Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 452

P.1 The power density is a real function.
From ( 7.30 ), we have:

1

R XX ðtÞ e jot d t; S XX ðoÞ¼

1

1

¼ R XX ðtÞð cos ot j sin otÞ d t;

1

1 1

¼ R XX ðtÞ cos ot d t j R XX ðtÞ sin ot d t: ð 7 : 33 Þ

1 1

The second integral in ( 7.33 ) is zero, because the integral over a symmetric
range of the product of R XX ( t ) (which is an even function) and sin ot (which is an
odd function) is zero.
Therefore, we get:

1

S XX ðoÞ¼ R XX ðtÞ cos ot d t; (7.34)

1

which is a real function.

P.2 A PSD is an even function in a radian frequency :

o

S XX ðoÞ¼S XX ðoÞ: (7.35)

From ( 7.34 ), we have:

1 1

S XX ðoÞ¼ R XX ðtÞ cos ðotÞ d t ¼ R XX ðtÞ cos ot d t ¼ S XX ðoÞ: (7.36)

1 1

P.3 The average power of a random process is obtained as the area below the PSD,
as shown in Fig. 7.3 .

1 1

ðtÞ ¼ P ¼

1
2 p

EX 2 S XX ðoÞ d o ¼ S XX ðf Þ d f ; (7.37)

1 1

where o ¼ 2 pf and f is a frequency in Hz.
From ( 7.31 ), we have:

1 1

1
2 p

S XX ðoÞ e jo 0 d o ¼ R XX ð 0 Þ¼EXðtÞXðt þ 0 Þ f g ¼ S XX ðf Þ d f : (7.38)

1 1