Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 188

where an unique transformation of the input variables ( 3.189 ) is defined as:

Y 1 ¼ g 1 ðX 1 ; ... ; X N Þ
Y 2 ¼ g 2 ðX 1 ; ... ; X N Þ
.
Y N ¼ g N ðX 1 ; ... ; X N Þ

(3.190)

Similarly as in ( 3.157 ), we have:

d y N
Jðx 1 ; ... ; x N Þ ;

d y 1 ; ... ;

d x 1 ; ... ; d x N ¼ (3.191)

where J is the Jacobian of the transformation ( 3.190 ), defined as:

@g 1
@x 1

@g 1
@x N

:::

. . .

Jðx 1 ; ... ; x N Þ¼ : (3.192)

@g N
@x 1

@g N
@x N

:::

Similarly, like ( 3.159 ), using ( 3.190 )-( 3.192 ), we get:

f X 1 X 2 ; ... ; X N ð x 1 ; x 2 ; ... ; x N Þ
Jðx 1 ; x 2 ; ... ; x N Þ

f Y 1 Y 2 ; ... ;Y N ðy 1 ; y 2 ; ... ; y N Þ¼ : (3.193)

x 1 ¼g 1

1 ðy 1 ;y 2 ; ... ;y N Þ
x 2 ¼g 1

j j

2 ðy 1 ;y 2 ; ... ;y N Þ
.

x N ¼g 1

N ðy 1 ;y 2 ; ... ;y N Þ

3.5 Characteristic Functions

3.5.1 Definition

The joint characteristic function of two random variables X 1 and X 2 , denoted as
f X 1 X 2 ðo 1 ; o 2 Þ , is defined as the expected value of the complex function

e jðo 1 X 1 þo 2 X 2 Þ ,

n o

: f X 1 X 2 ðo 1 ; o 2 Þ¼E e jðo 1 X 1 þo 2 X 2 Þ (3.194)

According to ( 3.87 ), the expression ( 3.194 ) is equal to:

1 1

e jðo 1 x 1 þo 2 x 2 Þ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : f X 1 X 2 ðo 1 ; o 2 Þ¼ (3.195)

1 1

The obtained expression ( 3.195 ) can be interpreted with the exception of
the exponent sign, as a two-dimensional Fourier transform of the joint PDF