Random Processes - Random Signals and Processes Primer with MATLAB - page 392

R XX ðtÞ¼EXðtÞXðt þ tÞ f g; R YY ðtÞ¼EYðtÞYðt þ tÞ f g: (6.100)

(b) The cross-correlation functions ( 6.92 ) and ( 6.93 ) depend only on the time
difference t :

R XY ðtÞ¼EXðtÞYðt þ tÞ f g (6.101)

or

R YX ðtÞ¼EYðtÞXðt þ tÞ f g: (6.102)

Example 6.6.1 Consider two random processes

XðtÞ¼x cos o 0 t þ r sin o 0 t;
YðtÞ¼r cos o 0 t x sin o 0 t;

(6.103)

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

Efxrg¼EfxgEfrg; (6.104)

with zero mean values;

Efxg¼Efrg¼ 0 ; (6.105)

and equal variances

s x ¼ s r ¼ s 2

: (6.106)

Determine whether the processes are jointly WS stationary.

Solution We first determine whether both processes are WS stationary.

Both processes have mean values of zero and, therefore, the first condition ( 6.99 )
is satisfied.
Next we will investigate whether both autocorrelation functions are only depen-
dent on a time difference t .
To this end we find:

R XX ðt; t þ tÞ¼EXðtÞXðt þ tÞ f g

¼ E x cos o 0 t þ r sin o 0 t ½ x cos o 0 ðt þ tÞþr sin o 0 ðt þ tÞ ½

f g

¼ Efx 2

g cos o 0 t cos o 0 ðt þ tÞþEfr 2

g sin o 0 t sin 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:

(6.107)