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It is illuminating to look at matrix representations of digital filters.F.1Every 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.,

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

For simplicity (in this appendix 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$ [84, p. 111].F.2Even 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.

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General Causal Linear Filter Matrix
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Analog Allpass Filters