Compensating the Group Delay of an Analog Filter using DSP

Question

I am having a problem in my current project where I need to measure different biological signals (like ECG) simultaneously.

As the instrumentation hardware is developed by different people over time, they are not identical to each other. All of them has different orders of low-pass and high-pass filters before their final amplification stage. These filters are mostly concentrated around the passband of [0.5-35] Hz. I want to compensate for the phase dispersion caused by those analog filters by designing my own digital filters. I have measured the transfer functions of those channels using chirp signals, to get an idea, please check the following figure. There might be small numerical errors in the calculation of the transfer function.

Not only eliminating the phase dispersion in the individual channels but also compensating the different amount of delay of those channels is important so that I can analyze fully "synchronized" signals. So far, I have been trying to design an all-pass filter in Matlab using iirgrpdelay function. However, the created compensation filters are not satisfying. The reason for that might be the huge variation in group delay (see the first figure) between [1 15] Hz, the region I want to compensate. Increasing the order of the filter doesn't help since I mostly get the following error from Matlab:

Poorly conditioned Hessian matrix. Cannot accurately compute the optimization because either the approximation error is extremely small (try reducing the number of poles or zeros) or the filter specifications yield huge magnitude variations, such as mag=[1 1e9 0 0].

I don't want to leave the question this broad but I am asking for suggestions to design such compensation filters in the digital domain so that I can neutralize this phase dispersion and delay in the analog instrumentation.

Answer 1

It's often easier to design FIR filters for compensating group delay. At the same time they could also compensate the magnitude if necessary. The easiest method is to use a complex least squares method, which boils down to solving a system of linear equations for the filter coefficients.

The difficult part is to choose an appropriate bulk delay in the definition of the desired phase response. A good starting point is to choose the average delay to be close to half the filter length.

Take a look at this answer for an example of FIR group delay equalization.

You can use the Matlab/Octave function lslevin.m for designing FIR filters with prescribed magnitude and phase responses.