Solution of Eigenvalue Integral Equation with Exponentially Oscillating Covariance Function

SUMMARY Karhunen-Loeve(KL)transformisoptimalformanysignaldetection,communicationandfilteringapplications.AnexplicitsolutionoftheKLintegralequationforapracticalcasewhenthecovariancefunctionofastationaryprocessisex-ponentiallyoscillatingisproposed. key words: Karhunen-Loeve transform, eigenvalue integralequation 1. IntroductionIntheoriesofsignaldetectionandfiltering[1]–[3],therefrequentlyoccurtheKLeigenvalueintegralequationortheFredholmequationofthefirsttypewhosekernelisthecovariancefunctionofthestationaryprocess, λϕ ( t )= T−T K ( |t−s| ) ϕ ( s ) ds, (1)where K ( x )representsthecovariancefunctionofacon-tinuousstationarysecondorderprocesspossessingacontinuousspectraldensity S ( w ), −T ≤ t≤ T .Denoteensembleaverageby E [.]andlet m ( t )= E [ n ( t,v )]and K ( t,s )= E [( n ( t,v ) −m ( t ))( n ( s,v ) −m ( s ))].Here K ( t , s )isthecovariancefunctionoftheprocess n ( t,v ). Ifthecovariancefunctioniscontinuousinsquare −T ≤t , s ≤ T , theprocessissecond-ordercontinuousin −T ≤ t ≤ T ,andifinaddition