Hello Everyone,

As is well-known, regular filters (e.g. analog elliptic filters) used to attain a certain magnitude response usually tend to distort the phase and hence the pulse response to varying degrees. Let's call such a filter H_mag(s)

Since all-pass filters supposedly allow one to choose a certain phase response, it might be possible to equalize the distortion resulting from the above so that one may get a much flatter group delay and hence a better pulse response.

I was wondering how one goes about designing such a filter. All the texts that I have seen only deal with magnitude response.

1) Could someone kindly suggest links/references that deal with design of all-pass filters to equalize phase distortion?

2) What sort of responses are realizable in principle?, in practice? e.g. Is it realistic to expect that one can make the lousy step response of an elliptic or chebyshev filter look nice and well-behaved like that of a bessel-thompson with suitable all-pass filtering down the chain?

3) Are all-pass filters really used in analog domain to make such phase corrections?

4) Any links to toolboxes etc. that permit one to do some of the work in MATLAB would be welcome. Trying to use hand-calculated expressions for group delay and mapping this back to a transfer function isn't really very easy I fear.

Note: I don't have the luxury of using digital signal processing as I am dealing with a continuous-time analog system.

Thanks for your comments!
Vivek

Ken Kundert wrote on Dec 30^th, 2013, 4:43pm:


Equiripple filters are another possibility to consider. Very flat in phase across the passband.

A linear time invariant system can (***in theory***) can do (pretty much) anything to the gain and phase response and then introduce a function to cancel the gain and phase response of the first section. (with some overall group delay)

However, I am ignoring a lot more subtle things by saying that.

Practical considerations of linearty noise dynamic range and real world circuit implementations are getting ignored in that statement.

Thanks again Ken & Jerry,

I suppose you are referring to the uncertainity principle for filters which states that one cannot simultaneously achieve an arbitrarily narrow response in time and frequency domains.

As far as I recall, the product of the widths of the two responses improves slightly with filter order, but not too much.

I was however intrigued by what I read in the text from Oppenheim and Schafer and was thinking that there might be some room for improvement since my filter might be operating away from the lower limit, i.e. it might not yet be optimized for minimum-width time- and frequency response.

My attempts at trying to linearize the phase with allpass filters placed after the main filter were not very helpful as the group delay variation could not be corrected sufficiently. This made me wonder how the so-called equalization referred to in the above text was actually done.

Perhaps I need to look into the details to see what the hard limits are and see how far I am from those, once.

Regards
Vivek

Thanks Jerry!

I do agree with your point that it is better to try and design the filter of interest to have as linear a phase as possible to begin with and not try to correct the phase of a lousy filter.

However, I am curious after having read a few articles and papers (most of them date back to the 50s and 60s). Once I have figured out what they are trying to do, I will get back and post the results.

Regards,
Vivek