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.,
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.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.