Definition:Convolution Integral

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This page is about Convolution Integral in the context of Integral Calculus. For other uses, see Definition:Convolution.

Definition

Let $f$ and $g$ be real functions which are integrable.

The convolution integral of $f$ and $g$ is defined as:

$\displaystyle \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$


Positive Real Domain

Let $f$ and $g$ be supported on the positive real numbers $\R_{\ge 0}$ only.

The convolution integral of $f$ and $g$ may be defined as:

$\displaystyle \map f t * \map g t := \int_0^t \map f u \map g {t - u} \rd u$


Cross-Correlation

The cross-correlation of $f$ and $g$ is defined as:

$\displaystyle \map f t \star \map g t := \int_{-\infty}^\infty \map f u \map g {t + u} \rd u$


Also see

  • Results about convolution integrals can be found here.


Sources