0
443views
Find the DFT of the sequence x(n)={1 for 0n20 otherwise  for N=4 and compute the corresponding amplitude and phase spectrum
1 Answer
0
11views

Solution:

X(k)=N1n=0x(n)ej2πknN

The DFT of the sequence x(n)={1 for 0n20 otherwise 

Here x(0)=1,x(1)=1,x(2)=1,x(3)=0;N=4.

For k=0:

X(0)=3n=0x(n)=x(0)+x(1)+x(2)+x(3)=3 Therefore |X(0)|=3,X(0)=0 For k=1 : X(1)=3n=0x(n)ejπn2=x(0)+x(1)ejπ2+x(2)ejπ+x(3)ej3π2=1+cosπ2jsinπ2+cosπjsinπ+0=1jl=j Therefore |X(1)|=1,X(1)=π2

For k=2

X(2)=3n=0x(n)ejπn=x(0)+x(1)ejπ+x(2)ej2π+x(3)ej3π=1+cosπjsinπ+cos2πjsin2π+0=11+1=1

Therefore |X(2)|=1,X(2)=0

For k=3

X(3)=3n=0x(n)ej3πn2=x(0)+x(1)ej3π2+x(2)ej3π+x(3)ej9π2=1+cos3π2jsin3π2+cos3πjsin3π+0=1+jl=j

Therefore |X(3)|=1,X(3)=π2

X(k)={3,1,1,1}X(k)={0,π2,0,π2}

enter image description here

Please log in to add an answer.