GATE 2016 EC (EC2) – Question 43
The 4-point sequence $x[n]=\{3,2,3,4\}$ has DFT $X[k]=\{12,2j,0,-2j\}$. Let $X_1[k]$ be the DFT of the 12-point sequence
$$x_1[n]=\{3,0,0,2,0,0,3,0,0,4,0,0\}.$$
The value of $|X_1[8]/X_1[11]|$ is ________.
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Show answer and explanation
Correct answer: 5.9 to 6.1
Explanation
Only indices n=3m are nonzero. Hence
$$X_1[k]=\sum_{m=0}^3x[m]e^{-j2\pi k m/4}=X[k\bmod4].$$
Thus $X_1[8]=X[0]=12$ and $X_1[11]=X[3]=-2j$. The magnitude ratio is $12/2=$ **6**.