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I am practising for upcoming exams and came across this question.

Let $h[n]$ be an FIR filter such that $h[n] = 0$ when $|n| > M$ and $h[n] = h[−n]$. A plot of $H(e^{j\omega})$ (DTFT of $h[n]$) is shown below. The eventual goal is to implement this filter digitally. Then, with the given information, is it possible to determine the minimum value of $M$? Yes or no, with a suitable technical explanation.

What I though was that if I knew the order of this filter I could have found the length of the FIR filter as well. However order cannot be extracted from this graph.

DTFT

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HINT: How many zeros of $H(e^{j\omega})$ do you count? Does that say anything about the (minimum) order of the corresponding polynomial?

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  • $\begingroup$ @PulkitJoshi: I think there's a double zero at $\omega=0$. And yes you're right, you can only see the zeros on the unit circle, but note that due to the symmetry of $h[n]$ you can only have zeros either on the unit circle or mirrored at the unit circle (the latter you wouldn't see directly in the frequency response plot). $\endgroup$
    – Matt L.
    Commented Jul 31, 2020 at 15:36
  • $\begingroup$ @PulkitJoshi: I think in this problem you can assume that all zeros are on the unit circle, i.e., they are visible in the frequency response. $\endgroup$
    – Matt L.
    Commented Jul 31, 2020 at 15:37
  • $\begingroup$ @PulkitJoshi: If the curve touches zero at $\omega=0$ then it's a double zero. $\endgroup$
    – Matt L.
    Commented Jul 31, 2020 at 16:19
  • $\begingroup$ @Tiklu: No, it's type I. $\endgroup$
    – Matt L.
    Commented Jul 31, 2020 at 20:23
  • $\begingroup$ Did you confirm it by checking if any zero is at pi ? Or is their any other way around? Sorry for so many consistent doubts. $\endgroup$
    – Tiklu
    Commented Jul 31, 2020 at 20:55

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