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Definition of a Signal

Definition. A real discrete-time signal is defined as any time-ordered sequence of real numbers. Similarly, a complex discrete-time signal is any time-ordered sequence of complex numbers.
Mathematically, we typically denote a signal as a real- or complex-valued function of an integer, e.g., $ x(n)$, $ n=0,1,2,\ldots$. Thus, $ x(n)$ is the $ n$th real (or complex) number in the signal, and $ n$ represents time as an integer sample number.

Using the set notation $ {\bf Z},{\bf R}$, and $ {\bf C}$ to denote the set of all integers, real numbers, and complex numbers, respectively, we can express that $ x$ is a real, discrete-time signal by expressing it as a function mapping every integer (optionally in a restricted range) to a real number:

$\displaystyle x:{\bf Z}\rightarrow {\bf R}

Alternatively, we can write $ x(n)\in{\bf R}$ for all $ n\in{\bf Z}$.

Similarly, a discrete-time complex signal is a mapping from each integer to a complex number:

$\displaystyle w:{\bf Z}\rightarrow {\bf C}

i.e., $ w(n)\in{\bf C}, \forall n\in{\bf Z}$ ($ w(n)$ is a complex number for every integer $ n$).

It is useful to define $ {\cal S}$ as the signal space consisting of all complex signals $ x(n)\in{\bf C}$, $ n\in{\bf Z}$.

We may expand these definitions slightly to include functions of the form $ x(nT)$, $ w(nT)$, where $ T\in{\bf R}$ denotes the sampling interval in seconds. In this case, the time index has physical units of seconds, but it is isomorphic to the integers. For finite-duration signals, we may prepend and append zeros to extend its domain to all integers $ {\bf Z}$.

Mathematically, the set of all signals $ x$ can be regarded a vector space5.2 $ {\cal S}$ in which every signal $ x$ is a vector in the space ( $ x\in{\cal S}$). The $ n$th sample of $ x$, $ x(n)$, is regarded as the $ n$th vector coordinate. Since signals as we have defined them are infinitely long (being defined over all integers), the corresponding vector space $ {\cal S}$ is infinite-dimensional. Every vector space comes with a field of scalars which we may think of as constant gain factors that can be applied to any signal in the space. For purposes of this book, ``signal'' and ``vector'' mean the same thing, as do ``constant gain factor'' and ``scalar''. The signals and gain factors (vectors and scalars) may be either real or complex, as applications may require.

By definition, a vector space is closed under linear combinations. That is, given any two vectors $ x_1\in{\cal S}$ and $ x_2\in{\cal S}$, and any two scalars $ \alpha$ and $ \beta$, there exists a vector $ y\in{\cal S}$ which satisfies $ y = \alpha x_1 + \beta x_2$, i.e.,

$\displaystyle y(n) = \alpha x_1(n) + \beta x_2(n)

for all $ n\in{\bf Z}$.

A linear combination is what we might call a mix of two signals $ x_1$ and $ x_2$ using mixing gains $ \alpha$ and $ \beta$ ( $ y = \alpha x_1 + \beta x_2$). Thus, a signal mix is represented mathematically as a linear combination of vectors. Since signals in practice can overflow the available dynamic range, resulting in clipping (or ``wrap-around''), it is not normally true that the space of signals used in practice is closed under linear combinations (mixing). However, in floating-point numerical simulations, closure is true for most practical purposes.5.3

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Next: Definition of a Filter

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About the Author: 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 for details.


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