## Definition of a Signal

Mathematically, we typically denote a signal as a real- or complex-valued function of an integer,

Definition.Areal discrete-time signalis defined as any time-ordered sequence of real numbers. Similarly, acomplex discrete-time signalis any time-ordered sequence of complex numbers.

*e.g.*, , . Thus, is the th real (or complex) number in the signal, and represents time as an integer

*sample number*. Using the

*set notation*, and to denote the set of all integers, real numbers, and complex numbers, respectively, we can express that is a real, discrete-time signal by expressing it as a function mapping every integer (optionally in a restricted range) to a real number:

*complex*signal is a mapping from each integer to a complex number:

*i.e.*, ( is a complex number for every integer ). It is useful to define as the

*signal space*consisting of all complex signals , . We may expand these definitions slightly to include functions of the form , , where 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 . Mathematically, the set of all signals can be regarded a

*vector space*

^{5.2}in which every signal is a vector in the space ( ). The th sample of , , is regarded as the th

*vector coordinate*. Since signals as we have defined them are infinitely long (being defined over all integers), the corresponding vector space 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 and , and any two scalars and , there exists a vector which satisfies ,

*i.e.*,

*mix*of two signals and using mixing gains and ( ). 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}

**Next Section:**

Definition of a Filter

**Previous Section:**

Summary