##

Introduction

It is illuminating to look at *matrix representations* of digital
filters.^{F.1}Every *linear* digital filter can be expressed as a
*constant matrix*
multiplying the input signal
(the
*input vector*) to produce the output signal (vector)
, *i.e.*,

*finite-length*inputs (to avoid infinite matrices), and the output signal will also be length . Thus, the filter matrix is a square matrix, and the input/output signal vectors are 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''
, where the column index and row index range from 0
to [84, p. 111].^{F.2}Even 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