Causal Recursive Filters

Equation (5.1) does not cover all LTI filters, for it represents only causal LTI filters. A filter is said to be causal when its output does not depend on any ``future'' inputs. (In more colorful terms, a filter is causal if it does not ``laugh'' before it is ``tickled.'') For example, $y(n) = x(n + 1)$ is a non-causal filter because the output anticipates the input one sample into the future. Restriction to causal filters is quite natural when the filter operates in real time. Many digital filters, on the other hand, are implemented on a computer where time is artificially represented by an array index. Thus, noncausal filters present no difficulty in such an ``off-line'' situation. It happens that the analysis for noncausal filters is pretty much the same as that for causal filters, so we can easily relax this restriction.