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