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Sec. 6–1 Some Basic Definitions 415
v(t, E 1 )
t 1 t 2
t
v(t, E 2 )
Noise
source
v(t)
v(t, E 3 )
t 1 t 2
t
v(t, E 4 )
t 1 t 2
t
t 1
t 2
t
Figure 6–1
v 1 =v(t 1 )={v(t 1 , E i ), all i}
v 2 =v(t 2 )={v(t 2 , E i ), all i}
Random-noise source and some sample functions of the random-noise process.
6–1 SOME BASIC DEFINITIONS
Random Processes
DEFINITION. A real random process (or stochastic process) is an indexed set of real
functions of some parameter (usually time) that has certain statistical properties.
Consider voltage waveforms that might be emitted from a noise source. (See Fig. 6–1.)
One possible waveform is v(t, E 1 ). Another is v(t, E 2 ). In general, v(t, E i ) denotes the waveform
that is obtained when the event E i of the sample space occurs. v(t, E i ) is said to be a
sample function of the sample space. The set of all possible sample functions {v(t, E i )} is
called the ensemble and defines the random process v(t) that describes the noise source. That
is, the events {E i } are mapped into a set of time functions {v(t, E i )}. This collection of functions
is the random process v(t). When we observe the voltage waveform generated by the
noise source, we see one of the sample functions.
Sample functions can be obtained by observing simultaneously the outputs of many
identical noise sources. To obtain all of the sample functions in general, an infinite number of
noise sources would be required.
The definition of a random process may be compared with that of a random variable. A
random variable maps events into constants, whereas a random process maps events into
functions of the parameter t.