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Faltung

Definition

Corresponds to multiplication in the frequncy domain.

x(t)*h(t)=x(τ)h(tτ)dτx(t) * h(t) = \int\limits_{-\infty}^{\infty} x(\tau) \cdot h(t-\tau) \,\text{d}\tau

with signals x(t)x(t), h(t)h(t).

Discrete: (f*g)[n]=k=f[k]g[nk](f*g)[n] = \sum\limits_{k = -\infty}^{\infty} {f[k] g[n-k]}

Comparison of convolution, cross-correlation, and auto-correlation. [From Wikimedia](https://de.wikipedia.org/wiki/Datei:Comparison_convolution_correlation_de.svg)
Comparison of convolution, cross-correlation, and auto-correlation. From Wikimedia
Convolution of $f(t) * g(t)$ showing $f(\tau)$ and $g(t - \tau)$
Convolution of f(t)*g(t)f(t) * g(t) showing f(τ)f(\tau) and g(tτ)g(t - \tau)

Properties

Refrences