Convolution is the correlation function of f(τ) with the reversed function g(t-τ).
The convolution operator is the asterisk symbol * .
The convolution of f(t) and g(t) is equal to the integral of f(τ) times f(t-τ):
Convolution of 2 discrete functions is defined as:
2 dimensional discrete convolution is usually used for image processing.
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)
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}
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)
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)
ℒ{f (t) * g(t)} = ℒ{f (t)} ⋅ ℒ{g(t)} = F(s) ⋅ G(s)