4. Let
$x_1[n] = \{1, 1, 1, 1, 0, 0, 0, 0\}$ (n=0, 1, 2...7)
$x_2[n] = \{2, 2, 2, 2, 0, 0, 0, 2\}$ (n =-4, -3,...,3)
and their N = 8 point DFTs are $X_1[k]$ and $X_2[k]$ respectively.
Without computing any DFTs or IDFTS
a. Express (and explain your answer) the DFT of $x_3[n]$
in terms of $X_1[k]$ and $X_2[k]$ and the DFT of a mystery sequence $x_5[n]$
$x_3[n] = \{-2, -2, -2, 1, 6, 6, 6, 6\}$ (10 points)
b. What is $x_5[n]$? (5 points)
c. $X_1[0] = X_2[0]$. Is this statement true or false? Show all calculations and reasoning. (5 points)
d. $x_4[n] = \{1, 0, 0, 0, 0, 1, 1, 1\}$ can be expressed in terms of $x_1[n]$. It can also be expressed in terms of Fourier transform only. Express $x_4[n]$ in terms of DFT of $x_1[n]$, i.e. $\mathcal{F}\{x_1[n]\}$. Explain your reasoning. (5 points)
