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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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written 2.3 years ago by |
Solution:
where N=4⇒ej2πN=ej2π4=ejπ2=j
H(k)=∑3n=0h[n]e−jπ2nk for k=0,1,2,3
H(0)=h[0]+h[1]+h[2]+h[3]=1H(1)=h[0]+h[1]e−jπ/2+h[2]⋅e−jπ+h[3]⋅e−j3π/2=2−j5H(2)=h[0]+h[1]e−jπ+h[2]⋅e−j2π+h[3]⋅e−j3π=−1H(3)=h[0]+h[1]e−j3π/2+h[2]⋅e−j3π+h[3]⋅e−j9π/2=2+j5
X(0)=x[0]+x[1]+x[2]+x[3]=2X(1)=x[0]+x[1]e−jπ/2+x[2]⋅e−jπ+x[3]⋅e−j3π/2=1−j3X(2)=x[0]+x[1]e−jπ+x[2]⋅e−j2π+x[3]⋅e−j3π=0X(3)=x[0]+x[1]e−j3π/2+x[2]⋅e−j3π+x[3]⋅e−j9π/2=1+j3