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General Causal Linear Filter Matrix

To be causal, the filter output at time $ n\in[0,N-1]$ cannot depend on the input at any times $ m$ greater than $ n$ . This implies that a causal filter matrix must be lower triangular. That is, it must have zeros above the main diagonal. Thus, a causal linear filter matrix $ \mathbf{h}$ will have entries that satisfy $ h_{mn}=0$ for $ n>m$ .

For example, the general $ 3\times 3$ causal, linear, digital-filter matrix operating on three-sample sequences is

$\displaystyle \mathbf{h}= \left[\begin{array}{ccc}
< & t & e\\ [2pt]
x & 2 & h\\ [2pt]
t & m & l
\end{array}\right]_comment_mark>
{h_{00}}{0}{0}%
{h_{10}}{h_{11}}{0}%
{h_{20}}{h_{21}}{h_{22}}
$

and the input-output relationship is of course

$\displaystyle \left[\begin{array}{c} y_0 \\ [2pt] y_1 \\ [2pt] y_2\end{array}\right] = \left[\begin{array}{ccc} < & t & e\\ [2pt] x & 2 & h\\ [2pt] t & m & l \end{array}\right]_comment_mark> {h_{00}}{0}{0}<tex2html_comment_mark> {h_{10}}{h_{11}}{0}<tex2html_comment_mark> {h_{20}}{h_{21}}{h_{22}} \left[\begin{array}{c} x_0 \\ [2pt] x_1 \\ [2pt] x_2\end{array}\right], \protect$ (F.1)

or, more explicitly,
$\displaystyle y_0$ $\displaystyle =$ $\displaystyle h_{00} x_0$  
$\displaystyle y_1$ $\displaystyle =$ $\displaystyle h_{10} x_0 + h_{11}x_1$  
$\displaystyle y_2$ $\displaystyle =$ $\displaystyle h_{20} x_0 + h_{21}x_1 + h_{22} x_2.
\protect$ (F.2)

While Eq.$ \,$ (F.2) covers the general case of linear, causal, digital filters operating on the space of three-sample sequences, it includes time varying filters, in general. For example, the gain of the ``current input sample'' changes over time as $ h_{00}, h_{11}, h_{22}$ .


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2016-01-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA