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Complex Digital Signal Processing in Telecommunications

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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:

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.*,

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}

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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