GATE 2024 IN – Question 53
The 4-point DFTs of $x[n],y[n]$ are $X[k]=[1,-j,1,j]$ and $Y[k]=[1,3j,1,-3j]$. If $z[n]$ is their 4-point circular convolution, $z[0]$ is ______ (nearest integer).
The DFT convention is $X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}$.
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Show answer and explanation
Correct answer: 2 to 2
Explanation
Circular convolution becomes DFT multiplication: $Z[k]=X[k]Y[k]=[1,(-j)(3j),1,j(-3j)]=[1,3,1,3]$. At $n=0$, the inverse DFT is the mean of these coefficients: $z[0]=(1+3+1+3)/4=\boxed2$.