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/*Description:
Compute the group delay of a filter.
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.
Calling Sequence
[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 (…, 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 (…, n, "whole") : evaluates the group delay at n frequencies between 0 and 2*pi.
[g, f] = grpdelay (…, n, Fs) : evaluates the group delay at n frequencies between 0 and Fs/2.
[g, f] = grpdelay (…, n, "whole", Fs) : evaluates the group delay at n frequencies between 0 and Fs.
[g, w] = grpdelay (…, w) : evaluates the group delay at frequencies w (radians per sample).
[g, f] = grpdelay (…, 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.)
Parameters:
b : Coefficients of the numerator polynomial of the filter (FIR or IIR).
a : (optional): Coefficients of the denominator polynomial of the filter (only for IIR filters).
n : (optional): Number of angular frequencies at which to evaluate the group delay. For efficient computation, n should be a product of small primes.
"whole" : (optional): Specifies frequency range. If specified: Frequencies range from 0 to 2π radians per sample.
Fs : (numeric value): Frequencies range from 0 to Fs Hz or Fs radians per sample.
w : (optional): Vector of specific frequencies (in radians per sample) at which to evaluate the group delay.
f : (optional): Vector of specific frequencies (in Hz) at which to evaluate the group delay.
Fs : (optional): Sampling rate in Hz. Used to specify the frequency range when f or "whole" is specified.
Dependencies: fft1
*/
function [gd,w] = grpdelay (b, a, n, whole, Fs)
lhs=argn(1);
rhs= argn(2);
if (rhs < 1 | rhs > 5) then
error("Invalid number of inputs")
end
HzFlag= %F;
select rhs
case 1 then
Fs=1; whole = "" ; n = 512 ; a = 1 ;
case 2 then
Fs=1; whole = "" ; n = 512 ;
case 3 then
Fs=1; whole = "" ;
case 4 then
Fs=1;
end
if (max(size(n)) > 1)
if (rhs > 4)
error("invalid inputs");
elseif (rhs > 3)
// grpdelay (B, A, F, Fs)
Fs = whole;
HzFlag = %T;
else
// grpdelay (B, A, W)
Fs = 1;
end
w = 2*%pi*n/Fs;
n = max(size (w) ) * 2;
whole = "";
else
if (rhs < 5)
Fs = 1; // return w in radians per sample
if (rhs < 4)
whole = "" ;
elseif (~type(whole)==10)
Fs = whole;
HzFlag = %T;
whole = "";
end
if (rhs < 3)
n = 512;
end
if (rhs < 2)
a = 1;
end
else
HzFlag = %T;
end
if (isempty (n))
n = 512;
end
if ( type(whole) == 10 && strcmp (whole, "whole"))
n = 2*n;
end
w = Fs*[0:n-1]/n;
end
if (~ HzFlag)
w = w * 2 * %pi;
end
a = a(:).';
b = b(:).';
oa = max(size(a)) -1; // order of a(z)
if (oa < 0) // a can be []
a = 1;
oa = 0;
end
ob = max(size(b)) -1; // order of b(z)
if (ob < 0) // b can be [] as well
b = 1;
ob = 0;
end
oc = oa + ob; // order of c(z)
c = conv (b, flipdim(conj (a),2)); // c(z) = b(z)*conj(a)(1/z)*z^(-oa)
cr = c.*(0:oc); // cr(z) = derivative of c wrt 1/z
num = fft1 (cr, n);
den = fft1 (c, n);
minmag = 10*%eps;
polebins = find (abs (den) < minmag);
for b = polebins
warning ("signal:grpdelay-singularity", "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 ( type(whole) == 10 && strcmp (whole, "whole"))
ns = n/2; // Matlab convention ... should be n/2 + 1
gd = gd(1:ns);
w = w(1:ns);
else
ns = n; // used in plot below
end
//// compatibility
gd = gd(:);
w = w(:);
if (lhs == 1)
if (HzFlag)
funits = "Hz";
else
funits = "radian/sample";
end
disp(" Plotting ")
plot (w(1:ns), gd(1:ns));
xlabel (["Frequency (" funits ")"]);
ylabel ("Group delay (samples)");
end
endfunction
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