563489578934

Sec. 6–7 Bandpass Processes 459
Proofs of Some Properties
Giving proofs for all 17 properties listed previously would be a long task. We therefore
present detailed proofs only for some of them. Proofs that involve similar mathematics will be
left to the reader as exercise problems.
Properties 1 through 3 have already been proven in the discussion preceding
Eq. (6–131). Property 4 follows readily from property 3 by taking the Fourier transform of
Eq. (6–133c). The mathematics is identical to that used in obtaining Eq. (4–25). Property
5 also follows directly from property 3. Property 6 will be shown subsequently. Property 7
follows from properties 3 and 8. As we will see later, properties 8 through 11 follow from
properties 12 and 13.
Properties 6 and 12 are obtained with the aid of Fig. 6–11. That is, as shown by
Eq. (4–77) in Sec. 4–13, x(t) and y(t) can be recovered by using product detectors. Thus,
x(t) = [2v(t) cos(v c t + u 0 )] * h(t)
(6–139)
and
y(t) = -[2v(t) sin(v c t + u 0 )] * h(t)
(6–140)
where h(t) is the impulse response of an ideal LPF that is bandlimited to B 0 hertz and u 0 is an
independent random variable that is uniformly distributed over (0, 2p) and corresponds to
the random start-up phase of a phase-incoherent receiver oscillator. Property 6 follows
from Eq. (6–139) by taking the ensemble average since cos (v c t + u 0 ) = 0 and
sin (v c t + u 0 ) = 0. Property 12, which is the PSD for x(t), can be evaluated by first evaluating
the autocorrelation for w 1 (t) of Fig. 6–11:
w 1 (t) = 2v(t) cos(v c t + u 0 )
R w1 (t) = w 1 (t)w 1 (t + t)
= 4v(t)v(t + t) cos(v c t + u 0 ) cos(v c (t + t) + u 0 )
But u 0 is an independent random variable; hence, using a trigonometric identity, we have
R w1 (t) = 4v(t)v(t + t) C 1 2 cos v ct + 1 2 cos(2v ct) + v c t + 2u 0 )D
However,
cos(2v c t + v c t + 2u 0 ) = 0, so
R w1 (t) = 2R v (t) cos v c t
(6–141)
The PSD of w 1 (t) is obtained by taking the Fourier transform of Eq. (6–141).
or
w1 (f) = 2 v (f) * C 1 2 d(f - f c) + 1 2 d(f + f c)D
w1 (f) = v (f - f c ) + v (f + f c )
Finally, the PSD of x(t) is
x (f) = ƒ H(f) ƒ 2 w1 (f)