## Convolution

Convolution is the correlation function of f(τ) with the reversed function g(t-τ).

The convolution operator is the asterisk symbol *** .**

- Continuous convolution
- Discrete convolution
- 2D discrete convolution
- Filter implementation with convolution
- Convolution theorem

### Continuous convolution

The convolution of f(t) and g(t) is equal to the integral of f(τ) times f(t-τ):

### Discrete convolution

Convolution of 2 discrete functions is defined as:

### 2D discrete convolution

2 dimensional discrete convolution is usually used for image processing.

### Filter implementation with convolution

We can filter the discrete input signal x(n) by convolution with the impulse response h(n) to get the output signal y(n).

*y*(*n*) = *x*(*n*) * *h*(*n*)

### Convolution theorem

The Fourier transform of a multiplication of 2 functions is equal to the convolution of the Fourier transforms of each function:

ℱ{*f * ⋅* g*} = ℱ{*f*} * ℱ{*g*}

The Fourier transform of a convolution of 2 functions is equal to the multiplication of the Fourier transforms of each function:

ℱ{*f * ** g*} = ℱ{*f*} ⋅ ℱ{*g*}

#### Convolution theorem for continuous Fourier transform

ℱ{*f *(*t*) ⋅
* g*(*t*)} = ℱ{*f*(*t*)} * ℱ{*g*(*t*)} = *F*(*ω*) * *G*(*ω*)

ℱ{*f *(*t*) *
* g*(*t*)} = ℱ{*f*(*t*)} ⋅ ℱ{*g*(*t*)} = *F*(*ω*) ⋅ *G*(*ω*)

#### Convolution theorem for discrete Fourier transform

ℱ{*f *(*n*) ⋅
* g*(*n*)} = ℱ{*f*(*n*)} * ℱ{*g*(*n*)} = *F*(*k*) * *G*(*k*)

ℱ{*f *(*n*) *
* g*(*n*)} = ℱ{*f*(*n*)} ⋅ ℱ{*g*(*n*)} = *F*(*k*) ⋅ *G*(*k*)

#### Convolution theorem for Laplace transform

ℒ{*f *(*t*) *
* g*(*t*)} = ℒ{*f*(*t*)} ⋅ ℒ{*g*(*t*)} = *F*(*s*) ⋅ *G*(*s*)