Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 305

5.3 Rician Random Variable

5.3.1 Relation of Rician, Rayleigh, and Normal Variables

Consider two independent normal random variables X 1 and X 2 with equal mean
values and variances, as defined in ( 5.37 ). Additionally, imagine that the constant A
is added to the variable X 1 , resulting in the normal variable X 1 0
X 1 0 ¼ X 1 þ A:

(5.48)

The mean value of the variable X 1 0 is equal to A .
EfX 1 0 g¼A:

(5.49)

The joint density function is given as:

2

2 ps 2 e ð x 1 0 A Þ

x 2
2 s 2

1

e

f X 1 0 X 2 ðx 1 0 ; x 2 Þ¼ f X 1 0 ðx 1 0 Þf X 2 ðx 2 Þ¼

2 s 2

(5.50)

2

2 ps 2 e ð x 1 0 A Þ

x 0 1 þ x 2 2 x 1 0 A þ A 2
2 s 2

þ x 2

1 1

2 ps 2 e

2 s 2

¼ ¼ :

Next we introduce the polar coordinates, as shown in Fig. 5.6 .
The transformation of X 1 and X 2 into polar coordinates X and y is defined as:

p

x 0 1 þ x 2

x ¼ ;

: y ¼ tan 1 x 2

x 1 0

(5.51)

Fig. 5.6 Transformation
of X 1 0 and X 2 to polar
coordinates X and y