GATE ECE 2008 | Question: 37

Question

$\{x(n)\}$ is a real-valued periodic sequence with a period N. $x(n)$ and $X(k)$ form $\mathrm{N}$-point Discrete Fourier Transform (DFT) pairs. The DFT $Y(k)$ of the sequence $\displaystyle{}y(n)=\frac{1}{N} \sum_{r=0}^{N-1} x(r) x(n+r)$ is

• A. $|X(k)|^{2}$
• B. $\displaystyle{}\frac{1}{N} \sum_{r=0}^{N-1} X(r) X^{*}(k+r)$
• C. $\displaystyle{}\frac{1}{N} \sum_{r=0}^{N-1} X(r) X(k+r)$
• D. $0$