Convolution Example 3: Matched Filtering

Figure 7.5 illustrates convolution of

to get

For example, could be a ``rectangularly windowed signal, zero-padded by a factor of 2,'' where the signal happened to be dc (all s). For the convolution, we need

which is the same as . When , we say that is a matched filter for .^7.7 In this case, is matched to look for a ``dc component,'' and also zero-padded by a factor of . The zero-padding serves to simulate acyclic convolution using circular convolution. Note from Eq.(7.3) that the maximum is obtained in the convolution output at time 0. This peak (the largest possible if all input signals are limited to in magnitude), indicates the matched filter has ``found'' the dc signal starting at time 0. This peak would persist in the presence of some amount of noise and/or interference from other signals. Thus, matched filtering is useful for detecting known signals in the presence of noise and/or interference [32].

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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.