Topic: Digital Signal Processing
If $z_0$ is a zero of a (real-valued) linear-phase FIR filter then following is/are also zero/zeros of a (real-valued) linear-phase FIR filter,
- A. $Z_0^*$
- B. $1/Z_0$
- C. $1/Z_0$, $Z_0^*$ and $1/z_0^*$
- D. $1/Z_0$ and $1/Z_0^*$
Answer: C. $1/Z_0$, $Z_0^*$ and $1/z_0^*$
Explanation: For a linear phase FIR filter, $h(n)$ should be symmetric. Using the symmetry condition $h(n)=h(N-1-n)$ in z-domain: $H(z)=z^{-(N-1)}H(1/z)$. If $H(z_0)=0$, then $H(1/z_0)=0$. Also, roots of a polynomial with real coefficients exist in complex conjugate pairs. Thus, $H(Z_0^*)=0$ and $H(1/Z_0^*)=0$.