Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 446

However, a mean power P

T=

ð

2

W
T ¼ lim

1
T

x 2

P ¼ lim
T!1

ðtÞ d t (7.6)

T!1

T= 2

is finite. This conclusion suggests the application of the Fourier transform to a
power rather than to amplitude of a realization. This idea is developed below.
Consider the part of a realization x ( t ) of a process X ( t ) in the interval [ T /2, T /2],
as shown in Fig. 7.1 .

xðtÞ for T= 2 t T= 2 ;

x T ðtÞ¼ (7.7)

0 otherwise :

For the finite interval T , x T will satisfy the condition ( 7.4 ) and, thus, x T will have
the Fourier transform:

T=

ð

2

1

T=

ð

2

x T ðtÞ e jot d t ¼ x T ðtÞ e jot d t ¼ xðtÞ e jot d t: X T ðoÞ¼ (7.8)

1

T= 2 T= 2

The energy of the signal X T ( t ) is:

T= 2

ð 1

x T ðtÞ d t ¼ x T ðtÞ d t:

W T ¼ (7.9)

1

T= 2

Fig. 7.1 Realization x ( t )in
the interval [- T /2, T /2]