Convolution Integral

The convolution integral of two $x_1(t)$ and $x_2(t)$ is denoted symbolically as:

$$x_1(t)\ast x_2(t)={\int }_{-\infty }^{\infty }{x}_1(\tau )x_1(t-\tau )d\tau$$

Discrete version:

$$y[n]=x[n]\ast h[n]=\sum _{m=-\infty }^{\infty }x[m]h[n-m]$$

Properties

1. Commutative property

$$x[n]\ast h[n]=h[n]\ast x[n]$$

2. Distributive property

$$x[n]\ast (h_1[n]+h_2[n])=x[n]\ast h_1[n]+x[n]\ast h_2[n]$$

3. Associative property

$$x[n]\ast (h_1[n]\ast h_2[n])=(x[n]\ast h_1[n])\ast h_2[n]$$

4. Sifting property

$$\begin{array}{rl}x(t)\ast h(t)& =y(t)\\ x(t)\ast h(t-\tau )& =x(t-\tau )\ast h(t)=y(t)\end{array}$$

5. Convolution with an impulse

$$\begin{array}{rl}x[n]\ast \delta [n]& =x[n]\\ x[n]\ast \delta [n-n_0]& =x[n-n_0]\end{array}$$