Matrix Filter Representations

It is illuminating to look at matrix representations of digital filters.^6.7Every linear digital filter can be expressed as a constant matrix $\mathbf{h}$ multiplying the input signal ${\underline{x}}$ (the input vector) to produce the output signal (vector) $\underline{y}$, i.e.,

$$\underline{y}= \mathbf{h}{\underline{x}}.$$

For simplicity (in this section only), we will restrict attention to finite-length inputs ${\underline{x}}^T = [x_0,\ldots,x_{N-1}]$ (to avoid infinite matrices), and the output signal will also be length $N$. Thus, the filter matrix $\mathbf{h}$ is a square $N\times N$ matrix, and the input/output signal vectors are $N\times 1$ column vectors.

More generally, any finite-order linear operator can be expressed as a matrix multiply. For example, the Discrete Fourier Transform (DFT) can be represented by the ``DFT matrix'' $[e^{-j2\pi kn/N}]$, where the column index $n$ and row index $k$ range from 0 to $N-1$ [83, p. 111].^6.8Even infinite-order linear operators are often thought of as matrices having infinite extent. In summary, if a digital filter is linear, it can be represented by a matrix.