stem/Signal Proc/Convolution.md
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Markdown

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. $x_1(t)\circledast x_2(t)=x_2(t)\circledast x_1(t)$
1. Commutativity
2. $(x_1(t)\circledast x_2(t))\circledast x_3(t)=x_1(t)\circledast (x_2(t)\circledast x_3(t))$
1. Associativity
3. $x_1(t)\circledast [x_2(t)+x_3(t)]=x_1(t)\circledast x_2(t)+ x_1(t)\circledast x_3(t)$
1. Distributivity
4. $Ax_1(t)\circledast Bx_2(t)=AB[x_1(t)\circledast x_2(t)]$
1. 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