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Design >> Analog Design >> designing all-pass filters to equalize phase response of a filter
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Message started by vivkr on Dec 30^th, 2013, 1:37pm

Title: designing all-pass filters to equalize phase response of a filter
Post by vivkr on Dec 30^th, 2013, 1:37pm

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

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by Ken Kundert on Dec 30^th, 2013, 4:43pm

If you want a filter with reasonable pulse response, you should use a filter that has reasonable flat delay characteristics. You should consider Bessel or Gaussian filters.
Bessel filters have maximally flat envelope delay (MFED).
Gaussian filters have impulse responses that approximate Gaussians, and so have no overshoot.

The reference I use on this is Handbook of Filter Synthesis by Anatol Zverev.

-Ken

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by loose-electron on Dec 31^st, 2013, 10:40am

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

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

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by vivkr on Jan 1^st, 2014, 1:00pm

Ken, Jerry,

Thanks for your answers. I am aware that one can achieve decent pulse response by using Bessel filters. But that is not what I am actually asking.

I am wondering if it is possible to "fix" the pulse response of say an elliptic filter with post-filtering.

If there were some halfway decent solution (one obviously cannot achieve the same response as a Bessel), then one might achieve good rejection up-front and fix the pulse response sufficiently.

The reason for asking is that I have heard from people that such things are possible (unreliable evidence), but also read in some DSP texts that one can do such a thing.

Admittedly, that's DSP, but here's what I have from "Discrete-Time Signal Processing" by Oppenheim & Schafer, 1989 edition, Sec. 7.3.2 in the chapter on "Computer-Aided Design of Discrete-Time IIR Filters":

"As an example of the utility of ...., consider the design of an allpass filter to compensate for phase nonlinearity of a linear system. This design problem is common in communications systems, where it is often necessary to "equalize" the group delay of a communications channel":

In a nutshell, is such an "equalization" of phase response of a highly selective but lousy filter (from pulse response point of view) possible using post-filtering?

Regards
Vivek

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by loose-electron on Jan 3^rd, 2014, 2:57am

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.

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by Ken Kundert on Jan 4^th, 2014, 4:01pm

Jerry says you can do almost anything with LTI filters in terms of magnitude and phase responses, but important is the 'almost'. There are somethings you cannot do. There is a fundamental tradeoff between the steepness of the transition between the passband and the reject band, and using an allpass filter will not get around that.

-Ken

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by vivkr on Jan 6^th, 2014, 6:24am

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

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by loose-electron on Jan 10^th, 2014, 4:21pm

Keep in mind there is the definition of a filter as a math exercise, vs. the practical real world implementations.

In the world of math if you can define a LTI function you should be able to define the complementary function as well.

However that ignores a lot of practical considerations of dynamic range, linearity, additive noise etc.

As a general rule, it is wiser to design a filter with the desired response rather than cascade two devices that have been independently designed and connected together.

If you need good passband phase, you may need a higher order device to get the roll off rejection that you could get with a different filter definition that has steep rolloff but poor passband phase.

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by vivkr on Jan 13^th, 2014, 5:49am

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

Title: Re: designing all-pass filters to equalize phase response of a filter
Post by loose-electron on Jan 13^th, 2014, 6:17pm

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

The reference I use on this is Handbook of Filter Synthesis by Anatol Zverev.

-Ken

Ah yes the big red book. Mine has been on the shelf for quite few years.
Seems the S-plane has not been updated recently. :)

There have been a few updates on filter configurations since that was written (1967) but the fundamentals have not changed.