0
626views
Given a sequence x(n) for 0n3, where x(0)=1,x(1)=3,x(2)=3, and x(3)=4. Evaluate its DFT X(k).
1 Answer
0
26views

Solution:

Since N=4,W4=ejπ/2, then using:

X(k)=3n=0x(n)Wkn4=3n=0x(n)ejπkn2

Thus, for k=0

X(0)=3n=0x(n)ej0=x(0)ej0+x(1)ej0+x(2)ej0+x(3)ej0=x(0)+x(1)+x(2)+x(3)=1+2+3+4=10 for k=1XX(1)=3n=0x(n)ejπn2=x(0)ej0+x(1)ejπ2+x(2)ejπ+x(3)ej3π2=x(0)jx(1)x(2)+jx(3)=1j23+j4=2+j2 for k=2X(2)=3n=0x(n)ejπn=x(0)ej0+x(1)ejπ+x(2)ej2π+x(3)ej3π=x(0)x(1)+x(2)x(3)=12+34=2 and for k=3X(3)=3n=0x(n)ej3πn2=x(0)ej0+x(1)ej3π2+x(2)ej3π+x(3)ejjπ2=x(0)+jx(1)x(2)jx(3)=1+j23j4=2j2

Please log in to add an answer.