Is there any resource or literature where I can find derivations/proofs for all four linear phase filter types starting from an arbitrary real valued impulse response? I looked at some of the literature such as Introduction to Digital Filters but couldn't find anything concrete.

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    $\begingroup$ T. W. Parks and C. S. Burrus, Digital Filter Design, New York: John Wiley and Sons, Inc., June 1987 $\endgroup$ May 3, 2023 at 20:24
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    $\begingroup$ Should be in any textbook on digital signal processing. But what do you mean by "starting from an arbitrary real valued impulse response"? $\endgroup$
    – Matt L.
    May 3, 2023 at 20:49
  • $\begingroup$ (1/2) Here is what I mean by an arbitrary real impulse response: Starting with $h[n]$ we can get $H(e^{j\omega})$ by taking its DTFT: $$H(e^{j\omega}) = \sum_{n=-\infty}^{\infty}h[n] e^{-j\omega n}$$ Now, Linear-phase filters have a symmetric impulse response, e.g., $$ h(n) = h(N-1-n), \quad n=0,1,2,\ldots,N-1.$$ $\endgroup$
    – user64710
    May 4, 2023 at 13:00
  • $\begingroup$ (2/2) The symmetric-impulse-response constraint means that linear-phase filters must be FIR filters, because a causal recursive filter cannot have a symmetric impulse response. But it also means other things for the frequency response which we can realise by taking advantage of that symmetry. Now where can I find the complete proof for all four cases starting with these steps? $\endgroup$
    – user64710
    May 4, 2023 at 13:01
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    $\begingroup$ I'm not sure what you're trying to prove, but perhaps the discussion at the following web page would be of interest to you: dsprelated.com/showarticle/808.php $\endgroup$ May 4, 2023 at 22:32


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