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will be WSS if and only if
and
Random Processes and Spectral Analysis Chap. 6
1. x(t) = y(t) = 0
(6–129b)
2. R x (t) = R y (t)
(6–129c)
3. R xy (t) = -R yx (t)
(6–129d)
Proof. The requirements for WSS are that v(t) be constant and R v (t, t + t) be a
function only of t. We see that v(t) = x(t) cos v c t - y(t) sin v c t is a constant for any value
of t only if x(t) = y(t) = 0. Thus, condition 1 is required.
The conditions required to make R v (t, t + t) only a function of t are found as follows:
R v (t, t + t)
or
= v(t)v(t + t)
= [x(t) cos v c t - y(t) sin v c t][x(t + t) cos v c (t + t) - y(t + t) sin v c (t + t)]
= x(t)x(t + t) cos v c t cos v c (t + t) - x(t)y(t + t) cos v c t sin v c (t + t)
- y(t)x(t + t) sin v c t cos v c (t + t) + y(t)y(t + t) sin v c t sin v c (t + t)
R v (t, t + t) = R x (t) cos v c t cos v c (t + t) - R xy (t) cos v c t sin v c (t + t)
- R yx (t) sin v c t cos v c (t + t) + R y (t) sin v c t sin v c (t + t)
When we use trigonometric identities for products of sines and cosines, this equation reduces to
R v (t, t + t) = 1 2 [R x(t) + R y (t)] cos v c t + 1 2 [R x(t) - R y (t)] cos v c (2t + t)
- 1 2 [R xy(t) - R xy (t)] sin v c t - 1 2 [R xy(t) + R xy (t)] sin v c (2t + t)
The autocorrelation for v(t) can be made to be a function of t only if the terms involving t are
set equal to zero. That is, [R x (t) - R y (t)] = 0 and [R xy (t) + R yx (t)] = 0. Thus, conditions 2 and
3 are required.
If conditions 1 through 3 of Eq. (6–129) are satisfied so that v(t) is WSS, properties 1
through 5 of Eqs. (6–133a) through (6–133e) are valid. Furthermore, the x(t) and y(t) components
of v(t) = x(t) cos w c t – y(t) sin w c t satisfy properties 6 through 14, as described by
Eqs. (6–133f) through (6–133n), when conditions 1 through 3 are satisfied. These properties are
highly useful in analyzing the random processes at various points in communication systems.
For a given bandpass waveform, the description of the complex envelope g(t) is not
unique. This is easily seen in Eq. (6–126), where the choice of the value for the parameter f c
is left to our discretion. Consequently, in the representation of a given bandpass waveform
v(t), the frequency components that are present in the corresponding complex envelope g(t)
depend on the value of f c that is chosen in the model. Moreover, in representing random
processes, one is often interested in having a representation for a WSS bandpass process with