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I know that it's possible to design an IIR with specific poles and zeroes to create specific frequency responses.

Is it also possible to design an IIR such that when you give it an impulse, it generates values of a desired sequence? Or at least a sequence within some tolerance of those values?

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  • $\begingroup$ A FIR won't do? Is your desired sequence of infinite length? $\endgroup$
    – Juancho
    Mar 1, 2017 at 18:18
  • $\begingroup$ This is a little strange and obscure but in looking at recurrent neural networks, all the ones I've seen use N layers for N past history items so work like FIRs. I'm starting with the basics but trying to understand how and if you could make a RNN that worked more like an IIR with internal state. $\endgroup$
    – Alan Wolfe
    Mar 1, 2017 at 19:09

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Yes, there are time-domain design methods for IIR filters. One of the best-known ones is Prony's method. It is well described in the book Digital Filter Design by T.W. Parks and C.S.Burrus (ch. 7.5).

If the desired sequence is (right-sided) periodic, then there's a very simple and exact solution possible (which is usually an exercise for DSP students).

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