Q.$x(0)=-(-1)^{n}$ for $N=4$ find $D F T$ .

$x(0)=(-1)^{0}=1$

$x(1)=(-1)^{1}=-1$

$x(2)=(-1)^{2}=1$

$x(3)=(-1)^{3}=-1$

$x(k)$

$\left[\begin{array}{l}{x(0)} \\ {x(t)} \\ {x(2)} \\ {x(3)}\end{array}\right]=\left[\begin{array}{cccc}{1} & {t} & {1} & {1} \\ {1} & {-j} & {-1} & {j} \\ {1} & {-1} & {1} & {-1} \\ {1} & {j} & {-1} & {-j}\end{array}\right]\left[\begin{array}{l}{x(0)} \\ {x(1)} \\ {x(2)} \\ {x(3)}\end{array}\right]$

$=\left[\begin{array}{cccc}{1} & {1} & {1} & {1} \\ {1} & {-j} & {-1} & {j} \\ {1} & {-1} & {1} & {-1} \\ {1} & {1} & {i} & {-j}\end{array}\right]\left[\begin{array}{c}{1} \\ {-1} \\ {1} \\ {-1}\end{array}\right]$

$x(0)=1-1+1+1=0$

$x(1)=1+j-1-j=0$

$x(2)=1+1+1+1 \leq 4$

$x(3)=1-j-1+j=0$