Question

What is the value of x(n)*h(n), 0≤n≤11 for the sequences x(n)={1,2,0,-3,4,2,-1,1,-2,3,2,1,-3} and h(n)={1,1,1} if we perform using overlap add fast convolution technique?

(a) {1,3,3,1,1,3,5,2,2,2,3,6}

(b) {1,2,0,-3,4,2,-1,1,-2,3,2,1,-3}

(c) {1,2,0,3,4,2,1,1,2,3,2,1,3}

(d) {1,3,3,-1,1,3,5,2,-2,2,3,6}

The question was posed to me in an online quiz.

The query is from Linear Filtering Methods Based on DFT in portion Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Correct option is (d) {1,3,3,-1,1,3,5,2,-2,2,3,6}

Easiest explanation: Since M=3, we chose the transform length for DFT and IDFT computations as L=2^M=2^3=8.

Since L=M+N-1, we get N=6.

According to Overlap add method, we get

x1‘(n)={1,2,0,-3,4,2,0,0} and h'(n)={1,1,1,0,0,0,0,0}

y1(n)=x1‘(n)*N h'(n) (circular convolution)={1,3,3,-1,1,3,6,2}

x2‘(n)={-1,1,-2,3,2,1,0,0} and h'(n)={1,1,1,0,0,0,0,0}

y2(n) = x2‘(n)*N h'(n)={-1,0,-2,2,3,6,3,1}

Thus we get, y(n) = {1,3,3,-1,1,3,5,2,-2,2,3,6}.