/usr/share/octave/packages/signal-1.2.2/grpdelay.m is in octave-signal 1.2.2-1build1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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## Copyright (C) 2004 Julius O. Smith III <jos@ccrma.stanford.edu>
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## Compute the group delay of a filter.
##
## [g, w] = grpdelay(b)
## returns the group delay g of the FIR filter with coefficients b.
## The response is evaluated at 512 angular frequencies between 0 and
## pi. w is a vector containing the 512 frequencies.
## The group delay is in units of samples. It can be converted
## to seconds by multiplying by the sampling period (or dividing by
## the sampling rate fs).
##
## [g, w] = grpdelay(b,a)
## returns the group delay of the rational IIR filter whose numerator
## has coefficients b and denominator coefficients a.
##
## [g, w] = grpdelay(b,a,n)
## returns the group delay evaluated at n angular frequencies. For fastest
## computation n should factor into a small number of small primes.
##
## [g, w] = grpdelay(b,a,n,'whole')
## evaluates the group delay at n frequencies between 0 and 2*pi.
##
## [g, f] = grpdelay(b,a,n,Fs)
## evaluates the group delay at n frequencies between 0 and Fs/2.
##
## [g, f] = grpdelay(b,a,n,'whole',Fs)
## evaluates the group delay at n frequencies between 0 and Fs.
##
## [g, w] = grpdelay(b,a,w)
## evaluates the group delay at frequencies w (radians per sample).
##
## [g, f] = grpdelay(b,a,f,Fs)
## evaluates the group delay at frequencies f (in Hz).
##
## grpdelay(...)
## plots the group delay vs. frequency.
##
## If the denominator of the computation becomes too small, the group delay
## is set to zero. (The group delay approaches infinity when
## there are poles or zeros very close to the unit circle in the z plane.)
##
## Theory: group delay, g(w) = -d/dw [arg{H(e^jw)}], is the rate of change of
## phase with respect to frequency. It can be computed as:
##
## d/dw H(e^-jw)
## g(w) = -------------
## H(e^-jw)
##
## where
## H(z) = B(z)/A(z) = sum(b_k z^k)/sum(a_k z^k).
##
## By the quotient rule,
## A(z) d/dw B(z) - B(z) d/dw A(z)
## d/dw H(z) = -------------------------------
## A(z) A(z)
## Substituting into the expression above yields:
## A dB - B dA
## g(w) = ----------- = dB/B - dA/A
## A B
##
## Note that,
## d/dw B(e^-jw) = sum(k b_k e^-jwk)
## d/dw A(e^-jw) = sum(k a_k e^-jwk)
## which is just the FFT of the coefficients multiplied by a ramp.
##
## As a further optimization when nfft>>length(a), the IIR filter (b,a)
## is converted to the FIR filter conv(b,fliplr(conj(a))).
## For further details, see
## http://ccrma.stanford.edu/~jos/filters/Numerical_Computation_Group_Delay.html
function [gd,w] = grpdelay(b,a=1,nfft=512,whole,Fs)
if (nargin<1 || nargin>5)
print_usage;
end
HzFlag=0;
if length(nfft)>1
if nargin>4
print_usage();
elseif nargin>3 % grpdelay(B,A,F,Fs)
Fs = whole;
HzFlag=1;
else % grpdelay(B,A,W)
Fs = 1;
end
w = 2*pi*nfft/Fs;
nfft = length(w)*2;
whole = '';
else
if nargin<5
Fs=1; % return w in radians per sample
if nargin<4, whole='';
elseif ~ischar(whole)
Fs = whole;
HzFlag=1;
whole = '';
end
if nargin<3, nfft=512; end
if nargin<2, a=1; end
else
HzFlag=1;
end
if isempty(nfft), nfft = 512; end
if ~strcmp(whole,'whole'), nfft = 2*nfft; end
w = Fs*[0:nfft-1]/nfft;
end
if ~HzFlag, w = w * 2*pi; end
oa = length(a)-1; % order of a(z)
if oa<0, a=1; oa=0; end % a can be []
ob = length(b)-1; % order of b(z)
if ob<0, b=1; ob=0; end % b can be [] as well
oc = oa + ob; % order of c(z)
c = conv(b,fliplr(conj(a))); % c(z) = b(z)*conj(a)(1/z)*z^(-oa)
cr = c.*[0:oc]; % cr(z) = derivative of c wrt 1/z
num = fft(cr,nfft);
den = fft(c,nfft);
minmag = 10*eps;
polebins = find(abs(den)<minmag);
for b=polebins
warning('grpdelay: setting group delay to 0 at singularity');
num(b) = 0;
den(b) = 1;
% try to preserve angle:
% db = den(b);
% den(b) = minmag*abs(num(b))*exp(j*atan2(imag(db),real(db)));
% warning(sprintf('grpdelay: den(b) changed from %f to %f',db,den(b)));
end
gd = real(num ./ den) - oa;
if strcmp(whole,'whole')==0
ns = nfft/2; % Matlab convention ... should be nfft/2 + 1
gd = gd(1:ns);
w = w(1:ns);
else
ns = nfft; % used in plot below
end
% compatibility
gd = gd(:); w = w(:);
if nargout==0
unwind_protect
grid('on'); % grid() should return its previous state
if HzFlag
funits = 'Hz';
else
funits = 'radian/sample';
end
xlabel(['Frequency (', funits, ')']);
ylabel('Group delay (samples)');
plot(w(1:ns), gd(1:ns), ';;');
unwind_protect_cleanup
grid('on');
end_unwind_protect
end
end
% ------------------------ DEMOS -----------------------
%!demo % 1
%! %--------------------------------------------------------------
%! % From Oppenheim and Schafer, a single zero of radius r=0.9 at
%! % angle pi should have a group delay of about -9 at 1 and 1/2
%! % at zero and 2*pi.
%! %--------------------------------------------------------------
%! title ('Zero at z = -0.9');
%! grpdelay([1 0.9],[],512,'whole',1);
%! hold on;
%! xlabel('Normalized Frequency (cycles/sample)');
%! stem([0, 0.5, 1],[0.5, -9, 0.5],'*b;target;');
%! hold off;
%!
%!demo % 2
%! %--------------------------------------------------------------
%! % confirm the group delays approximately meet the targets
%! % don't worry that it is not exact, as I have not entered
%! % the exact targets.
%! %--------------------------------------------------------------
%! b = poly([1/0.9*exp(1i*pi*0.2), 0.9*exp(1i*pi*0.6)]);
%! a = poly([0.9*exp(-1i*pi*0.6), 1/0.9*exp(-1i*pi*0.2)]);
%! title ('Two Zeros and Two Poles');
%! grpdelay(b,a,512,'whole',1);
%! hold on;
%! xlabel('Normalized Frequency (cycles/sample)');
%! stem([0.1, 0.3, 0.7, 0.9], [9, -9, 9, -9],'*b;target;');
%! hold off;
%!demo % 3
%! %--------------------------------------------------------------
%! % fir lowpass order 40 with cutoff at w=0.3 and details of
%! % the transition band [.3, .5]
%! %--------------------------------------------------------------
%! subplot(211);
%! Fs = 8000; % sampling rate
%! Fc = 0.3*Fs/2; % lowpass cut-off frequency
%! nb = 40;
%! b = fir1(nb,2*Fc/Fs); % matlab freq normalization: 1=Fs/2
%! [H,f] = freqz(b,1,[],1);
%! [gd,f] = grpdelay(b,1,[],1);
%! title(sprintf('b = fir1(%d,2*%d/%d);',nb,Fc,Fs));
%! xlabel('Normalized Frequency (cycles/sample)');
%! ylabel('Amplitude Response (dB)');
%! grid('on');
%! plot(f,20*log10(abs(H)));
%! subplot(212);
%! del = nb/2; % should equal this
%! title(sprintf('Group Delay in Pass-Band (Expect %d samples)',del));
%! ylabel('Group Delay (samples)');
%! axis([0, 0.2, del-1, del+1]);
%! plot(f,gd);
%! axis();
%!demo % 4
%! %--------------------------------------------------------------
%! % IIR bandstop filter has delays at [1000, 3000]
%! %--------------------------------------------------------------
%! Fs = 8000;
%! [b, a] = cheby1(3, 3, 2*[1000, 3000]/Fs, 'stop');
%! [H,f] = freqz(b,a,[],Fs);
%! [gd,f] = grpdelay(b,a,[],Fs);
%! subplot(211);
%! title('[b,a] = cheby1(3, 3, 2*[1000, 3000]/Fs, \'stop\');');
%! xlabel('Frequency (Hz)');
%! ylabel('Amplitude Response');
%! grid('on');
%! plot(f,abs(H));
%! subplot(212);
%! title('[gd,f] = grpdelay(b,a,[],Fs);');
%! ylabel('Group Delay (samples)');
%! plot(f,gd);
% ------------------------ TESTS -----------------------
%!test % 00
%! [gd1,w] = grpdelay([0,1]);
%! [gd2,w] = grpdelay([0,1],1);
%! assert(gd1,gd2,10*eps);
%!test % 0A
%! [gd,w] = grpdelay([0,1],1,4);
%! assert(gd,[1;1;1;1]);
%! assert(w,pi/4*[0:3]',10*eps);
%!test % 0B
%! [gd,w] = grpdelay([0,1],1,4,'whole');
%! assert(gd,[1;1;1;1]);
%! assert(w,pi/2*[0:3]',10*eps);
%!test % 0C
%! [gd,f] = grpdelay([0,1],1,4,0.5);
%! assert(gd,[1;1;1;1]);
%! assert(f,1/16*[0:3]',10*eps);
%!test % 0D
%! [gd,w] = grpdelay([0,1],1,4,'whole',1);
%! assert(gd,[1;1;1;1]);
%! assert(w,1/4*[0:3]',10*eps);
%!test % 0E
%! [gd,f] = grpdelay([1 -0.9j],[],4,'whole',1);
%! gd0 = 0.447513812154696; gdm1 =0.473684210526316;
%! assert(gd,[gd0;-9;gd0;gdm1],20*eps);
%! assert(f,1/4*[0:3]',10*eps);
%!test % 1A:
%! gd= grpdelay(1,[1,.9],2*pi*[0,0.125,0.25,0.375]);
%! assert(gd, [-0.47368;-0.46918;-0.44751;-0.32316],1e-5);
%!test % 1B:
%! gd= grpdelay(1,[1,.9],[0,0.125,0.25,0.375],1);
%! assert(gd, [-0.47368;-0.46918;-0.44751;-0.32316],1e-5);
%!test % 2:
%! gd = grpdelay([1,2],[1,0.5,.9],4);
%! assert(gd,[-0.29167;-0.24218;0.53077;0.40658],1e-5);
%!test % 3
%! b1=[1,2];a1f=[0.25,0.5,1];a1=fliplr(a1f);
%! % gd1=grpdelay(b1,a1,4);
%! gd=grpdelay(conv(b1,a1f),1,4)-2;
%! assert(gd, [0.095238;0.239175;0.953846;1.759360],1e-5);
%!test % 4
%! Fs = 8000;
%! [b, a] = cheby1(3, 3, 2*[1000, 3000]/Fs, 'stop');
%! [h, w] = grpdelay(b, a, 256, 'half', Fs);
%! [h2, w2] = grpdelay(b, a, 512, 'whole', Fs);
%! assert (size(h), size(w));
%! assert (length(h), 256);
%! assert (size(h2), size(w2));
%! assert (length(h2), 512);
%! assert (h, h2(1:256));
%! assert (w, w2(1:256));
%!test % 5
%! a = [1 0 0.9];
%! b = [0.9 0 1];
%! [dh, wf] = grpdelay(b, a, 512, 'whole');
%! [da, wa] = grpdelay(1, a, 512, 'whole');
%! [db, wb] = grpdelay(b, 1, 512, 'whole');
%! assert(dh,db+da,1e-5);
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