GATE 2018 EC – Question 65
Let $X[k]=k+1$, $0\le k\le7$ be 8-point DFT of a sequence $x[n]$, where $X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}$.
The value (correct to two decimal places) of $\sum_{n=0}^{3}x[2n]$ is ________.
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Correct answer: 3
Explanation
The sum of the even-indexed samples is $\sum_{n\ \text{even}}x[n]=\frac{X[0]+X[4]}{2}$, since $\frac12\left[1+(-1)^n\right]$ selects even $n$ and $\sum x[n](-1)^n=X[4]$. So the sum is $\frac{1+5}{2}=3$.