Definition of a Filter

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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

is the output signal at time

. (We will also refer to $x(\cdot)$ as simply

.) The general filter is denoted by ${\cal T}_n\{x\}$ , which stands for any transformation from a signal

to a sample value at time

. 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

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 (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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written by 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.

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Introduction to Digital Filters
Linear Time-Invariant Digital Filters
Definition of a Filter