Convolution - Wikipedia

Abstraction: Integral operation computing weighted overlap of two functions at all shifts

Key points:

• Defined as (f * g)(t) = integral of f(τ)g(t−τ) dτ; equivalent to cross-correlation with one function reflected
• Satisfies commutative, associative, and distributive properties; L1 functions under convolution form a commutative associative Banach algebra
• Convolution theorem: Fourier transform of convolution equals pointwise product of Fourier transforms; enables O(N log N) fast convolution via FFT
• LTI (linear time-invariant) systems are characterized by convolution: every bounded translation-invariant linear operator on Lp is a convolution with a tempered distribution
• Discrete circular convolution diagonalized by DFT matrix; convolution operators on cyclic groups represented by circulant matrices
• Generalizes to locally compact groups via Haar measure; on compact groups, Peter-Weyl theorem provides analogous spectral decomposition

Connections: Convolution · Fourier Analysis · Signal Processing

Source: http://en.wikipedia.org/wiki/Convolution