Converting the Desired Amplitude Response to Minimum Phase

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Fitting Filters to Measured Amplitude Responses

Desired Impulse Response

Further Reading on Digital Filter Design

The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.html

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The phase spectrum of a signal is given by the phase (complex angle) of its Fourier transform. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Fourier_Transform_DFT.html

Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

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The magnitude spectrum of a signal is given by the magnitude (complex modulus) of its Fourier transform. — Click for https://ccrma.stanford.edu/~jos/mdft/Discrete_Fourier_Transform_DFT.html

Click for https://ccrma.stanford.edu/~jos/sasp/Minimum_Phase_Filter_Design.html#13817

Matlab is a high-level scripting language for scientific computing — Click for https://ccrma.stanford.edu/~jos/matlab/

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

A causal filter is any filter whose impulse response is zero prior to time zero. — Click for https://ccrma.stanford.edu/~jos/filters/Causal_Recursive_Filters.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

The invfreqz function (in matlab) computes the coefficients of a recursive digital filter based on specified frequency-response samples. — Click for http://www.dsprelated.com/showcode/20.php

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Minimum-phase polynomials have all their zeros inside the unit circle of the complex plane. — Click for https://ccrma.stanford.edu/~jos/filters/Minimum_Phase_Polynomials.html

The amplitude response of an LTI filter is simply the magnitude of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Amplitude_Response_I_I.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Fitting Filters to Measured Amplitude Responses

Desired Impulse Response

Further Reading on Digital Filter Design

Converting the Desired Amplitude Response to Minimum Phase

Phase-sensitive filter-design methods such as the equation-error method implemented in invfreqz are normally constrained to produce filters with causal impulse responses.^9.15 In cases such as this (phase-sensitive filter design when we don't care about phase--or don't have it), it is best to compute the minimum phase corresponding to the desired amplitude response [452].

As detailed in Fig.8.8, the minimum phase is constructed by the cepstral method [452].^9.16

The four-pole, one-zero filter fit using invfreqz is shown in Fig.8.7.

**Figure 8.7:** Overlay of desired amplitude response (solid) and that of a fourth-order filter fit (dashed) using `invfreqz`.
$\includegraphics[width=\twidth]{eps/tmps2-Hh}$

c = ifft(Sdb); % compute real cepstrum from log magnitude spectrum
% Check aliasing of cepstrum (in theory there is always some):
caliaserr = 100*norm(c(round(Ns*0.9:Ns*1.1)))/norm(c);
disp(sprintf(['Cepstral time-aliasing check: Outer 20%% of ' ...
    'cepstrum holds %0.2f %% of total rms'],caliaserr));
% = 0.09 percent
if caliaserr>1.0 % arbitrary limit
  error('Increase Nfft and/or smooth Sdb to shorten cepstrum');
end
% Fold cepstrum to reflect non-min-phase zeros inside unit circle:
% If complex:
% cf=[c(1),c(2:Ns-1)+conj(c(Nfft:-1:Ns+1)),c(Ns),zeros(1,Nfft-Ns)];
cf = [c(1), c(2:Ns-1)+c(Nfft:-1:Ns+1), c(Ns), zeros(1,Nfft-Ns)];
Cf = fft(cf); % = dB_magnitude + j * minimum_phase
Smp = 10 .^ (Cf/20); % minimum-phase spectrum

Smpp = Smp(1:Ns); % nonnegative-frequency portion
wt = 1 ./ (fk+1); % typical weight fn for audio
wk = 2*pi*fk/fs;
[B,A] = invfreqz(Smpp,wk,NZ,NP,wt);
Hh = freqz(B,A,Ns);

figure(3);
plot(fk,db([Smpp(:),Hh(:)])); grid('on');
xlabel('Frequency (Hz)');
ylabel('Magnitude (dB)');
title('Magnitude Frequency Response');
% legend('Desired','Filter');

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Converting the Desired Amplitude Response to Minimum Phase

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University