##

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