Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 447

According to the Parseval's theorem (see Appendix E ), the energy W T can be
expressed as:

1

1
2 p

2 d o:

W T ¼ j X T ðoÞ j (7.10)

1

The obtained expression is also the energy of a signal x ( t ) in the interval [ T /2,
T /2]. Consequently, the energy of the whole signal x ( t ) is obtained from ( 7.10 ),
when T !1 ,

W ¼ lim
T!1

W T : (7.11)

From ( 7.11 ), the power of the signal x ( t ) is:

W T
T :

P xx ¼ lim
T!1

(7.12)

Using ( 7.10 ), we finally arrive at:

2 3

1

1
T

1
2 p

4

2 d o

5 :

P xx ¼ lim
T!1

j X T ðoÞ j (7.13)

1

This expression can be rewritten as:

1

2

1
2 p

j X T ðoÞ j

P xx ¼ lim

T!1

d o: (7.14)

T

1

The left side of ( 7.14 ) is a power. Consequently, the expression in the integral,
lim
T!1

2

=T ,isa power spectral density (PSD) with a unit [Power/Hertz].
The PSD is denoted by S xx ( o )

j X T ðoÞ j

2

j X T ðoÞ j

S xx ðoÞ¼ lim
T!1

: (7.15)

T

Using ( 7.14 ) and ( 7.15 ), we have:

1

1
2 p

P xx ¼ S xx ðoÞ d o:

(7.16)

1

From ( 7.15 ), we can conclude that the obtained PSD exists and presents the
spectral content of a particular realization of a process X ( t ).