SIGNALS AND SYSTEMS-V PROPERTIES OF SYSTEMS CIRCULAR CONVOLUTION

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

Consider the situation in which both x(n) and h(n) are periodic sequences. The resulting convolution is then called the periodic or circular convolution.

We have already seen that conventional convolution is the output of a given system resulting from its reaction (or impulse response) to an external input. In this case, even though the external input x(n) can be periodic, the impulse response h(n) of the system can not and will not be periodic. For example, the gain of an amplifier or the attenuation of a filter will not be periodic for obvious reasons. Therefore, we conclude that in conventional convolution, the impulse response of the system will not be periodic. Then what is periodic or circular convolution?

The answer to the above question is obvious: circular convolution is just a convolution-like operation between two periodic sequences of equal lengths. To distinguish it from the conventional convolution, we use the following rules:

·        Circular convolution of two sequences x(n) and h(n) is given by the expression

where x₁(n) and x₂(n), or x(n) and  h(n), respectively, are the input sequences and y(n) is the convolution sum.

·        To distinguish circular convolution from linear convolution, we use the notation

                                                  y(n) = x(n) © h(n)  (2)                                                    

·      The convolution sum of two periodic sequences is also found to be periodic. Hence the convolution need be done over one period only. 

It can be seen that the circular or periodic convolution can also be obtained through our tabular method quite easily, even though in many other textbooks, they use a circular shift method using circles to represent the sequences. In this blog, only the tabular method of circular convolution will be described, as this is much faster, clearer, and easily understandable than other methods. Further, it can be implemented easily using computer algorithms. We now illustrate the computation using a numerical example.

ILLUSTRATIVE EXAMPLE 1: Obtain the circular convolution of the sequences

                     x(n) = (…, 1, 2, 1, 2, 1, 2, …)  (1)                                                    

                              h(n) = (…, 2, 1, 2, 1, 2, 1, …)   (2)                                       

Solution: The tabulation for the example is shown in Tables 1, 2, and 3, respectively. Table 1 shows x(k), h(k), h(‒k).

Table 2 shows x(k), h(‒k), and their columnwise product y(0). Notice that we have shown only one period (containing two terms only) of the product y(0). This will be periodically repeated in other columns.

Table 3 shows the column entries of Table 2 shifted by one cell to the right to yield h(1‒k), and using this we find y(1). Since there are only two terms each in x(k and h(k), we stop the computation at this point. It can be seen that the convolution sum is periodic with the terms (8, 7) in it, i.e., (..., 8,7, 8, 7, 8, 7, ...).

It can be seen that tabulation method is very general and easy to use in all forms of convolution. It can be used to solve problems in cyclic (periodic) shift.