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fdesign.arbmagnphase is a MATLAB built-in function that will create a filter that best fits the frequency-response that it is given.

I'd like to implement a similar function in Python. What algorithms are used to synthesize filters based on an arbitrary frequency response?

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For FIR filter design, common algorithms include Remez exchange or Parks-McClellan : What is the difference between Remez exchange and Parks-McClellan filter design?

For IIR filters, one possibility is the Differential Evolution algorithm, as described in some IEEE journals. If you have a realistic desired phase response and can guess the filter order, then Berchin's FDLS method may be suitable.

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