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Filters and Convolution

A reason for the importance of convolution (defined in §7.2.4) is that every linear time-invariant system8.7can be represented by a convolution. Thus, in the convolution equation

$\displaystyle y = h\ast x \protect$ (8.1)

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

Figure 8.12: The filter interpretation of convolution.

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

$\displaystyle \delta(n) \isdef \left\{\begin{array}{ll}
1, & n=0 \\ [5pt]
0, & n\neq 0. \\
\end{array} \right.

For example, for sequences of length $ N=4$ , $ \delta = [1,0,0,0]$ .

The impulse signal is the identity element under convolution, since

$\displaystyle (x\ast \delta)_n \isdef \sum_{m=0}^{N-1}x(m) \delta(n-m) = x(n).

If we set $ x=\delta$ in Eq.(8.1) above, we get

$\displaystyle y = h\ast \delta = h.

Thus, $ h$ , 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 [71]. No matter what the LTI system is, we can feed it an impulse, record what comes out, call it $ h(n)$ , and implement the system by convolving the input signal $ x$ with the impulse response $ h$ . In other words, every LTI system has a convolution representation in terms of its impulse response.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University