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grpdelay.m

## Copyright (C) 2004 Julius O. Smith III
##
## 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 2, 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.  The GNU
## General Public License has more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.
##
## Plot, demos, and help info copied and adapted from 
## grpdelay.m, Copyright (C) 2000 Paul Kienzle

## 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)
    usage("[g, w]=grpdelay(b [, a [, n [, 'whole' [, Fs]]]])");
  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

% ------------------------ 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(); oneplot();

%!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);
%! oneplot();

% ------------------------ 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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