Convolution

Corresponds to multiplication in the frequncy domain.

\[x(t) * h(t) = \int\limits_{-\infty}^{\infty} x(\tau) \cdot h(t-\tau) \diff \tau\]

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

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

Comparison of convolution, cross-correlation, and auto-correlation. From Wikimedia

Convolution of \(f(t) * g(t)\) showing \(f(\tau)\) and \(g(t - \tau)\)

Properties

• Kommutativität: \(f(t)*g(t) = g(t)*f(t)\)

• Assoziativität: \(f(t)*(g(t)*h(t)) = (f(t)*g(t))*h(t)\)

• Distributivität: \(f(t)*(g(t) + h(t)) = f(t)*g(t) + f(t)*h(t)\)

• Faltung mit Dirac: \(x(t)*\delta(t-b) = x(t-b)\) (Gleiches Signal verschoben)

• Kausalität: \(h(t - \tau) = 0\) für \(\tau > t\)
  \(h[n - l] = 0\) für \(l > n\)

Refrences