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Sec. 6–7 Bandpass Processes 457
12. x (f) = y (f) = e [ v(f - f c ) + v (f + f c )], |f| 6 B 0
(6–133l)
0, f elsewhere
13. xy (f) = e j[ v(f - f c ) - v (f + f c )],
0,
|f| 6 B 0
f elsewhere
(6–133m)
14. xy (f) = - xy (-f) = - yx (f).
(6–133n)
Additional properties can be obtained when v(t) is a wide-sense stationary single-sideband
(SSB) random process. If v(t) is SSB about f =;f c , then, from Chapter 5, we have
g(t) = x(t) ; jxN(t)
(6–134)
where the upper sign is used for USSB and the lower sign is used for LSSB. xN(t) is the Hilbert
transform of x(t). Using Eq. (6–134), we obtain some additional properties:
15. When v(t) is an SSB process about f =;f c ,
R g (t) = 2[R x (t) ; jRN x (t)]
(6–135a)
where RN x (t) = [1>(pt)] * R x (t).
16. For USSB processes,
g (f) = e 4 x(f), f 7 0
0, f 6 0
(6–135b)
17. For LSSB processes,
g (f) = e 0, f 7 0
4 x (f), f 6 0
(6–135c)
From property 9, we see that if the PSD of v(t) is even about f = f c , f 7 0, then
R xy (t) 0 for all t. Consequently, x(t) and y(t) will be orthogonal processes when v ( f ) is
even about f = f c , f 7 0. Furthermore, if, in addition, v(t) is Gaussian, x(t) and y(t) will be
independent Gaussian random processes.
Example 6–11 SPECTRA FOR THE QUADRATURE COMPONENTS
OF WHITE BANDPASS NOISE
Assume that v(t) is an independent bandlimited white-noise process. Let the PSD of v(t) be N 0 2
over the frequency band f 1 | f | f 2 , as illustrated in Fig. 6–10a. Using property 12, we can
evaluate the PSD for x(t) and y(t). This is obtained by summing the translated spectra v ( f - f c )
and v ( f+f c ), illustrated in Fig. 6–10b and c to obtain x (f ) shown in Fig. 6–10d. Note that the
spectrum x ( f ) is zero for | f | 7 B 0 . Similarly, the cross-spectrum, xy (f ), can be obtained by
using property 13. This is shown in Fig. 6–10e. It is interesting to note that, over the frequency
range where the cross-spectrum is nonzero, it is completely imaginary, since v ( f ) is a real function.
In addition, the cross-spectrum is always an odd function.