Convolution Integrals

From Qwiki

Convolution Integrals are used extensively for solving differential equations in terms of impulse responses. The convolution of two functions $f 1 (t)$ and $f 2 (t)$ , denoted by $*$ , is usually defined as

$\left(f_1 * f_2\right)(\tau) = \int_{-\infty}^\infty f_1(t) f_2(\tau-t) dt.$

Convolutions are commutatitve

$\left(f_1 * f_2\right)(\tau) = \left(f_2 * f_1\right)(\tau)$
and associative $\left(f_1 * \left(f_2 * f_3\right)\right)(\tau) = \left(\left(f_1 * f_2 \right)*f_3\right)(\tau).$

An alternative, and sometimes very useful, way to write the convolution integral is explicitly symmetric

$\left(f_1 * f_2\right)(\tau) = \int_{-\infty}^\infty dt_1 \int_{-\infty}^\infty dt_2 f_1(t_1) f_2(t_2) \delta\left(\tau-t_1-t_2\right) .$

In this form, the multiconvolution is a straightforward generalization of the simple convolution

$(f_1*\cdots *f_n)(\tau) = \int dt_1\cdots \int dt_n f_1(t_1) \cdots f_n(t_n) \delta(\tau-t_1-\cdots - t_n) .$

Retrieved from "http://qwiki.stanford.edu/wiki/Convolution_Integrals"

Categories: Concept | Mathematics

Convolution Integrals

From Qwiki

Views

Personal tools

Navigation

Search

Toolbox