6.4 Continuous time circular convolution and the ctfs

Introduction

This module relates circular convolution of periodic signals in the time domain to multiplication in the frequency domain.

Signal circular convolution

Given a signal $f(t)$ with Fourier coefficients $c_n$ and a signal $g(t)$ with Fourier coefficients $d_n$ , we can define a new signal, $v(t)$ , where $v(t)=⊛(f(t), g(t))$ We find that the Fourier Series representation of $v(t)$ , $a_n$ , is such that $a_n=c_n{d}_n$ . $⊛(f(t), g(t))$ is the circular convolution of two periodic signals and is equivalent to the convolution over one interval, i.e. $⊛(f(t), g(t))=\int_0^T \int_0^T f(\tau )g(t-\tau )\,d \tau \,d t$ .

Exercise

Take a look at a square pulse with a period of T.

For this signal $$c_n=\begin{cases}\frac{1}{T} & \text{if $n=0$}\\ \frac{1}{2}\frac{\sin (\frac{\pi }{2}n)}{\frac{\pi }{2}n} & \text{otherwise}\end{cases}$$

Take a look at a triangle pulse train with a period of T.

This signal is created by circularly convolving the square pulse with itself. The Fourier coefficients for this signal are $a_n=c_n^2=\frac{1}{4}\frac{\sin (\frac{\pi }{2}n)^2}{(\frac{\pi }{2}n)^2}$

Conclusion

Circular convolution in the time domain is equivalent to multiplication of the Fourier coefficients in the frequency domain.