General Causal Linear Filter Matrix

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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About the Author: 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.