## 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

*banded Toeplitz filter matrix*:

(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:

*linear operators*[56]. Thus, we may say that every LTI filter corresponds to a

*Toeplitz linear operator*.

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General Causal Linear Filter Matrix