26 lines
935 B
Markdown
26 lines
935 B
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
|