Random Processes - Random Signals and Processes Primer with MATLAB - page 414

Finally, placing ( 6.226 ) into ( 6.223 ), and using ( 6.221 ) and ( 6.222 ), we arrive at:

h i

1
4

1 þ e 2 ljtj R XX ðtÞ ¼ : (6.227)

Exercise 6.15 Consider two random processes:

XðtÞ ¼ x sin o 0 t þ r cos o 0 t
YðtÞ ¼ x sin o 0 tr cos o 0 t :

(6.228)

where o 0 is a constant, and x and r are uncorrelated random variables:

Efxrg ¼ EfxgEfrg: (6.229)

The random variables x and r have means of zero,

Efxg ¼ Efrg ¼ 0 ; (6.230)

and equal variances

s x ¼ s r ¼ s 2 :

(6.231)

Determine whether the processes are jointly WS stationary.

Answer We must first determine whether both processes are WS stationary.

Both processes have a mean value of 0 and, therefore, the first condition for WS
is satisfied.
Next we investigate if both autocorrelation functions depend only on the time
difference t .
To this end we find:

R XX ðt; t þ tÞ ¼ EfXðtÞXðt þ tÞg
¼ E x sin o 0 t þ r cos o 0 t f ½ x sin o 0 ðt þ tÞþr cos o 0 ðt þ tÞ ½ g

¼ Efx 2 g sin o 0 t sin o 0 ðt þ tÞþEfr 2

g cos o 0 t cos o 0 ðt þ tÞ
þ Efxrg sin o 0 t cos o 0 ðt þ tÞþEfxrg cos o 0 t sin o 0 ðt þ tÞ:
(6.232)

From ( 6.230 ) and ( 6.231 ), we have:

Efx 2 g ¼ Efr 2 g ¼ s 2 : (6.233)

Using the conditions ( 6.229 ) and ( 6.230 ), we can write:

Efxrg ¼ 0 : (6.234)