Convolution

#Math
3b1b Video

$\displaystyle p_{X+Y}(s)=p_{X}*p_{Y}=p_{Y}*p_{X}$

Discrete Case

$\displaystyle p_{X}*p_{Y}=\sum_{x = 1}^{N}p_{X}(x)\cdot p_{Y}(s-x)$

• $\displaystyle p_{X}$ and $\displaystyle p_{Y}$ are probability mass functions
• $\displaystyle s$ is the value of the sum of the two variables we are trying to find the probability of occurring
• $\displaystyle N$ is the number of possible sums

Continuous Case

$\displaystyle f*g=\int_{-\infty}^{\infty} f(x)g(s-x) , \mathrm{d}x$

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

Signal Processing

$\displaystyle x(t)=\int_{-\infty}^{\infty} x(\tau)\delta(t-\tau) , \mathrm{d}\tau$

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

$\displaystyle y(t)=x*h=\int_{-\infty}^{\infty} x(\tau)h(t-\tau) , \mathrm{d}\tau$

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

$\displaystyle x(t)\rightarrow \underset{\text{LTI}}{\fbox{h(t)}}\rightarrow y(t)$

• Equivalent way of saying the above equation in block notation

Properties

$\displaystyle fg=gf$

• Commutative

$\displaystyle f*(gh)=(fg)*h$

• Associative

$\displaystyle f*(g+h)=fg+fh$

A system is stable if $\displaystyle \int_{-\infty}^{\infty} \lvert h(t)\rvert , \mathrm{d}t<\infty$ ($\displaystyle h(t)$ is absolutely integrable)

$\displaystyle h*(\alpha x_{1}+\beta x_{2})=\alpha(hx_{1})+\beta(hx_{2})$

$\displaystyle f*g=h\rightarrow f(t-a)*g(t-b)=h(t-(a+b))$

• Time-shifting property of convolution

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

• Derivatives distribute to just one of the functions