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 .
General LTI Filter Matrix