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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}

Figure: Script (matlab) for converting the (real) desired amplitude response to minimum-phase form for invfreqz. This script generated Fig.8.7.

 
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 © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA