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Compute the DFT of the following two sequences, h[n]={1,3,1,2} and x[n]={1,2,0,1}
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Solution:

where N=4ej2πN=ej2π4=ejπ2=j

H(k)=3n=0h[n]ejπ2nk for k=0,1,2,3

H(0)=h[0]+h[1]+h[2]+h[3]=1H(1)=h[0]+h[1]ejπ/2+h[2]ejπ+h[3]ej3π/2=2j5H(2)=h[0]+h[1]ejπ+h[2]ej2π+h[3]ej3π=1H(3)=h[0]+h[1]ej3π/2+h[2]ej3π+h[3]ej9π/2=2+j5

X(0)=x[0]+x[1]+x[2]+x[3]=2X(1)=x[0]+x[1]ejπ/2+x[2]ejπ+x[3]ej3π/2=1j3X(2)=x[0]+x[1]ejπ+x[2]ej2π+x[3]ej3π=0X(3)=x[0]+x[1]ej3π/2+x[2]ej3π+x[3]ej9π/2=1+j3

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