On The Duality BetweenRateAnd Power
Rajesh Sundaresan · 2007
Sequence design problems areconsidered inthis paper. Theproblem ofsum powerminimization ina spread spectrum system canbereduced totheproblem ofsumcapacity maximization, andviceversa. A solution tooneoftheproblems yields asolution totheother. Subsequently, conceptually simple sequence design algorithms knowntoholdforthewhite-noise caseareextended tothecolored noise case. Thealgorithms yield anupperboundof2N -L onthenumberofsequences where N istheprocessing gainandL thenumberofnon-interfering subsets ofusers. Ifsomeusers(atmostN -1)areallowed to signal alongalimited numberofmultiple dimensions, thenN orthogonal sequences suffice. I.INTRODUCTION Consider asymbol-synchronous code-division multiple ac- cess(CDMA)system. Thekthuserisassigned anN-sequence SkC RN ofunitenergy, i.e., SkSk = 1.Theprocessing gain isN chips, andthenumberofusers isK.Userkmodulates thevector skbyitsdatasymbol XkCR andtransmits XkSk overN chips. Thistransmission interferes withother users' transmissions andiscorrupted bynoise. Thereceived signal ismodeled by K Y Z SkXk+ Z, k=1 whereZ isa zero-mean Gaussian randomvector witha covariance matrix E.We willconsider thefollowing two sequence design problems. ProblemI:Userkhasapowerconstraint Pkunits perchip, i.e., E(X2) < Npk.Thegoalthenistoassign sequences and datarates touserssothat thesumoftheindividual rates at whichtheuserscantransmit datareliably (inanasymptotic sense) ismaximized. Themaximumvalue CSLLm iscalled the sumcapacity. ProblemII, adual toProblem I,isonewhereuserkdemands reliable transmission ataminimumraterkbits/chip. Thegoal istoassign sequences andpowers tousers sothat despite their mutual interference andnoise, eachoftheusers cantransmit reliably atorgreater thantheir required rates, andthesum ofthereceived powers(energy/chip) atthebase-station is minimized. Viswanath andAnantharam (1) havesolved Problem Iand provided anexplicit characterization forthesumcapacity. Guess(2)hassolved Problem IIfortheparticular casewhen