563489578934

Sec. 6–6 The Gaussian Random Process 449
or
y Hx
(6–115)
where the elements of the N × N matrix H are related to the impulse response of the linear
network by
h ij = [h(t i - l j )] ¢l
(6–116)
We will now show that y is described by an N-dimensional Gaussian PDF when x is described
by an N-dimensional Gaussian PDF. This may be accomplished by using the theory for a multivariate
functional transformation, as given in Appendix B. From Eq. (B–99), the PDF of y is
The Jacobian is
f y (y) = f x(x)
|J(y/x)|
`
x = H -1 y
dy 1 dy 1
Á dy 1
dx 1 dx 2 dx N
Ja y dy 2 dy 2
Á dy 2
x b = Det G dx 1 dx 2 dx N W = Det [H] ! K
o o o
dy N dy N
Á dy N
dx 1 dx 2 dx N
(6–117)
(6–118)
where K is a constant. In this problem, J (y/x) is a constant (not a function of x) because
y = Hx is a linear transformation. Thus,
f y (y) = 1
|K| f x(H -1 y)
or
1
f y (y) =
(6–119)
(2p) N/2 1/2
|K| |Det C x |
e-(1/2) [(H-1 y -m x ) T -1
C x (H -1 y-m x )]
where the subscript x has been appended to the quantities that are associated with x(t). But we
know that m y = Hm x , and from matrix theory, we have the property [AB] T = B T A T , so that
the exponent of Eq. (6–119) becomes
where
or
- 1 2 [(y - m y) T (H -1 ) T ]C x -1 [ H -1 (y - m y )] = - 1 2 [(y - m y) T C ȳ 1 (y - m y )]
C ȳ 1 = (H -1 ) T C x -1 H -1
C y = HC x H T
(6–120)
(6–121)
(6–122)