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
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 . 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:
Cyclic Convolution Matrix
General Causal Linear Filter Matrix