The general linear, time-invariant (LTI) matrix is Toeplitz. A Toeplitz matrix is constant along all its diagonals. For example, the general LTI matrix is given by
and restricting to causal LTI filters yields
Note that the gain of the ``current input sample'' is now fixed at for all time. Also note that we can handle only length 3 FIR filters in this representation, and that the output signal is ``cut off'' at time . The cut-off time is one sample after the filter is fully ``engaged'' by the input signal (all filter coefficients see data). Even if the input signal is zero at time and beyond, the filter should be allowed to ``ring'' for another two samples. We can accommodate this by appending two zeros to the input and going with a banded Toeplitz filter matrix:
(F.3) |
In general, if a causal FIR filter is length , then its order is , so to avoid ``cutting off'' the output signal prematurely, we must append at least zeros to the input signal. Appending zeros in this way is often called zero padding, and it is used extensively in spectrum analysis [84]. As a specific example, an order 5 causal FIR filter (length 6) requires 5 samples of zero-padding on the input signal to avoid output truncation.
If the FIR filter is noncausal, then zero-padding is needed before the input signal in order not to ``cut off'' the ``pre-ring'' of the filter (the response before time ).
To handle arbitrary-length input signals, keeping the filter length at 3 (an order 2 FIR filter), we may simply use a longer banded Toeplitz filter matrix:
A complete matrix representation of an LTI digital filter (allowing for infinitely long input/output signals) requires an infinite Toeplitx matrix, as indicated above. Instead of working with infinite matrices, however, it is more customary to speak in terms of linear operators [56]. Thus, we may say that every LTI filter corresponds to a Toeplitz linear operator.