GATE 2025 IN – Question 54
A length-4 sequence has $x[0]=1$, $x[1]=2$, $x[2]=2$, $x[3]=4$ and DTFT $X(e^{j\omega})$. Sampling at $\omega_k=2\pi k/3$ gives a sequence periodic in integer $k$ with period 3. Another sequence $y[n]$ has DFT $Y[k]=X(e^{j\omega_k})$ for $0\le k\le2$. The value of $y[0]$ is ______ (in integer).
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Correct answer: 5 to 5
Explanation
Three equally spaced DTFT samples correspond to a 3-point DFT of the original sequence folded modulo 3: $y[n]=\sum_m x[n+3m]$.
For $n=0$, only samples 0 and 3 contribute, so $y[0]=x[0]+x[3]=1+4=\boxed5$.