General LTI Filter Matrix

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

(F.3) |

We could add more rows to obtain more output samples, but the additional outputs would all be zero.

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

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