GATE 2022 EC – Question 55

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

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

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