Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 451

Solution Using ( 7.30 ) and ( 6.77 ), we write:

1

U 2 e 2 ljtj e jot d t;

S XX ðoÞ¼

1

2
4

3
5 ;

ð 1

0

¼ U 2 e ð 2 ljoÞt d t þ e ð 2 lþjoÞt d t

1 0

2 l

¼ 2 U 2

þ o 2 :

ð 7 : 32 Þ

2

ð 2 lÞ

The spectral density is shown in Fig. 7.2 for l ¼ 1 and U ¼ 1.
As expected, one may note from Fig. 7.2 , that the spectrum band becomes wider
for higher values of l , i.e. average number of zero crossing in a unit of time.

7.1.4 Properties of PSD

The PSD has a number of useful properties. Some of them are presented below.
We will consider only a real process which is at least WS stationary.

1

U 2 / l

0.9

0.8

0.7

0.6

U 2 /2 l

0.5

0.4

0.3

0.2

- 2 l 2 l

0.1

0
- 20 - 15 - 10 - 5 0 5 10 15 20

w

Fig. 7.2 Power spectral density (PSD) in Example 7.1.1