Convolution

 

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

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

f(t)*g(t)=\int_{-\infty }^{\infty }f(\tau )g(t-\tau )d\tau

 

Discrete convolution

Convolution of 2 discrete functions is defined as:

f(n)*g(n)=\sum_{k=-\infty }^{\infty }f(k)\: g(n-k)

 

2D discrete convolution

2 dimensional discrete convolution is usually used for image processing.

f(n,m)*g(n,m)=\sum_{j=-\infty }^{\infty }\sum_{k=-\infty }^{\infty }f(j,k)\: g(n-j,m-k)

 

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)

 

See also

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