Random Processes - Random Signals and Processes Primer with MATLAB - page 387

• The probability that more than one zero crossing in an infinitesimal interval
appear is infinitely small compared to the probability of only one zero crossing.

• If a zero crossing occurs in the interval t , then there is an equal probability of its

occurring in any of the infinitesimal intervals d t , in which the interval t is
divided.

Find the autocorrelation function and demonstrate its properties.
Solution According to the aforementioned properties of the given process, we can
conclude that the process is a Poisson process (see Sect. 5.7 ).

Therefore, the number of the zero crossing in a given interval t is described by
the Poisson formula:

k

Pfk zero crossing in tg¼ ð lt Þ

e lt

; (6.71)

k !

where l is an average number of zero crossing in a unit of time.
Random variables in time instants t 1 ¼ t and t 2 ¼t + t , are denoted as X 1 and X 2 ,
respectively, as shown in Fig. 6.9 . The autocorrelation function is:

R XX ðtÞ¼X 1 X 2 ¼ X

X

2 2

x 1 i x 2 j PfðX 1 ¼ x 1 i Þ\ðX 2 ¼ x 2 j Þ ; tg: (6.72)

i¼ 1 j¼ 1

The joint probability from ( 6.72 ) can be expressed using the conditional proba-
bility, as shown in the following equation:

PfðX 1 ¼ x 1 i Þ\ðX 2 ¼ x 2 j Þ ; t; t þ tg
¼ PfX 1 ¼ x 1 i ; tgPfX 2 ¼ x 2 j ; t þ tjX 1 ¼ x 1 i ; tg:

(6.73)

The random variables X 1 and X 2 have an equal probability of taking the values
U and U :

PfX 1 ¼ Ug¼PfX 1 ¼Ug¼PfX 2 ¼ Ug¼PfX 2 ¼Ug¼ 1 = 2 : (6.74)

We find the conditional probabilities by taking into account the following
considerations.
The amplitudes in time instants t and t + t , will be the same if there is either no
zero crossing or an even number of zero crossing in the time interval t :

PfX 2 ¼ UjX 1 ¼ U ; tg¼PfX 2 ¼UjX 1 ¼U ; tg
¼ 1
k¼ 0

(6.75)

P X ðk ; tÞ;

; 2 ; 4 ; ...