i)

$x(n)= 1, 0 \lt n \lt3$

$x(n)= 0, 4\lt n \lt7$

ii)

$x_1(n)=1, n=0$

$x_1(n)=0, 1 \lt n \lt4$

$x_1(n)=1, 5 \lt n \lt7$

$x_2(n)=0, 0 \lt n \lt1$

$x_2(n)=1, 2 \lt n \lt 5$

$x_2(n)=1, 6 \lt n \lt7$

x(n)={1,1,1,1,0,0,0,0}

DFT by DIF-FFT

Output of stage-1

$S_1 (0)=x(0)+x(4)=1+0=1$

$S_1 (1)=x(1)+x(5)=1+0=1$

$S_1 (2)=x(2)+x(6)=1+0=1$

$S_1 (3)=x(3)+x(7)=1+0=1$

$S_1 (4)=[x(0)-x(4) ] W_8^0=(1-0)(1)=1$

$S_1 (5)=[x(1)-x(5)]W_8^1=(1-0)(0.707-j0.707)$

$S_1 (6)=[x(2)-x(6) ] W_8^2=(1-0)(-j)=-j$

$S_1 (7)=[x(3)-x(7)W_8^3=(1-0)(-0.707-j0.707)$

Output of stage-2

$S_2 (0)=S_1 (0)+S_1 (2)=1+1=2$

$S_2 (1)=S_1 (1)+S_1 (3)=1+1=2$

$S_2 (2)=S_1 (0)+S_1 (2)]W_8^0=(1-1)W_8^0=0$

$S_2 (3)=S_1 (1)+S_1 (3)]W_8^2=(1-1) W_8^2=0=0$

$S_2 (4)=S_1 (4)+S_1 (6)=1-j$

$S_2 (5)=S_1 (5)+S_1 (7)=0.707-j0.707-0.707-j0.707=-1.141j$

$S_2 (6)=[S_1 (4)-S_1 (6) ] W_8^0=(1+j)(1)=1j=1+j$

$S_2 (7)=[S_1 (5)-S_1 (7) ] W_8^2=(0.707-j0.707+0.707+j0.707)(-j)=-1.414j$

Final Output

$X(0)=S_2 (0)+S_2 (1)=2+2=4$

$X(4)=S_2 (0)-S_2 (1)=2-2=0$

$X(2)=S_2 (2)+S_2 (3)=0+0=0$

$X(6)=S_2 (2)-S_2 (3)=0-0=0$

$X(1)=S_2 (4)+S_2 (5)=(1-j)+(-1.414j)=1-2.414j$

$X(5)=S_2 (4)-S_2 (5)=(1-j)-(-1.414j)=1+0.414j$

$X(3)=S_2 (6)+S_2 (7)=(1+j)+(-1.414j)=1-0.414j$

$X(7)=S_2 (6)-S_2 (7)=(1+j)-(-1.414j)=1+2.414j$

$X(K)={4,1-2.414j,0,1-0.414j,0,1+0.414j,0,1+2.414j}$

$ii) x_1 (n)={1,0,0,0,0,1,1,}$

$x_1 (n)=x(n+3)$

By the shifting property

$x(n+3)(↔)^{DFT} X(k).) W_8^{-3K}$

$X_1 (0)=X(0) W_8^0=4(1)=4$

$X_1 (1)=X(1) W_8^{-.3}=(1-2.414j)(-0.707+j0.707)$

$X_1 (2)=X(2) W_8^{-6}=0$

$X_1 (3)=X(3) W_8^{-9}=(1-0.414j)(0.707+0.707j)$

$X_1 (4)=X(4) W_8^{-12}=0$

$X_1 (5)=X(5) W_8^{-15}=(1+0.414j)(0.707-0.707j)$

$X_1 (6)=X(6) W_8^{-18}=0$

$X_1 (7)=X(7) W_8^{-21}=(1+2.414j)(-0.707-0.707j)$

$X(k)={4,1+2.414j,0,1+0.414j,0,1-0.414j,0,1-2.414j}$

$ii) x_2 (n)={0,0,1,1,1,1,0,0}$

$x_2 (n)=x(n-2)$

By time shift property,

$x(n-2)(↔)^{DFT} X(k).) W_8^{2K}$

$X_2 (0)=X(0) W_8^0=4(1)=4$

$X_2 (1)=X(1) W_8^2=(1-2.414j)(-j)=-2.414-j$

$X_2 (2)=X(2) W_8^4=0$

$X_2 (3)=X(3) W_8^6=(1-0.414j)(j)=0.414+j$

$X_2 (4)=X(4) W_8^8=(0)(1)=0$

$X_2 (5)=X(5) W_8^{10}=(1+0.414j)(-j)=0.414-j$

$X_2 (6)=X(5) W_8^{12}=0$

$X_2 (7)=X(7) W_8^{14}=-2.414+j$

$X_2 (k)={4,-2.414-j,0,0.414+j,0,0.414-j,0,-2.414+j}$