Random Processes - Random Signals and Processes Primer with MATLAB - page 408

or equivalently

1
2 EfX 2

R XX ðt; t þ tÞ¼ g

cos o 0 t þ y o 0 ðt þ tÞy ½ ð Þ þ cos o 0 t þ y þ o 0 ðt þ tÞþy ð Þ

1
2 EfX 2

¼ g cos ðo 0 tÞþ cos ð 2 o 0 t þ 2 y þ o 0 tÞ ½

1
2 EfX 2

¼ g cos ðo 0 tÞþ cos 2 ðo 0 t þ yÞþo 0 t ½ ð Þ :

(6.188)

The received result indicates that the autocorrelation function depends not only
on t but also on t . Therefore, the second condition is not satisfied and the process is
not a WS stationary.

Exercise 6.8 Verify and explain whether or not each of the following can be an
autocorrelation function.

2 t
1 þ t 2 :

gðtÞ¼

(a) (6.189)

2 t 2
1 þ t 2 :

(b) gðtÞ¼ (6.190)

(c) gðtÞ¼ sin o 0 t; o 0 ¼ constant : (6.191)

(d) gðtÞ¼ cos o 0 t; o 0 ¼ constant : (6.192)

Answer An autocorrelation function which is a function of a time difference t must
have the properties listed in Sect. 6.5.4 .

(a) This function cannot be an autocorrelation function because it does not have the
property ( 6.45 )

j gðtÞ j gð 0 Þ: (6.193)

> g (0).
Additionally, this function does not have the property ( 6.51 ) (i.e., it is not an
even function).
(b) This function satisfies the property ( 6.51 ), i.e., it is an even function. However,
it cannot be an autocorrelation function because it does not have the property
( 6.45 ). For example, g (0) ¼ 0, g (1) ¼ 1, and g (1)

For example, g (0) ¼ 0, g (2) ¼ 4/5; and thus g (2)

> g (0).
(c) This function cannot be an autocorrelation function because it is not an even
function.