Question

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}

This question was posed to me in a job interview.

My doubt stems from Properties of DFT in section Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Right option is (d) {14,16,14,16}

The explanation: 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}.