Convolution

Convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. It is a fundamental tool in mathematics, physics, and engineering.

Definition

In mathematics, the convolution of two functions f and g is defined as the integral of the product of the two functions after one is reversed and shifted. The integral is evaluated for all possible values of shift, producing the convolution function.

Properties

Convolution has several important properties. It is commutative, meaning that the order of the two functions does not matter. It is also associative, which means that when convolving three functions, it does not matter which pair is convolved first. Furthermore, convolution is distributive, so it can be distributed over addition of functions.

Applications

Convolution has wide applications in various fields. In physics, it describes the distribution of a quantity that is spread or smeared out. In engineering, it is used in signal processing to analyze and manipulate signals. In computer science, it is used in image and audio processing, among other applications.

See also

• Fourier transform
• Laplace transform
• Z-transform
• Signal processing
• Image processing

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  Comparison of convolution and correlation

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  Convolution illustration

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  Convolution of box signal with itself

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  Convolution of spiky function with box

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  2D Convolution Animation

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  Halftone, Gaussian Blur