Convolution

Integral operator

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

Properties

$x_1(t)\circledast x_2(t)=x_2(t)\circledast x_1(t)$ $x_{1} (t) ⊛ x_{2} (t) = x_{2} (t) ⊛ x_{1} (t)$
- Commutativity
$(x_1(t)\circledast x_2(t))\circledast x_3(t)=x_1(t)\circledast (x_2(t)\circledast x_3(t))$ $(x_{1} (t) ⊛ x_{2} (t)) ⊛ x_{3} (t) = x_{1} (t) ⊛ (x_{2} (t) ⊛ x_{3} (t))$
- Associativity
$x_1(t)\circledast [x_2(t)+x_3(t)]=x_1(t)\circledast x_2(t)+ x_1(t)\circledast x_3(t)$ $x_{1} (t) ⊛ [x_{2} (t) + x_{3} (t)] = x_{1} (t) ⊛ x_{2} (t) + x_{1} (t) ⊛ x_{3} (t)$
- Distributivity
$Ax_1(t)\circledast Bx_2(t)=AB[x_1(t)\circledast x_2(t)]$ $A x_{1} (t) ⊛ B x_{2} (t) = A B [x_{1} (t) ⊛ x_{2} (t)]$
- Associativity with Scalar
Symmetrical graph about origin
$y(t)=x_1(t-a)\circledast x_2(t-b)$ $y (t) = x_{1} (t - a) ⊛ x_{2} (t - b)$
- $x(t)=x_1(t)\circledast x_2(t)$
- $y(t)=x(t-a-b)$
$x(t)=x_1(t)\circledast x_2(t)$ $x (t) = x_{1} (t) ⊛ x_{2} (t)$
- $x_1$ 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 $x(t)=b_1+b_2$
$\overline{x \circledast y}=\bar x \circledast \bar y$
$(x \circledast y)'=x'\circledast y=x\circledast y'$

Applications

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

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]