# Derivation of the 4 types of real-valued linear-phase FIR filters

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.

• T. W. Parks and C. S. Burrus, Digital Filter Design, New York: John Wiley and Sons, Inc., June 1987 May 3, 2023 at 20:24
• Should be in any textbook on digital signal processing. But what do you mean by "starting from an arbitrary real valued impulse response"? May 3, 2023 at 20:49
• (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., $$h(n) = h(N-1-n), \quad n=0,1,2,\ldots,N-1.$$
– user64710
May 4, 2023 at 13:00
• (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?
– user64710
May 4, 2023 at 13:01
• 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 May 4, 2023 at 22:32