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Definition of a Filter

Definition. A real digital filter $ {\cal T}_n$ is defined as any real-valued function of a real signal for each integer $ n\in{\bf Z}$.
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.

We may express the input-output relation of a digital filter by the notation

$\displaystyle y(n)={\cal T}_n\{x(\cdot)\} \protect$ (5.1)

where $ x(\cdot)$ denotes the entire input signal, and $ y(n)$ is the output signal at time $ n$. (We will also refer to $ x(\cdot)$ as simply $ x$.) The general filter is denoted by $ {\cal T}_n\{x\}$, which stands for any transformation from a signal $ x$ to a sample value at time $ n$. The filter $ {\cal T}$ can also be called an operator on the space of signals $ {\cal S}$. The operator $ {\cal T}$ maps every signal $ x\in{\cal S}$ to some new signal $ y\in{\cal S}$. (For simplicity, we take $ {\cal S}$ to be the space of complex signals whenever $ {\cal T}$ is complex.) If $ {\cal T}$ is linear, it can be called a linear operator on $ {\cal S}$. If, additionally, the signal space $ {\cal S}$ consists only of finite-length signals, all $ N$ samples long, i.e., $ {\cal S}\subset{\bf R}^N$ or $ {\cal S}\subset{\bf C}^N$, then every linear filter $ {\cal T}$ may be called a linear transformation, which is representable by constant $ N\times N$ matrix.

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 $ n$ (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].

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Examples of Digital Filters
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Definition of a Signal