Convolution is a mathematical operation of combining two functions
Definition of Convolution
For continuous functions, we have
For discrete signals, we have convolution defined as
Subtopics
- visual explanation of convolution
- convolution as smoothing
- convolution theorem - relation of convolution and Fourier-like transforms
Properties of Convolution
Convolution has several nice properties:
(commutativity) (associativity) (distributivity) (impulse convolution) - If
, then (time shift property) - If
, (time scaling property)
Applications
Convolutions has various applications. For example:
- multiplication of polynomials as convolution
- sum of probability distributions as convolution and the central limit theorem
- convolution can be used to express the output of a linear time-invariant system when zero-state response and input is given
- Convolution is often used to express filters such as the Gaussian filtering