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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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.