or Laplacian
+% filters, which are simple convolutions that act as edge detectors/HPFs.
diff --git a/lessons/lesson06/handel.mat b/lessons/lesson06/handel.mat
new file mode 100644
index 0000000..6154ae5
Binary files /dev/null and b/lessons/lesson06/handel.mat differ
diff --git a/lessons/lesson06/jacob-boot.jpg b/lessons/lesson06/jacob-boot.jpg
new file mode 100644
index 0000000..4045db9
Binary files /dev/null and b/lessons/lesson06/jacob-boot.jpg differ
diff --git a/lessons/lesson06/octave-log.txt b/lessons/lesson06/octave-log.txt
new file mode 100644
index 0000000..92a0105
--- /dev/null
+++ b/lessons/lesson06/octave-log.txt
@@ -0,0 +1,1791 @@
+octave:7>
+^[[200~p = [2 -2 0 -3]; % 2x^3 - 2x^2 - 3^[[201~octave:7>
+octave:7>
+octave:7>
+octave:7>
+octave:7> p = [2 -2 0 -3]; % 2x^3 - 2x^2 - 3
+octave:8> y = polyval(p, -5:.05:5);
+octave:9> plot(y)
+octave:10> figure
+octave:11> plot(y)
+octave:12> close all
+octave:13> r = roots(p)
+r =
+
+ 1.5919 + 0i
+ -0.2960 + 0.9245i
+ -0.2960 - 0.9245i
+
+octave:14> polyval(p, r)
+ans =
+
+ 8.4377e-15 + 0i
+ 2.6645e-15 + 1.1102e-16i
+ 2.6645e-15 - 1.1102e-16i
+
+octave:15> abs(polyval(p, r))
+ans =
+
+ 8.4377e-15
+ 2.6668e-15
+ 2.6668e-15
+
+octave:16> p1 = poly(r)
+p1 =
+
+ 1.0000e+00 -1.0000e+00 -5.5511e-16 -1.5000e+00
+
+octave:17>
+^[[200~b = [-4 8];
+a = [1 6 8];^[[201~octave:17>
+octave:17>
+octave:17>
+octave:17>
+octave:17> b = [-4 8];
+a = [1 6 8];
+octave:19> help residue
+'residue' is a function from the file /usr/share/octave/7.3.0/m/polynomial/re
+sidue.m
+
+ -- [R, P, K, E] = residue (B, A)
+ -- [B, A] = residue (R, P, K)
+ -- [B, A] = residue (R, P, K, E)
+ The first calling form computes the partial fraction expansion for
+ the quotient of the polynomials, B and A.
+
+ The quotient is defined as
+
+ B(s) M r(m) N
+ ---- = SUM ------------- + SUM k(i)*s^(N-i)
+ A(s) m=1 (s-p(m))^e(m) i=1
+
+ where M is the number of poles (the length of the R, P, and E), the
+ K vector is a polynomial of order N-1 representing the direct
+ contribution, and the E vector specifies the multiplicity of the
+ m-th residue's pole.
+
+ For example,
+
+ b = [1, 1, 1];
+ a = [1, -5, 8, -4];
+ [r, p, k, e] = residue (b, a)
+ => r = [-2; 7; 3]
+ => p = [2; 2; 1]
+ => k = [](0x0)
+ => e = [1; 2; 1]
+
+ which represents the following partial fraction expansion
+
+ s^2 + s + 1 -2 7 3
+ ------------------- = ----- + ------- + -----
+ s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1)
+
+ The second calling form performs the inverse operation and computes
+ the reconstituted quotient of polynomials, B(s)/A(s), from the
+ partial fraction expansion; represented by the residues, poles, and
+ a direct polynomial specified by R, P and K, and the pole
+ multiplicity E.
+
+ If the multiplicity, E, is not explicitly specified the
+ multiplicity is determined by the function 'mpoles'.
+
+ For example:
+
+ r = [-2; 7; 3];
+ p = [2; 2; 1];
+ k = [1, 0];
+ [b, a] = residue (r, p, k)
+ => b = [1, -5, 9, -3, 1]
+ => a = [1, -5, 8, -4]
+
+ where mpoles is used to determine e = [1; 2; 1]
+
+ Alternatively the multiplicity may be defined explicitly, for
+ example,
+
+ r = [7; 3; -2];
+ p = [2; 1; 2];
+ k = [1, 0];
+ e = [2; 1; 1];
+ [b, a] = residue (r, p, k, e)
+ => b = [1, -5, 9, -3, 1]
+ => a = [1, -5, 8, -4]
+
+ which represents the following partial fraction expansion
+
+ -2 7 3 s^4 - 5s^3 + 9s^2 - 3s + 1
+ ----- + ------- + ----- + s = --------------------------
+ (s-2) (s-2)^2 (s-1) s^3 - 5s^2 + 8s - 4
+
+ See also: mpoles, poly, roots, conv, deconv.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:20> [r, p, k] = residue(b, a)
+r =
+
+ -12
+ 8
+
+p =
+
+ -4
+ -2
+
+k = [](0x0)
+octave:21> [r, p, k, e] = residue(b, a)
+r =
+
+ -12
+ 8
+
+p =
+
+ -4
+ -2
+
+k = [](0x0)
+e =
+
+ 1
+ 1
+
+octave:22> b
+b =
+
+ -4 8
+
+octave:23> a
+a =
+
+ 1 6 8
+
+octave:24>
+octave:24>
+octave:24>
+octave:24>
+octave:24> a = [1 2 -3 4];
+b = [5 0 3 -2];
+octave:26> conv(a, b)
+ans =
+
+ 5 10 -12 24 -13 18 -8
+
+octave:27>
+^[[201~octave:27>
+octave:27>
+octave:27>
+octave:27>
+octave:27> b1 = [1]; % numerator
+a1 = [1 -0.5]; % denominator
+figure;
+zplane(b1, a1);
+error: 'zplane' undefined near line 1, column 1
+
+The 'zplane' function belongs to the signal package from Octave Forge
+which you have installed but not loaded. To load the package, run 'pkg
+load signal' from the Octave prompt.
+
+Please read <https://www.octave.org/missing.html> to learn how you can
+contribute missing functionality.
+octave:31> pkg load signal
+octave:32> b1 = [1]; % numerator
+a1 = [1 -0.5]; % denominator
+figure;
+zplane(b1, a1);
+octave:36> close all
+octave:37>
+
+a = [1 1/sqrt(2) 1/4];^[[201~octave:37>
+octave:37>
+octave:37>
+octave:37>
+octave:37> b = [2 3 0];
+a = [1 1/sqrt(2) 1/4];
+octave:39>
+^[[200~[z, p, k] = tf2zp(b, a);^[[201~octave:39>
+octave:39>
+octave:39>
+octave:39>
+octave:39> [z, p, k] = tf2zp(b, a);
+octave:40> z
+z =
+
+ -1.5000
+ 0
+
+octave:41> p
+p =
+
+ -0.3536 + 0.3536i
+ -0.3536 - 0.3536i
+
+octave:42> figure
+octave:43> zplane(z, p)
+octave:44> close all
+octave:45>
+octave:45>
+octave:45>
+octave:45>
+octave:45>
+octave:45> b2 = [1];
+a2 = [1 -1.5];
+b3 = [1 -1 0.25];
+a3 = [1 -23/10 1.2];
+octave:49>
+octave:49>
+octave:49>
+octave:49>
+octave:49>
+octave:49> subplot(3, 1, 2);
+zplane(b2, a2);
+title('$$H_2(z)=\frac{z}{z-\frac{3}{2}}$$', 'interpreter', 'latex');
+sh: 1: dvipng: not found
+warning: latex_renderer: a run-time test failed and the 'latex' interpreter h
+as been disabled.
+warning: called from
+ __axis_label__ at line 36 column 6
+ title at line 64 column 8
+
+octave:52> close all
+octave:53>
+^[[200~zplane(b2, a2);^[[201~octave:53>
+octave:53>
+octave:53>
+octave:53>
+octave:53> zplane(b2, a2);
+octave:54> close all
+octave:55>
+octave:55>
+octave:55>
+octave:55>
+octave:55>
+octave:55> [h1, t] = impz(b1, a1, 8); % h is the IMPULSE RESPONSE
+octave:56> plot(t, h1)
+octave:57> close all
+octave:58> stem(t, h1);
+octave:59> close all
+octave:60>
+n = 0:1:5;
+x1 = (1/2).^n;
+y1 = conv(x1, h1);
+
+figure;
+subplot(2, 1, 1);
+stem(y1);
+title('Convolution between x_1 and h_1');
+subplot(2, 1, 2);
+y2 = filter(b1, a1, x1);
+stem(y2);
+title('Filter with b_1,a_1 and x_1');
+xlim([0 14]);^[[201~octave:60>
+octave:60>
+octave:60>
+octave:60>
+octave:60> n = 0:1:5;
+x1 = (1/2).^n;
+y1 = conv(x1, h1);
+
+figure;
+subplot(2, 1, 1);
+stem(y1);
+title('Convolution between x_1 and h_1');
+subplot(2, 1, 2);
+y2 = filter(b1, a1, x1);
+stem(y2);
+title('Filter with b_1,a_1 and x_1');
+xlim([0 14]);
+octave:72> close all
+octave:73> b1
+b1 = 1
+octave:74> a1
+a1 =
+
+ 1.0000 -0.5000
+
+octave:75>
+octave:75>
+octave:75>
+octave:75>
+octave:75>
+octave:75> freqz(b1, a1);
+octave:76>
+octave:76>
+octave:76>
+octave:76>
+octave:76> [H, w] = freqz(b1, a1);
+octave:77> close all
+octave:78> H
+H =
+
+ 2.0000 + 0i
+ 1.9999 - 0.0123i
+ 1.9995 - 0.0245i
+ 1.9990 - 0.0368i
+ 1.9982 - 0.0490i
+ 1.9972 - 0.0612i
+ 1.9959 - 0.0734i
+ 1.9945 - 0.0856i
+ 1.9928 - 0.0977i
+ 1.9909 - 0.1097i
+ 1.9888 - 0.1217i
+ 1.9865 - 0.1337i
+ 1.9839 - 0.1456i
+ 1.9812 - 0.1574i
+ 1.9782 - 0.1691i
+ 1.9750 - 0.1808i
+ 1.9717 - 0.1923i
+ 1.9681 - 0.2038i
+ 1.9643 - 0.2152i
+ 1.9603 - 0.2265i
+ 1.9562 - 0.2377i
+ 1.9519 - 0.2487i
+ 1.9473 - 0.2597i
+ 1.9426 - 0.2706i
+ 1.9378 - 0.2813i
+ 1.9327 - 0.2919i
+ 1.9275 - 0.3024i
+ 1.9221 - 0.3127i
+ 1.9166 - 0.3229i
+ 1.9109 - 0.3330i
+ 1.9050 - 0.3429i
+ 1.8991 - 0.3527i
+ 1.8929 - 0.3623i
+ 1.8867 - 0.3718i
+ 1.8803 - 0.3812i
+ 1.8738 - 0.3904i
+ 1.8671 - 0.3994i
+ 1.8604 - 0.4083i
+ 1.8535 - 0.4170i
+ 1.8465 - 0.4255i
+ 1.8394 - 0.4339i
+ 1.8323 - 0.4422i
+ 1.8250 - 0.4503i
+ 1.8176 - 0.4582i
+ 1.8102 - 0.4659i
+ 1.8026 - 0.4735i
+ 1.7950 - 0.4809i
+ 1.7873 - 0.4882i
+ 1.7796 - 0.4953i
+ 1.7718 - 0.5022i
+ 1.7639 - 0.5090i
+ 1.7560 - 0.5155i
+ 1.7480 - 0.5220i
+ 1.7400 - 0.5283i
+ 1.7320 - 0.5344i
+ 1.7239 - 0.5403i
+ 1.7157 - 0.5461i
+ 1.7076 - 0.5517i
+ 1.6994 - 0.5572i
+ 1.6912 - 0.5625i
+ 1.6829 - 0.5676i
+ 1.6747 - 0.5726i
+ 1.6664 - 0.5775i
+ 1.6582 - 0.5822i
+ 1.6499 - 0.5867i
+ 1.6416 - 0.5911i
+ 1.6333 - 0.5954i
+ 1.6250 - 0.5995i
+ 1.6168 - 0.6034i
+ 1.6085 - 0.6072i
+ 1.6002 - 0.6109i
+ 1.5920 - 0.6144i
+ 1.5838 - 0.6178i
+ 1.5756 - 0.6211i
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+ 1.5592 - 0.6272i
+ 1.5511 - 0.6301i
+ 1.5430 - 0.6328i
+ 1.5349 - 0.6355i
+ 1.5268 - 0.6380i
+ 1.5188 - 0.6403i
+ 1.5108 - 0.6426i
+ 1.5029 - 0.6448i
+ 1.4949 - 0.6468i
+ 1.4871 - 0.6487i
+ 1.4792 - 0.6505i
+ 1.4714 - 0.6522i
+ 1.4637 - 0.6538i
+ 1.4560 - 0.6553i
+ 1.4483 - 0.6567i
+ 1.4407 - 0.6580i
+ 1.4331 - 0.6592i
+ 1.4256 - 0.6603i
+ 1.4181 - 0.6613i
+ 1.4107 - 0.6622i
+ 1.4033 - 0.6630i
+ 1.3960 - 0.6637i
+ 1.3887 - 0.6644i
+ 1.3815 - 0.6649i
+ 1.3743 - 0.6654i
+ 1.3672 - 0.6658i
+ 1.3602 - 0.6661i
+ 1.3532 - 0.6664i
+ 1.3462 - 0.6665i
+ 1.3393 - 0.6666i
+ 1.3325 - 0.6667i
+ 1.3257 - 0.6666i
+ 1.3190 - 0.6665i
+ 1.3123 - 0.6663i
+ 1.3057 - 0.6661i
+ 1.2991 - 0.6658i
+ 1.2926 - 0.6654i
+ 1.2862 - 0.6650i
+ 1.2798 - 0.6645i
+ 1.2735 - 0.6640i
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+ 1.2610 - 0.6627i
+ 1.2548 - 0.6620i
+ 1.2487 - 0.6613i
+ 1.2427 - 0.6605i
+ 1.2367 - 0.6596i
+ 1.2307 - 0.6587i
+ 1.2249 - 0.6578i
+ 1.2190 - 0.6568i
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+ 1.2019 - 0.6536i
+ 1.1963 - 0.6524i
+ 1.1907 - 0.6512i
+ 1.1853 - 0.6500i
+ 1.1798 - 0.6487i
+ 1.1744 - 0.6475i
+ 1.1691 - 0.6461i
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+ 1.1586 - 0.6434i
+ 1.1534 - 0.6419i
+ 1.1483 - 0.6405i
+ 1.1432 - 0.6390i
+ 1.1382 - 0.6375i
+ 1.1332 - 0.6359i
+ 1.1283 - 0.6344i
+ 1.1235 - 0.6328i
+ 1.1186 - 0.6312i
+ 1.1139 - 0.6295i
+ 1.1092 - 0.6278i
+ 1.1045 - 0.6262i
+ 1.0999 - 0.6244i
+ 1.0953 - 0.6227i
+ 1.0908 - 0.6210i
+ 1.0863 - 0.6192i
+ 1.0819 - 0.6174i
+ 1.0775 - 0.6156i
+ 1.0731 - 0.6138i
+ 1.0688 - 0.6120i
+ 1.0646 - 0.6101i
+ 1.0604 - 0.6082i
+ 1.0562 - 0.6063i
+ 1.0521 - 0.6044i
+ 1.0480 - 0.6025i
+ 1.0440 - 0.6006i
+ 1.0400 - 0.5987i
+ 1.0361 - 0.5967i
+ 1.0322 - 0.5948i
+ 1.0283 - 0.5928i
+ 1.0245 - 0.5908i
+ 1.0207 - 0.5888i
+ 1.0169 - 0.5868i
+ 1.0132 - 0.5848i
+ 1.0096 - 0.5828i
+ 1.0060 - 0.5807i
+ 1.0024 - 0.5787i
+ 0.9988 - 0.5767i
+ 0.9953 - 0.5746i
+ 0.9918 - 0.5726i
+ 0.9884 - 0.5705i
+ 0.9850 - 0.5684i
+ 0.9816 - 0.5663i
+ 0.9783 - 0.5643i
+ 0.9750 - 0.5622i
+ 0.9717 - 0.5601i
+ 0.9685 - 0.5580i
+ 0.9653 - 0.5559i
+ 0.9622 - 0.5538i
+ 0.9590 - 0.5517i
+ 0.9560 - 0.5496i
+ 0.9529 - 0.5475i
+ 0.9499 - 0.5453i
+ 0.9469 - 0.5432i
+ 0.9439 - 0.5411i
+ 0.9410 - 0.5390i
+ 0.9381 - 0.5369i
+ 0.9352 - 0.5347i
+ 0.9324 - 0.5326i
+ 0.9296 - 0.5305i
+ 0.9268 - 0.5284i
+ 0.9240 - 0.5262i
+ 0.9213 - 0.5241i
+ 0.9186 - 0.5220i
+ 0.9159 - 0.5198i
+ 0.9133 - 0.5177i
+ 0.9107 - 0.5156i
+ 0.9081 - 0.5134i
+ 0.9055 - 0.5113i
+ 0.9030 - 0.5092i
+ 0.9005 - 0.5071i
+ 0.8980 - 0.5049i
+ 0.8956 - 0.5028i
+ 0.8931 - 0.5007i
+ 0.8907 - 0.4986i
+ 0.8884 - 0.4964i
+ 0.8860 - 0.4943i
+ 0.8837 - 0.4922i
+ 0.8814 - 0.4901i
+ 0.8791 - 0.4880i
+ 0.8768 - 0.4859i
+ 0.8746 - 0.4837i
+ 0.8724 - 0.4816i
+ 0.8702 - 0.4795i
+ 0.8680 - 0.4774i
+ 0.8659 - 0.4753i
+ 0.8638 - 0.4732i
+ 0.8617 - 0.4711i
+ 0.8596 - 0.4690i
+ 0.8575 - 0.4670i
+ 0.8555 - 0.4649i
+ 0.8535 - 0.4628i
+ 0.8515 - 0.4607i
+ 0.8495 - 0.4586i
+ 0.8475 - 0.4566i
+ 0.8456 - 0.4545i
+ 0.8437 - 0.4524i
+ 0.8418 - 0.4504i
+ 0.8399 - 0.4483i
+ 0.8380 - 0.4462i
+ 0.8362 - 0.4442i
+ 0.8344 - 0.4421i
+ 0.8326 - 0.4401i
+ 0.8308 - 0.4380i
+ 0.8290 - 0.4360i
+ 0.8273 - 0.4340i
+ 0.8255 - 0.4319i
+ 0.8238 - 0.4299i
+ 0.8221 - 0.4279i
+ 0.8204 - 0.4259i
+ 0.8188 - 0.4239i
+ 0.8171 - 0.4219i
+ 0.8155 - 0.4198i
+ 0.8139 - 0.4178i
+ 0.8123 - 0.4158i
+ 0.8107 - 0.4138i
+ 0.8091 - 0.4119i
+ 0.8075 - 0.4099i
+ 0.8060 - 0.4079i
+ 0.8045 - 0.4059i
+ 0.8030 - 0.4039i
+ 0.8015 - 0.4020i
+ 0.8000 - 0.4000i
+ 0.7985 - 0.3980i
+ 0.7971 - 0.3961i
+ 0.7956 - 0.3941i
+ 0.7942 - 0.3922i
+ 0.7928 - 0.3902i
+ 0.7914 - 0.3883i
+ 0.7900 - 0.3864i
+ 0.7887 - 0.3844i
+ 0.7873 - 0.3825i
+ 0.7860 - 0.3806i
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+ 0.7833 - 0.3767i
+ 0.7820 - 0.3748i
+ 0.7807 - 0.3729i
+ 0.7795 - 0.3710i
+ 0.7782 - 0.3691i
+ 0.7769 - 0.3672i
+ 0.7757 - 0.3653i
+ 0.7745 - 0.3635i
+ 0.7732 - 0.3616i
+ 0.7720 - 0.3597i
+ 0.7708 - 0.3578i
+ 0.7697 - 0.3560i
+ 0.7685 - 0.3541i
+ 0.7673 - 0.3522i
+ 0.7662 - 0.3504i
+ 0.7650 - 0.3485i
+ 0.7639 - 0.3467i
+ 0.7628 - 0.3449i
+ 0.7617 - 0.3430i
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+ 0.7595 - 0.3394i
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+ 0.7574 - 0.3357i
+ 0.7563 - 0.3339i
+ 0.7553 - 0.3321i
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+ 0.7532 - 0.3285i
+ 0.7522 - 0.3267i
+ 0.7512 - 0.3249i
+ 0.7502 - 0.3231i
+ 0.7492 - 0.3213i
+ 0.7482 - 0.3195i
+ 0.7472 - 0.3177i
+ 0.7463 - 0.3159i
+ 0.7453 - 0.3142i
+ 0.7444 - 0.3124i
+ 0.7435 - 0.3106i
+ 0.7425 - 0.3089i
+ 0.7416 - 0.3071i
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+ 0.7389 - 0.3019i
+ 0.7380 - 0.3001i
+ 0.7372 - 0.2984i
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+ 0.7346 - 0.2932i
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+ 0.7313 - 0.2863i
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+ 0.7297 - 0.2829i
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+ 0.7273 - 0.2778i
+ 0.7266 - 0.2762i
+ 0.7258 - 0.2745i
+ 0.7250 - 0.2728i
+ 0.7243 - 0.2711i
+ 0.7235 - 0.2694i
+ 0.7228 - 0.2678i
+ 0.7221 - 0.2661i
+ 0.7214 - 0.2644i
+ 0.7206 - 0.2628i
+ 0.7199 - 0.2611i
+ 0.7192 - 0.2595i
+ 0.7185 - 0.2578i
+ 0.7178 - 0.2562i
+ 0.7172 - 0.2545i
+ 0.7165 - 0.2529i
+ 0.7158 - 0.2512i
+ 0.7152 - 0.2496i
+ 0.7145 - 0.2480i
+ 0.7139 - 0.2464i
+ 0.7132 - 0.2447i
+ 0.7126 - 0.2431i
+ 0.7119 - 0.2415i
+ 0.7113 - 0.2399i
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+ 0.7101 - 0.2367i
+ 0.7095 - 0.2350i
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+ 0.7083 - 0.2318i
+ 0.7077 - 0.2303i
+ 0.7071 - 0.2287i
+ 0.7065 - 0.2271i
+ 0.7060 - 0.2255i
+ 0.7054 - 0.2239i
+ 0.7048 - 0.2223i
+ 0.7043 - 0.2207i
+ 0.7037 - 0.2192i
+ 0.7032 - 0.2176i
+ 0.7026 - 0.2160i
+ 0.7021 - 0.2145i
+ 0.7016 - 0.2129i
+ 0.7010 - 0.2113i
+ 0.7005 - 0.2098i
+ 0.7000 - 0.2082i
+ 0.6995 - 0.2067i
+ 0.6990 - 0.2051i
+ 0.6985 - 0.2036i
+ 0.6980 - 0.2020i
+ 0.6975 - 0.2005i
+ 0.6970 - 0.1989i
+ 0.6966 - 0.1974i
+ 0.6961 - 0.1959i
+ 0.6956 - 0.1943i
+ 0.6952 - 0.1928i
+ 0.6947 - 0.1913i
+ 0.6942 - 0.1897i
+ 0.6938 - 0.1882i
+ 0.6933 - 0.1867i
+ 0.6929 - 0.1852i
+ 0.6925 - 0.1837i
+ 0.6920 - 0.1822i
+ 0.6916 - 0.1807i
+ 0.6912 - 0.1791i
+ 0.6908 - 0.1776i
+ 0.6904 - 0.1761i
+ 0.6899 - 0.1746i
+ 0.6895 - 0.1731i
+ 0.6891 - 0.1716i
+ 0.6887 - 0.1701i
+ 0.6884 - 0.1687i
+ 0.6880 - 0.1672i
+ 0.6876 - 0.1657i
+ 0.6872 - 0.1642i
+ 0.6868 - 0.1627i
+ 0.6865 - 0.1612i
+ 0.6861 - 0.1597i
+ 0.6857 - 0.1583i
+ 0.6854 - 0.1568i
+ 0.6850 - 0.1553i
+ 0.6847 - 0.1539i
+ 0.6843 - 0.1524i
+ 0.6840 - 0.1509i
+ 0.6836 - 0.1494i
+ 0.6833 - 0.1480i
+ 0.6830 - 0.1465i
+ 0.6826 - 0.1451i
+ 0.6823 - 0.1436i
+ 0.6820 - 0.1422i
+ 0.6817 - 0.1407i
+ 0.6814 - 0.1392i
+ 0.6811 - 0.1378i
+ 0.6808 - 0.1363i
+ 0.6805 - 0.1349i
+ 0.6802 - 0.1335i
+ 0.6799 - 0.1320i
+ 0.6796 - 0.1306i
+ 0.6793 - 0.1291i
+ 0.6790 - 0.1277i
+ 0.6787 - 0.1263i
+ 0.6785 - 0.1248i
+ 0.6782 - 0.1234i
+ 0.6779 - 0.1220i
+ 0.6777 - 0.1205i
+ 0.6774 - 0.1191i
+ 0.6771 - 0.1177i
+ 0.6769 - 0.1162i
+ 0.6766 - 0.1148i
+ 0.6764 - 0.1134i
+ 0.6761 - 0.1120i
+ 0.6759 - 0.1106i
+ 0.6757 - 0.1091i
+ 0.6754 - 0.1077i
+ 0.6752 - 0.1063i
+ 0.6750 - 0.1049i
+ 0.6747 - 0.1035i
+ 0.6745 - 0.1021i
+ 0.6743 - 0.1007i
+ 0.6741 - 0.0992i
+ 0.6739 - 0.0978i
+ 0.6737 - 0.0964i
+ 0.6735 - 0.0950i
+ 0.6733 - 0.0936i
+ 0.6731 - 0.0922i
+ 0.6729 - 0.0908i
+ 0.6727 - 0.0894i
+ 0.6725 - 0.0880i
+ 0.6723 - 0.0866i
+ 0.6721 - 0.0852i
+ 0.6720 - 0.0838i
+ 0.6718 - 0.0824i
+ 0.6716 - 0.0810i
+ 0.6714 - 0.0796i
+ 0.6713 - 0.0783i
+ 0.6711 - 0.0769i
+ 0.6710 - 0.0755i
+ 0.6708 - 0.0741i
+ 0.6706 - 0.0727i
+ 0.6705 - 0.0713i
+ 0.6703 - 0.0699i
+ 0.6702 - 0.0685i
+ 0.6701 - 0.0672i
+ 0.6699 - 0.0658i
+ 0.6698 - 0.0644i
+ 0.6697 - 0.0630i
+ 0.6695 - 0.0616i
+ 0.6694 - 0.0602i
+ 0.6693 - 0.0589i
+ 0.6691 - 0.0575i
+ 0.6690 - 0.0561i
+ 0.6689 - 0.0547i
+ 0.6688 - 0.0533i
+ 0.6687 - 0.0520i
+ 0.6686 - 0.0506i
+ 0.6685 - 0.0492i
+ 0.6684 - 0.0478i
+ 0.6683 - 0.0465i
+ 0.6682 - 0.0451i
+ 0.6681 - 0.0437i
+ 0.6680 - 0.0424i
+ 0.6679 - 0.0410i
+ 0.6678 - 0.0396i
+ 0.6678 - 0.0382i
+ 0.6677 - 0.0369i
+ 0.6676 - 0.0355i
+ 0.6675 - 0.0341i
+ 0.6675 - 0.0328i
+ 0.6674 - 0.0314i
+ 0.6673 - 0.0300i
+ 0.6673 - 0.0287i
+ 0.6672 - 0.0273i
+ 0.6672 - 0.0259i
+ 0.6671 - 0.0246i
+ 0.6671 - 0.0232i
+ 0.6670 - 0.0218i
+ 0.6670 - 0.0205i
+ 0.6669 - 0.0191i
+ 0.6669 - 0.0177i
+ 0.6669 - 0.0164i
+ 0.6668 - 0.0150i
+ 0.6668 - 0.0136i
+ 0.6668 - 0.0123i
+ 0.6668 - 0.0109i
+ 0.6667 - 0.0095i
+ 0.6667 - 0.0082i
+ 0.6667 - 0.0068i
+ 0.6667 - 0.0055i
+ 0.6667 - 0.0041i
+ 0.6667 - 0.0027i
+ 0.6667 - 0.0014i
+
+octave:79> w
+w =
+
+ 0
+ 0.0061
+ 0.0123
+ 0.0184
+ 0.0245
+ 0.0307
+ 0.0368
+ 0.0430
+ 0.0491
+ 0.0552
+ 0.0614
+ 0.0675
+ 0.0736
+ 0.0798
+ 0.0859
+ 0.0920
+ 0.0982
+ 0.1043
+ 0.1104
+ 0.1166
+ 0.1227
+ 0.1289
+ 0.1350
+ 0.1411
+ 0.1473
+ 0.1534
+ 0.1595
+ 0.1657
+ 0.1718
+ 0.1779
+ 0.1841
+ 0.1902
+ 0.1963
+ 0.2025
+ 0.2086
+ 0.2148
+ 0.2209
+ 0.2270
+ 0.2332
+ 0.2393
+ 0.2454
+ 0.2516
+ 0.2577
+ 0.2638
+ 0.2700
+ 0.2761
+ 0.2823
+ 0.2884
+ 0.2945
+ 0.3007
+ 0.3068
+ 0.3129
+ 0.3191
+ 0.3252
+ 0.3313
+ 0.3375
+ 0.3436
+ 0.3497
+ 0.3559
+ 0.3620
+ 0.3682
+ 0.3743
+ 0.3804
+ 0.3866
+ 0.3927
+ 0.3988
+ 0.4050
+ 0.4111
+ 0.4172
+ 0.4234
+ 0.4295
+ 0.4357
+ 0.4418
+ 0.4479
+ 0.4541
+ 0.4602
+ 0.4663
+ 0.4725
+ 0.4786
+ 0.4847
+ 0.4909
+ 0.4970
+ 0.5031
+ 0.5093
+ 0.5154
+ 0.5216
+ 0.5277
+ 0.5338
+ 0.5400
+ 0.5461
+ 0.5522
+ 0.5584
+ 0.5645
+ 0.5706
+ 0.5768
+ 0.5829
+ 0.5890
+ 0.5952
+ 0.6013
+ 0.6075
+ 0.6136
+ 0.6197
+ 0.6259
+ 0.6320
+ 0.6381
+ 0.6443
+ 0.6504
+ 0.6565
+ 0.6627
+ 0.6688
+ 0.6750
+ 0.6811
+ 0.6872
+ 0.6934
+ 0.6995
+ 0.7056
+ 0.7118
+ 0.7179
+ 0.7240
+ 0.7302
+ 0.7363
+ 0.7424
+ 0.7486
+ 0.7547
+ 0.7609
+ 0.7670
+ 0.7731
+ 0.7793
+ 0.7854
+ 0.7915
+ 0.7977
+ 0.8038
+ 0.8099
+ 0.8161
+ 0.8222
+ 0.8283
+ 0.8345
+ 0.8406
+ 0.8468
+ 0.8529
+ 0.8590
+ 0.8652
+ 0.8713
+ 0.8774
+ 0.8836
+ 0.8897
+ 0.8958
+ 0.9020
+ 0.9081
+ 0.9143
+ 0.9204
+ 0.9265
+ 0.9327
+ 0.9388
+ 0.9449
+ 0.9511
+ 0.9572
+ 0.9633
+ 0.9695
+ 0.9756
+ 0.9817
+ 0.9879
+ 0.9940
+ 1.0002
+ 1.0063
+ 1.0124
+ 1.0186
+ 1.0247
+ 1.0308
+ 1.0370
+ 1.0431
+ 1.0492
+ 1.0554
+ 1.0615
+ 1.0677
+ 1.0738
+ 1.0799
+ 1.0861
+ 1.0922
+ 1.0983
+ 1.1045
+ 1.1106
+ 1.1167
+ 1.1229
+ 1.1290
+ 1.1351
+ 1.1413
+ 1.1474
+ 1.1536
+ 1.1597
+ 1.1658
+ 1.1720
+ 1.1781
+ 1.1842
+ 1.1904
+ 1.1965
+ 1.2026
+ 1.2088
+ 1.2149
+ 1.2210
+ 1.2272
+ 1.2333
+ 1.2395
+ 1.2456
+ 1.2517
+ 1.2579
+ 1.2640
+ 1.2701
+ 1.2763
+ 1.2824
+ 1.2885
+ 1.2947
+ 1.3008
+ 1.3070
+ 1.3131
+ 1.3192
+ 1.3254
+ 1.3315
+ 1.3376
+ 1.3438
+ 1.3499
+ 1.3560
+ 1.3622
+ 1.3683
+ 1.3744
+ 1.3806
+ 1.3867
+ 1.3929
+ 1.3990
+ 1.4051
+ 1.4113
+ 1.4174
+ 1.4235
+ 1.4297
+ 1.4358
+ 1.4419
+ 1.4481
+ 1.4542
+ 1.4603
+ 1.4665
+ 1.4726
+ 1.4788
+ 1.4849
+ 1.4910
+ 1.4972
+ 1.5033
+ 1.5094
+ 1.5156
+ 1.5217
+ 1.5278
+ 1.5340
+ 1.5401
+ 1.5463
+ 1.5524
+ 1.5585
+ 1.5647
+ 1.5708
+ 1.5769
+ 1.5831
+ 1.5892
+ 1.5953
+ 1.6015
+ 1.6076
+ 1.6137
+ 1.6199
+ 1.6260
+ 1.6322
+ 1.6383
+ 1.6444
+ 1.6506
+ 1.6567
+ 1.6628
+ 1.6690
+ 1.6751
+ 1.6812
+ 1.6874
+ 1.6935
+ 1.6997
+ 1.7058
+ 1.7119
+ 1.7181
+ 1.7242
+ 1.7303
+ 1.7365
+ 1.7426
+ 1.7487
+ 1.7549
+ 1.7610
+ 1.7671
+ 1.7733
+ 1.7794
+ 1.7856
+ 1.7917
+ 1.7978
+ 1.8040
+ 1.8101
+ 1.8162
+ 1.8224
+ 1.8285
+ 1.8346
+ 1.8408
+ 1.8469
+ 1.8530
+ 1.8592
+ 1.8653
+ 1.8715
+ 1.8776
+ 1.8837
+ 1.8899
+ 1.8960
+ 1.9021
+ 1.9083
+ 1.9144
+ 1.9205
+ 1.9267
+ 1.9328
+ 1.9390
+ 1.9451
+ 1.9512
+ 1.9574
+ 1.9635
+ 1.9696
+ 1.9758
+ 1.9819
+ 1.9880
+ 1.9942
+ 2.0003
+ 2.0064
+ 2.0126
+ 2.0187
+ 2.0249
+ 2.0310
+ 2.0371
+ 2.0433
+ 2.0494
+ 2.0555
+ 2.0617
+ 2.0678
+ 2.0739
+ 2.0801
+ 2.0862
+ 2.0923
+ 2.0985
+ 2.1046
+ 2.1108
+ 2.1169
+ 2.1230
+ 2.1292
+ 2.1353
+ 2.1414
+ 2.1476
+ 2.1537
+ 2.1598
+ 2.1660
+ 2.1721
+ 2.1783
+ 2.1844
+ 2.1905
+ 2.1967
+ 2.2028
+ 2.2089
+ 2.2151
+ 2.2212
+ 2.2273
+ 2.2335
+ 2.2396
+ 2.2457
+ 2.2519
+ 2.2580
+ 2.2642
+ 2.2703
+ 2.2764
+ 2.2826
+ 2.2887
+ 2.2948
+ 2.3010
+ 2.3071
+ 2.3132
+ 2.3194
+ 2.3255
+ 2.3317
+ 2.3378
+ 2.3439
+ 2.3501
+ 2.3562
+ 2.3623
+ 2.3685
+ 2.3746
+ 2.3807
+ 2.3869
+ 2.3930
+ 2.3991
+ 2.4053
+ 2.4114
+ 2.4176
+ 2.4237
+ 2.4298
+ 2.4360
+ 2.4421
+ 2.4482
+ 2.4544
+ 2.4605
+ 2.4666
+ 2.4728
+ 2.4789
+ 2.4850
+ 2.4912
+ 2.4973
+ 2.5035
+ 2.5096
+ 2.5157
+ 2.5219
+ 2.5280
+ 2.5341
+ 2.5403
+ 2.5464
+ 2.5525
+ 2.5587
+ 2.5648
+ 2.5710
+ 2.5771
+ 2.5832
+ 2.5894
+ 2.5955
+ 2.6016
+ 2.6078
+ 2.6139
+ 2.6200
+ 2.6262
+ 2.6323
+ 2.6384
+ 2.6446
+ 2.6507
+ 2.6569
+ 2.6630
+ 2.6691
+ 2.6753
+ 2.6814
+ 2.6875
+ 2.6937
+ 2.6998
+ 2.7059
+ 2.7121
+ 2.7182
+ 2.7243
+ 2.7305
+ 2.7366
+ 2.7428
+ 2.7489
+ 2.7550
+ 2.7612
+ 2.7673
+ 2.7734
+ 2.7796
+ 2.7857
+ 2.7918
+ 2.7980
+ 2.8041
+ 2.8103
+ 2.8164
+ 2.8225
+ 2.8287
+ 2.8348
+ 2.8409
+ 2.8471
+ 2.8532
+ 2.8593
+ 2.8655
+ 2.8716
+ 2.8777
+ 2.8839
+ 2.8900
+ 2.8962
+ 2.9023
+ 2.9084
+ 2.9146
+ 2.9207
+ 2.9268
+ 2.9330
+ 2.9391
+ 2.9452
+ 2.9514
+ 2.9575
+ 2.9637
+ 2.9698
+ 2.9759
+ 2.9821
+ 2.9882
+ 2.9943
+ 3.0005
+ 3.0066
+ 3.0127
+ 3.0189
+ 3.0250
+ 3.0311
+ 3.0373
+ 3.0434
+ 3.0496
+ 3.0557
+ 3.0618
+ 3.0680
+ 3.0741
+ 3.0802
+ 3.0864
+ 3.0925
+ 3.0986
+ 3.1048
+ 3.1109
+ 3.1170
+ 3.1232
+ 3.1293
+ 3.1355
+
+octave:80>
+Hph = rad2deg(unwrap(angle(H)));^[[201~octave:80>
+octave:80>
+octave:80>
+octave:80>
+octave:80> Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+octave:82>
+octave:82>
+octave:82>
+octave:82>
+octave:82>
+octave:82> figure;
+subplot(2, figure;
+subplot(2, 1, 1);
+plot(w, Hdb);ency (rad)");
+xlabel("Frequency (rad)");
+ylabel("|H| (dB)");
+xlim([0 pi]);2 pi]);
+xticks([0 pi/2 pi]);pi/2', '\pi'});
+xticklabels({'0', '\pi/2', '\pi'});
+title("Magnitude Response");
+subplot(2, 1, 2);
+subplot(2, 1, 2);
+plot(w, Hph);ency (rad)");
+xlabel("Frequency (rad)");
+ylabel("Phase (deg)");
+xlim([0 pi]);2 pi]);
+xticks([0 pi/2 pi]);pi/2', '\pi'});
+xticklabels({'0', '\pi/2', '\pi'});
+title("Phase Response");
+octave:99> close all
+octave:100>
+
+
+
+
+^[[200~zer = -0.5;
+pol = 0.9*exp(j*2*pi*[-0.3 0.3]');^[[201~octave:100>
+octave:100>
+octave:100>
+octave:100>
+octave:100> zer = -0.5;
+pol = 0.9*exp(j*2*pi*[-0.3 0.3]');
+octave:102> zer
+zer = -0.5000
+octave:103> pol
+pol =
+
+ -0.2781 - 0.8560i
+ -0.2781 + 0.8560i
+
+octave:104>
+octave:104>
+octave:104>
+octave:104>
+octave:104>
+octave:104> figure;
+zplane(zer, pol);
+octave:106> close all
+octave:107>
+octave:107>
+octave:107>
+octave:107>
+octave:107>
+octave:107> [b4,a4] = zp2tf(zer, pol, 1);
+octave:108>
+^[[200~[H,w] = freqz(b4, a4);^[[201~octave:108>
+octave:108>
+octave:108>
+octave:108>
+octave:108> [H,w] = freqz(b4, a4);
+octave:109>
+Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));^[[201~octave:109>
+octave:109>
+octave:109>
+octave:109>
+octave:109> Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+octave:111>
+
+plot(w, Hdb);
+xlabel("Frequency (rad)");
+ylabel("|H| (dB)");
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+xticklabels({'0', '\pi/2', '\pi'});
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+plot(w, Hph);
+xlabel("Frequency (rad)");
+ylabel("Phase (deg)");
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+xticklabels({'0', '\pi/2', '\pi'});
+title("Phase Response");^[[201~octave:111>
+octave:111>
+octave:111>
+octave:111>
+octave:111> figure;
+subplot(2, 1figure;
+subplot(2, 1, 1);
+plot(w, Hdb);ency (rad)");
+xlabel("Frequency (rad)");
+ylabel("|H| (dB)");
+xlim([0 pi]);2 pi]);
+xticks([0 pi/2 pi]);pi/2', '\pi'});
+xticklabels({'0', '\pi/2', '\pi'});
+title("Magnitude Response");
+subplot(2, 1, 2);
+subplot(2, 1, 2);
+plot(w, Hph);ency (rad)");
+xlabel("Frequency (rad)");
+ylabel("Phase (deg)");
+xlim([0 pi]);2 pi]);
+xticks([0 pi/2 pi]);pi/2', '\pi'});
+xticklabels({'0', '\pi/2', '\pi'});
+title("Phase Response");
+octave:128> close all
+octave:129>
+^[[200~a = [1 0.4 1];
+b = [0.2 0.3 1];^[[201~octave:129>
+octave:129>
+octave:129>
+octave:129>
+octave:129> a = [1 0.4 1];
+b = [0.2 0.3 1];
+octave:131>
+^[[200~w = logspace(-1, 1);
+H = freqs(b, a, w);^[[201~octave:131>
+octave:131>
+octave:131>
+octave:131>
+octave:131> w = logspace(-1, 1);
+H = freqs(b, a, w);
+octave:133>
+octave:133>
+octave:133>
+xlabel("Frequency (rad/s)");
+ylabel("|H| (dB)");
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+semilogx(w, Hph);
+xlabel("Frequency (rad/s)");
+ylabel("Phase (deg)");
+title("Phase Response");^[[201~octave:133>
+octave:133>
+octave:133> Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+
+figure;
+subplot(2, 1, 1);
+semilogx(w, Hdb);
+xlabel("Frequency (rad/s)");
+ylabel("|H| (dB)");
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+semilogx(w, Hph);
+xlabel("Frequency (rad/s)");
+ylabel("Phase (deg)");
+title("Phase Response");
+octave:146> close all
+octave:147>
+^[[200~[z, p, k] = tf2zp(b, a);
+figure;
+splane(z, p);^[[201~octave:147>
+octave:147>
+octave:147>
+octave:147>
+octave:147> [z, p, k] = tf2zp(b, a);
+figure;
+splane(z, p);
+octave:150> close all
+octave:151> load handel
+octave:152> soundsc(y, Fs)
+ALSA lib pcm_dsnoop.c:566:(snd_pcm_dsnoop_open) unable to open slave
+ALSA lib pcm_dmix.c:999:(snd_pcm_dmix_open) unable to open slave
+ALSA lib pcm.c:2666:(snd_pcm_open_noupdate) Unknown PCM cards.pcm.rear
+ALSA lib pcm.c:2666:(snd_pcm_open_noupdate) Unknown PCM cards.pcm.center_lfe
+ALSA lib pcm.c:2666:(snd_pcm_open_noupdate) Unknown PCM cards.pcm.side
+Cannot connect to server socket err = No such file or directory
+Cannot connect to server request channel
+jack server is not running or cannot be started
+JackShmReadWritePtr::~JackShmReadWritePtr - Init not done for -1, skipping un
+lock
+JackShmReadWritePtr::~JackShmReadWritePtr - Init not done for -1, skipping un
+lock
+Cannot connect to server socket err = No such file or directory
+Cannot connect to server request channel
+jack server is not running or cannot be started
+JackShmReadWritePtr::~JackShmReadWritePtr - Init not done for -1, skipping un
+lock
+JackShmReadWritePtr::~JackShmReadWritePtr - Init not done for -1, skipping un
+lock
+ALSA lib pcm_oss.c:397:(_snd_pcm_oss_open) Cannot open device /dev/dsp
+ALSA lib pcm_oss.c:397:(_snd_pcm_oss_open) Cannot open device /dev/dsp
+ALSA lib pcm_a52.c:1001:(_snd_pcm_a52_open) a52 is only for playback
+ALSA lib confmisc.c:160:(snd_config_get_card) Invalid field card
+ALSA lib pcm_usb_stream.c:482:(_snd_pcm_usb_stream_open) Invalid card 'card'
+ALSA lib confmisc.c:160:(snd_config_get_card) Invalid field card
+ALSA lib pcm_usb_stream.c:482:(_snd_pcm_usb_stream_open) Invalid card 'card'
+ALSA lib pcm_dmix.c:999:(snd_pcm_dmix_open) unable to open slave
+Cannot connect to server socket err = No such file or directory
+Cannot connect to server request channel
+jack server is not running or cannot be started
+JackShmReadWritePtr::~JackShmReadWritePtr - Init not done for -1, skipping un
+lock
+JackShmReadWritePtr::~JackShmReadWritePtr - Init not done for -1, skipping un
+lock
+octave:153> size(y)
+ans =
+
+ 73113 1
+
+octave:154> ufs
+error: 'ufs' undefined near line 1, column 1
+octave:155> Fs
+Fs = 8192
+octave:156>
+^[[200~T = 1/Fs;
+t = 0:T:(length(y)-1)*T;^[[201~octave:156>
+octave:156>
+octave:156>
+octave:156>
+octave:156> T = 1/Fs;
+t = 0:T:(length(y)-1)*T;
+octave:158>
+
+noise = 0.2*sin(2*pi*t*Fnoise).'; % additive "noise" with freq. 2.5kHz
+y = noise + y;^[[201~octave:158>
+octave:158>
+octave:158>
+octave:158>
+octave:158> Fnoise = 2500;
+noise = 0.2*sin(2*pi*t*Fnoise).'; % additive "noise" with freq. 2.5kHz
+y = noise + y;
+octave:161> soundsc(y, Fs)
+octave:162>
+octave:162>
+octave:162>
+octave:162>
+octave:162>
+octave:162> N = 2^15;
+S = fft(y, N); % N-point DFT (best to use power of 2)
+octave:164> N
+N = 32768
+octave:165>
+^[[200~S = fftshift(abs(S)) / N;^[[201~octave:165>
+octave:165>
+octave:165>
+octave:165>
+octave:165> S = fftshift(abs(S)) / N;
+octave:166>
+^[[200~F = Fs .* (-N/2:N/2-1) / N;^[[201~octave:166>
+octave:166>
+octave:166>
+octave:166>
+octave:166> F = Fs .* (-N/2:N/2-1) / N;
+octave:167>
+octave:167>
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';^[[201~octave:167>
+octave:167>
+octave:167>
+octave:167> figure;
+plot(F, S);
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
+octave:172> close all
+octave:173>
+^[[200~figure;
+plot(F2, abs(S));
+% plot(F3, abs(S));
+% plot(F4, abs(S));
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';^[[201~octave:173>
+octave:173>
+octave:173>
+octave:173>
+octave:173> figure;
+plot(F2, abs(S));
+% plot(F3, abs(S));
+% plot(F4, abs(S));
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
+error: 'F2' undefined near line 1, column 6
+octave:178>
+^[[200~ * Fs / (2 * pi);^[[201~octave:178>
+octave:178>
+octave:178>
+octave:178>
+octave:178> * Fs / (2 * pi);
+wd = linspace(-pi, pi, N+1);
+wd = wd(1:end-1);
+F2 = wd * Fs / (2 * pi);error: parse error:
+
+ syntax error
+
+>>> * Fs / (2 * pi);
+ ^
+octave:178>
+octave:178>
+octave:178>
+octave:178>
+octave:178> wd = linspace(-pi, pi, N+1);
+octave:179> wd = wd(1:end-1);
+octave:180> F2 = wd * Fs / (2 * pi);
+octave:181> close all
+octave:182>
+^[[200~figure;
+plot(F2, abs(S));
+% plot(F3, abs(S));
+% plot(F4, abs(S));
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';^[[201~octave:182>
+octave:182>
+octave:182>
+octave:182>
+octave:182> figure;
+plot(F2, abs(S));
+% plot(F3, abs(S));
+% plot(F4, abs(S));
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
+octave:187> close all
+octave:188>
+octave:188>
+octave:188>
+octave:188>
+octave:188>
+octave:188> x = imread('bw-cat.jpg');
+imshow(x);
+octave:190> close all
+octave:191>
+octave:191>
+octave:191>
+octave:191>
+octave:191>
+octave:191> F = fftshift(fft2(x));
+octave:192>
+^[[200~lpf_mask = zeros(size(F));
+[H, W] = size(F);
+lpf_mask(floor(H/2-H/10):floor(H/2+H/10), ...
+ floor(W/2-W/10):floor(W/2+W/10)) = 1;^[[201~octave:192>
+octave:192>
+octave:192>
+octave:192>
+octave:192> lpf_mask = zeros(size(F));
+[H, W] = size(F);
+lpf_mask(floor(H/2-H/10):floor(H/2+H/10), ...
+ floor(W/2-W/10):floor(W/2+W/10)) = 1;
+octave:195>
+^[[200~figure;
+subplot(121);
+% Normalize FFT to range [0, 1]
+imshow(log(abs(F)) / max(log(abs(F(:)))));
+title 'Fourier transform';^[[201~octave:195>
+octave:195>
+octave:195>
+octave:195>
+octave:195> figure;
+subplot(121);
+% Normalize FFT to range [0, 1]
+imshow(log(abs(F)) / max(log(abs(F(:)))));
+title 'Fourier transform';
+octave:199>
+octave:199>
+octave:199>
+octave:199>
+octave:199>
+octave:199> % Show the LPF mask
+subplot(122);
+imshow(lpf_mask);
+title 'LPF mask';
+octave:202> close all
+octave:203>
+octave:203>
+octave:203>
+octave:203>
+octave:203>
+octave:203> im_filtered_fft = lpf_mask .* F; % high frequencies remove
+d im_filtered_fft = lpf_mask .* F; % high frequencies remove
+d = ifft2(ifftshift(im_filtered_fft)); % ifft2 == ifft 2-D
+f = ifft2(ifftshift(im_filtered_fft)); % ifft2 == ifft 2-Dfft(:)))));
+imshow(log(abs(im_filtered_fft)) / max(log(abs(im_filtered_fft(:)))));
+title 'Filtered FFT';
+octave:207> close all
+octave:208>
+^[[200~figure;
+subplot(121);
+fnorm = abs(f) / max(abs(f(:)));
+imshow(fnorm);
+title 'High frequencies removed';
+
+subplot(122);
+xnorm = abs(double(x)) / max(double(x(:)));
+imshow(xnorm);
+title 'Original image';^[[201~octave:208>
+octave:208>
+octave:208>
+octave:208>
+octave:208> figure;
+subplot(121);
+fnorm = abs(f) / max(abs(f(:)));
+imshow(fnorm);
+title 'High frequencies removed';
+
+subplot(122);
+xnorm = abs(double(x)) / max(double(x(:)));
+imshow(xnorm);
+title 'Original image';
+octave:217> close all
+octave:218>
+octave:218>
+octave:218>
+octave:218>
+octave:218>
+octave:218> sharp_image = abs(fnorm - xnorm);
+sharp_image_norm = sharp_image / max(sharp_image(:));
+imshow(sharp_image_norm);
+
+%% Pinpoints these changes
+edge_image = zeros(size(sharp_image_norm));
+edge_image(sharp_image_norm > 3.5*std(sharp_image_norm(:))) = 1;
+imshow(edge_image);
+octave:224> close all
+octave:225>
diff --git a/lessons/lesson06/polynomials.m b/lessons/lesson06/polynomials.m
new file mode 100644
index 0000000..11dc4e3
--- /dev/null
+++ b/lessons/lesson06/polynomials.m
@@ -0,0 +1,66 @@
+%% Lesson 6a: Polynomials
+% Before we step into z transform, we should take a look at what functions
+% are there in MATLAB to evaluate polynomials. There are 4 major functions
+% to know about -- polyval, root/poly and conv.
+clc; clear; close all;
+
+%% polyval
+% Allows you to evaluate polynomials specified as an array of coefficients.
+p = [2 -2 0 -3]; % 2x^3 - 2x^2 - 3
+x = -5:.05:5;
+a = polyval(p,x);
+a2 = 2*x.^3 - 2*x.^2 - 3;
+
+figure;
+subplot(2, 1, 1);
+plot(x, a);
+title('Using polyval');
+
+subplot(2, 1, 2);
+plot(x, a2);
+title('Using algebraic expression');
+% The graphs are the same!
+
+%% roots/poly
+% These two functions are ~inverses! They allow you
+% to find the roots / coefficients of a vector of numbers
+r = roots(p);
+p1 = poly(r);
+
+% `poly` will always return a monic polynomial (the highest coefficient
+% will be 1).
+
+%% residue
+% Will apply later in the Signals course. Pretty much partial fractions
+% from Calc I.
+b = [-4 8];
+a = [1 6 8];
+[r, p, k] = residue(b, a);
+
+[bb, aa] = residue(r, p, k); % can go both ways
+
+% Note that a, b are row vectors, whereas r, p are column vectors! This
+% convention is true when we look at tf/zpk forms.
+
+%% conv
+% You've probably heard this a lot from Prof. Fontaine.
+% Convolution in the time domain is multiplication in the frequency domain
+% (applying a LTI system to a signal is convolution in the time domain).
+%
+% Convolution is also used to multiply two polynomials together (if
+% specified as an array of coefficients.)
+a = [1 2 -3 4];
+b = [5 0 3 -2];
+p2 = conv(a, b); % Use convolution
+
+x = -5:.05:5;
+
+% a little experiment
+figure;
+subplot(2, 1, 1);
+plot(x, polyval(a, x) .* polyval(b, x));
+title('Multiplying polynomial values elementwise');
+
+subplot(2, 1, 2);
+plot(x, polyval(p2, x));
+title('Using conv on convolved polynomials');
diff --git a/lessons/lesson06/splane.m b/lessons/lesson06/splane.m
new file mode 100644
index 0000000..d5113c5
--- /dev/null
+++ b/lessons/lesson06/splane.m
@@ -0,0 +1,16 @@
+% `splane` creates a pole-zero plot using pole and zero inputs with the
+% real and imaginary axes superimposed
+function splane(z,p)
+ x1 = real(z);
+ x2 = real(p);
+ y1 = imag(z);
+ y2 = imag(p);
+
+ plot(x1, y1, 'o', x2, y2, 'x');
+ xlabel('Real Part');
+ ylabel('Imaginary Part');
+ axis('equal');
+ line([0 0], ylim(), 'LineStyle', ':');
+ line(xlim(), [0 0], 'LineStyle', ':');
+end
+
diff --git a/lessons/lesson06/ztlesson.pdf b/lessons/lesson06/ztlesson.pdf
new file mode 100644
index 0000000..b77626e
Binary files /dev/null and b/lessons/lesson06/ztlesson.pdf differ
diff --git a/lessons/lesson06/ztransform.m b/lessons/lesson06/ztransform.m
new file mode 100644
index 0000000..7c2a105
--- /dev/null
+++ b/lessons/lesson06/ztransform.m
@@ -0,0 +1,237 @@
+%% Lesson 6b: The Z-transform
+% * Have a overview idea of the z transform
+% * Know how to plot pole zero plots of the z transform, and what to infer
+% * Know how to input a signal to the transfer function of z transform
+clc; clear; close all;
+
+%% Review
+% Plot all discrete-time signals with `stem()`! Don't use `plot()`.
+x = [2 4 7 2 0 8 5];
+stem(x);
+title("Discrete Time signal");
+
+%% Overview of the z transform
+% The z transform converts a discrete time signal into a complex frequency
+% domain representation (a series of complex exponentials).
+% There are two key motivations of the z transform:
+%
+% * By identifying the impulse response of the system we can determine how
+% it would perform in different frequencies
+% * By identifying the presence of increasing or decreasing oscillations in
+% the impulse response of a system we can determine whether the system
+% could be stable or not
+
+%% Z transform
+% One of the things we usually evaluate in the z transform is the pole zero
+% plot, since it can tell us about stability and causality in
+% different regions of convergence. Let us just look at one simple example
+% of a pole zero plot.
+
+%% Z transform - generate a pole zero plot
+% There are multiple ways to generate the pole zero plot. If you are
+% interested, you should look up tf2zp (and its inverse, zp2tf), pzplot,
+% etc. We are going to introduce the most simple function -- zplane. In the
+% example, b1 and a1 and coefficients of the numerator and denominator,
+% i.e, the transfer function. The transfer function is the
+% the ratio between output (denominator) and input (numerator).
+
+b1 = [1]; % numerator
+a1 = [1 -0.5]; % denominator
+figure;
+zplane(b1, a1);
+
+%% BE CAREFUL!
+% If the input vectors are COLUMN vectors, they would be interpreted as
+% poles and zeros, and if they are ROW vectors, they would be interpreted
+% as coefficients of the transfer function.
+
+%% BE CAREFUL!
+% zplane takes negative power coefficients, while tf2zp (the one used
+% in the homework) uses positive power coefficients.
+
+% In particular, note that polynomials with negative power efficients are
+% LEFT-ALIGNED, while polynomials with positive power coefficients are
+% RIGHT-ALIGNED. E.g., for positive coefficients b1 and [0 b1] are the
+% same polynomial, but not so if negative coefficients. See the following
+% example.
+
+% $1/(1-0.5z^-1) = z/(z-0.5)$
+figure;
+zplane(b1, a1);
+
+% $(0+z^-1)/(1-z^-1) = 1/(z-0.5)$
+figure;
+zplane([0 b1], a1);
+
+%% tf2zp and zp2tf
+b = [2 3 0];
+a = [1 1/sqrt(2) 1/4];
+
+[z, p, k] = tf2zp(b, a);
+% alternatively, tf2zpk
+
+figure;
+zplane(z, p);
+[b, a] = zp2tf(z, p, k);
+
+%% Z transform - More examples!
+
+b2 = [1];
+a2 = [1 -1.5];
+b3 = [1 -1 0.25];
+a3 = [1 -23/10 1.2];
+
+% Example 1 : One pole, outer region is stable and causal
+figure;
+subplot(3, 1, 1);
+zplane(b1, a1);
+title('$$H_1(z)=\frac{z}{z-\frac{1}{2}}$$', 'interpreter', 'latex');
+
+% Example 2 : One pole, outer region is causal but not stable, inner region
+% is stable but not causal
+subplot(3, 1, 2);
+zplane(b2, a2);
+title('$$H_2(z)=\frac{z}{z-\frac{3}{2}}$$', 'interpreter', 'latex');
+
+% Example 3 : Two poles, outer region is causal, middle region is stable
+subplot(3, 1, 3);
+zplane(b3, a3);
+title('$$H_3(z)=\frac{(z-\frac{1}{2})^2}{(z-\frac{4}{5})(z-\frac{3}{2})}$$', ...
+ 'interpreter', 'latex');
+
+%% On Stability and Causality
+% * The system is stable iff the region contains the unit circle.
+% * The system is causal iff the region contains infinity.
+% * If there is an infinity to the pole, then it's not stable.
+% * If all the poles are in the unit circle, then the system could be both
+% stable and causal.
+
+%% Z transform - Impulse Response
+% The impulse response is useful because its Fourier transform
+% (the frequency response) allows us to observe how an input sinusoid
+% at different frequencies will be transformed.
+%
+% The impulse response is obtained by applying the filter to a delta.
+% There is a custom function to do this (`impz`), but you can also obtain
+% it other ways.
+%
+% The impulse response for the running example is the following (can you
+% derive this by hand?)
+% $h[n] = (1/2)^n$
+[h1, t] = impz(b1, a1, 8); % h is the IMPULSE RESPONSE
+figure;
+impz(b1, a1, 32); % for visualization don't get the return value
+
+%% Z transform - Convolution
+% You can apply a system on a signal in two ways:
+% * Convolve with the impulse response (which we obtained just now).
+% * Use the `filter` functions (which uses tf coefficients).
+n = 0:1:5;
+x1 = (1/2).^n;
+y1 = conv(x1, h1);
+
+figure;
+subplot(2, 1, 1);
+stem(y1);
+title('Convolution between x_1 and h_1');
+subplot(2, 1, 2);
+y2 = filter(b1, a1, x1);
+stem(y2);
+title('Filter with b_1,a_1 and x_1');
+xlim([0 14]);
+% The above are the same! Except note that convolution creates a longer
+% "tail."
+
+%% freqz: digital frequency response
+% The frequency response is the z-transform of the impulse response. It
+% tells us how the system will act on each frequency of the input signal
+% (scaling and phase-shifting).
+%
+% This is a somewhat complicated setup -- you may want to save it to a
+% function if you have to do this repeatedly.
+[H, w] = freqz(b1, a1);
+
+Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+
+% Note the above, especially the `unwrap` function! Make sure to understand
+% what each of the functions above is for.
+
+figure;
+subplot(2, 1, 1);
+plot(w, Hdb);
+xlabel("Frequency (rad)");
+ylabel("|H| (dB)");
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+xticklabels({'0', '\pi/2', '\pi'});
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+plot(w, Hph);
+xlabel("Frequency (rad)");
+ylabel("Phase (deg)");
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+xticklabels({'0', '\pi/2', '\pi'});
+title("Phase Response");
+
+%% Another example of freqz
+% By default, `freqz` knows nothing about the sampling rate, so the
+% frequency scale is digital. We can make it in analog frequency by
+% specifying the sampling rate of the digital signal.
+
+n = 1024; % number of samples in frequency response
+fs = 20000; % sampling frequency of input signal
+[H, f] = freqz(b1, a1, n, fs);
+
+Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+
+figure;
+subplot(2, 1, 1);
+plot(f, Hdb);
+xlabel("Frequency (Hz)");
+ylabel("|H| (dB)")
+xlim([0 fs/2]);
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+plot(f, Hph);
+xlabel("Frequency (Hz)");
+ylabel("Phase (deg)");
+xlim([0 fs/2]);
+title("Phase Response");
+
+%% Yet another example of freqz
+zer = -0.5;
+pol = 0.9*exp(j*2*pi*[-0.3 0.3]');
+
+figure;
+zplane(zer, pol);
+
+[b4,a4] = zp2tf(zer, pol, 1);
+
+[H,w] = freqz(b4, a4);
+
+Hdb = 20*log10(abs(H));
+Hph = rad2deg(unwrap(angle(H)));
+
+figure;
+subplot(2, 1, 1);
+plot(w, Hdb);
+xlabel("Frequency (rad)");
+ylabel("|H| (dB)");
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+xticklabels({'0', '\pi/2', '\pi'});
+title("Magnitude Response");
+
+subplot(2, 1, 2);
+plot(w, Hph);
+xlabel("Frequency (rad)");
+ylabel("Phase (deg)");
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+xticklabels({'0', '\pi/2', '\pi'});
+title("Phase Response");
diff --git a/lessons/lesson06/ztransform.pdf b/lessons/lesson06/ztransform.pdf
new file mode 100644
index 0000000..6653735
Binary files /dev/null and b/lessons/lesson06/ztransform.pdf differ
diff --git a/lessons/lesson07/bandstop2500.m b/lessons/lesson07/bandstop2500.m
new file mode 100644
index 0000000..f14457a
--- /dev/null
+++ b/lessons/lesson07/bandstop2500.m
@@ -0,0 +1,35 @@
+function Hd = bandstop2500
+%BANDSTOP2500 Returns a discrete-time filter object.
+
+% MATLAB Code
+% Generated by MATLAB(R) 9.9 and Signal Processing Toolbox 8.5.
+% Generated on: 24-Mar-2021 17:32:55
+
+% Elliptic Bandstop filter designed using FDESIGN.BANDSTOP.
+
+% All frequency values are in Hz.
+Fs = 8192; % Sampling Frequency
+
+Fpass1 = 2000; % First Passband Frequency
+Fstop1 = 2300; % First Stopband Frequency
+Fstop2 = 2700; % Second Stopband Frequency
+Fpass2 = 3000; % Second Passband Frequency
+Apass1 = 0.5; % First Passband Ripple (dB)
+Astop = 60; % Stopband Attenuation (dB)
+Apass2 = 1; % Second Passband Ripple (dB)
+match = 'both'; % Band to match exactly
+
+% Construct an FDESIGN object and call its ELLIP method.
+h = fdesign.bandstop(Fpass1, Fstop1, Fstop2, Fpass2, Apass1, Astop, ...
+ Apass2, Fs);
+Hd = design(h, 'ellip', 'MatchExactly', match);
+
+% Get the transfer function values.
+[b, a] = tf(Hd);
+
+% Convert to a singleton filter.
+Hd = dfilt.df2(b, a);
+
+
+
+% [EOF]
diff --git a/lessons/lesson07/filter_design.m b/lessons/lesson07/filter_design.m
new file mode 100644
index 0000000..4bad32f
--- /dev/null
+++ b/lessons/lesson07/filter_design.m
@@ -0,0 +1,131 @@
+%% Lesson 7a. Designing filters the modern way
+% * Learn how to use `filterDesigner`
+% * Learn how to use `fdesign`
+close all; clear; clc;
+
+%% Filter Overview
+% Remember order = # of delay elements, want to minimize to meet
+% some spec.
+%
+% There are 4 main types of filters:
+% * Lowpass - passes low frequencies through
+% * Highpass - passes high frequencies through
+% * Bandpass - passes a specific band
+% * Bandstop - stops a specific band from going through
+%
+% There are four primary implementations of filters that optimize
+% different properties:
+% * Butterworth - maxflat passband, monotonic passband & stopband
+% * Chebychev I - equiripple passband, monotonic stopband
+% * Chebychev II - monotonic passband, equiripple stopband
+% * Elliptic - minimum order, equiripple passband & stopband
+%
+% In this lesson, you will be able to very clearly see these
+% properties and filter types!
+
+%% Filter design (GUI)
+% Note: Ask Prof. Keene about `filterDesigner`, it'll make him
+% very happy.
+filterDesigner; % then export to workspace as numerator and
+ % denominator (not SOS for now)
+
+%% Applying a GUI-designed filter
+% Note that if a filter is generated using a `filterDesigner` function,
+% it will be a filter object. You can apply a filter object using the
+% `filter()` function. If your filter is given as an impulse response, you
+% can either use the `filter()` function or convolve the impulse response
+% and the input signal.
+
+% Load handel again
+load handel;
+
+% Get a generated filter (this should decimate a narrow frequency band
+% centered at 2.5kHz, assuming a 8192Hz sampling frequency).
+flt = bandstop2500;
+
+% Perform the filter
+y1 = filter(flt, y);
+
+%% Filter design via `fdesign`
+% Filter specifications
+Fsample = 44100;
+Apass = 1;
+Astop = 80;
+Fpass = 1e3;
+Fstop = 1e4;
+
+% Use the appropriate `fdesign.*` object to create the filter; in this case,
+% we're designing a lowpass filter.
+specs = fdesign.lowpass(Fpass, Fstop, Apass, Astop, Fsample);
+
+% Note: the arguments to `fdesign.*` can be altered by passing in a SPEC option
+% -- the following call to `fdesign.lowpass` will produce the same filter. If
+% you want to do more complex stuff (like setting the 3 dB point directly or
+% setting the filter order), you can change this argument. See `doc
+% fdesign.lowpass` for more info.
+equivalent_specs = fdesign.lowpass('Fp,Fst,Ap,Ast', ...
+ Fpass, Fstop, Apass, Astop, Fsample);
+
+% For a given filter type, there are a myriad of different ways to design it
+% depending on what properties you want out of the filter (equiripple
+% passband/stopband, monotonic passband/stopband, minimum-order, max-flat,
+% etc.), which our lowpass filter object supports.
+%
+% Note: `'SystemObject'` should be set to `true` by Mathworks' recommendation
+% for these calls, even though they will often work without it.
+designmethods(specs, 'SystemObject', true);
+
+% There are also various options we can set when designing the filter. In this
+% example, we'll tell it to match the passband amplitude exactly.
+designoptions(specs, 'butter', SystemObject=true);
+
+% Finally, let's make this thing!
+butter_filter = design(specs, 'butter', ...
+ MatchExactly='passband', SystemObject=true);
+
+%% Examining a filter
+% Filters can be visualized with `fvtool`, drawing a nice box around the
+% transition zone and lines at the edge frequencies & important amplitudes:
+h = fvtool(butter_filter); % note no call to `figure`!
+
+% `measure` is another way to check:
+measurements = measure(butter_filter);
+
+% `h` is a handle to the graphics object we've created with `fvtool`, and we
+% can change its properties (listed with `get(h)`) if we wish:
+h.FrequencyScale = 'Log';
+ax = h.Children(15); % the axes object, found by looking at `h.Children`
+ax.YLim = [-60 10]; % in dB
+
+%% Using a filter
+% Filter objects have a lot of methods to them (things like `freqz`, `impz`,
+% `islinphase`, `isallpass`, etc -- check `methods(butter_filter)` if you want
+% the list) that are rather useful. One method they *don't* have is `filter`,
+% so if you want to apply a filter to a signal, you do it like this:
+load handel; % of course; loads `y` and `Fs` into the workspace
+y1 = butter_filter(y); % use like a function!
+sound(y, Fs);
+sound(y1, Fs);
+
+%% Plotting the result
+% Obtain the FFT of the original and the filtered signal.
+shifted_fft_mag = @(sig, len) fftshift(abs(fft(sig, len))) / len;
+N = 2^15;
+S1 = shifted_fft_mag(y, N);
+S2 = shifted_fft_mag(y1, N);
+F = linspace(-Fsample/2, Fsample/2, N); % not quite accurate but close
+
+% Plot the FFTs.
+figure;
+
+subplot(2, 1, 1);
+plot(F, S1);
+title 'Fourier Transform of Original Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
+
+subplot(2, 1, 2);
+plot(F, S2);
+title 'Fourier Transform of Filtered Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
diff --git a/lessons/lesson07/old_filter_design.m b/lessons/lesson07/old_filter_design.m
new file mode 100644
index 0000000..207f63a
--- /dev/null
+++ b/lessons/lesson07/old_filter_design.m
@@ -0,0 +1,103 @@
+%% Lesson 7c. Designing filters the old fashioned way
+% (notes kept for historical reasons; the new notes are in `filter_design.m`.
+%
+% * Learn how to use the `filterDesigner` GUI.
+% * Learn how to manually design a filter.
+clc; clear; close all;
+
+%% Filter Overview
+% Remember order = # of delay elements, want to minimize to meet
+% some spec.
+%
+% There are 4 main types of filters:
+% * Lowpass - passes low frequencies through
+% * Highpass - passes high frequencies through
+% * Bandpass - passes a specific band
+% * Bandstop - stops a specific band from going through
+%
+% There are four primary implementations of filters that optimize
+% different properties:
+% * Butterworth - maxflat passband, monotonic passband & stopband
+% * Chebychev I - equiripple passband, monotonic stopband
+% * Chebychev II - monotonic passband, equiripple stopband
+% * Elliptic - minimum order, equiripple passband & stopband
+%
+% In this lesson, you will be able to very clearly see these
+% properties and filter types!
+
+%% Filter Design (GUI)
+% Note: Ask Prof. Keene about `filterDesigner`, it'll make him
+% very happy.
+filterDesigner; % then export to workspace as numerator and
+ % denominator (not SOS for now)
+
+%% Filter Design (Code)
+% Manually specify specifications
+Fs = 44100;
+Apass = 1;
+Astop = 80;
+Fpass = 1e3;
+Fstop = 1e4;
+wpass = Fpass / (Fs/2); % normalized to nyquist freq.
+wstop = Fstop / (Fs/2); % normalized to nyquist freq.
+
+% Compute the minimum order for Butterworth filter
+% that meets the specifications.
+% Alternatively, see `butter1ord`, `cheby1ord`,
+% `cheby2ord`, `ellipord`.
+n = buttord(wpass, wstop, Apass, Astop);
+
+% Generate Butterworth filter with order `n` (which is
+% designed to meet the specification).
+% Alternatively, see `cheby1`, `cheby2`, `ellip`.
+[b, a] = butter(n, wpass);
+[H, W] = freqz(b, a);
+
+% Plot the frequency response. (At this point, recall that
+% the frequency response is the FFT of the impulse response.
+% Can you generate the frequency response without using
+% the `freqz` function (instead using the `impz` and `fft`
+% functions)?
+f = W .* Fs/(2*pi);
+figure;
+semilogx(f, 20*log10(abs(H)));
+xlim([1e2 1e5]);
+ylim([-100 5]);
+grid on;
+title 'Butterworth Lowpass Filter';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude (dB)';
+
+%% Applying a filter
+% Note that if a filter is generated using a `filterDesigner` function,
+% it will be a filter object. You can apply a filter object using the
+% `filter()` function. If your filter is given as an impulse response, you
+% can either use the `filter()` function or convolve the impulse response
+% and the input signal.
+
+% Load handel again
+load handel;
+
+% Get a generated filter (this should decimate a narrow frequency band
+% centered at 2.5kHz, assuming a 8192Hz sampling frequency).
+flt = bandstop2500;
+
+% Perform the filter
+y1 = filter(flt, y);
+
+%% How does it sound now?
+sound(y1, Fs);
+
+%% Plotting the result
+% Obtain the FFT of the filtered signal.
+N = 2^15;
+S = fft(y1, N);
+S = fftshift(abs(S)) / N;
+F = linspace(-Fs/2, Fs/2, N);
+
+% Plot the FFT.
+figure;
+plot(F, S);
+title 'Fourier Transform of Audio';
+xlabel 'Frequency (Hz)';
+ylabel 'Magnitude';
diff --git a/lessons/lesson07/transfer.m b/lessons/lesson07/transfer.m
new file mode 100644
index 0000000..e75ac6a
--- /dev/null
+++ b/lessons/lesson07/transfer.m
@@ -0,0 +1,93 @@
+%% Lesson 7c. Transfer function objects
+close all; clear; clc;
+
+%% Transfer function overview
+% Transfer functions are abstract representations of LTI systems in terms of
+% rational functions (numerator and denominator). We used to work with transfer
+% functions in terms of MATLAB-style polynomials, but Mathworks has come up
+% with a new, supposedly improved way: transfer function objects! These are
+% created via `tf` in a variety of ways. For more info, see Mathworks'
+% documentation -- these models are most helpful when working with cascades of
+% systems, such as for controls modeling.
+
+%% Digital transfer function objects
+% To create a digital tf. object, specify a numerator, denominator, and
+% sampling time (which may be set to -1 if you want to leave it undefined). The
+% numerator and denominator are specified in decreasing power order, with s^0
+% or z^0 on the right.
+dig_num = [1]; % 1z⁰ = 1
+dig_den = [2 -1]; % 2z¹ - 1z⁰ = 2z - 1
+timestep = 0.1; % 0.1 s
+dig_sys = tf(dig_num, dig_den, timestep);
+
+%% Analog transfer function objects
+% Analog transfer functions are created the same way as digital ones, just
+% without the timestep.
+an_num = [1 -1]; % 1s¹ - 1s⁰ = 1s - 1
+an_den = [1 2 1]; % s² + 2s + 1
+an_sys = tf(an_num, an_den);
+
+%% Alternate ways of specifying
+% MATLAB provides a syntax for creating transfer functions out of coded
+% rational expressions, as this can be clearer:
+s = tf('s'); % create an s variable that may then be used mathematically
+an_sys_2 = (s - 1)/(s + 1)^2; % same as above
+
+%% Investigating the transfer function
+% One property of interest is the step response, which we can view thus:
+figure;
+stepplot(an_sys);
+
+% MATLAB has a built-in function to get a Bode plot, too (though you should
+% modify this in post if you want to highlight specific details in the plot).
+figure;
+bodeplot(an_sys);
+
+%% One application
+% This example is stolen from Mathworks' doc page "Control System Modeling with
+% Model Objects," as it's an interesting perspective on what you can do with
+% these. In it, we model the following system:
+%
+% x ---> F(s) ---> +( ) ---> C(s) ---> G(s) ---+---> y
+% - |
+% ^ |
+% \---------- S(s) ---------/
+
+% specify the components of a system
+G = zpk([], [-1, -1], 1); % no zeroes, double pole at -1, analog
+C = pid(2, 1.3, 0.3, 0.5); % PID controller object
+S = tf(5, [1 4]);
+F = tf(1, [1 1]);
+
+% find the open-loop transfer function (by breaking the loop at the subtract)
+open_loop = F*S*G*C; % multiplication in the "frequency" domain
+
+% check out where the poles & zeroes are
+figure;
+pzplot(open_loop);
+title('Open loop pole/zero plot');
+xlim([-6 1]);
+ylim([-2 2]);
+
+% see what this thing does with a step input
+figure;
+stepplot(open_loop);
+title('Open-loop system step response');
+% note that this is *not* stable with a step input, as the PID controller loses
+% its shit!
+
+% find the closed-loop, whole_system response using `feedback` to specify a
+% feedback connection
+full_system = F*feedback(G*C, S);
+
+% see what this thing looks like now in the s-plane
+figure;
+pzplot(full_system);
+title('Full system pole/zero plot');
+xlim([-6 1]);
+ylim([-2 2]);
+
+% now stable with a step input, as we have feedback
+figure;
+stepplot(full_system);
+title('Full system step response');
diff --git a/lessons/lesson08/octave-log.txt b/lessons/lesson08/octave-log.txt
new file mode 100644
index 0000000..78f0076
--- /dev/null
+++ b/lessons/lesson08/octave-log.txt
@@ -0,0 +1,810 @@
+octave:5> pkg load symbolic
+octave:6> syms x z
+Symbolic pkg v3.0.1: Python communication link active, SymPy v1.11.1.
+octave:7> syms
+Symbolic variables in current scope:
+ x
+ z
+octave:8> x
+x = (sym) x
+octave:9> syms x_0
+octave:10> xu-0
+error: 'xu' undefined near line 1, column 1
+octave:11> x_0
+x_0 = (sym) x₀
+octave:12> v = x + 2
+v = (sym) x + 2
+octave:13> w = x * z
+w = (sym) x⋅z
+octave:14> help subs
+error: help: 'subs' not found
+octave:15> help @sym/subs
+'@sym/subs' is a function from the file /usr/share/octave/packages/symbolic-3.0.1/@sym
+/subs.m
+
+ -- Method on @sym: subs (F, X, Y)
+ -- Method on @sym: subs (F, Y)
+ -- Method on @sym: subs (F)
+ Replace symbols in an expression with other expressions.
+
+ Example substituting a value for a variable:
+ syms x y
+ f = x*y;
+ subs(f, x, 2)
+ ⇒ ans = (sym) 2⋅y
+ If X is omitted, ‘symvar’ is called on F to determine an
+ appropriate variable.
+
+ X and Y can also be vectors or lists of syms to replace:
+ subs(f, {x y}, {sin(x) 16})
+ ⇒ ans = (sym) 16⋅sin(x)
+
+ F = [x x*y; 2*x*y y];
+ subs(F, {x y}, [2 sym(pi)])
+ ⇒ ans = (sym 2×2 matrix)
+
+ ⎡ 2 2⋅π⎤
+ ⎢ ⎥
+ ⎣4⋅π π ⎦
+
+ With only one argument, ‘subs(F)’ will attempt to find values for
+ each symbol in F by searching the workspace:
+ f = x*y
+ ⇒ f = (sym) x⋅y
+
+ x = 42;
+ f
+ ⇒ f = (sym) x⋅y
+ Here assigning a numerical value to the variable ‘x’ did not change
+ the expression (because symbols are not the same as variables!)
+ However, we can automatically update ‘f’ by calling:
+ subs(f)
+ ⇒ ans = (sym) 42⋅y
+
+ *Warning*: ‘subs’ cannot be easily used to substitute a ‘double’
+ matrix; it will cast Y to a ‘sym’. Instead, create a “function
+ handle” from the symbolic expression, which can be efficiently
+ evaluated numerically. For example:
+ syms x
+ f = exp(sin(x))
+ ⇒ f = (sym)
+
+ sin(x)
+ ℯ
+
+ fh = function_handle(f)
+ ⇒ fh =
+
+ @(x) exp (sin (x))
+
+ fh(linspace(0, 2*pi, 700)')
+ ⇒ ans =
+ 1.0000
+ 1.0090
+ 1.0181
+ 1.0273
+ 1.0366
+ ...
+
+ *Note*: Mixing scalars and matrices may lead to trouble. We
+ support the case of substituting one or more symbolic matrices in
+ for symbolic scalars, within a scalar expression:
+ f = sin(x);
+ g = subs(f, x, [1 sym('a'); pi sym('b')])
+ ⇒ g = (sym 2×2 matrix)
+
+ ⎡sin(1) sin(a)⎤
+ ⎢ ⎥
+ ⎣ 0 sin(b)⎦
+
+ When using multiple variables and matrix substitions, it may be
+ helpful to use cell arrays:
+ subs(y*sin(x), {x, y}, {3, [2 sym('a')]})
+ ⇒ ans = (sym) [2⋅sin(3) a⋅sin(3)] (1×2 matrix)
+
+ subs(y*sin(x), {x, y}, {[2 3], [2 sym('a')]})
+ ⇒ ans = (sym) [2⋅sin(2) a⋅sin(3)] (1×2 matrix)
+
+ *Caution*, multiple interdependent substitutions can be ambiguous
+ and results may depend on the order in which you specify them. A
+ cautionary example:
+ syms y(x) A B
+ u = y + diff(y, x)
+ ⇒ u(x) = (symfun)
+ d
+ y(x) + ──(y(x))
+ dx
+
+ subs(u, {y, diff(y, x)}, {A, B})
+ ⇒ ans = (sym) A
+
+ subs(u, {diff(y, x), y}, {B, A})
+ ⇒ ans = (sym) A + B
+
+ Here it would be clearer to explicitly avoid the ambiguity by
+ calling ‘subs’ twice:
+ subs(subs(u, diff(y, x), B), y, A)
+ ⇒ ans = (sym) A + B
+
+ See also: @sym/symfun.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:16> syms r
+octave:17> w
+w = (sym) x⋅z
+octave:18> subs(w, z, r)
+ans = (sym) r⋅x
+octave:19> solve(x + 3 == 5, x)
+ans = (sym) 2
+octave:20>
+octave:20> syms a b c
+octave:21> solve(a*x^2 + b*x + c, x)
+ans = (sym 2×1 matrix)
+
+ ⎡ _____________⎤
+ ⎢ ╱ 2 ⎥
+ ⎢ b ╲╱ -4⋅a⋅c + b ⎥
+ ⎢- ─── - ────────────────⎥
+ ⎢ 2⋅a 2⋅a ⎥
+ ⎢ ⎥
+ ⎢ _____________⎥
+ ⎢ ╱ 2 ⎥
+ ⎢ b ╲╱ -4⋅a⋅c + b ⎥
+ ⎢- ─── + ────────────────⎥
+ ⎣ 2⋅a 2⋅a ⎦
+
+octave:22> help @sym/solve
+'@sym/solve' is a function from the file /usr/share/octave/packages/symbolic-3.0.1/@sy
+m/solve.m
+
+ -- Method on @sym: SOL = solve (EQN)
+ -- Method on @sym: SOL = solve (EQN, VAR)
+ -- Method on @sym: SOL = solve (EQN1, ..., EQNN)
+ -- Method on @sym: SOL = solve (EQN1, ..., EQNN, VAR1, ..., VARM)
+ -- Method on @sym: SOL = solve (EQNS, VARS)
+ -- Method on @sym: [S1, ..., SN] = solve (EQNS, VARS)
+ Symbolic solutions of equations, inequalities and systems.
+
+ Examples
+ syms x
+ solve(x == 2*x + 6, x)
+ ⇒ ans = (sym) -6
+ solve(x^2 + 6 == 5*x, x);
+ sort(ans)
+ ⇒ ans = (sym 2×1 matrix)
+ ⎡2⎤
+ ⎢ ⎥
+ ⎣3⎦
+
+ Sometimes its helpful to assume an unknown is real:
+ syms x real
+ solve(abs(x) == 1, x);
+ sort(ans)
+ ⇒ ans = (sym 2×1 matrix)
+ ⎡-1⎤
+ ⎢ ⎥
+ ⎣1 ⎦
+
+ In general, the output will be a list of dictionaries. Each entry
+ of the list is one a solution, and the variables that make up that
+ solutions are keys of the dictionary (fieldnames of the struct).
+ syms x y
+ d = solve(x^2 == 4, x + y == 1);
+
+ % the first solution
+ d{1}.x
+ ⇒ (sym) -2
+ d{1}.y
+ ⇒ (sym) 3
+
+ % the second solution
+ d{2}.x
+ ⇒ (sym) 2
+ d{2}.y
+ ⇒ (sym) -1
+
+ But there are various special cases for the output (single versus
+ multiple variables, single versus multiple solutions, etc). FIXME:
+ provide a ’raw_output’ argument or something to always give the
+ general output.
+
+ Alternatively:
+ [X, Y] = solve(x^2 == 4, x + y == 1, x, y)
+ ⇒ X = (sym 2×1 matrix)
+ ⎡-2⎤
+ ⎢ ⎥
+ ⎣2 ⎦
+ ⇒ Y = (sym 2×1 matrix)
+ ⎡3 ⎤
+ ⎢ ⎥
+ ⎣-1⎦
+
+ You can solve inequalities and systems involving mixed inequalities
+ and equations. For example:
+ solve(x^2 == 4, x > 0)
+ ⇒ ans = (sym) x = 2
+
+ syms x
+ solve(x^2 - 1 > 0, x < 10)
+ ⇒ ans = (sym) (-∞ < x ∧ x < -1) ∨ (1 < x ∧ x < 10)
+
+ See also: @sym/eq, @sym/dsolve.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:23> eqs = [x + z == 14; x - z == 6]
+<stdin>:7: SymPyDeprecationWarning:
+
+non-Expr objects in a Matrix is deprecated. Matrix represents
+a mathematical matrix. To represent a container of non-numeric
+entities, Use a list of lists, TableForm, NumPy array, or some
+other data structure instead.
+
+See https://docs.sympy.org/latest/explanation/active-deprecations.html#deprecated-non-
+expr-in-matrix
+for details.
+
+This has been deprecated since SymPy version 1.9. It
+will be removed in a future version of SymPy.
+
+eqs = (sym 2×1 matrix)
+
+ ⎡x + z = 14⎤
+ ⎢ ⎥
+ ⎣x - z = 6 ⎦
+
+octave:24> soln = solve(eqs, [x z])
+soln =
+
+ scalar structure containing the fields:
+
+ x =
+
+ <class sym>
+
+ z =
+
+ <class sym>
+
+
+octave:25> soln.x
+ans = (sym) 10
+octave:26> soln.z
+ans = (sym) 4
+octave:27> help subs
+error: help: 'subs' not found
+octave:28> help @sym/subs
+'@sym/subs' is a function from the file /usr/share/octave/packages/symbolic-3.0.1/@sym
+/subs.m
+
+ -- Method on @sym: subs (F, X, Y)
+ -- Method on @sym: subs (F, Y)
+ -- Method on @sym: subs (F)
+ Replace symbols in an expression with other expressions.
+
+ Example substituting a value for a variable:
+ syms x y
+ f = x*y;
+ subs(f, x, 2)
+ ⇒ ans = (sym) 2⋅y
+ If X is omitted, ‘symvar’ is called on F to determine an
+ appropriate variable.
+
+ X and Y can also be vectors or lists of syms to replace:
+ subs(f, {x y}, {sin(x) 16})
+ ⇒ ans = (sym) 16⋅sin(x)
+
+ F = [x x*y; 2*x*y y];
+ subs(F, {x y}, [2 sym(pi)])
+ ⇒ ans = (sym 2×2 matrix)
+
+ ⎡ 2 2⋅π⎤
+ ⎢ ⎥
+ ⎣4⋅π π ⎦
+
+ With only one argument, ‘subs(F)’ will attempt to find values for
+ each symbol in F by searching the workspace:
+ f = x*y
+ ⇒ f = (sym) x⋅y
+
+ x = 42;
+ f
+ ⇒ f = (sym) x⋅y
+ Here assigning a numerical value to the variable ‘x’ did not change
+ the expression (because symbols are not the same as variables!)
+ However, we can automatically update ‘f’ by calling:
+ subs(f)
+ ⇒ ans = (sym) 42⋅y
+
+ *Warning*: ‘subs’ cannot be easily used to substitute a ‘double’
+ matrix; it will cast Y to a ‘sym’. Instead, create a “function
+ handle” from the symbolic expression, which can be efficiently
+ evaluated numerically. For example:
+ syms x
+ f = exp(sin(x))
+ ⇒ f = (sym)
+
+ sin(x)
+ ℯ
+
+ fh = function_handle(f)
+ ⇒ fh =
+
+ @(x) exp (sin (x))
+
+ fh(linspace(0, 2*pi, 700)')
+ ⇒ ans =
+ 1.0000
+ 1.0090
+ 1.0181
+ 1.0273
+ 1.0366
+ ...
+
+ *Note*: Mixing scalars and matrices may lead to trouble. We
+ support the case of substituting one or more symbolic matrices in
+ for symbolic scalars, within a scalar expression:
+ f = sin(x);
+ g = subs(f, x, [1 sym('a'); pi sym('b')])
+ ⇒ g = (sym 2×2 matrix)
+
+ ⎡sin(1) sin(a)⎤
+ ⎢ ⎥
+ ⎣ 0 sin(b)⎦
+
+ When using multiple variables and matrix substitions, it may be
+ helpful to use cell arrays:
+ subs(y*sin(x), {x, y}, {3, [2 sym('a')]})
+ ⇒ ans = (sym) [2⋅sin(3) a⋅sin(3)] (1×2 matrix)
+
+ subs(y*sin(x), {x, y}, {[2 3], [2 sym('a')]})
+ ⇒ ans = (sym) [2⋅sin(2) a⋅sin(3)] (1×2 matrix)
+
+ *Caution*, multiple interdependent substitutions can be ambiguous
+ and results may depend on the order in which you specify them. A
+ cautionary example:
+ syms y(x) A B
+ u = y + diff(y, x)
+ ⇒ u(x) = (symfun)
+ d
+ y(x) + ──(y(x))
+ dx
+
+ subs(u, {y, diff(y, x)}, {A, B})
+ ⇒ ans = (sym) A
+
+ subs(u, {diff(y, x), y}, {B, A})
+ ⇒ ans = (sym) A + B
+
+ Here it would be clearer to explicitly avoid the ambiguity by
+ calling ‘subs’ twice:
+ subs(subs(u, diff(y, x), B), y, A)
+ ⇒ ans = (sym) A + B
+
+ See also: @sym/symfun.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:29> eqn(1)
+error: 'eqn' undefined near line 1, column 1
+octave:30> eqs(1)
+ans = (sym) x + z = 14
+octave:31> subs(eqs(1), {x, z}, {soln.x, soln.z})
+ans = (sym) True
+octave:32> syms n
+octave:33> solve(n^2 == 4, n)
+ans = (sym 2×1 matrix)
+
+ ⎡-2⎤
+ ⎢ ⎥
+ ⎣2 ⎦
+
+octave:34> assume n positive
+octave:35> assumptions
+ans =
+{
+ [1,1] = n: positive
+}
+
+octave:36> solve(n^2 == 4, n)
+ans = (sym) 2
+octave:37> assume n
+error: assume: general algebraic assumptions are not supported
+error: called from
+ assert at line 109 column 11
+ assume at line 65 column 3
+octave:38> syms n
+octave:39> assumptions
+ans = {}(0x0)
+octave:40> help @sym/assume
+'@sym/assume' is a function from the file /usr/share/octave/packages/symbolic-3.0.1/@s
+ym/assume.m
+
+ -- Method on @sym: X = assume (X, COND, COND2, ...)
+ -- Method on @sym: X = assume (X, 'clear')
+ -- Method on @sym: [X, Y] = assume ([X Y], ...)
+ -- Method on @sym: assume (X, COND, COND2, ...)
+ -- Method on @sym: assume (X, 'clear')
+ -- Method on @sym: assume ([X Y], ...)
+ New assumptions on a symbolic variable (replace old if any).
+
+ This function has two different behaviours depending on whether it
+ has an output argument or not. The first form is simpler; it
+ returns a new sym with assumptions given by COND, for example:
+ syms x
+ x1 = x;
+ x = assume(x, 'positive');
+ assumptions(x)
+ ⇒ ans =
+ {
+ [1,1] = x: positive
+ }
+ assumptions(x1) % empty, x1 still has the original x
+ ⇒ ans = {}(0x0)
+
+ Another example to help clarify:
+ x1 = sym('x', 'positive')
+ ⇒ x1 = (sym) x
+ x2 = assume(x1, 'negative')
+ ⇒ x2 = (sym) x
+ assumptions(x1)
+ ⇒ ans =
+ {
+ [1,1] = x: positive
+ }
+ assumptions(x2)
+ ⇒ ans =
+ {
+ [1,1] = x: negative
+ }
+
+ The second form—with no output argument—is different; it attempts
+ to find *all* instances of symbols with the same name as X and
+ replace them with the new version (with COND assumptions). For
+ example:
+ syms x
+ x1 = x;
+ f = sin(x);
+ assume(x, 'positive');
+ assumptions(x)
+ ⇒ ans =
+ {
+ [1,1] = x: positive
+ }
+ assumptions(x1)
+ ⇒ ans =
+ {
+ [1,1] = x: positive
+ }
+ assumptions(f)
+ ⇒ ans =
+ {
+ [1,1] = x: positive
+ }
+
+ To clear assumptions on a variable use ‘assume(x, 'clear')’, for
+ example:
+ syms x positive
+ f = sin (x);
+ assume (x, 'clear')
+ isempty (assumptions (f))
+ ⇒ ans = 1
+
+ *Warning*: the second form operates on the caller’s workspace via
+ evalin/assignin. So if you call this from other functions, it will
+ operate in your function’s workspace (and not the ‘base’
+ workspace). This behaviour is for compatibility with other
+ symbolic toolboxes.
+
+ FIXME: idea of rewriting all sym vars is a bit of a hack, not well
+ tested (for example, with global vars.)
+
+ See also: @sym/assumeAlso, assume, assumptions, sym, syms.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:41> assume n positive
+octave:42> assumptions
+ans =
+{
+ [1,1] = n: positive
+}
+
+octave:43> assume n clear
+octave:44> assumptions
+ans = {}(0x0)
+octave:45> exp(log(x))
+ans = (sym) x
+octave:46> e^(log(x))
+ans = (sym) x
+octave:47> x + 3
+ans = (sym) x + 3
+octave:48> (x + 3)*(1/x + 3)
+ans = (sym)
+
+ ⎛ 1⎞
+ ⎜3 + ─⎟⋅(x + 3)
+ ⎝ x⎠
+
+octave:49> (x + 3)*(1/(x + 3))
+ans = (sym) 1
+octave:50> help @sym/simplify
+'@sym/simplify' is a function from the file /usr/share/octave/packages/symbolic-3.0.1/
+@sym/simplify.m
+
+ -- Method on @sym: simplify (X)
+ Simplify an expression.
+
+ Example:
+ syms x
+ p = x^2 + x + 1
+ ⇒ p = (sym)
+ 2
+ x + x + 1
+ q = horner (p)
+ ⇒ q = (sym) x⋅(x + 1) + 1
+
+ d = p - q
+ ⇒ d = (sym)
+ 2
+ x - x⋅(x + 1) + x
+
+ isAlways(p == q)
+ ⇒ 1
+
+ simplify(p - q)
+ ⇒ (sym) 0
+
+ Please note that ‘simplify’ is not a well-defined mathematical
+ operation: its precise behaviour can change between software
+ versions (and certainly between different software packages!)
+
+ See also: @sym/isAlways, @sym/factor, @sym/expand, @sym/rewrite.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:51> (x + 3)^2 + (x + 3) + (x + 3)^2
+ans = (sym)
+
+ 2
+ x + 2⋅(x + 3) + 3
+
+octave:52> simplify((x + 3)^2 + (x + 3) + (x + 3)^2)
+ans = (sym)
+
+ 2
+ x + 2⋅(x + 3) + 3
+
+octave:53> expand((x + 3)^2 + (x + 3) + (x + 3)^2)
+ans = (sym)
+
+ 2
+ 2⋅x + 13⋅x + 21
+
+octave:54> collect((x + 3)^2 + (x + 3) + (x + 3)^2)
+error: 'collect' undefined near line 1, column 1
+octave:55> combine
+error: 'combine' undefined near line 1, column 1
+octave:56> factor
+error: Invalid call to factor. Correct usage is:
+
+ -- PF = factor (Q)
+ -- [PF, N] = factor (Q)
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:57> expand((x + 3)^2 + (x + 3) + (x + 3)^2)
+ans = (sym)
+
+ 2
+ 2⋅x + 13⋅x + 21
+
+octave:58> factor(ans)
+ans = (sym) (x + 3)⋅(2⋅x + 7)
+octave:59> partfrac
+error: 'partfrac' undefined near line 1, column 1
+octave:60> help @sym/partfrac
+'@sym/partfrac' is a function from the file /usr/share/octave/packages/symbolic-3.0.1/
+@sym/partfrac.m
+
+ -- Method on @sym: partfrac (F)
+ -- Method on @sym: partfrac (F, X)
+ Compute partial fraction decomposition of a rational function.
+
+ Examples:
+ syms x
+ f = 2/(x + 4)/(x + 1)
+ ⇒ f = (sym)
+ 2
+ ───────────────
+ (x + 1)⋅(x + 4)
+
+ partfrac(f)
+ ⇒ ans = (sym)
+ 2 2
+ - ───────── + ─────────
+ 3⋅(x + 4) 3⋅(x + 1)
+
+ Other examples:
+ syms x y
+ partfrac(y/(x + y)/(x + 1), x)
+ ⇒ ans = (sym)
+ y y
+ - ─────────────── + ───────────────
+ (x + y)⋅(y - 1) (x + 1)⋅(y - 1)
+
+ partfrac(y/(x + y)/(x + 1), y)
+ ⇒ ans = (sym)
+ x 1
+ - ─────────────── + ─────
+ (x + 1)⋅(x + y) x + 1
+
+ See also: @sym/factor.
+
+Additional help for built-in functions and operators is
+available in the online version of the manual. Use the command
+'doc <topic>' to search the manual index.
+
+Help and information about Octave is also available on the WWW
+at https://www.octave.org and via the help@octave.org
+mailing list.
+octave:61>
+octave:61>
+octave:61>
+octave:61>
+octave:61>
+octave:61> partfrac((x^7+x^2)/(x^3 - 3*x + 2))
+ans = (sym)
+
+ 4 2 124 25 2
+ x + 3⋅x - 2⋅x + 9 - ───────── + ───────── + ──────────
+ 9⋅(x + 2) 9⋅(x - 1) 2
+ 3⋅(x - 1)
+
+octave:62> (x^7+x^2)/(x^3 - 3*x + 2)
+ans = (sym)
+
+ 7 2
+ x + x
+ ────────────
+ 3
+ x - 3⋅x + 2
+
+octave:63> p = [1 2 3 4]
+p =
+
+ 1 2 3 4
+
+octave:64> poly2sim(p, x)
+error: 'poly2sim' undefined near line 1, column 1
+octave:65> syms f(y)
+octave:66> syms
+Symbolic variables in current scope:
+ a
+ ans
+ b
+ c
+ eqs
+ f (symfun)
+ n
+ r
+ v
+ w
+ x
+ x_0
+ y
+ z
+octave:67> f
+f(y) = (symfun) f(y)
+octave:68> f(y) = y^2
+f(y) = (symfun)
+
+ 2
+ y
+
+octave:69> f
+f(y) = (symfun)
+
+ 2
+ y
+
+octave:70> finverse
+error: 'finverse' undefined near line 1, column 1
+
+The 'finverse' function belongs to the symbolic package from Octave
+Forge but has not yet been implemented.
+
+Please read <https://www.octave.org/missing.html> to learn how you can
+contribute missing functionality.
+octave:71> syms t
+octave:72> int(1/t, 1, x)
+ans = (sym) log(x)
+octave:73> int(1/t, .5, x)
+warning: passing floating-point values to sym is dangerous, see "help sym"
+warning: called from
+ double_to_sym_heuristic at line 50 column 7
+ sym at line 379 column 13
+ int at line 138 column 7
+
+ans = (sym) log(x) + log(2)
+octave:74> x^2
+ans = (sym)
+
+ 2
+ x
+
+octave:75> x^2 + 5
+ans = (sym)
+
+ 2
+ x + 5
+
+octave:76> diff(x^2 + 5)
+ans = (sym) 2⋅x
+octave:77> diff(x^2 + z^3 + 5)
+ans = (sym) 2⋅x
+octave:78> diff(x^2 + z^3 + 5, z)
+ans = (sym)
+
+ 2
+ 3⋅z
+
+octave:79> syms y(t) a
+octave:80> diff(y) == a*y
+ans = (sym)
+
+ d
+ ──(y(t)) = a⋅y(t)
+ dt
+
+octave:81> dsolve(diff(y) == a*y, t)
+error: Python exception: AttributeError: 'Symbol' object has no attribute 'lhs'
+ occurred at line 7 of the Python code block:
+ ics = {ics.lhs: ics.rhs}
+error: called from
+ pycall_sympy__ at line 179 column 7
+ dsolve at line 198 column 8
+octave:82> dirac(x)
+ans = (sym) δ(x)
+octave:83> laplace(dirac(x))
+ans = (sym) 1
+octave:84> laplace(dirac(x - 1))
+ans = (sym)
+
+ -s
+ ℯ
+
+octave:85> (1:5)(2)
+ans = 2
+octave:86>
diff --git a/lessons/lesson08/octave-workspace b/lessons/lesson08/octave-workspace
new file mode 100644
index 0000000..e70527a
Binary files /dev/null and b/lessons/lesson08/octave-workspace differ
diff --git a/lessons/lesson08/symbolic_toolbox.m b/lessons/lesson08/symbolic_toolbox.m
new file mode 100644
index 0000000..8084666
--- /dev/null
+++ b/lessons/lesson08/symbolic_toolbox.m
@@ -0,0 +1,183 @@
+%% Symbolic Toolbox
+% MATLAB is typically associated with computing *numerical* solutions
+% (i.e., estimating solutions using floating point calculations), we
+% may often want to compute *analytic* solutions (exact solutions,
+% often represented by formulas).
+%
+% Wolfram Mathematica is a premier example of technical software with
+% support for symbolic computation, but we can achieve many useful
+% symbolic calculations in MATLAB using the *symbolic toolbox*.
+%
+% The symbolic toolbox is useful if you want to represent an abstract
+% expression. You can think of a symbolic calculation as storing the
+% structure of expressions to operate on rather than the values
+% themselves.
+clear; clc; close all;
+
+%% Create symbolic variables
+syms x z;
+
+%% Creating expressions
+v = x + 2;
+w = x * z;
+
+%% Substituting expressions into a symbolic expression
+subs(v, x, 3);
+
+% You can also substitute symbolic expressions, not only values.
+syms r;
+subs(w, z, r);
+
+%% Solve a symbolic equation for a variable
+% You may want to look at the documentation page for the `solve` function.
+% If the expression cannot be solved analytically, MATLAB will default
+% to using a numerical solver.
+solve(x + 3 == 5, x);
+
+syms a b c;
+solve(a*x^2 + b*x + c, x);
+
+% Solving multiple variables simultaneously
+eqs = [x + z == 14, x - z == 6];
+soln = solve(eqs, [x z]);
+soln.x;
+soln.z;
+
+%% Set "assumptions" (constraints) on a variable
+% This is the shorthand syntax for setting assumptions on symbolic objects.
+% See also the `assume` function.
+syms n positive;
+solve(n^2 == 4, n);
+
+%% Simplify an expression
+% This attempts to simplify an expression using known
+% mathematical identities.
+simplify(exp(log(x)));
+
+% See also `simplifyFraction` for more efficient simplification
+% of fractions only.
+
+%% Collect
+% This collects coefficients of like terms together.
+collect((x + 2) * (x + 3), x);
+
+%% Combine
+% Tries to rewrite products of powers in the expression as a single power
+combine(x^4 * x^3);
+
+% Has other options such as 'log' and 'sincos'
+assume(x > 0);
+combine(log(x) + log(2*x), 'log');
+combine(sin(x)*cos(x), 'sincos');
+
+%% Expand
+% Similar to `collect`, but expands all variables. More or less the
+% inverse of `combine`.
+expand((x + 5)^3);
+
+%% Partial fraction decomposition
+partfrac((x^7+x^2)/(x^3 - 3*x + 2));
+
+%% poly2sym and sym2poly
+p = [1 2 3 4];
+psym = poly2sym(p, x);
+q = sym2poly(psym);
+
+%% Symbolic functions
+% Symbolic functions are like symbolic expressions but with a formal
+% parameter rather than all free parameters. This may be useful for
+% substituting for a symbolic variable or for differential equations.
+syms f(y);
+f(y) = y^2;
+f(4);
+
+%% Compose symbolic functions
+syms g(y)
+g(y) = sin(y);
+h = compose(f, g);
+
+%% Symbolic function inverse
+f(y) = exp(y);
+finverse(f)
+
+%% Symbolic differentiation
+% Note that this `diff` is different from the `diff` from before.
+% This one is actually `sym.diff`.
+syms x y z;
+v = x^2 + 16*x + 5;
+dvdx = diff(v);
+dvdx_3 = subs(dvdx, x,3);
+
+%% Partial differentiation
+% This function assumes derivatives commute (i.e., second partials exist
+% and are continuous).
+w = -x^3 + x^2 + 3*y + 25*sin(x*y);
+dwdx = diff(w, x);
+dwdy = diff(w, y);
+
+% third order derivative w.r.t. x
+dw3dx3 = diff(w, x, 3);
+
+% Vector operations exist as well such as: curl, divergence, gradient,
+% laplacian, jacobian, hessian...
+
+%% Differential equations
+syms y(t) a;
+dsolve(diff(y) == a*y, t);
+dsolve(diff(y) == a*y, y(0) == 5); % with initial conditions
+
+%% Symbolic integration
+v = x^2 + 16*x + 5;
+V = int(v);
+
+%% Definite integral
+syms a b;
+Vab = int(v, a, b);
+V01 = int(v, 0, 1);
+
+%% z transform
+syms a n z;
+f = a^n;
+ztrans(f);
+
+%% Symbolic matrices
+syms m;
+n = 4;
+A = m.^((0:n)'*(0:n));
+D = diff(A);
+
+%% Laplace Transform
+syms t a s;
+f = exp(-a*t);
+laplace(f);
+
+%% Inverse Laplace Transform
+ilaplace(-4*s+8/(s^2+6*s+8));
+
+%% Fourier Transform
+f = dirac(x) + sin(x);
+fourier(f);
+
+%% Inverse Fourier Transform
+syms w;
+
+% fourier transform of cosine(t)
+expr = dirac(w+1) + dirac(w-1);
+ift = ifourier(expr);
+simplify(ift);
+
+%% Note: performance
+syms x v;
+v = x^3 + 5*x^2 - 3*x + 9;
+t = 1:1e4;
+
+tic;
+X = t.^3 + 5*t.^2 - 3*t + 9;
+toc;
+
+tic;
+Y = subs(v,x,t);
+toc;
+
+% Symbolic computation and data structures are relatively expensive
+% when compared to numerical operations! Use sparingly!
diff --git a/lessons/lesson09/stoch.m b/lessons/lesson09/stoch.m
new file mode 100644
index 0000000..d365abf
--- /dev/null
+++ b/lessons/lesson09/stoch.m
@@ -0,0 +1,182 @@
+%% INTRODUCTION TO STOCHASTICITY
+
+% Electrical engineers often use MATLAB to simulate random process, and the
+% reason that it is so popular is that a lot of problems in EE are
+% stochastic problems. What happens when there's random noise in a channel,
+% what is the probability of corrupting a signal to the point that it
+% cannot be recovered?
+
+% Today, we are going to explore the idea of randomness with case studies.
+% We are first going to show simple examples of stochastic processes, and
+% will conclude with a more elaborate exploration of stochasticity in
+% communication channels
+clc; clear; close all;
+
+%% EXAMPLE 1 : MONTY HALL
+% The classic monty hall problem -- do you switch the door or not. The
+% correct answer is you always switch. But why?? Let's find out
+
+% Let's do a simulation to experimentally see the results, let's say there
+% are two contestants, the first contestant never switches, the second
+% contestant switches.
+
+% There are different ways to do it. Let's assume that the prize is
+% always behind door 3 and we choose at random which door the contestant
+% chooses.
+
+% If contestant 1 picks 3, then the contestant wins. For contestant 2, if
+% the contestant chooses 3 then we choose one of the two other doors at
+% random to show. If contestant 2 chose door 1 or 2, we show the
+% contestant the other empty door.
+
+N = 5e5;
+choose_door = unidrnd(3,1,N);
+result = zeros(2,N);
+result(1, choose_door == 3) = 1;
+
+% Inefficient but for readability. It's clear that contestant 2 has a 2/3
+% success rate
+for i = 1:N
+ if choose_door(i) == 3
+ choose_door(i) = unidrnd(2,1);
+ else
+ choose_door(i) = 3;
+ end
+end
+
+result(2, choose_door == 3) = 1;
+
+% take the running mean
+win_rate = cumsum(result, 2);
+x = 1:1:N;
+win_rate = win_rate ./ [x;x];
+
+%% Plotting the results of example 1
+figure;
+hold on;
+plot(x, win_rate(1,:), 'DisplayName', 'no switch');
+plot(x, win_rate(2,:), 'DisplayName', 'switch');
+hold off;
+legend show;
+title 'Monty Hall Problem Simulation Results';
+
+%% EXAMPLE 2 : CASINO
+
+% A casino offers a game in which a fair coin is tossed repeatedly until
+% the first heads comes up, for a total of k+1 tosses. The casino will pay
+% out k dollars
+
+% Let's see how much money you are going to get if you play the game
+N = 5e5;
+num_tosses = zeros(1,N);
+for i = 1:N
+ res = randi([0,1]);
+ while (res ~= 0)
+ num_tosses(i) = num_tosses(i) + 1;
+ res = randi([0,1]);
+ end
+end
+
+figure;
+hold on;
+histogram(num_tosses, unique(num_tosses));
+hold off;
+xlabel Frequency;
+ylabel Winnings;
+
+%% Radar/Signal Detection Example
+N = 10000;
+i = 1:N;
+rdm = rand(1,N);
+X = rdm > 0.5; % Signal: 0 or 1 equiprobable
+N = randn(1,N); % Noise: standard normal
+
+Y = X + N; % Recieved
+
+% For each observation, either the sample comes from N(0,1) or N(1,1)
+
+% To guess if the true signal for each observation, x_i is 0 or 1 we
+% compute P[true = 0 | x_i] and P[true = 1 | x_i] and choose the larger
+% one.
+
+% By Bayes' Law,
+% P[x_i | true = S]*P[true = S]
+% P[true = S | x_i] = ----------------------------- , S = 0,1
+% P[x_i]
+
+% Since P[x_i] is common to both P[true = 0 | x_i] and P[true = 1 | x_i],
+% we only have to maximize the numerator. Also since
+% P[true = 0] = P[true = 1], we need to maximize P[x_i | true = S], S = 0,1
+
+%% Showing the decision boundary visually
+x = -5:0.001:5;
+figure;
+plot(x, [normpdf(x,0,1); normpdf(x,1,1)]);
+xline(0.5, '--');
+
+legend 'Sent 0', 'Sent 1';
+xlabel 'Received value';
+ylabel 'Likelihood';
+
+%% Modeling the distribution boundary
+prob0 = normpdf(Y, 0, 1);
+prob1 = normpdf(Y, 1, 1);
+
+decision = prob1 > prob0;
+result = decision == X;
+
+figure;
+plot(i, cumsum(result)./i);
+xlabel 'Number of Observations';
+ylabel 'Percent Correct';
+title 'Signal Detection';
+
+%%
+% What happens if the prior probabilities are not equal?
+%
+% If the priors are known, then we can multiply them to the probabilities
+% we used earlier according to Bayes' Law. This technique is called MAP
+% (Maximum a posteriori) estimation.
+%
+% But if the priors are NOT known, then we can assume equiprobable and use
+% the same technique as before. This technique is called ML (Maximum
+% likelihood) estimation. Clearly this performs worse than MAP if the
+% priors are not equal, i.e., the equiprobable assumption is false.
+
+P0 = 0.8;
+X = rdm > P0; % Now the signal is 1 with probability 0.2
+
+Y = X + N;
+
+%% Graphing this new problem
+figure;
+subplot(211);
+plot(x, [normpdf(x,0,1); normpdf(x,1,1)]);
+xline(0.5, '--');
+title ML;
+xlabel 'Received value';
+ylabel 'Likelihood';
+
+subplot(212);
+plot(x, [normpdf(x,0,1)*P0; normpdf(x,1,1)*(1 - P0)]);
+xline(0.5 + log(P0/(1-P0)), '--');
+title 'MAP';
+xlabel 'Received value';
+ylabel 'Likelihood';
+
+%% MAP decision rule
+% Now we include the priors
+prob0MAP = normpdf(Y, 0, 1) * P0;
+prob1MAP = normpdf(Y, 1, 1) * (1 - P0);
+
+decisionMAP = prob1MAP > prob0MAP;
+
+resultML = decision == X; % previous example was ML
+resultMAP = decisionMAP == X; % current MAP estimate
+
+figure;
+plot(i, [cumsum(resultML)./i; cumsum(resultMAP)./i]);
+xlabel 'Number of Observations';
+ylabel 'Percent Correct';
+title 'Signal Detection: MAP vs. ML';
+legend ML MAP;
\ No newline at end of file
diff --git a/lessons/lesson10/chaos.m b/lessons/lesson10/chaos.m
new file mode 100644
index 0000000..630871e
--- /dev/null
+++ b/lessons/lesson10/chaos.m
@@ -0,0 +1,256 @@
+%% Lesson 10, extra -- Chaos!
+% Based on work for PH429: Chaos & Nonlinear Dynamics. The following was
+% translated from the original NumPy/Matplotlib code. See Strogatz' Nonlinear
+% Dynamics and Chaos (the source for much of this information) if you want to
+% know more!
+
+close all; clear; clc;
+
+% Simple systems of nonlinear differential equations (also iterated nonlinear
+% maps) often exhibit surprisingly complex behavior. Lorenz's "strange
+% attractor," discovered in 1963, was an example of this: he discovered that an
+% exceedingly simplified (and entirely deterministic!) model of the weather was
+% still able to produce unpredictable fluctuations in temperature, wind speed,
+% etc. -- the system exponentially amplified miniscule parameter changes, so
+% every decimal place mattered. Further work with analog computers and the
+% then-emerging digital computers led to the field we now call chaos theory.
+
+% Chaos has much to do with fractal geometry, so we'll start by generating a
+% few fractals. Because... why not?
+
+%% The Chaos Game & Sierpinski Triangles
+% The game is simple:
+% 1. Place three vertices in the plane, corresponding to the vertices of an
+% equilateral triangle.
+% 2. Randomly choose a point P(0).
+% 3. Each turn, choose a random vertex, and plot a point P(n) halfway between
+% that vertex and point P(n - 1).
+% Let's see what happens.
+
+n1 = 5e5; % points
+vertices = [0, 0; .5, .5*sqrt(3); 1, 0];
+r = .5; % fraction of the way between points and vertices (here halfway)
+
+% point coordinates
+points = zeros(n1, 2);
+choices = randi([1 3], n1, 1);
+
+% randomly choose the first point somewhere in the plane
+points(1, :) = 100*randn(1, 2);
+
+% choose the rest
+% note: the loop here is actually necessary, like it was in G-S -- each
+% iteration depends upon the previous one.
+for i = 2:n1
+ choice = choices(i);
+ vertex = vertices(choice, :);
+ last_point = points(i - 1, :);
+
+ points(i, :) = last_point + r*(vertex - last_point);
+end
+
+% plot the thing
+figure;
+scatter(points(:, 1), points(:, 2), 1);
+xlim([-.1 1.1]);
+ylim([-.2 1]);
+xticks([]);
+yticks([]);
+pbaspect([1 1 1]); % make the plot box square
+title('The Sierpinski Triangle, by Chance', 'Interpreter', 'latex');
+
+% A rather important property of fractals is their fractional dimension -- this
+% triangle, for example, is not quite two-dimensional but is clearly more than
+% one-dimensional. So how do we measure this dimension?
+
+% One simple measure, known as self-similarity dimension, is relatively easy to
+% find for the Sierpinski triangle. The idea is we simply examine how many
+% smaller versions of a given shape could fit into that shape, and what size
+% they would be in relation. For example, a square with scale 1 can be made up
+% of 4 smaller squares scaled by 1/2, or 9 smaller squares with scale 1/3:
+% ----------------- -----------------
+% | | | | |
+% | | | 1/2 | 1/2 |
+% | | | | |
+% | 1 | = |-------|-------|
+% | | | | |
+% | | | 1/2 | 1/2 |
+% | | | | |
+% ----------------- -----------------
+% A square thus has self-similarity dimension log₂4 = log₃9 = 2 (as one might
+% expect). The Sierpinski triangle is composed of 3 copies of itself scaled by
+% 1/2, and thus has self-similarity dimension log₂3 ≈ 1.585.
+
+% I wonder if we could have MATLAB calculate these...
+
+%% Iterated Maps
+% One way to create chaotic behavior is to iterate a nonlinear map: e.g., plot
+% the set {x, sin(x), sin(sin(x)), sin(sin(sin(x))), ...}. The values will
+% generally hop unpredictably around, but will often stay closer to some values
+% than others...
+
+% The following plot is a visual representation of these iterations, called a
+% cobweb. Cobwebs are often used to qualitatively analyze nonlinear maps. The
+% idea is, since the output of the map will be fed back into the input, we plot
+% a point on the map's curve, translate horizontally to the line y = x, then
+% translate vertically back to the map and repeat. This gives a visual
+% representation of plugging the output back into the input.
+
+% When you see this, think about which points on the map will "attract" the
+% cobweb, which will "repel" it, and how one might go about making that
+% distinction.
+
+n2 = 10;
+map = @(x) 2*sin(x);
+
+% create the map curve and the center line
+x = linspace(0, pi, 100).';
+y_map = map(x);
+y_line = x;
+
+% create the cobweb
+map_pts = zeros(2*n2 + 1, 2);
+map_pts(1, :) = [.2, 0];
+for i = 1:n2
+ last_pt = map_pts(2*i - 1, :);
+ mapped = [last_pt(1), map(last_pt(1))];
+ on_the_line = [mapped(2), mapped(2)];
+
+ map_pts(2*i, :) = mapped;
+ map_pts(2*i + 1, :) = on_the_line;
+end
+
+% plot it
+figure;
+hold on;
+plot(x, y_map, 'b');
+plot(x, y_line, 'k');
+plot(map_pts(:, 1), map_pts(:, 2), 'r');
+
+start = map_pts(1, :);
+scatter(start(1), start(2), 'r');
+text(start(1) + .03, start(2) + .03, 'Start Point');
+
+xlim([0 pi]);
+xticks([0 pi/2 pi]);
+ylim([0 2.1]);
+xticklabels({'0', '\pi/2', '\pi'});
+
+title('Cobweb of \(2 \sin x\)', 'Interpreter', 'latex');
+
+%% Orbit Diagrams
+% The above example was... not too crazy. The map x → α sin x *can* exhibit
+% more complex behavior, but it requires α to be greater than π (so that we can
+% escape the [0, π] interval). The map will first stay close to two points,
+% then a few, then... altogether too many. This is usually plotted with what is
+% called an orbit diagram, showing roughly how much time the map spends at
+% certain values for a given α.
+
+map_alpha = @(alpha, x) alpha.*sin(x);
+
+n3 = 6e2; % iterations to go through
+frames = 9e2; % run this many parallel alpha computations
+toss = 2e2; % delete this many points from the start
+
+min_alpha = 2.2;
+max_alpha = 3.1;
+alphas = linspace(min_alpha, max_alpha, frames);
+
+% at least we can run the frames in parallel...
+values = zeros(n3, frames);
+values(1, :) = pi/4; % the first input each time
+for i = 2:n3
+ values(i, :) = map_alpha(alphas, values(i - 1, :));
+end
+
+% plot this crazy thing
+values = values((toss + 1):end, :);
+figure;
+scatter(alphas.', values, 1, [.7 0 0], 'MarkerEdgeAlpha', .3);
+
+title('Orbit of \(\alpha \sin x\)', 'Interpreter', 'latex');
+xlabel('\alpha');
+ylabel('map value');
+
+%% The Phase Plane
+% In continuous time, we have differential equations rather than iterated maps.
+% One way to analyze higher-order equations is using the so-called phase plane,
+% which is a plot of our state variable and its derivative. The idea is that a
+% second-order differential equation really acts like a system in two
+% dimensions, and is best analyzed directly as such.
+
+% We'll analyze the motion of an damped pendulum, which has the governing
+% equation d²θ/dt² + b dθ/dt + (g/L) sin θ = 0. For simplicity (it won't change
+% the qualitative behavior), we'll let g/L = 1. Then (the phase plane bit)
+% we break this into two differential equations,
+% * dθ/dt = ω and
+% * dω/dt = -(sin θ + bω),
+% and make a plot of the *vector* (dθ/dt, dω/dt) in θ-ω space. We can easily
+% plot the vector field with `quiver`, and MATLAB's `streamline` is rather
+% effective at plotting either single trajectories or a whole suite of possible
+% solutions.
+
+b = .2;
+min_theta = -3*pi;
+max_theta = 3*pi;
+min_omega = -3*pi;
+max_omega = 3*pi;
+theta_range = linspace(min_theta, max_theta, 300);
+omega_range = linspace(min_omega, max_omega, 200);
+[theta, omega] = meshgrid(theta_range, omega_range);
+theta_dot = omega;
+omega_dot = -(sin(theta) + b*omega);
+
+% to cut down on the # of points we plot
+d = 10;
+[sparse_x, sparse_y] = ...
+ meshgrid(d:d:length(theta_range), d:d:length(omega_range));
+sparse = sub2ind(size(theta), sparse_y(:), sparse_x(:));
+
+% plot this thing
+figure;
+hold on;
+quiver(theta(sparse), omega(sparse), ...
+ theta_dot(sparse), omega_dot(sparse), 'm');
+streamline(theta, omega, theta_dot, omega_dot, ...
+ theta(sparse_y, [1 end]), omega(sparse_y, [1 end]));
+
+xlim([min_theta max_theta]);
+xticks(linspace(min_theta, max_theta, 7));
+ylim([min_omega max_omega]);
+
+xlabel('pendulum angle');
+ylabel('angular velocity');
+title('Behavior of a Damped Pendulum', 'Interpreter', 'latex');
+
+%% Strange Attractors
+% I won't go into detail here -- suffice it to say the behavior of this system
+% would be quite opaque without numerical integration. Fortunately, we have
+% just that! Lorenz managed to prove analytically that this system had no
+% closed cycles and yet all its trajectories eventually stayed within a bounded
+% region of space, but beyond that... well, we'll see.
+sigma = 10;
+r = 28;
+b = 8/3;
+
+x_dot = @(x, y, z) sigma*(y - x);
+y_dot = @(x, y, z) r*x - y - x.*z;
+z_dot = @(x, y, z) x.*y - b*z;
+
+[x, y, z] = meshgrid(linspace(-50, 50, 100));
+u = x_dot(x, y, z);
+v = y_dot(x, y, z);
+w = z_dot(x, y, z);
+
+figure;
+hold on;
+streamline(x, y, z, u, v, w, 0, 1, 0, [.1 2e4]);
+scatter3(0, 1, 0);
+view(3);
+grid on;
+
+text(0, 0, -2, 'Start Point');
+xlabel('x');
+ylabel('y');
+zlabel('z');
+title("Lorenz's Strange Attractor", 'Interpreter', 'latex');