General Linear, Time-Invariant, Filter Matrix

The general linear, time-invariant matrix is Toeplitz. A Toeplitz matrix is constant along all its diagonals. For example, the general $3\times 3$ LTI matrix is given by

$$\mathbf{h}= \left[\begin{array}{ccc} h_{0} & h_{-1} & h_{-2}\\ [2pt] h_{1} & h_{0} & h_{-1}\\ [2pt] h_{2} & h_{1} & h_{0} \end{array}\right]$$

and restricting to causal LTI filters yields

$$\mathbf{h}= \left[\begin{array}{ccc} h_{0} & 0 & 0\\ [2pt] h_{1} & h_{0} & 0\\ [2pt] h_{2} & h_{1} & h_{0} \end{array}\right].$$

Note that the gain of the ``current input sample'' is now fixed at $h_0$ 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 $n=3$. 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 $n=3$ 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 $5\times 5$ banded Toeplitz filter matrix:

$$\left[\begin{array}{c} y_0 \\ [2pt] y_1 \\ [2pt] y_2 \\ [2pt] y_3... ...{array}{c} x_0 \\ [2pt] x_1 \\ [2pt] x_2 \\ [2pt] 0\\ [2pt] 0\end{array}\right]$$ (6.11)

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 $N_h$, then its order is $N_h-1$, so to avoid ``cutting off'' the output signal prematurely, we must append at least $N_h-1$ zeros to the input signal. Appending zeros in this way is often called zero padding, and it is used extensively in spectrum analysis [83]. 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 $n = 0$).

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:

$$\mathbf{h}= \left[\begin{array}{ccccccc} h_0 & 0 & 0 & 0 & 0 & 0... ...& \\ \vdots & \vdots & \vdots & & \ddots & \ddots & \ddots \end{array}\right]$$

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.