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
6. y(t)=x_1(t-a)\circledast x_2(t-b)
   • x(t)=x_1(t)\circledast x_2(t)
   • y(t)=x(t-a-b)
7. x(t)=x_1(t)\circledast x_2(t)
   • x_1 between a_1 and b_1
   • x_2 between a_2 and b_2
   • Starting point of x(t)=a_1+a_2
   • Ending point of x(t)=b_1+b_2
8. \overline{x \circledast y}=\bar x \circledast \bar y
9. (x \circledast y)'=x'\circledast y=x\circledast y'

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]