Random Processes - Random Signals and Processes Primer with MATLAB - page 394

P.2 In general, a cross-correlation function is neither even nor odd . The following
relation holds:

R XY ðtÞ ¼ R YX ðtÞ: (6.113)

From the definition of a cross-correlation function ( 6.94 )-( 6.95 ), we write

R XY ðtÞ ¼ E XðtÞYðttÞ

g : f (6.114)

Introducing

t 0 ¼ tt; (6.115)

into ( 6.114 ), we can write:

R XY ðtÞ ¼ E Yðt 0 ÞXðt 0 þ tÞ

f g ¼ R YX ðtÞ: (6.116)

P.3 This property establishes the upper limit for a cross correlation function as a
geometric mean of the corresponding autocorrelation functions at the origin:

j

p

j R XY ðtÞ R XX ð 0 ÞR YY ð 0 Þ : (6.117)

For any two random variables, we can write:

2

EfX 2 gEfY 2

½ EfXYg g: (6.118)

Then we have:

ðtÞ E Y 2

2

E X 2

½ E XðtÞYðt þ tÞ f g ðt þ tÞ (6.119)

resulting in the desired result ( 6.117 ).

P.4 This property establishes another upper limit for a cross-correlation function as
an arithmetic mean of the corresponding autocorrelation functions in the origin.

This is a stronger limit than that of ( 6.117 ) because a geometric mean of two
positive numbers cannot exceed an arithmetic mean of the same numbers:

1
2 R XX ð 0 ÞþR YY ð 0 Þ

j R XY ðtÞ j ½ :

(6.120)

To establish this property, we write:

2

¼ X 2 ðtÞþY 2 ½ XðtÞYðt þ tÞ ðtÞ 2 XðtÞYðt þ tÞ 0 : (6.121)