# Estimation of Group Delay from Noisy Frequency Response Measurements

Consider a linear, time-invariant system. Noisy measurements of the magnitude and phase associated with the frequency response of the system are available over some dense generally non-uniform grid of frequencies.

I need to use these measurements to estimate group delay. Direct approximation of the derivative of phase via successive differences in the phase measurements is extremely noisy.

Some form of smoothing seems appropriate (e.g., spline smoothing). If I know the maximum rate of change of group delay as a function of frequency, how can I incorporate that side-information into a smoothing procedure? More generally, is anyone aware of other techniques (beyond fitting the frequency response measurements to some rational filter model and computing group delay from the associated parametric model)?

• i think that if you compute the group delay from the finite differences (representing the derivative) and low-pass filter that noisy difference with an LPF of gain 1 (or 0 dB) at DC, that the result (after dividing $\Delta\phi$ with $\Delta\omega$ and negating) will be the group delay. – robert bristow-johnson May 15 '20 at 6:51
• @robertbristow-johnson That is certainly a valid approach. This made me think of using an FIR differentiator as well but the phase response samples are non-uniformly spaced. The question then really becomes one of bandwidth selection, which I don't know how to do in some systematic way. For example, it would be great to know that by choosing a particular BW, we are able to capture variations in group delay up to some maximum. – rhz May 15 '20 at 17:43
• i.e., some maximum rate of change... – rhz May 15 '20 at 20:31