Question 24.28: (a) Calculate the circular convolution, h[n] = f ⊛ g, of the......
(a) Calculate the circular convolution, h[n]=f \circledast g, of the two periodic sequences f[n] = 9,-1, 3 and g[n] = 7,2,-4.
(b) Develop a graphical representation of this process.
(a) The sequence g[n] is depicted in Figure 24.27.
We use the formula given above. In this Example, N = 3. First let n = 0.
Next let n = 1.
\begin{aligned} h[1] & =\sum_{m=0}^2 f[m] g[1-m] \\ & =f[0] g[1]+f[1] g[0]+f[2] g[-1] \\ & =(9)(2)+(-1)(7)+(3)(-4) \\ & =-1 \end{aligned}Finally, let n = 2.
\begin{aligned} h[2] & =\sum_{m=0}^2 f[m] g[2-m] \\ & =f[0] g[2]+f[1] g[1]+f[2] g[0] \\ & =(9)(-4)+(-1)(2)+(3)(7) \\ & =-17 \end{aligned}So the circular convolution f \circledast g=73,-1,-17 .
If required, a table can be constructed which summarizes all the necessary information as was shown in Example 24.27. The sequences must be extended to show their periodicity, and this time we are only interested in generating the convolution sequence over one period, namely that section of the table for m = 0, 1,2.
(b) A graphical representation can be developed by listing the fixed sequence f[m], for m = 0, 1, 2, anticlockwise around an inner circle as shown in Figure 24.28. We list g[-m] around an outer circle but do so clockwise to take account of the folding. By rotating the outer circle anticlockwise we obtain g[1 -m] and g[2 -m]. By multiplying neighbouring terms and adding we obtain the required convolution. The result is 73, -1, -17 as obtained in part (a).
