1. Research Institute for Future Media Computing
未来媒体技术与计算研究所
Research Institute for Future Media Computing
Tutorial
3. Key Operation on Signals: Convolution
江健民，国家千人计划特聘教授
深圳大学未来媒体技术与计算研究所所长
Office Room: 409

2. Problem: Given two sequences x(n)={1, 3, 2, -3, 0, 2, -1, 0, -2, 3, -2, 1 …}(N1=?), and h(n)={1, 0, 1} (N2=3), calculate the convolution of these two sequences.
Solution via Overlap-add:
Divide x(n) into sections of length N1=6;
x1(n)={1, 3, 2, -3, 0, 2}; x2(n)={-1, 0, -2, 3, -2, 1};
N=N1+N2-1 = 8 = 2d (d=3), satisfying the requirements for use of a radix-2 FFT;
Apply FFT to calculate circular convolution, x1(n)*h(n) and x2(n)*h(n), and then overlap-add to derive the linear convolution of x(n)*h(n).

3. Specific Solutions:
Augmenting zeros to x1(n), x2(n), and h(n) as follows:
x’1(n)={1, 3, 2, -3, 0, 2, 0, 0};
x’2(n)={-1, 0, -2, 3, -2, 1, 0, 0};
h’(n)={1, 0, 1, 0, 0, 0, 0, 0}
y1(0)= h’(0)x’1(0)=1; y1(1)=h’(0)x’1(1)+h’(1)x’1(0)=3;
y1(2)=h’(0)x’1(2)+h’(1)x’1(1)+h’(0)x’1(0)=3;
y1(3)=h’(0)x1(3)+h’(1)x1(2)+h’(2)x1(1)+h’(1)x1(2)…=0; …
Y1(n)={1, 3, 3, 0, 2, -1, 0, 2}; y2(n)={-1, 0, -3, 1, -4, 4, -2, -1}

4. Y1(n)={1, 3, 3, 0, 2, -1, 0, 2};
y2(n)={0, 0, 0, 0, 0, 0, -1, 0, -3, 1, -4, 4, -2, -1}
Y(n)=y1(n)+y2(n) = {1, 3, 3, 0, 2, -1, -1, 2, -3, 1, -4, 4, -2, -1}
Exercise: Use linear convolution to verify that the above results are correct.