Definition:Convolution Integral
Jump to navigation
Jump to search
This page is about Convolution Integral in the context of Integral Calculus. For other uses, see Convolution.
Definition
Let $f$ and $g$ be real functions which are integrable.
The convolution integral of $f$ and $g$ is defined as:
- $\ds \map f t * \map g t := \int_{-\infty}^\infty \map f u \map g {t - u} \rd u$
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:
- $\ds \map f t * \map g t := \int_0^t \map f u \map g {t - u} \rd u$
Cross-Correlation
The cross-correlation of $f$ and $g$ is defined as:
- $\ds \map f t \star \map g t := \int_{-\infty}^\infty \map f u \map g {t + u} \rd u$
Also known as
A convolution integral is also generally known just as a convolution.
However, it has to be borne in mind that the term convolution in general has a wider definition.
Also see
- Results about convolution integrals can be found here.
Sources
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Frontispiece
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Chapter $4$: Notation for some useful Functions: Summary of special symbols: Table $4.1$ Special symbols
- 1978: Ronald N. Bracewell: The Fourier Transform and its Applications (2nd ed.) ... (previous) ... (next): Inside Back Cover