## 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., , . 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:

Alternatively, we can write for all .

Similarly, a discrete-time 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 space5.2 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.,

for all .

A linear combination is what we might call a 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

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