Definition of a Signal
Mathematically, we typically denote a signal as a real- or complex-valued function of an integer, e.g.,
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.
![$ x(n)$](http://www.dsprelated.com/josimages_new/filters/img88.png)
![$ n=0,1,2,\ldots$](http://www.dsprelated.com/josimages_new/filters/img382.png)
![$ x(n)$](http://www.dsprelated.com/josimages_new/filters/img88.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
Using the set notation
, and
to denote
the set of all integers, real numbers, and complex numbers,
respectively, we can express that
is a real, discrete-time signal
by expressing it as a function mapping every integer (optionally in
a restricted range) to a real number:
![$\displaystyle x:{\bf Z}\rightarrow {\bf R}
$](http://www.dsprelated.com/josimages_new/filters/img385.png)
![$ x(n)\in{\bf R}$](http://www.dsprelated.com/josimages_new/filters/img386.png)
![$ n\in{\bf Z}$](http://www.dsprelated.com/josimages_new/filters/img387.png)
Similarly, a discrete-time complex signal is a mapping from each integer to a complex number:
![$\displaystyle w:{\bf Z}\rightarrow {\bf C}
$](http://www.dsprelated.com/josimages_new/filters/img388.png)
![$ w(n)\in{\bf C}, \forall n\in{\bf Z}$](http://www.dsprelated.com/josimages_new/filters/img389.png)
![$ w(n)$](http://www.dsprelated.com/josimages_new/filters/img390.png)
![$ n$](http://www.dsprelated.com/josimages_new/filters/img89.png)
It is useful to define as the signal space consisting
of all complex signals
,
.
We may expand these definitions slightly to include functions of the
form ,
, where
denotes the sampling
interval in seconds. In this case, the time index has physical units
of seconds, but it is isomorphic to the integers. For finite-duration
signals, we may prepend and append zeros to extend its domain to all
integers
.
Mathematically, the set of all signals can be regarded a
vector space5.2
in
which every signal
is a vector in the space (
). The
th sample of
,
, is regarded as the
th vector
coordinate. Since signals as we have defined them are infinitely
long (being defined over all integers), the corresponding vector space
is infinite-dimensional. Every vector space comes with
a field of scalars which we may think of as constant gain
factors that can be applied to any signal in the space. For purposes
of this book, ``signal'' and ``vector'' mean the same thing, as do
``constant gain factor'' and ``scalar''. The signals and gain factors
(vectors and scalars) may be either real or complex, as applications
may require.
By definition, a vector space is closed under linear
combinations. That is, given any two vectors
and
, and any two scalars
and
, there exists a
vector
which satisfies
, i.e.,
![$\displaystyle y(n) = \alpha x_1(n) + \beta x_2(n)
$](http://www.dsprelated.com/josimages_new/filters/img409.png)
![$ n\in{\bf Z}$](http://www.dsprelated.com/josimages_new/filters/img387.png)
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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Definition of a Filter
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Summary