Wiley signals and systems e book TLFe BO 442

17.5. Power Derrsity Spectra

427

cannot be given because of the delta impulse. In the best case we ctm form

as a deterministic desc*riptionof the rancloiii process. Forming t hc cxpected values
first in thc time-domain and then transforining thein yield? the power densitv
spect r i m
7r

@:z;:,:(jO)
= -6(0 - 0 0)

2

I

IT

4--qw
2

+U{,)

pg:r

( t)= cos U ( )t
2

as the Fourier transform of the ACF. It i s similar to E:{IX,(p)I} and likewise

indimtm that the random signal oiily contailis frccpeiicy tompoiimts at &WO.

-

The mean square of a xandorii process c m also be calcrrlated direct,ly from the
power density spectrum. First, the auttrwrrelstion function is expressed as tlic
i n ~ ~ r Fourier
se
transform of the power density spec%rum:

A s the mean square is equal to the value of the auto-correlation fimctiou at
== 0, we can obtain the relalioiiship between the power tfeiisity spectrum arid
mean square by putting z = 0 into (17.71):
z‘

The iiiean square is therefore equal to the integral of the power deusity sprctruni, multiplied by a factor 1/2n.
area is equal to the
mean square of the
signal (“power“)

Figuie 17.9: Thc arm uadcr the power dcusity q)ec.ti.um
proportional to thr meati square of the crignnl

@,,(JOJ)

or a signal r ( t ) is