The definition of group delay,
does not give an immediately useful recipe for computing group delay numerically. In this section, we describe the theory of operation behind the matlab function for group-delay computation given in §J.8.
A more useful form of the group delay arises from the logarithmic derivative of the frequency response. Expressing the frequency response in polar form as
yields the following logarithmic decomposition of magnitude and phase:
Thus, the real part of the logarithm of the frequency response equals the log amplitude response, while the imaginary part equals the phase response.
Since differentiation is linear, the logarithmic derivative becomes
where and denote the derivatives of and , respectively, with respect to . We may therefore express the group delay as
Consider first the FIR case in which , with
where denotes `` ramped'', 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
In matlab, the group delay, in samples, can be computed simply as
D = 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
where is given in Eq. (7.9), and
may be called the ``flip-conjugate'' or ``Hermitian conjugate'' of the polynomial .^{8.4}In matlab, the C polynomial is given by
C = conv(B,fliplr(conj(A)));It is straightforward to show (Problem 11) that
The phase of the IIR filter is therefore
and the group delay computation thus reduces to the FIR case:
This method is implemented in §J.8.