0
443views
Find the DFT of the sequence x(n)={1 for 0≤n≤20 otherwise for N=4 and compute the corresponding amplitude and phase spectrum
1 Answer
written 2.3 years ago by |
Solution:
X(k)=∑N−1n=0x(n)e−j2πknN
The DFT of the sequence x(n)={1 for 0≤n≤20 otherwise
Here x(0)=1,x(1)=1,x(2)=1,x(3)=0;N=4.
For k=0:
X(0)=3∑n=0x(n)=x(0)+x(1)+x(2)+x(3)=3 Therefore |X(0)|=3,∠X(0)=0 For k=1 : X(1)=3∑n=0x(n)e−jπn2=x(0)+x(1)e−jπ2+x(2)e−jπ+x(3)e−j3π2=1+cosπ2−jsinπ2+cosπ−jsinπ+0=1−j−l=−j Therefore |X(1)|=1,∠X(1)=−π2
For k=2
X(2)=3∑n=0x(n)e−jπn=x(0)+x(1)e−jπ+x(2)e−j2π+x(3)e−j3π=1+cosπ−jsinπ+cos2π−jsin2π+0=1−1+1=1
Therefore |X(2)|=1,∠X(2)=0
For k=3
X(3)=3∑n=0x(n)e−j3πn2=x(0)+x(1)e−j3π2+x(2)e−j3π+x(3)e−j9π2=1+cos3π2−jsin3π2+cos3π−jsin3π+0=1+j−l=j
Therefore |X(3)|=1,∠X(3)=π2
∣X(k)={3,1,1,1}∠X(k)={0,−π2,0,π2}