OPTIMALRESTORATIONOFA RANDOM SIGNALFROM ITS PROJECTIONINTOEUCLIDIANSPACE
Andrey V. Kovalenko, Vitaliy N. Kurashov · 2007
Theproblem ofstatistically optimal signal restoration fromitsarbitrary finite dimensional linear projection isconsidered. Thegeneral solution isobtained in theformofthesetoftheoptimal interpolators, whichdepend onsignal second order statistics andprojecting technique. Inmanybranches ofradio engineering andoptics there exists theproblem of signal restoration, wherethesignal isarandom process orafield. Initial dataforthe restoration areanobservation result, i.e. asetofN values, whichdefine thesignal mapontothespace RN.Mathematical relation between thesignal andits mapdepends onthemethodofobservation, whichcaninclude suchoperations asfiltering, heterodyning, hardware derivation, averaging andsampling. Whenitispossible, they trytouselinear observation channels, whichproduce theoutput projection ofthe desired signal. Theproblem ofoptimal restoration consists inconstructing the inverse operation: theexperimentally taken projection mustbeassociated withthe estimate, whichbelongs tothesamespace astheoriginal signal, andisthebestsignal approximation forsomecertain criterion. Properly chosen criterion ofoptimality should takeinto account themaximumofapriori knowledge about thesignal. Usually, theinformation ontherandomprocess isrestricted tothespecified correlation function. Thenthesignal canberestored fortheminimal variance criterion. Inthis paper wepresent thegeneral solution oftheproblem oftheoptimal signal restoration fromitsarbitrary finite dimensional linear projection. We also consider relation ofthis solution totheknownspecial cases. Lettheobservation ofthesignal 0(t), te(0; T)consists intaking thesetof values {un },n=1,N ,andisexpressed as un=Hn{0(t)}+r