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:

$$x:\mathbb{Z}\rightarrow \mathbb{R}$$

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:

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

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 space^5.2 ${\cal S}$ 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.,

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

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