Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 468

Example 7.3.1 Consider the random processes from Example 6.6.1, where the
cross-correlation function was given as:

R XY ¼s 2 sin o 0 t; (7.87)

where o 0 is a particular deterministic frequency. Find the cross-spectral density and
demonstrate its properties.

Solution From ( 7.75 ), the cross-PSD is:

S XY ðoÞ¼s 2 1
1

1

sin o 0 t e jot d t ¼ s 2
2 j

e jot d t;

e jo 0 t e jo 0 t

1

1

d t

s 2
2 j

e ðjo 0 þjoÞt e ðjo 0 þjoÞt

¼

1

1

d t;

¼ j s 2
2

e ðjo 0 þjoÞt þ e ðjo 0 þjoÞt

1

¼ j s 2

2 dðo o 0 Þj s 2 2 dðo þ o 0 Þ:

(7.88)

P.1. From ( 7.88 ), the cross-spectral density is imaginary, i.e., it is not a real function.

P.2. To verify the property in ( 7.79 ), we find a cross-spectral power density S YX ( o ).
Using ( 7.80 ) and ( 7.87 ), we get the cross-correlation function:

R YX ðtÞ¼s 2 sin o 0 t: (7.89)

From ( 7.77 ), we get:

S YX ðoÞ¼s 2 1
1

1

s 2
2 j

e jot d t;

sin o 0 t e jot d t ¼ e jo 0 t e jo 0 t

1

1

d t

s 2
2 j

e ðjo 0 þjoÞt þ e ðjo 0 þjoÞt

¼

1

1

d t;

¼ j s 2
2

þ e ðjo 0 þjoÞt e ðjo 0 þjoÞt

1

¼j s 2

2 dðo o 0 Þþj s 2 2 dðo þ o 0 Þ:

(7.90)