Random Processes - Random Signals and Processes Primer with MATLAB - page 443

called an average power and this quantity is thus representative of the
process. An average normalized power is an average power measured in
the 1 O impedance:

ðtÞ :

P av ¼ EX 2 (6.266)

Note that this value is related to an autocorrelation function for a WS
stationary process:

ðtÞ ¼ R XX ð 0 Þ:

P av ¼ EX 2 (6.267)

A.6.10. Considering that this random process is a measured voltage, and r is the
impedance in Ohms, then the average power is:

ðtÞ =r:

P av ; v ¼ EX 2 (6.268)

Similarly, if the random process is a measured current, and r is the
impedance in Ohms, then the average power is:

ðtÞ : P av ; c ¼ rE X 2

(6.269)

However, it is often desirable not to be concerned with whether a signal is a
voltage or a current [MIL04, p. 300]. Consequently, average normalized
power which is a power measured in a 1 O ( 6.267 ) is used rather than

average powers ( 6.268 )or( 6.269 ).
A.6.11. An autocorrelation does not provide a complete description of the process.
It describes a process only in two points. As such, it is possible that
different realizations result in a same autocorrelation function.
A.6.12. There is no such restriction for an autocorrelation function to be a nonneg-
ative (see, for example, the autocorrelation function in ( 6.160 ))
A.6.13. Two WS stationary processes are not necessarily jointly WS stationary
processes because the additional conditions ( 6.101 )and( 6.102 )mustbe
satisfied (see Exercise 6.15).
A.6.14. From ( 6.143 ), a time-averaged mean value cannot be a function of time but
only a constant. Therefore, if a statistical mean is not a constant cannot be
equal to a time-averaged mean value. Consequently, the process cannot be
mean ergodic.
A.6.15. From ( 6.151 ), a time-averaged autocorrelation function cannot be a func-
tion of time. Therefore, a statistical autocorrelation function which
depends on time t cannot be equal to a time-averaged autocorrelation
function and, consequently, a process cannot be autocorrelation ergodic.
A.6.16. It is not possible. There are different processes and thus different joint
PDFs which have equal autocorrelation functions. Therefore, knowledge
of an autocorrelation function is not equivalent to knowledge of a joint
PDF and is less informative.