Circular Convolution¶
In a previous post, we have explained the importance of the convolution operation for signal processing and signal analysis. We have described the convolution integral and illustrated the involved functions.
In this post we will focus on an operation called Circular convolution which is strongly related to the conventional convolution (also called linear convolution) we have described before. Let us reconsider the normal, linear convolution in the discrete domain. Given two sequences x[n] and h[n], their convolution is given by
(x*h)[n] = \sum_{n'=-\infty}^{\infty}x[n']\cdot h[n-n'], \quad n=-\infty,\dots,\infty.The linear convolution lets one one sequence slide over the other and sums the overlapping parts. The circular convolution of two sequences x[n], h[n] is now considering a wrap-around of the sequences after a period of N samples. So, the circular convolution is defined by
(x\otimes h)[n]=\sum_{n'=0}^{N-1}x[n']\cdot h[(n'-n)_N], \quad n=0,\dots,N-1,where (n'-n)_N gives the remainder of n'-n divided by N. For example, (-1)_N=N-1. This means, if the index for x[(n'-n)_N] would leave the range 0,\dots,N-1 to the left, it would wrap around and come in from the right again. This means that the circular convolution is periodic with length N.
In discrete domain, the convolution theorem actually holds only for the circular ...