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Determine the 8-point DFT of the sequence x(n)={1,1,1,1,1,1,0,0}
written 2.3 years ago by gravatar for prajapatijaimin prajapatijaimin • 3.3k •   modified 2.3 years ago
discrete time signal processing
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written 2.3 years ago by gravatar for prajapatijaimin prajapatijaimin • 3.3k

Solution:

X(k)=∑N−1n=0x(n)e−j2πkn/Nk=0,1,…N−1

For N=8

X(k)=∑7n=0x(n)e−jπkn/4k=0,1,2…..N−1

For k=0

X(0)=7∑n=0x(n)X(0)=x(0)+x(1)+x(2)+x(3)+x(4)+x(5)+x(6)+x(7)=1+1+1+1+1+1+0+0=6

For k=1

X(1)=7∑n=0x(n)e−jπn/4X(1)=x(0)+x(1)e−jπ/4+x(2)e−jπ/2+x(3)e−j3π/4+x(4)e−jπ+x(5)e−j5π/4+x(6)e−j3π/2+x(7)e−j7π/4=1+0.707−j0.707−j−0.707−j0.707−1−0.707+j0.707=−0.707−j1.707

For k=2

X(2)=7∑n=0x(n)e−jπn/2X(2)=x(0)+x(1)e−jπ/2+x(2)e−jπ+x(3)e−j3π/2+x(4)e−j2π+x(5)e−j5π/2+x(6)e−j3π+x(7)e−j7π/2=1−j−1+j+1−j=1−j

For k=3

X(3)=7∑n=0x(n)e−j3πn/4X(3)=x(0)+x(1)e−j3π/4+x(2)e−jπ/2+x(3)e−j9π/4+x(4)e−j3π+x(5)e−j15π/4+x(6)e−j9//4+x(7)e−j21π/4=1−0.707−j0.707+j+0.707−j0.707−1+0.707+j0.707=0.707+j0.293

For k=4

X(4)=7∑n=0x(n)e−jπnX(4)=x(0)+x(1)e−jπ+x(2)e−jπ2+x(3)e−jπ3+x(4)e−jπ4+x(5)e−jπ5+x(6)e−jπ6+x(7)e−jπ7=1−1+1−1+1−1=0

For k=5

X(5)=7∑n=0x(n)e−j5πn/4X(5)=x(0)+x(1)e−j5π/4+x(2)e−j5π/2+x(3)e−j5πn/4+x(4)e−j5π+x(5)e−j25π/4+x(6)e−j15π/2+x(7)e−j35π/4=1−0.707+j0.707−j+0.707+j0.707−1+0.707−j0.707=0.707−j0.293

For k=6

X(6)=7∑n=0x(n)e−j3πn/2X(6)=x(0)+x(1)e−j3π/2+x(2)e−j3π+x(3)e−j9π/2+x(4)e−j6π+x(5)e−j15π+x(6)e−j9π+x(7)e−j21π/2=1+j−1−j+1+j⇒=1+j

For k=7

X(7)=∑7n=0x(n)e−j7πn/4

X(7)=x(0)+x(1)e−j7π/4+x(2)e−j7π/2+x(3)e−j21π/4+x(4)e−j7π+x(5)e−j35π/4+x(6)e−j21π/2+x(7)e−j49π/4=1+0.707+j0.707+j−0.707+j0.707−1−0.707−j0.707=−0.707+j1.707X(K)={6,0.707−j1.707,1−j,0.707+j0.293,0,0.707−j0.293,1+j,−0.707+j1.707}
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