Question

If H(z) is the z-transform of the impulse response of an FIR filter, then which of the following relation is true?

(a) z^M+1.H(z^-1)=±H(z)

(b) z^-(M+1).H(z^-1)=±H(z)

(c) z^(M-1).H(z^-1)=±H(z)

(d) z^-(M-1).H(z^-1)=±H(z)

I have been asked this question in an interview for internship.

Asked question is from Design of FIR Filters in division Digital Filters Design of Digital Signal Processing

Answer 1

The correct choice is (d) z^-(M-1).H(z^-1)=±H(z)

Easy explanation: We know that H(z)=\(\sum_{k=0}^{M-1} h(k)z^{-k}\) and h(n)=±h(M-1-n) n=0,1,2…M-1

When we incorporate the symmetric and anti-symmetric conditions of the second equation into the first equation and by substituting z^-1 for z, and multiply both sides of the resulting equation by z^-(M-1) we get z^-(M-1).H(z^-1)=±H(z)