446 18 Hurzcloln Sigiials and LTI-Systems Llze corivoliition integral is irisertcd (18.44) then jntertahnnging the sequence of integration and expected va1iir.s yields 111 the expectcd value we again recogniw the auto-correlation function of t hc input signal ~ ( t ) : 4Sd4 =/ (18.46) fwPrJ(T+P:dP LL By substituting 71 = /A we finally rcach the desired expressrioii ibr the c'rosscorrela\,ion function of the input and output of nn LTI-system a i d i t s inipulsc- response: (PTI,(T) = / h" ( - v ) pr5(z - a,) dv = 11" (- r) * p, , (z) (18.47) U Tlz~fornirila obtained for thc cross-corrclntion function pg,( r ) betmmm the input and outpiit, is easier t,o rcmcmber To derive it from (18.47) we fi~stn i a k ~ iise of t,hc synimetry relatioriships for correlation fiinctions from Chapter 17.4.2 Froin (17.54) in cwnbimtion with (18.46) we obtain (18.38) The cwrijngate syininetry of the anto-torrrlation function ( 17.58) yields (18.49) Both possible cross-correlation functions between the input mid outpit of tm EH-system arc casily obtained by convolution of the auto- correltition fiirictiori at, the input with the imprtlse response h( r) m c l h* (- z): (18.50) (18.51) (18.52) I I