stem/Signal Proc/Convolution.md

Integral operator
-   Satisfies mathematical properties of integral operator
-   Product of two after one has been reversed and shifted

$$x(t)=x_1(t)\circledast x_2(t)=\int_{-\infty}^\infty x_1(t-\tau)\cdot x_2(\tau)d\tau$$

# Properties
1. $x_1(t)\circledast x_2(t)=x_2(t)\circledast x_1(t)$
	- Commutativity
2. $(x_1(t)\circledast x_2(t))\circledast x_3(t)=x_1(t)\circledast (x_2(t)\circledast x_3(t))$
	- Associativity
3.  $x_1(t)\circledast [x_2(t)+x_3(t)]=x_1(t)\circledast x_2(t)+ x_1(t)\circledast x_3(t)$
	- Distributivity
4. $Ax_1(t)\circledast Bx_2(t)=AB[x_1(t)\circledast x_2(t)]$
	- Associativity with Scalar
5. Symmetrical graph about origin

# Applications

1. Communications systems
	- Shift signal in frequency domain (Frequency modulation)
2. System analysis
	- Find system output given input and transfer function

# Polynomial Multiplication
-   Convolving coefficients of two poly gives coefficients of product

# Discrete
$$G[i,j]=H[u,v]\circledast F[i,j]$$
$$G[i,j]=\sum^k_{u=-k}\sum^k_{v=-k} H[u,v]F[i-u,j-v]$$