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I've recently encountered a rather clever low-pass filter that appears to have been designed to sacrifice uniform group delay in the pass-band to get better frequency response relative to the filter length. What methods are available to design FIR filters that maximize frequency performance relative to length while allowing phase to become nonlinear (but still continuous) in the pass-band?

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  • $\begingroup$ Making a FIR have non-linear phase, with the same number of taps, will not improve its frequency characteristics, while halving the number will only make it worse (it approximates the square of the magnitude). IIRs have (much) better frequency response and they're more varied. I would recommend sticking to IIRs if non-linear phase is not an issue, but that's just me. $\endgroup$ – a concerned citizen May 28 '17 at 6:33
  • $\begingroup$ Are you talking about filters with a prescribed non-linear phase (e.g., an approximately linear in the pass band, but with reduced delay), or don't you care about the phase as long as magnitude specs are satisfied? $\endgroup$ – Matt L. May 28 '17 at 21:10
  • $\begingroup$ @Matt As long as there are no discontinuities, any systematic phase response in the passband is okay, as it can be compensated for with offline prefiltering. $\endgroup$ – Justin Olbrantz May 29 '17 at 0:38
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Although not a comprehensive list, the following are filter algorithms available in Matlab for designing non-linear phase FIR filters:

firlpnorm: Nonlinear phase equiripple or least squares, uses Lp norm: for equiripple filters p = infinity, for least squares p =2.

cfirmpm: complex and nonlinear phase equiripple filters.

See the Matlab documentation for full details on how to design non-linear phase using these algorithms.

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One way to make a non-linear out of a FIR means solving for the zeroes, keeping only the inside/outside ones, then rebuilding the impulse response, and all this with equiripple filters due to the method of adding a DC to the middle coefficient (this for both same-N, and half-N filters). But this can be extended to non-equiripple filters, as well, see this, for example. My recommendation remains: IIRs are a better investment in terms of computing power, for the same, or better results, particularly since you, yourself, say that you will compensate the phase distortion afterwards. Of course, you have the last saying.

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check this out: DSP Trick: Using Parks-McClellan to Design a Non-Linear Phase FIR Filter

if you need me to explain it, lemme know in a comment and i will come back and try to be explicit about it.

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