### Numerical Computation of Group Delay

The definition of group delay,*logarithmic derivative*of the frequency response. Expressing the frequency response in polar form as

imim

Consider first the FIR case in which , with
In this case, the derivative is simply

*i.e.*, the th coefficient of the polynomial is , for . In matlab, we may compute

`Br`from

`B`via the following statement:

Br = B .* [0:M]; % Compute ramped B polynomialThe group delay of an FIR filter can now be written as

imimre

In matlab, the group delay, in samples, can be computed simply asD = real(fft(Br) ./ fft(B))where the

`fft`, of course, approximates the Discrete Time Fourier Transform (DTFT). Such sampling of the frequency axis by this approximation is information-preserving whenever the number of samples (FFT length) exceeds the polynomial order . The

*ratio*of sampled DTFTs, however, is

*undersampled*, in general. In fact, we may have at some frequencies (``zeros on the unit circle''). The

`grpdelay`matlab utility in §J.8 watches out for division by zero, and simply sets the group delay to zero at such frequencies. Note that the true group delay approaches infinite magnitude as either a zero or pole approaches the unit circle. Finally, when there are both poles and zeros, we have

Straightforward differentiation yields

and this can be implemented analogous to the FIR case discussed above. However, a faster algorithm (usually) results from converting the IIR case to the FIR case:

where

^{8.4}In matlab, the C polynomial is given by

C = conv(B,fliplr(conj(A)));It is straightforward to show (Problem 11) that

re

This method is implemented in §J.8.
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