
38
2 Mathematical Preliminaries
Let this system have a single real solution z
l9
z
2
, ...,z„; then it can be shown
that
p
01
,...,e„(yi>y2>--->y*)
=
Pft
/
w
(*l>*2,
··-,*„)
\J(z
l9
z
29
...
9
z
n
)\
where J
9
called the Jacobian of transformation (80), is given by
J(z
i9
z
2
,...,z„)
dz
x
3z
2
dz„
^ζ
λ
dz
2
(91)
(92)
If Eqs. (90) have more than one solution, then we add in the right-hand
side (91) the corresponding expressions resulting from all the solutions. For
a proof of (91), see, for example, Papoulis [9].
2.4 RANDOM FIELDS
2.4.1 Definition of a Random Field
In Section 2.3.4, we discussed sets of random variables over the set Ω.
We will now generalize this discussio ...