Introduction

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

For simplicity (in this appendix only), we will restrict attention to 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.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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.