General Causal Linear Filter Matrix

To be causal, the filter output at time $n\in[0,N-1]$ cannot depend on the input at any times $m$ greater than $n$ . This implies that a causal filter matrix must be lower triangular. That is, it must have zeros above the main diagonal. Thus, a causal linear filter matrix $\mathbf{h}$ will have entries that satisfy $h_{mn}=0$ for $n>m$ .

For example, the general $3\times 3$ causal, linear, digital-filter matrix operating on three-sample sequences is

$$\mathbf{h}= \left[\begin{array}{ccc} h_{00} & 0 & 0\\ [2pt] h_{10} & h_{11} & 0\\ [2pt] h_{20} & h_{21} & h_{22} \end{array}\right]$$

and the input-output relationship is of course

$$\left[\begin{array}{c} y_0 \\ [2pt] y_1 \\ [2pt] y_2\end{array}\right] = \left[\begin{array}{ccc} h_{00} & 0 & 0\\ [2pt] h_{10} & h_{11} & 0\\ [2pt] h_{20} & h_{21} & h_{22} \end{array} \right] \left[\begin{array}{c} x_0 \\ [2pt] x_1 \\ [2pt] x_2\end{array}\right], \protect$$ (F.1)

or, more explicitly,

$$y_0 = h_{00} x_0$$

$$y_1 = h_{10} x_0 + h_{11}x_1$$

$$y_2 = h_{20} x_0 + h_{21}x_1 + h_{22} x_2. \protect$$ (F.2)

While Eq.(F.2) covers the general case of linear, causal, digital filters operating on the space of three-sample sequences, it includes time varying filters, in general. For example, the gain of the ``current input sample'' changes over time as $h_{00}, h_{11}, h_{22}$ .