Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 307

Using the Bessel function ( 5.58 ), we get the PDF of the variable X as:

x 2 þ A 2

2 s 2

:

x xA

s 2 s 2 e

f X ðxÞ¼ I 0 (5.59)

The PDF ( 5.59 ) is said to be a Rician PDF .
The Rician PDFs for different values of A and s 2 ¼ 1 are shown in Fig. 5.7 .

For A ¼ 0, the Bessel function I 0 ¼ 1 and the Rician PDF becomes the Rayleigh
PDF, as shown in Fig. 5.7 . This is also obvious from ( 5.48 ) where the variable X 1 0
becomes X 1 .
For higher values of A , the Rician PDF becomes more symmetrical and it
approaches normal random variable.

5.4 Exponential Random Variable

5.4.1 Density and Distribution Function

The exponential density function is given as:

l e lx

for x 0 ;

f X ðxÞ¼ (5.60)

0 otherwise,

where l is a constant parameter.

RICIAN PDF

0.7

A=0

0.6

A=1

A=0.5

A=2

0.5

A=3

0.4

A=4

0.3

0.2

0.1

0 0

1 2 3 4 5

Range

6 7 8 9 10

Fig. 5.7 Rician PDFs for s 2 ¼ 1 and different values A