8. If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), X2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then what is the expression for x3(m)?

A. \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m+n)\)

B. \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)\)

C. \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N \)

D. \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m+n)_N \)

Answer: C

If X1(n), x2(n) and x3(m) are three sequences each of length N whose DFTs are given as X1(k), x2(k) and X3(k) respectively and X3(k)=X1(k).X2(k), then according to the multiplication property of DFT we have x3(m) is the circular convolution of X1(n) and x2(n).

That is x3(m) = \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N \).

9. What is the circular convolution of the sequences X1(n)={2,1,2,1} and x2(n)={1,2,3,4}?

A. {14,14,16,16}
B. {16,16,14,14}
C. {2,3,6,4}
D. {14,16,14,16}

Answer: D

We know that the circular convolution of two sequences is given by the expression

x(m)= \(\sum_{n=0}^{N-1}x_1 (n) x_2 (m-n)_N\)

For m=0, x2((-n))4={1,4,3,2}
For m=1, x2((1-n))4={2,1,4,3}
For m=2, x2((2-n))4={3,2,1,4}
For m=3, x2((3-n))4={4,3,2,1}
Now we get x(m)={14,16,14,16}.

10. What is the circular convolution of the sequences X1(n)={2,1,2,1} and x2(n)={1,2,3,4}, find using the DFT and IDFT concepts?

A. {16,16,14,14}
B. {14,16,14,16}
C. {14,14,16,16}
D. None of the mentioned

Answer: B

Given X1(n)={2,1,2,1}=>X1(k)=[6,0,2,0]

Given x2(n)={1,2,3,4}=>X2(k)=[10,-2+j2,-2,-2-j2]

when we multiply both DFTs we obtain the product X(k)=X1(k).X2(k)=[60,0,-4,0]

By applying the IDFT to the above sequence, we get x(n)={14,16,14,16}.