GATE 2024 IN Q.38: solution and answer

Official answer: CSource: Official answer key, GATE 2024 IN (IISc Bengaluru)

1. Write the $z$-transform directly from the definition. Since $x[n]$ is non-zero only for $0\le n\le 7$: $$X(z)=\sum _{n=0}^7(0.2{)}^n{z}^{-n}=1+0.2z^{-1}+(0.2{)}^2{z}^{-2}+\cdots +(0.2{)}^7{z}^{-7}$$ This is a finite sum of 8 terms, so it converges wherever every term is finite. No condition like $|z|>0.2$ comes in, because there is no infinite series to converge.

2. Check each term for finite, non-zero $z$: the $n=0$ term is the constant 1, and every term with $n\ge 1$ contains $z^{-n}=1/z^n$, which is finite for every $z\ne 0$. At $z=0$ these terms are infinite, so $z=0$ is excluded.

3. Check $z=\infty$: every $z^{-n}$ with $n\ge 1$ goes to 0, so $X(z)\to 1$, which is finite. The sequence has no terms with $n<0$, so $X(z)$ contains no positive powers of $z$, and $z=\infty$ belongs to the ROC.

4. Confirm with the closed form (finite geometric series, ratio $0.2z^{-1}$): $$X(z)=\frac{1-(0.2{)}^8{z}^{-8}}{1-0.2z^{-1}}=\frac{z^8-(0.2{)}^8}{z^7(z-0.2)}$$

5. The factor $z-0.2$ in the denominator is not a real pole. The numerator vanishes at $z=0.2e^{j2\pi k/8}$, $k=0,1,\dots ,7$, and the $k=0$ zero is exactly $z=0.2$, so it cancels. This must happen, because the sum in step 1 is plainly finite at $z=0.2$ (it equals $1+1+\cdots +1=8$).

6. The only poles left are the 7 poles at $z=0$ (from $z^7$). So the ROC is the whole $z$-plane except $z=0$, which is option (C).