Convolution¶
Definition¶
Basic 1D Form¶
For complex-valued function \(f, g: \mathbb{R} \mapsto \mathbb{C}\), the convolution of \(f\) and \(g\) is defined as
\[
(f*g)(t) \triangleq \int _{-\infty}^{\infty}f(\tau)g(t-\tau)\,d\tau = \int _{-\infty}^{\infty}f(t-\tau)g(\tau)\,d\tau
\]
For complex-valued discrete function \(f[n]\) and \(g[n]\), the discrete convolution of \(f\) and \(g\) is given by
\[
(f*g)[n] = \sum_{m=-\infty}^{\infty}f[m]g[n-m] = \sum_{m=-\infty}^{\infty}f[n-m]g[m]
\]
YYC Note: Here I assume the codomain of \(f\) and \(g\) are 1-dimensional because I don't know how to compute \(f(\tau)g(t-\tau)\) if their codomain is not 1-dimentional.
General multi-dimensional Form¶
For complex-valued function \(f, g: \mathbb{R}^d \mapsto \mathbb{C}\), the convolution of \(f\) and \(g\) is defined as
\[
(f*g)(x)=\int_{\mathbb{R}^d}f(y)g(x-y)\,dy=\int_{\mathbb{R}^d}f(x-y)g(y)\,dy
\]
\((f*g)(x)\) is well-defined only if \(f\) and \(g\) decay sufficiently rapidly at infinity in order for the integral to exist.
Properties¶
Algebraic properties¶
Commutativity
\[
f*g = g*f
\]
Associativity
\[
f*(g*h) = (f*g)*h
\]
Distributivity
\[
f*(g+h) = f*g + f*h
\]
Associativity with scalar multiplication
\[
a (f*g) = (af)*g, \quad \forall a \in \mathbb{C}
\]
Complex conjugation
\[
\overline{f*g}=\overline{f}*\overline{g}
\]