Wiley signals and systems e book TLFe BO 456

se Now that we have clefincd tlie statistiml description of iriptxt m t l o u i p i t procwsc~ that arisc by additioii 01 multiplir+ation of each other, we can consider the CO ymncling

For the cross spectrum the sarnr rule applics using t lie rcspcctiw po~ver spectra

These results also become particularly simple if both random proce- h5es 7 are uii- and a t least nnr has A mro mean 1 A c cross-corrclntion function pfcl( 7)

osspoxwr drnsiiy qict’trurn Q fi,(juj tlwn disapprar a n d we are Irft with:

(18.21) (18.22) All results in this scclion hold cctrrcsporiciirigly lor the cross-correlation funrtiou ancl the crosspower derisily spectrim I)etweeii g(1) and : / i t )

se

Now that we have clefincd tlie statistiml description of iriptxt m t l o u i p i t procwsc~ that arisc by additioii 01 multiplir+ation of each other, we can consider the CO ymncling rc+itionships tor input and out put signals of L7’1-hystenis A s dcscripti forms for LTI-sy5,fcms, we will clioosc tlie irrlpiilse rcsyon,.;e arid freqiiency respoiisc

No assumptions are rriade aLoii1 the iriner structure of the system Next it rrmst

be clarified wlietlicr the a stationary 01 ergodic iupril, signal brings al)out the same properties m the outpiit kignal 7’0 do t b h we fir& ( h i v e the conn

the cliffcwnt averages at the input, and oiitpitt of LTI-systems ixr detail

B8.2,I Stationaritgr an

We start wit11 an LTE-system as in Figure 18.3 and considcr if the input, process

is st:ttionary 01 an ergoclic random proccss, then does the outpiit process also

roperties? If that is the ciLq(A we can also use the correlatiou fitriction drrtsit,y dcscriptioii that was introducwl in Chapter 17 on the output

er [be condiLion of weak stationarily

If the inpiit process is stationary t h c ~ i the ;acond-order expected va11at.s do not

(’lliLwLgC2 when thc input, signal i.; shiftcd by time At (compare (1 7.12)):

E{d(.;v(tl), c ( t a ) ) } = E{ f ( r ( t 1 i- At), .r(l2 + A t ) ) } f 18.23) Beca,iisc the svstrxri time-invariant, for tlrr output sigiial y ( t j -= A5’{x(fj),

y ( t L 4- At) = s { z ( t z 4 At)}

E M y ( f L ) , d t a ) ) ) = E {dV(tl + At,,’y(t2 + A t ) ) } >

(18.24)

(18.25) holds and from (15.23) we obtain