Random Processes - Random Signals and Processes Primer with MATLAB - page 442

functions. In order to apply a statistical description of a process, we need a
theoretically infinite number of realizations (see Sect. 6.2 ).
A.6.2. The word “stochastic” is a synonym for the word “random” and thus the
two can be used interchangeably.
A.6.3. The terms “random process” and “random signal” are sometimes used
interchangeably. However, a random process is a collection of all
realizations in ensemble. In contrast, a random signal is only one realiza-
tion of a process.
If a process is ergodic then characteristics of the process can be obtained
observing one realization.
For example, if a process is mean ergodic, then one can find the mean
value by observing one realization which is equivalent to a statistical mean
value obtained observing process (all realizations) in one point.
A.6.4. If a WS process does not have a periodic component, then

j R XX ðtÞ j<R XX ð 0 Þ: (6.262)

However, in order to include a periodic WS stationary process (see, for
example, Exercise 6.1 where the autocorrelation function periodically
takes the value R XX (0)), in the inequality ( 6.262 ) is added sign of the
equality.
A.6.5. Yes they can, as explained in Sect. 6.5.3 .
A.6.6. The opposite is not necessarily true. A WS stationary process is not
necessarily a second-order stationary. See, for example, [THO71, p. 211].
A.6.7. If an autocorrelation function does not have a constant term, it follows that
the mean value of the process is zero. However, if an autocorrelation
function has a constant term c , this term is equal

2

c ¼ EXðtfg : (6.263)

From here,

EXðtfg¼ c

p

: (6.264)

Therefore, in this case only the magnitude of E { X ( t )} is determined and the
sign cannot be determined.
A.6.8. Yes it can. Such a process is called white noise and is explained in Chap. 7 .
A.6.9. For a particular realization x ( t ), we have an instantaneous normalized
power measured using a 1 O

impedance

pðtÞ¼x 2 ðtÞ: (6.265)

This value varies from one realization to another. As such, it cannot be
representative of the process. A power averaged for all realizations is