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460
Random Processes and Spectral Analysis Chap. 6
or
which is property 12.
Property 8 follows directly from property 12 by taking the inverse Fourier transform:
Making changes in the variables, let f 1 f f c in the first integral, and let f 1 = f f c in the
second integral. Then we have
R x (t) =
L
f c +B 0
x (f) = e [ v(f - f c ) + v ( f + f c )], | f | 6 B 0
0, otherwise
R x (t) = -1 [ x (f)]
=
L
B 0
-B 0
v (f - f c )e j2pft df +
L
B 0
f c -B 0
v (-f 1 ) e j2p(f c-f 1 )t df 1 +
L
f c +B 0
-B 0
v (f + f c )e j2pft df
f c -B 0
v (f 1 ) e j2p(f 1-f c )t df 1
But v (f) v (f), since v(t) is a real process. Furthermore, because v(t) is bandlimited, the
limits on the integrals may be changed to integrate over the interval (0, q). Thus,
q
R x (t) = 2 v (f 1 ) c ej2p(f 1-f c )t + e -j2p(f 1-f c )t
L
0
2
d df 1
which is identical to property 8.
In a similar way, properties 13 and 9 can be shown to be valid. Properties 10 and 14
follow directly from property 9.
For SSB processes, y(t) = ; xN(t). Property 15 is then obtained as follows:
; j[-xN(t)x(t + t) + [x(t)xN(t + t)]
(6–142)
Enploying the definition of a cross-correlation function and using property 10, we have
(6–143)
Furthermore, knowing that xN(t) is the convolution of x(t) with 1/(pt), we can demonstrate
(see Prob. 6–46) that
and
R gg (t) = g * (t)g(t + t)
= [x(t) < jxN(t)][x(t + t) ; j xN(t + t)]
= [x(t)x(t + t) + xN(t)xN(t + t)]
R xxN(t) = x(t)xN(t + t) =- R xx N (t) =-xN(t)x(t + t)
R xN (t) = R x (t)
R xxN(t) = RN xx (t)
(6–144)
(6–145)