Convolution

3b1b Video

$p_{X+Y}(s)=[p_X\ast p_Y](s)=[p_Y\ast p_X](s)$

Discrete Case

$[p_X\ast p_Y](s)=\sum _{x=1}^N{p}_X(x)\cdot p_Y(s-x)$

• $p_X$ and $p_Y$ are probability mass functions
• $s$ is the value of the sum of the two variables we are trying to find the probability of occurring
• $N$ is the number of possible sums

Continuous Case

$[f\ast g](s)={\int }_{-\infty }^{\infty }f(x)g(s-x)\mathrm{d}x$

• Essentially gives the probability of getting a sum $s$ when adding $f$ and $g$, which are two PDF’s of two different random variables

Signal Processing

$x(t)={\int }_{-\infty }^{\infty }x(\tau )\delta (t-\tau )\mathrm{d}\tau$

• $\delta (\cdot )$ is the delta function
• Desmos Demo

$y(t)=[x\ast h](t)={\int }_{-\infty }^{\infty }x(\tau )h(t-\tau )\mathrm{d}\tau$

• $h$ is the impulse response
• Called the flip and drag technique

$x(t)\to \underset{\text{LTI}}{\boxed{h(t)}}\to y(t)$

• Equivalent way of saying the above equation in block notation

Properties

$f\ast g=g\ast f$

• Commutative

$f\ast (g\ast h)=(f\ast g)\ast h$

• Associative

$f\ast (g+h)=f\ast g+f\ast h$

A system is stable if ${\int }_{-\infty }^{\infty }|h(t)|\mathrm{d}t<\infty$ ($h(t)$ is absolutely integrable)

$h\ast (\alpha x_1+\beta x_2)=\alpha (h\ast x_1)+\beta (h\ast x_2)$

$f\ast g=h\to f(t-a)\ast g(t-b)=h(t-(a+b))$

• Time-shifting property of convolution

$\frac{\mathrm{d}}{\mathrm{d}t}(f\ast g)=\left(\frac{\mathrm{d}}{\mathrm{d}t}f\right)\ast g=f\ast \left(\frac{\mathrm{d}}{\mathrm{d}t}g\right)$

• Derivatives distribute to just one of the functions