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35
lessons/lesson07/bandstop2500.m
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35
lessons/lesson07/bandstop2500.m
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function Hd = bandstop2500
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%BANDSTOP2500 Returns a discrete-time filter object.
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% MATLAB Code
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% Generated by MATLAB(R) 9.9 and Signal Processing Toolbox 8.5.
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% Generated on: 24-Mar-2021 17:32:55
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% Elliptic Bandstop filter designed using FDESIGN.BANDSTOP.
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% All frequency values are in Hz.
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Fs = 8192; % Sampling Frequency
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Fpass1 = 2000; % First Passband Frequency
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Fstop1 = 2300; % First Stopband Frequency
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Fstop2 = 2700; % Second Stopband Frequency
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Fpass2 = 3000; % Second Passband Frequency
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Apass1 = 0.5; % First Passband Ripple (dB)
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Astop = 60; % Stopband Attenuation (dB)
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Apass2 = 1; % Second Passband Ripple (dB)
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match = 'both'; % Band to match exactly
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% Construct an FDESIGN object and call its ELLIP method.
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h = fdesign.bandstop(Fpass1, Fstop1, Fstop2, Fpass2, Apass1, Astop, ...
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Apass2, Fs);
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Hd = design(h, 'ellip', 'MatchExactly', match);
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% Get the transfer function values.
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[b, a] = tf(Hd);
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% Convert to a singleton filter.
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Hd = dfilt.df2(b, a);
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% [EOF]
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131
lessons/lesson07/filter_design.m
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131
lessons/lesson07/filter_design.m
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%% Lesson 7a. Designing filters the modern way
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% * Learn how to use `filterDesigner`
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% * Learn how to use `fdesign`
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close all; clear; clc;
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%% Filter Overview
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% Remember order = # of delay elements, want to minimize to meet
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% some spec.
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%
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% There are 4 main types of filters:
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% * Lowpass - passes low frequencies through
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% * Highpass - passes high frequencies through
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% * Bandpass - passes a specific band
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% * Bandstop - stops a specific band from going through
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%
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% There are four primary implementations of filters that optimize
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% different properties:
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% * Butterworth - maxflat passband, monotonic passband & stopband
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% * Chebychev I - equiripple passband, monotonic stopband
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% * Chebychev II - monotonic passband, equiripple stopband
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% * Elliptic - minimum order, equiripple passband & stopband
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%
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% In this lesson, you will be able to very clearly see these
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% properties and filter types!
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%% Filter design (GUI)
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% Note: Ask Prof. Keene about `filterDesigner`, it'll make him
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% very happy.
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filterDesigner; % then export to workspace as numerator and
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% denominator (not SOS for now)
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%% Applying a GUI-designed filter
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% Note that if a filter is generated using a `filterDesigner` function,
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% it will be a filter object. You can apply a filter object using the
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% `filter()` function. If your filter is given as an impulse response, you
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% can either use the `filter()` function or convolve the impulse response
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% and the input signal.
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% Load handel again
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load handel;
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% Get a generated filter (this should decimate a narrow frequency band
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% centered at 2.5kHz, assuming a 8192Hz sampling frequency).
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flt = bandstop2500;
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% Perform the filter
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y1 = filter(flt, y);
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%% Filter design via `fdesign`
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% Filter specifications
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Fsample = 44100;
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Apass = 1;
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Astop = 80;
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Fpass = 1e3;
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Fstop = 1e4;
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% Use the appropriate `fdesign.*` object to create the filter; in this case,
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% we're designing a lowpass filter.
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specs = fdesign.lowpass(Fpass, Fstop, Apass, Astop, Fsample);
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% Note: the arguments to `fdesign.*` can be altered by passing in a SPEC option
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% -- the following call to `fdesign.lowpass` will produce the same filter. If
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% you want to do more complex stuff (like setting the 3 dB point directly or
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% setting the filter order), you can change this argument. See `doc
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% fdesign.lowpass` for more info.
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equivalent_specs = fdesign.lowpass('Fp,Fst,Ap,Ast', ...
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Fpass, Fstop, Apass, Astop, Fsample);
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% For a given filter type, there are a myriad of different ways to design it
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% depending on what properties you want out of the filter (equiripple
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% passband/stopband, monotonic passband/stopband, minimum-order, max-flat,
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% etc.), which our lowpass filter object supports.
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%
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% Note: `'SystemObject'` should be set to `true` by Mathworks' recommendation
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% for these calls, even though they will often work without it.
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designmethods(specs, 'SystemObject', true);
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% There are also various options we can set when designing the filter. In this
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% example, we'll tell it to match the passband amplitude exactly.
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designoptions(specs, 'butter', SystemObject=true);
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% Finally, let's make this thing!
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butter_filter = design(specs, 'butter', ...
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MatchExactly='passband', SystemObject=true);
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%% Examining a filter
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% Filters can be visualized with `fvtool`, drawing a nice box around the
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% transition zone and lines at the edge frequencies & important amplitudes:
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h = fvtool(butter_filter); % note no call to `figure`!
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% `measure` is another way to check:
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measurements = measure(butter_filter);
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% `h` is a handle to the graphics object we've created with `fvtool`, and we
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% can change its properties (listed with `get(h)`) if we wish:
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h.FrequencyScale = 'Log';
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ax = h.Children(15); % the axes object, found by looking at `h.Children`
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ax.YLim = [-60 10]; % in dB
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%% Using a filter
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% Filter objects have a lot of methods to them (things like `freqz`, `impz`,
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% `islinphase`, `isallpass`, etc -- check `methods(butter_filter)` if you want
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% the list) that are rather useful. One method they *don't* have is `filter`,
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% so if you want to apply a filter to a signal, you do it like this:
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load handel; % of course; loads `y` and `Fs` into the workspace
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y1 = butter_filter(y); % use like a function!
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sound(y, Fs);
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sound(y1, Fs);
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%% Plotting the result
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% Obtain the FFT of the original and the filtered signal.
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shifted_fft_mag = @(sig, len) fftshift(abs(fft(sig, len))) / len;
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N = 2^15;
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S1 = shifted_fft_mag(y, N);
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S2 = shifted_fft_mag(y1, N);
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F = linspace(-Fsample/2, Fsample/2, N); % not quite accurate but close
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% Plot the FFTs.
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figure;
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subplot(2, 1, 1);
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plot(F, S1);
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title 'Fourier Transform of Original Audio';
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xlabel 'Frequency (Hz)';
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ylabel 'Magnitude';
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subplot(2, 1, 2);
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plot(F, S2);
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title 'Fourier Transform of Filtered Audio';
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xlabel 'Frequency (Hz)';
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ylabel 'Magnitude';
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103
lessons/lesson07/old_filter_design.m
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103
lessons/lesson07/old_filter_design.m
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%% Lesson 7c. Designing filters the old fashioned way
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% (notes kept for historical reasons; the new notes are in `filter_design.m`.
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%
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% * Learn how to use the `filterDesigner` GUI.
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% * Learn how to manually design a filter.
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clc; clear; close all;
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%% Filter Overview
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% Remember order = # of delay elements, want to minimize to meet
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% some spec.
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%
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% There are 4 main types of filters:
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% * Lowpass - passes low frequencies through
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% * Highpass - passes high frequencies through
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% * Bandpass - passes a specific band
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% * Bandstop - stops a specific band from going through
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%
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% There are four primary implementations of filters that optimize
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% different properties:
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% * Butterworth - maxflat passband, monotonic passband & stopband
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% * Chebychev I - equiripple passband, monotonic stopband
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% * Chebychev II - monotonic passband, equiripple stopband
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% * Elliptic - minimum order, equiripple passband & stopband
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%
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% In this lesson, you will be able to very clearly see these
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% properties and filter types!
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%% Filter Design (GUI)
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% Note: Ask Prof. Keene about `filterDesigner`, it'll make him
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% very happy.
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filterDesigner; % then export to workspace as numerator and
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% denominator (not SOS for now)
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%% Filter Design (Code)
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% Manually specify specifications
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Fs = 44100;
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Apass = 1;
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Astop = 80;
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Fpass = 1e3;
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Fstop = 1e4;
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wpass = Fpass / (Fs/2); % normalized to nyquist freq.
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wstop = Fstop / (Fs/2); % normalized to nyquist freq.
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% Compute the minimum order for Butterworth filter
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% that meets the specifications.
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% Alternatively, see `butter1ord`, `cheby1ord`,
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% `cheby2ord`, `ellipord`.
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n = buttord(wpass, wstop, Apass, Astop);
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% Generate Butterworth filter with order `n` (which is
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% designed to meet the specification).
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% Alternatively, see `cheby1`, `cheby2`, `ellip`.
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[b, a] = butter(n, wpass);
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[H, W] = freqz(b, a);
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% Plot the frequency response. (At this point, recall that
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% the frequency response is the FFT of the impulse response.
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% Can you generate the frequency response without using
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% the `freqz` function (instead using the `impz` and `fft`
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% functions)?
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f = W .* Fs/(2*pi);
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figure;
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semilogx(f, 20*log10(abs(H)));
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xlim([1e2 1e5]);
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ylim([-100 5]);
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grid on;
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title 'Butterworth Lowpass Filter';
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xlabel 'Frequency (Hz)';
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ylabel 'Magnitude (dB)';
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%% Applying a filter
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% Note that if a filter is generated using a `filterDesigner` function,
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% it will be a filter object. You can apply a filter object using the
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% `filter()` function. If your filter is given as an impulse response, you
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% can either use the `filter()` function or convolve the impulse response
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% and the input signal.
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% Load handel again
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load handel;
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% Get a generated filter (this should decimate a narrow frequency band
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% centered at 2.5kHz, assuming a 8192Hz sampling frequency).
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flt = bandstop2500;
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% Perform the filter
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y1 = filter(flt, y);
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%% How does it sound now?
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sound(y1, Fs);
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%% Plotting the result
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% Obtain the FFT of the filtered signal.
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N = 2^15;
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S = fft(y1, N);
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S = fftshift(abs(S)) / N;
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F = linspace(-Fs/2, Fs/2, N);
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% Plot the FFT.
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figure;
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plot(F, S);
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title 'Fourier Transform of Audio';
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xlabel 'Frequency (Hz)';
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ylabel 'Magnitude';
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93
lessons/lesson07/transfer.m
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93
lessons/lesson07/transfer.m
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%% Lesson 7c. Transfer function objects
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close all; clear; clc;
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%% Transfer function overview
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% Transfer functions are abstract representations of LTI systems in terms of
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% rational functions (numerator and denominator). We used to work with transfer
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% functions in terms of MATLAB-style polynomials, but Mathworks has come up
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% with a new, supposedly improved way: transfer function objects! These are
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% created via `tf` in a variety of ways. For more info, see Mathworks'
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% documentation -- these models are most helpful when working with cascades of
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% systems, such as for controls modeling.
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%% Digital transfer function objects
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% To create a digital tf. object, specify a numerator, denominator, and
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% sampling time (which may be set to -1 if you want to leave it undefined). The
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% numerator and denominator are specified in decreasing power order, with s^0
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% or z^0 on the right.
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dig_num = [1]; % 1z⁰ = 1
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dig_den = [2 -1]; % 2z¹ - 1z⁰ = 2z - 1
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timestep = 0.1; % 0.1 s
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dig_sys = tf(dig_num, dig_den, timestep);
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%% Analog transfer function objects
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% Analog transfer functions are created the same way as digital ones, just
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% without the timestep.
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an_num = [1 -1]; % 1s¹ - 1s⁰ = 1s - 1
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an_den = [1 2 1]; % s² + 2s + 1
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an_sys = tf(an_num, an_den);
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%% Alternate ways of specifying
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% MATLAB provides a syntax for creating transfer functions out of coded
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% rational expressions, as this can be clearer:
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s = tf('s'); % create an s variable that may then be used mathematically
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an_sys_2 = (s - 1)/(s + 1)^2; % same as above
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%% Investigating the transfer function
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% One property of interest is the step response, which we can view thus:
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figure;
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stepplot(an_sys);
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% MATLAB has a built-in function to get a Bode plot, too (though you should
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% modify this in post if you want to highlight specific details in the plot).
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figure;
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bodeplot(an_sys);
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%% One application
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% This example is stolen from Mathworks' doc page "Control System Modeling with
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% Model Objects," as it's an interesting perspective on what you can do with
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% these. In it, we model the following system:
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%
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% x ---> F(s) ---> +( ) ---> C(s) ---> G(s) ---+---> y
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% - |
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% ^ |
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% \---------- S(s) ---------/
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% specify the components of a system
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G = zpk([], [-1, -1], 1); % no zeroes, double pole at -1, analog
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C = pid(2, 1.3, 0.3, 0.5); % PID controller object
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S = tf(5, [1 4]);
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F = tf(1, [1 1]);
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% find the open-loop transfer function (by breaking the loop at the subtract)
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open_loop = F*S*G*C; % multiplication in the "frequency" domain
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% check out where the poles & zeroes are
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figure;
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pzplot(open_loop);
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title('Open loop pole/zero plot');
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xlim([-6 1]);
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ylim([-2 2]);
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% see what this thing does with a step input
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figure;
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stepplot(open_loop);
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title('Open-loop system step response');
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% note that this is *not* stable with a step input, as the PID controller loses
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% its shit!
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% find the closed-loop, whole_system response using `feedback` to specify a
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% feedback connection
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full_system = F*feedback(G*C, S);
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% see what this thing looks like now in the s-plane
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figure;
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pzplot(full_system);
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title('Full system pole/zero plot');
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xlim([-6 1]);
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ylim([-2 2]);
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% now stable with a step input, as we have feedback
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figure;
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stepplot(full_system);
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title('Full system step response');
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