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Sec. 6–1 Some Basic Definitions 423
If the random processes x(t) and y(t) are jointly ergodic, the time average may be used
to replace the ensemble average. For correlation functions, this becomes
R xy (t) ! x(t)y(t + t) 8x(t)y(t + t)9
(6–29)
where
T/2
8[ #
1
]9 = lim [ # ] dt
(6–30)
T: q T L-T/2
when x(t) and y(t) are jointly ergodic. In this case, cross-correlation functions and autocorrelation
functions of voltage or current waveforms may be measured by using an electronic
circuit that consists of a delay line, a multiplier, and an integrator. The measurement technique
is illustrated in Fig. 6–3.
Complex Random Processes
In previous chapters, the complex envelope g(t) was found to be extremely useful in describing
bandpass waveforms. Bandpass random signals and noise can also be described in terms
of the complex envelope, where g(t) is a complex baseband random process.
DEFINITION. A complex random process is
g(t) ! x(t) + jy(t)
(6–31)
where x(t) and y(t) are real random processes and j = 2-1.
A complex process is strict-sense stationary if x(t) and y(t) are jointly strict-sense
stationary; that is,
f g (x(t 1 ), y(t œ 1), x(t 2 ), y(t 2 œ ), Á , x(t N ), y(t N œ ))
for any value of t 0 and any N : q.
= f g (x(t 1 + t 0 ), y(t 1 œ + t 0 ), Á , x(t N + t 0 ), y(t N
œ
+ t 0 ))
(6–32)
4-quadrant
multiplier
x (t)
Delay line
Delay =
Integrator
(low-pass filter)
≈ R xy ()
y (t)
(a) Cross-correlation
4-quadrant
multiplier
x (t)
Delay line
Delay =
Integrator
(low-pass filter)
≈ R x ()
(b) Autocorrelation
Figure 6–3
Measurement of correlation functions.