GATE 2018 EC Q.13: solution and answer

Official answer: BSource: Official answer key, GATE 2018 EC (IIT Guwahati)

1. All-pass pole-zero rule: for $|H(e^{j\omega })|$ to be constant, every pole at $p$ must have a paired zero at the conjugate-reciprocal point $1/p^{\ast }$. The zero has the same angle as the pole and the reciprocal radius.

2. Pole $0.25\angle 0^{\circ }$ pairs with a zero at $\frac{1}{0.25}\angle 0^{\circ }=4\angle 0^{\circ }$. Pole $2\angle 30^{\circ }$ pairs with a zero at $\frac{1}{2}\angle 30^{\circ }=0.5\angle 30^{\circ }$.

3. So $4\angle 0^{\circ }$ and $0.5\angle 30^{\circ }$ are zeros, not poles. Option (A) is false.

4. ROC and stability: an LTI system is stable exactly when the ROC of $H(z)$ contains the unit circle $|z|=1$. The ROC is a ring bounded by pole radii and contains no poles. Here one pole has radius $0.25$ (inside the unit circle) and one has radius $2$ (outside).

5. A right-sided $h[n]$ has ROC $|z|>r_{\max }$, where $r_{\max }\ge 2$ is the largest pole radius, so it cannot contain $|z|=1$. A left-sided $h[n]$ has ROC $|z|<r_{\min }$, where $r_{\min }\le 0.25$, so it cannot contain $|z|=1$ either. The only ROC that can contain the unit circle is an annulus lying between the pole at radius $0.25$ and the pole at radius $2$ (for the two given poles, $0.25<|z|<2$).

6. An annular ROC means the impulse response is two-sided. So the system is stable only when $h[n]$ is two-sided. Option (B) is true.

7. An all-pass system has constant magnitude, not constant phase. If the phase were constant as well, $H(e^{j\omega })$ would be one fixed complex number $K$ for every $\omega$. Then $h[n]=K\delta [n]$, which has no poles, but this system is given to have poles. So the phase varies with $\omega$ and (C) is false.

8. (D) is false for the same reason. $H(z)$ is a non-constant rational function (it has poles and zeros), so its angle changes from point to point in the $z$-plane, and it is not even defined at the zeros $4\angle 0^{\circ }$ and $0.5\angle 30^{\circ }$.

9. Answer: (B).