Random Processes - Random Signals and Processes Primer with MATLAB - page 396

P.4 In order to verify the property ( 6.120 ), we use the results ( 6.126 ) and ( 6.112 ).

From ( 6.120 ), we write:

1
2 R XX ð 0 ÞþR YY ð 0 Þ

1
2 2 s 2

jR XY ðtÞj ¼ s 2 ¼ s 2

j sin o 0 tj ½ ¼ (6.129)

resulting in ( 6.128 ), which is always true. Therefore, ( 6.120 ) is satisfied.

6.6.4 Cross-Covariance

A cross-covariance function for two processes X ( t ) and Y ( t ) is defined as:

C XY ðt; t þ tÞ¼EXðtÞEXðtfg f ½ Yðt þ tÞEYðt þ tÞ ½ f g g: (6.130)

This expression can be rewritten in the following form:

C XY ðt; t þ tÞ¼EXðtÞYðt þ tÞ f g EXðtf EYðt þ tÞ f g

EXðtf EYðt þ tÞ f g þEXðtf EYðt þ tÞ f g

¼ EXðtÞYðt þ tÞ f g EXðtf EYðt þ tÞ f g

¼ R XY ðt; t þ tÞEXðtf EYðt þ tÞ f g (6.131)

If processes are at least jointly WS stationary, then ( 6.131 ) reduces to:

C XY ðtÞ¼R XY ðtÞEfXgEfYg: (6.132)

If a cross-covariance is equal to zero,

C XY ðtÞ¼ 0 ; (6.133)

then processes X ( t ) and Y ( t ) are called uncorrelated . Using ( 6.131 ), the following
relation represents uncorrelated processes:

R XY ðt; t þ tÞ¼EXðtf EYðt þ tÞ f g: (6.134)

Similarly, if uncorrelated processes are at least jointly WS stationary, from
( 6.132 ) we have:

R XY ðtÞ¼EfXgEfYg: (6.135)

The same results ( 6.134 ) and ( 6.135 ) also stand for independent process. In other
words, independent processes are also uncorrelated . However, the opposite is not
necessarily true (except in the case of jointly Gaussian processes).