# How to find minimum length of a FIR symmetric filter if I am given DTFT

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.

HINT: How many zeros of $$H(e^{j\omega})$$ do you count? Does that say anything about the (minimum) order of the corresponding polynomial?
• @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). Jul 31, 2020 at 15:36
• @PulkitJoshi: If the curve touches zero at $\omega=0$ then it's a double zero. Jul 31, 2020 at 16:19