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.,
![$\displaystyle \underline{y}= \mathbf{h}{\underline{x}}.
$](http://www.dsprelated.com/josimages_new/filters/img1955.png)
![$ {\underline{x}}^T = [x_0,\ldots,x_{N-1}]$](http://www.dsprelated.com/josimages_new/filters/img1956.png)
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
![$ \mathbf{h}$](http://www.dsprelated.com/josimages_new/filters/img1952.png)
![$ N\times N$](http://www.dsprelated.com/josimages_new/filters/img418.png)
![$ N\times 1$](http://www.dsprelated.com/josimages_new/filters/img1639.png)
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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General Causal Linear Filter Matrix
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Analog Allpass Filters