18 Random Sinrials and LTI-Systems 44-0 aiid obtain for the power density s p e ~ ~ , ri@vv ui~ fjd), ~ the expression j @Y,@) = @'ff.(./") + =W@,,Ol.l>) -t-@,,(P) (18.15) which is obviously real in addition, the pomr density spectrim G g T j ( pis) m8de 111) of the spwtra of the &pills f ft) and q(t) wn td of their ~ i ~ t crosspower u x ~ density syectritm The results so far hold for the addition of any two riklltlom processes J ( $ ) atid g ( t ) Possible corrclatiou hetwwn the ru~idoxriprocesses i s takm into amount by the cross-correlation function pfg( r ) and crosspower density sp These relsi tionships become much simpler i f both raridorrr processes are iincortdatcct >~ndat least oire fia a XQPO iwan Then, fyorn (l'T.S.5) we obtain (@&) = 0 @p.t,{(j.i)= o (18.16) and the rclations (18.12) and (18.J.5) simplify to PYUb3 (18 17) = L7IS@) -I- P,W Q'ff 43,, (ju) Assiirnirig that the processes arc iincorieldteti holds in niariy c a m wliere g( t ) reprcsents EI signal interfering with the useful sigrial f ( t ) k:xamplm arr aniplificr noise independent of the input signal CIF atitmospheric effects 011 ratlio transmission that is 1ilce.cviscindependent of the t r ~ ~ i i s ~signal i i ~ t ~As, ~this ~ kind of interference aJso usudly has x zero mean, the simplc relationships (18.17) and (18.18) also hold T h ~ ysay that when two uiicorrelated renclom processes are added, snrl as long a ~ at j least one has a zero Irieiin the auto-eoruelation fiirtctioiis and power density spectra itre alqo added together to form the respective fiinctions tor the complete signal model