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.

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:

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.*,

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University