derivative of mutual information zero snr the gaussian-noise case噪声情况下零信噪比互信息导数.pdfVIP

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.57,NO.11,NOVEMBER20117307

DerivativeofMutualInformationatZeroSNR:

TheGaussian-NoiseCase

YihongWu,StudentMember,IEEE,DongningGuo,Member,IEEE,andSergioVerdú,Fellow,IEEE

—AssumingadditiveGaussiannoise,ageneralsuffi-ofPrelov’swork[9,Lemma2.2],whichrequireistenceof

cientconditionontheinputdistributionisestablishedtoguaranteeinputmomentsofthorderforsome.Furthermore,a

thattheratioofmutualinformationtosignal-to-noiseratio(SNR)simplifiedargumentwasgivenin[7]withoutdelvingintothe

goestoonehalfnatasSNRvanishes.TheresultallowsSNR-de-

pendentinputdistributionandsideinformation.technicalitiesrequiredtojustifytheinterchangeofexpectation

andlimit.

IndexTerms—Gaussiannoise,low-powerregime,minimumThegoalofthispaperistopresentarigorous,self-contained

mean-squareerror(MMSE),mutualinformation,signal-to-noise

ratio(SNR).proofoftheasymptoticresultallowingamoregeneralsetup

thanthoseconsideredbefore,inwhichtheinputdistributioncan

dependontheSNR,andanSNR-dependentsideinformation

I.INTRODUCTIONmaybeavailabletothereceiver.Unliketheproofin[10]for

[7,Lemma1],theproo