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

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

CTFT

$$\mathcal{F}\left(\Omega \right)={\int}_{-\infty}^{\infty}f\left(t\right){e}^{-(i\Omega t)}dt$$

1

Inverse CTFT

$$f\left(t\right)=\frac{1}{2\pi}{\int}_{-\infty}^{\infty}\mathcal{F}\left(\Omega \right){e}^{i\Omega t}d\Omega $$

2

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

$$y\left(t\right)={\int}_{-\infty}^{\infty}x\left(\tau \right)h\left(t-\tau \right)d\tau $$

3

$$y\left(t\right)=x\left(t\right)*h\left(t\right)$$

4

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

$$F(f*g)=F\left(f\right)\xb7F\left(g\right)$$

5

$$F(f\xb7g)=F\left(f\right)*F\left(g\right)$$

6

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

$$f*g={F}^{-1}(F\left(f\right)\xb7F\left(g\right))$$

7

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).