GATE 2021 EC – Question 51

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Numerical answer

Consider the signals $x[n]=2^{n-1}u[-n+2]$ and $y[n]=2^{-n+2}u[n+1]$, where $u[n]$ is the unit step sequence. Let $X(e^{j\omega})$ and $Y(e^{j\omega})$ be the discrete-time Fourier transform of $x[n]$ and $y[n]$, respectively. The value of the integral

$$\frac{1}{2\pi}\int_0^{2\pi}X(e^{j\omega})\,Y(e^{-j\omega})\,d\omega$$

(rounded off to one decimal place) is ________.

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Correct answer: 8

Explanation

Writing both transforms as sums, $\frac1{2\pi}\int_0^{2\pi}X(e^{j\omega})Y(e^{-j\omega})d\omega=\sum_nx[n]y[n]$ (a Parseval-type identity). $x[n]=2^{n-1}$ for $n\le2$ and $y[n]=2^{2-n}$ for $n\ge-1$, so the overlap is $n=-1,0,1,2$, where $x[n]y[n]=2^{n-1}2^{2-n}=2$. The sum is $4\times2=8$.