Question 24.27: Suppose f[n] is the sequence 3, 9, 2, -1, and g[n] is the se......
Suppose f[n] is the sequence 3, 9, 2, -1, and g[n] is the sequence -4, 8, 5.
(a) Find the linear convolution h[n] = f * g using the previous formula.
(b) Develop a graphical interpretation of this process.
(a) Note that both f and g are finite sequences of length 4 and 3 respectively. The convolution f * g will be a sequence of length 4 + 3 – 1 = 6. By definition,
h[n]=\sum_{m=0}^n f[m] g[n-m] \quad \text { for } n=0,1,2, \ldots, 5The first term in h[n] is obtained by letting n = 0:
\begin{aligned} h[0] & =\sum_{m=0}^0 f[m] g[0-m] \\ & =f[0] g[0] \\ & =(3)(-4) \\ & =-12 \end{aligned}The second term is obtained by letting n = 1:
\begin{aligned} h[1] & =\sum_{m=0}^1 f[m] g[1-m] \\ & =f[0] g[1]+f[1] g[0] \\ & =(3)(8)+(9)(-4) \\ & =24-36 \\ & =-12 \end{aligned}Subsequent terms are calculated in a similar manner. You should obtain these for yourself to ensure that you understand the process. The complete sequence is
h[n]=f * g=-12,-12,79,65,2,-5(b) The graphical interpretation is developed along the same lines as was done for the continuous convolution in Section 24.8.
The sequences f[m] and g[m] are shown in Figure 24.24. The sequence g[-m] is found by reflection in the vertical axis, that is folding as shown in Figure 24.25(a). Then the folded graph can be translated n units to the left or the right by changing the argument of g from g[m] to g[n -m]. If n is positive the graph moves to the right. Study Figures 24.25(b-g) to observe this. Convolution is the sum of products of f[m] with g[n – m]. The graph of f[m] is superimposed. We are only interested in values of n for which the graphs overlap – otherwise each product is zero. For each value of n the superimposed graphs make it easy to see which values must be multiplied together and added. If required, a table can be constructed which summarizes all the necessary information as shown below.
\begin{array}{rrrrrrrrrr} \hline & m & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\ & f[m] & – & – & 3 & 9 & 2 & -1 & – & – \\ n=0 & g[-m] & 5 & 8 & -4 & – & – & – & – & – \\ n=1 & g[1-m] & – & 5 & 8 & -4 & – & – & – & – \\ n=2 & g[2-m] & – & – & 5 & 8 & -4 & – & – & – \\ n=3 & g[3-m] & – & – & – & 5 & 8 & -4 & – & – \\ n=4 & g[4-m] & – & – & – & – & 5 & 8 & -4 & – \\ n=5 & g[5-m] & – & – & – & – & – & 5 & 8 & -4 \\ \hline \end{array}