## 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

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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