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) I IT qw +U{,) pg:r ( t)= cos U ( )t 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 = 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