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Given a desired filter magnitude response, plus acceptable linear phase and minimum phase FIR filter approximations, how can one design a filter with the given response and a delay specification of some value between minimum phase and linear phase?

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What do you mean by "linear phase and minimum phase FIR filter approximations?" And, what is the format of your delay specification? Are you looking to gain something (i.e. a lower filter order to get the magnitude response that you want) by allowing some degree of freedom for the phase response? – Jason R May 5 '12 at 13:14
@Jason R : For instance, filters created by some matlab toolbox. For r in [0..1], delay = r * linear_filter_delay + (1 - r) * min_phase_filter_delay. Looking for different reconstruction interpolation properties. – hotpaw2 May 5 '12 at 14:16

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Least square errors works well with FIR filters. General IIR filters are more difficult and typically require an iterative search algorithm. One specific type of IIR, named warped FIR filters, can also match arbitrary amplitude and phase response with a least square errors approach.

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IIR design should be trivial, just take the minimum phase filter design, and reflect some zeros to outside the unit circle or across to the other side of the complex plane. – hotpaw2 May 5 '12 at 14:19
Not trivial at all, I think. While that would certainly change the phase (by adding effectively a 1rst order allpass), it's not clear to me how you use that to fit an arbitrary phase vs. frequency specification – Hilmar May 6 '12 at 1:53

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