To be *causal*, the filter output at time
cannot
depend on the input at any times
greater than
. 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
will have entries that satisfy
for
.

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

and the input-output relationship is of course

or, more explicitly,

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
.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University