Filters and Convolution

A reason for the importance of convolution (defined in §7.2.4) is that every linear time-invariant system^8.4can be represented by a convolution. Thus, in the convolution equation

we may interpret as the input signal to a filter, as the output signal, and as the digital filter, as shown in Fig.8.12.

The impulse or ``unit pulse'' signal is defined by

For example, for sequences of length , .

The impulse signal is the identity element under convolution, since

If we set in Eq.(8.1) above, we get

Thus, , which we introduced as the convolution representation of a filter, has been shown to be more specifically the impulse response of the filter.

It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response [65]. No matter what the LTI system is, we can feed it an impulse, record what comes out, call it , and implement the system by convolving the input signal with the impulse response . In other words, every LTI system has a convolution representation in terms of its impulse response.

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Subsections

• Frequency Response
• Amplitude Response
• Phase Response

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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.