GATE ECE 2022 | Question: 45

Question

For a vector $\overline{x} = \left[ x [0], x[1], \dots, x[7] \right],$ the $\text{8-point}$ discrete Fourier transform $\text{(DFT)}$ is denoted by $\overline{X} = DFT (\overline{x}) = \left[ X [0], X[1], \dots, X[7] \right],$ where

$$X[k] = \displaystyle \sum_{n=0}^{7} x[n] \; \text{exp} \left(-j \frac{2 \pi}{8} nk \right).$$

Here, $j = \sqrt{-1}.$ If $\overline{x} = [1, 0, 0, 0, 2, 0, 0, 0]$ and $\overline{y} = DFT (DFT(\overline{x})),$ then the value of $y[0]$ is _____________ (rounded off to one decimal place).