x(n)={4,3,2,1,0,0}

i) Find p(n)=ifP(K)= $W_N^2K X(K)$

Above statement signifies, circular time shift properly.

$x(n-l)(↔)^{DFT} W_6^{2K} X(K)$

By comparison we have,

l=2 ,N=6

p(n)=x(n-2)

P(n)={0,0,4,3,2,1}

ii) If Q(K)=X(K-3),Find q(n)

Above statement signifies circular frequency shift.

$x(n) e^{\frac{-j 2πKl}{(N}} (↔)^{DFT} X(K-l)$

By comparison we have,

l=3

$q(n)=x(n) e^{\frac{-j 2πln}{6}}$

$q(n)=x(n).e^{-jπn}$

$q(0)=x(0) e^0 =4$

$q(1)=x(1) e^{-jπ}=-3$

$q(2)=x(2) e^{-2jπ}=2$

$q(3)=x(3) e^{-3jπ}=-1$

$q(4)=x(4) e^{-4jπ}=0$

$q(5)=x(5) e^{-5jπ}=0$

q(n)={4,-3,2,-1,0,0}