8.4 Continuous time convolution and the ctft

Introduction

This module discusses convolution of continuous signals in the time and frequency domains.

Continuous time fourier transform

The CTFT transforms a infinite-length continuous signal in the time domain into an infinite-length continuous signal in the frequency domain.

Convolution integral

The convolution integral expresses the output of an LTI system based on an input signal, $x(t)$ , and the system's impulse response, $h(t)$ . The convolution integral is expressed as

Demonstration

Convolution theorem

Let $f$ and $g$ be two functions with convolution $f*g$ .. Let $F$ be the Fourier transform operator. Then

By applying the inverse Fourier transform $F^{-1}$ , we can write:

Conclusion

The Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) corresponds to point-wise multiplication in the other domain (e.g., frequency domain).