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Random Processes and Spectral Analysis Chap. 6
where t = t 2 - t 1 .
Some properties of cross-correlation functions of jointly stationary real processes are
and
1. R xy (-t) = R yx (t)
2. ƒ R xy (t) ƒ … 3R x (0)R y (0)
3. ƒ R xy (t) ƒ … 1 2 [R x(0) + R y (0)]
(6–20)
(6–21)
(6–22)
The first property follows directly from the definition, Eq. (6–19). Property 2 follows from the
fact that
[x(t) + Ky(t + t)] 2 Ú 0
(6–23)
for any real constant K. Expanding Eq. (6–23), we obtain an equation that is a quadratic in K:
[R y (0)]K 2 + [2R xy (t)]K + [R x (0)] Ú 0
(6–24)
For K to be real, it can be shown that the discriminant of Eq. (6–24) has to be nonpositive. †
That is,
[2R xy (t)] 2 - 4[R y (0)][R x (0)] … 0
(6–25)
This is equivalent to property 2, as described by Eq. (6–21). Property 3 follows directly from
Eq. (6–24), where K =;1. Furthermore,
ƒ R xy (t) ƒ … 3R x (0)R y (0) … 1 (6–26)
2 [R x(0) + R y (0)]
because the geometric mean of two positive numbers R x (0) and R y (0) does not exceed their
arithmetic mean.
Note that the cross-correlation function of two random processes x(t) and y(t) is a generalization
of the concept of the joint mean of two random variables defined by Eq. (B–91).
Here, x 1 is replaced by x(t), and x 2 is replaced by y(t t). Thus, two random processes x(t)
and y(t) are said to be uncorrelated if
R xy (t) = [x(t)][y(t + t)] = m x m y
(6–27)
for all values of t. Similarly, two random processes x(t) and y(t) are said to be orthogonal
if
R xy (t) = 0
(6–28)
for all values of t.
As mentioned previously, if y(t) = x(t), the cross-correlation function becomes an
autocorrelation function. In this sense, the autocorrelation function is a special case of the crosscorrelation
function. Of course, when y(t) K x(t), all the properties of the cross-correlation
function reduce to those of the autocorrelation function.
† The parameter K is a root of this quadratic only when the quadratic is equal to zero. When the quadratic is
positive, the roots have to be complex.