Convolution

The convolution of two signals and in may be denoted `` '' and defined by

Note that this is circular convolution (or ``cyclic'' convolution).^7.4 The importance of convolution in linear systems theory is discussed in §8.3.

Cyclic convolution can be expressed in terms of previously defined operators as

where and . This expression suggests graphical convolution, discussed below in §7.2.4.

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Subsections

• Commutativity of Convolution
• Convolution as a Filtering Operation
• Convolution Example 1: Smoothing a Rectangular Pulse
• Convolution Example 2: ADSR Envelope
• Convolution Example 3: Matched Filtering
• Graphical Convolution
• Polynomial Multiplication
• Multiplication of Decimal Numbers

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Order a Hardcopy of Mathematics of the DFT

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.