## 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