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Random Processes and Spectral Analysis Chap. 6
But e j2u c
= 0 and R g (t) = g*(t)g(t + t), since g(t) is assumed to be wide-sense stationary.
Thus,
R v (t, t + t) = 1 2 Re{R g(t) e jv ct }
(6–131)
The right-hand side of Eq. (6–131) is not a function of t, so R v (t, t + t) = R v (t). Consequently,
Eq. (6–130) gives a model for a wide-sense stationary bandpass process.
Furthermore, for this model as described by Eq. (6–130), properties 1 through 5 of
Eq. (6–133a) through Eq. (6–133e) are valid, but all the properties 6 through 14
[Eqs. (6–133f) through (6–133h)] are not valid for the x(t) and y(t) components of v(t) = x(t)
cos (v c t + u c ) - y(t) sin (v c t + u c ), unless all the conditions of Eq. (6–129) are satisfied.
However, as will be proved later, the detected x(t) and y(t) components at the output of quadrature
product detectors (see Fig. 6–11) satisfy properties 6 through 14, provided that the
start-up phase of the detectors, u 0 , is independent of v(t). [Note that the x(t) and y(t) components
associated with v(t) at the input to the detector are not identical to the x(t) and y(t) quadrature
output waveforms unless u c = u 0 ; however, the PSDs may be identical.]
The complex-envelope representation of Eq. (6–130) is quite useful for evaluating the
output of detector circuits. For example, if v(t) is a signal-plus-noise process that is applied to
a product detector, x(t) is the output process if the reference is 2 cos (v c t + u c ) and y(t) is the
output process if the reference is -2 sin(v c t + u c ). (See Chapter 4.) Similarly, R(t) is the output
process for an envelope detector, and u(t) is the output process for a phase detector.
Properties of WSS Bandpass Processes
Theorems giving the relationships between the autocorrelation functions and the PSD of v(t),
g(t), x(t), and y(t) can be obtained. These and other theorems are listed subsequently as properties
of bandpass random processes. The relationships assume that the bandpass process v(t)
is real and WSS. † The bandpass nature of v(t) is described mathematically with the aid of
Fig. 6–10a, in which
v (f) = 0 for f 2 6 ƒ fƒ 6 f 1
(6–132)
where 0 6 f 1 f c f 2 . Furthermore, a positive constant B 0 is defined such that B 0 is the
largest frequency interval between f c and either band edge, as illustrated in Fig. 6–10a, and
B 0 6 f c .
The properties are as follows:
1. g(t) is a complex wide-sense-stationary baseband process.
(6–133a)
2. x(t) and y(t) are real jointly wide-sense stationary baseband processes. (6–133b)
3. R v (t) = 1 2 Re{R g(t)e jvct }.
(6–133c)
ƒ ƒ
4. v (f) = 1 4 [ g(f - f c ) + g (-f - f c )].
(6–133d)
5. v 2 = 1 2 g(t) 2 = R v (0) = 1 2 R g(0).
(6–133e)
† v(t) also has zero mean value, because it is a bandpass process.