Question 24.33: (a) Find the circular autocorrelation of the sequence f[n] =......
(a) Find the circular autocorrelation of the sequence f[n] = 3, 2, 1 using the formula.
(b) Develop a graphical method for performing this calculation.
(a) Here N = 3. From the definition
\begin{aligned} c[n] & =f \star f=\sum_{m=0}^2 f[m] f[m-n] \quad \text { for } n=0,1,2 \\ c[0] & =\sum_{m=0}^2 f[m] f[m] \\ & =(3)(3)+(2)(2)+(1)(1) \\ & =14 \\ c[1] & =\sum_{m=0}^2 f[m] f[m-1] \\ & =f[0] f[-1]+f[1] f[0]+f[2] f[1] \\ & =(3)(1)+(2)(3)+(1)(2) \\ & =11 \end{aligned}.
\begin{aligned} c[2] & =\sum_{m=0}^2 f[m] f[m-2] \\ & =(3)(2)+(2)(1)+(1)(3) \\ & =11 \end{aligned}Hence c[n] = 14, 11, 11.
(b) The graphical method involves listing the sequence f[m], m = 0, 1, 2, around an inner circle. Around an outer circle we list it again. This method is identical to that used in Example 24.28 for circular convolution, but because now there is no folding, the sequence on the outer circle is not reversed. The calculation can be seen in Figure 24.33.
