GATE 2024 EC – Question 59

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

The relationship between any $N$-length sequence $x[n]$ and its corresponding $N$-point discrete Fourier transform $X[k]$ is defined as

$$X[k]=\mathcal F\{x[n]\}.$$

Another sequence $y[n]$ is formed as below

$$y[n]=\mathcal F\Big\{\mathcal F\big\{\mathcal F\{\mathcal F\{x[n]\}\}\big\}\Big\}.$$

For the sequence $x[n]=\{1,2,1,3\}$, the value of $Y[0]$ is ______.

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Correct answer: 16

Explanation

Applying the DFT twice gives $\mathcal F\{\mathcal F\{x[n]\}\}=N\,x[(-n)\bmod N]$. Applying it four times gives $N^2x[n]$. With $N=4$, $y[n]=16\,x[n]$, so the value at $n=0$ is $16\times x[0]=16$. (The question's $Y[0]$ is the $n=0$ value of $y$.)