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GATE EC 2022 Set 1 — Question 55

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NAT+2 / -0MediumSequences, Series & ConvergenceCalculusEngineering Mathematics

Engineering Mathematics → Calculus → Sequences, Series & Convergence

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Question

For a vector xˉ=[x[0],x[1],…,x[7]]\bar{\boldsymbol{x}} = [x[0], x[1], \ldots, x[7]], the 8-point discrete Fourier transform (DFT) is denoted by Xˉ=DFT(xˉ)=[X[0],X[1],…,X[7]]\bar{\boldsymbol{X}} = DFT(\bar{\boldsymbol{x}}) = [X[0], X[1], \ldots, X[7]], whereX[k]=∑n=07x[n]exp⁡(−j2π8nk).X[k] = \sum_{n=0}^{7} x[n] \exp\left(-j\frac{2\pi}{8}nk\right).Here, j=−1j = \sqrt{-1}. If xˉ=[1,0,0,0,2,0,0,0]\bar{\boldsymbol{x}} = [1, 0, 0, 0, 2, 0, 0, 0] and yˉ=DFT(DFT(xˉ))\bar{\boldsymbol{y}} = DFT(DFT(\bar{\boldsymbol{x}})), then the value of y[0]y[0] is _________ (rounded off to one decimal place).
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