Definition of a Filter
Thus, a real digital filter maps every real, discrete-time signal to a real, discrete-time signal. A complex filter, on the other hand, may produce a complex output signal even when its input signal is real.
Definition. A real digital filteris defined as any real-valued function of a real signal for each integer
.
We may express the input-output relation of a digital filter by the notation
where
![$ x(\cdot)$](http://www.dsprelated.com/josimages_new/filters/img107.png)
![$ y(n)$](http://www.dsprelated.com/josimages_new/filters/img90.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
![$ x(\cdot)$](http://www.dsprelated.com/josimages_new/filters/img107.png)
![$ x$](http://www.dsprelated.com/josimages_new/filters/img101.png)
![$ {\cal T}_n\{x\}$](http://www.dsprelated.com/josimages_new/filters/img414.png)
![$ x$](http://www.dsprelated.com/josimages_new/filters/img101.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
![$ {\cal T}$](http://www.dsprelated.com/josimages_new/filters/img415.png)
![$ {\cal S}$](http://www.dsprelated.com/josimages_new/filters/img391.png)
![$ {\cal T}$](http://www.dsprelated.com/josimages_new/filters/img415.png)
![$ x\in{\cal S}$](http://www.dsprelated.com/josimages_new/filters/img401.png)
![$ y\in{\cal S}$](http://www.dsprelated.com/josimages_new/filters/img402.png)
![$ {\cal S}$](http://www.dsprelated.com/josimages_new/filters/img391.png)
![$ {\cal T}$](http://www.dsprelated.com/josimages_new/filters/img415.png)
![$ {\cal T}$](http://www.dsprelated.com/josimages_new/filters/img415.png)
![$ {\cal S}$](http://www.dsprelated.com/josimages_new/filters/img391.png)
![$ {\cal S}$](http://www.dsprelated.com/josimages_new/filters/img391.png)
![$ N$](http://www.dsprelated.com/josimages_new/filters/img278.png)
![$ {\cal S}\subset{\bf R}^N$](http://www.dsprelated.com/josimages_new/filters/img416.png)
![$ {\cal S}\subset{\bf C}^N$](http://www.dsprelated.com/josimages_new/filters/img417.png)
![$ {\cal T}$](http://www.dsprelated.com/josimages_new/filters/img415.png)
![$ N\times N$](http://www.dsprelated.com/josimages_new/filters/img418.png)
In this book, we are concerned primarily with single-input,
single-output (SISO) digital filters. For
this reason, the input and output signals of a digital filter are
defined as real or complex numbers for each time index (as opposed
to vectors). When both the input and output signals are
vector-valued, we have what is called a
multi-input, multi-out (MIMO) digital filter. We look at MIMO allpass filters in
§C.3 and MIMO state-space filter forms in Appendix G,
but we will not cover transfer-function analysis of MIMO filters using
matrix fraction descriptions [37].
Next Section:
Examples of Digital Filters
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Definition of a Signal