Output of stage 1

$S_1 (0)= x(0) +x(2)=2+4=6$

$S_1 (1)=x(0)-x(2)=2-4=-2$

$S_1 (2)=x(1)+x(3)=3+5=8$

$S_1 (3)=x(1)-x(3)=3-5=-2$

Final Output

$X(0)=S_1 (0)+S_1 (2) W_4^0=6+8$

$X(1)=S_1 (1)+S_1 (3)W_4^1=-2+2j$

$X(2)=S_1 (0)-S_1 (2)W_4^0=6-8$

$X(3)=S_1 (1)-S_1 (3)W_4^1=-2-2j$

$X(K)={14,-2+2j,-2,-2-2j}$

ii) y(n)=x(n-1), Find DFT of y(n)

by circular shifting property

$x(n-l) (↔)^{DFT} X(K).e^{\frac{-j2πkl}{N}}$

$x(n-1) (↔)^{DFT} X(k).W_4^K$

$Y(0)=X(0).W_4^0=(14×1)=14$

$Y(1)=X(1).W_4^1=(-2+2j)(-j)=+2+2j$

$Y(2)=X(2) W_4^2=(-2)(-1)=2$

$Y(3)=X(3) W_4^3=(-2-2j)(+j)=2-2j$

$Y(k)={14,2+2j,2 ,2-2j}$

iii) m(n)=x(n)+jy(n)

By linearly property

M(K)=X(K)+jY(K)

M(0)=14+14j

M(1)=(-2+2j)+j(+2+2j)=-4+4j

M(2)=-2+2j

M(3)=(-2-2j)+j(2-2j)=0

M(K)={14+14j,-4+4j,-2+2j,0}