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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 $ \mathbb{Z},\mathbb{R}$ , and $ \mathbb{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:\mathbb{Z}\rightarrow \mathbb{R}

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

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

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

i.e., $ w(n)\in\mathbb{C}, \forall n\in\mathbb{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\mathbb{C}$ , $ n\in\mathbb{Z}$ .

We may expand these definitions slightly to include functions of the form $ x(nT)$ , $ w(nT)$ , where $ T\in\mathbb{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 $ \mathbb{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\mathbb{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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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University