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Sec. 6–6 The Gaussian Random Process 447
The N-dimensional Gaussian PDF can be written compactly by using matrix notation.
Let x be the column vector denoting the N random variables:
The N-dimensional Gaussian PDF is
where the mean vector is
f x (x) =
x 1 x(t 1 )
x 2 x(t 2 )
x = D T = D T
o o
x N x(t N )
1
(2p) N>2 |Det C| 1>2 e-(1/2)[(x-m)T C -1 (x-m)]
(6–107)
(6–108)
xq 1
m 1
xq 2 m 2
m = x = D T = D T
o o
xq N m N
(6–109)
and where (x - m) T denotes the transpose of the column vector (x - m).
Det C is the determinant of the matrix C, and C -1 is the inverse of the matrix C. The
covariance matrix is defined by
c 11 c 12
Á c 1N
where the elements of the matrix are
c
C = D 21 c 22
Á c 2N
T
o o o
c N1 c N2
Á c NN
c ij = (x i - m i )(x j - m j ) = [x(t i ) - m i ][x(t j ) - m j ]
(6–110)
(6–111)
For a wide-sense stationary process, m i = x(t i ) = m j = x(t j ) = m. The elements of
the covariance matrix become
c ij = R x (t j - t i ) - m 2
(6–112)
If, in addition, the x i happen to be uncorrelated, x i x j = xq i xq j for i j, and the covariance
matrix becomes
C = D
s 2 0 Á 0
0 s 2 Á 0
T
o ∞ o
0 0 Á s
2
(6–113)
where s 2 = x 2 - m 2 = R x (0) - m 2 . That is, the covariance matrix becomes a diagonal matrix
if the random variables are uncorrelated. Using Eq. (6–113) in Eq. (6–108), we can conclude that
the Gaussian random variables are independent when they are uncorrelated.