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Find IDFT of the sequence X (K) = (5,0,1-j,0,1,0,1+j,0)
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Solution:

We have,

x(n)=1NN1k=0X(k)ej2πkn/Nn=0,1,.,N1

For N=8

x(n)=18N2k=0X(k)ejπn/4n=0,1,.7

For n=0;x(0)=7k=0X(k)=18[5+0+1j+0+1+j+0]=1

For n=1;x(1)=187k=0X(k)ejπk/4=18[5+(1j)(j)+1(1)+(1+j)(j)]=6/8=0.75

For n=2;x(2)=187k=0X(k)ejπk/2=18[5+(1j)(1)+1(1)+(1+j)(1)]=4/8=0.5

For n=3;x(3)=187k=0X(k)ej3πk/4=18[5+(1j)(j)+1(1)+(1+j)(j)]=2/8=0.25

For n=4;x(4)=187k=0X(k)ej5πk/4=18[5+(1j)(1)+1(1)+(1+j)(1)]=8/8=1

For n=5;x(5)=187k=0X(k)ej5πk/4=18[5+(1j)(j)+(1)(1)+(1+j)(j)]=6/8=0.75

For n=6;x(6)=187k=0X(k)ej3πk/2=18[5+(1j)(1)+1(1)+(1+j)(j)]=4/8=0.5

For n=7;x(7)=187k=0ej7πk/4=18[5+(1j)(j)+1(1)+(1+j)(j)]=2/8=0.25

x(n)={1,0.75,0.5,0.25,1,0.75,0.5,0.25}

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