5DigitalCommunicationThroughBand-LimitedChannels

南京理工大学电光学院通信工程系NanjingUniversityofScienceandTechnologyDepartmentofCommunicationEngineeringDigitalCommunicationsHanYuBingTel:84303169Office:TheEOEBuilding,RoomA311DigitalCommunicationsHanYuBing2DigitalCommunicationofChannelsofLinearFilteringInpreviouschapters,weconsideredthetransmissionofdigitalinformationthroughAWGNchannel.Ineffect,nobandwidthconstraintwasimposedonthesignaldesignandthecommunicationsystemdesign.Inthischapter,weconsidertheproblemofsignaldesignwhenthechannelisband-limitedtosomespecifiedbandwidthofWHz.SignalDesignforBand-LimitedChannelDataDetectionandProbabilityofErrorforControlledISIOptimumReceiverforChannelswithISIandAWGNEqualizerDigitalCommunicationsHanYuBing3SignalDesignforBand-LimitedChannelTransmittedsignalLinearfilterchannelIdealchannelornon-distortingchannel2()Re[()]Cjftstvteπ=])(Re[2)(2tfjCetcthπ=WffC=,0)(WfefCfCfj=,)()()(θ)()(0tttc−=αδ02)(ftjefCπα−⋅=⎪⎩⎪⎨⎧⇒≤≠=⇒≤≠distortiondelay||,)(21(f)distortionamplitude||,)(WfconstdffdWfconstfCθπτTheenvelopedelaycharacteristicDigitalCommunicationsHanYuBing4SignalDesignforBand-LimitedChannelReceivedsignalTheformoftransmittedsignal,g(t)istransmittedshapingfilter，Inisinformationsequence。Linearfilterchannelresponseofthetransmittedsignal()()nnvtIgtnT−∑＝WffG=,0)(()()()()lrtvtctzt=∗+()[()]CfFct=WffC=,0)()()()()(tnthtstr+∗=])(Re[)(2tfjlCetrtrπ=()()()()lrtvtctzt=∗+DigitalCommunicationsHanYuBing5SignalDesignforBand-LimitedChannelGeneralprocessingmodelofband-limiteddigitalcommunicationsystemz(t)iszero-mean，ThepowerspectraldensityfunctioniscomplexGaussianrandomprocessfortheN0{In}istransmittedcomplexinformationsequenceg(t)transmittedshapingfilter)(tg发送成形滤波器)(tc线性滤波器)(tgR接收滤波器)(tz0tkTt+=kykI()()nnvtIgtnT−∑＝transmittingfilterchannelimpulsereceivingfilterDigitalCommunicationsHanYuBing6SignalDesignforBand-LimitedChannelreceivedsignalOutputofreceivefilter)()()(tznTthItrnnl+−∑＝∫∞∞−−=∗=τττdtcgtctgth)()()()()(()()()()lrtvtctzt=∗+)()()(tnTtxItynnυ+−∑＝∫∞∞−−=∗=τττυdtgztgtztRR)()()()()(∫∞∞−−=∗=τττdtghtgthtxRR)()()()()(DigitalCommunicationsHanYuBing7SignalDesignforBand-LimitedChannelSamplevalueτ0isthetransmissiondelayx0issignalgainscalefactor，setx0=1DigitalCommunicationsHanYuBing8DesignofBand-LimitedSignalsforNoIntersymbolInterference(1)desiredinformationsymbol;(2)inter-symbolinterference(ISI);(3)additiveGaussiannoiseObjective:designx(t)，maketheoutputsignalsequencenoISIk1+k2+k1−k2−k0x1x2x1−x2−x1−kI2−kI1+kI2+kIknkkkxIyυ+=∑00,,01)(≠=⎩⎨⎧==ΔkkxkTxkDigitalCommunicationsHanYuBing9NyquistcriterionforzeroISI：ProofTfTTmfXnnnTxm21||,)(00,,01)(≤=+⇔≠=⎩⎨⎧=∑∫∫∑∑∫∑∫∫−−−+−∞∞−=+=+===TTfnTjTTfnTjmmTTfnTjmTTmTTmfnTjfnTjdfefBdfeTmfXdfeTmfXdfefXdfefXnTx2/12/122/12/122/12/122/1/2/1/22)()]([)()()()(πππππ∑+=mTmfXfB)()(B(f)hasperiodof1/TTTTiXiπωπω≤=+∑)2(DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing10Fouriertransform∫−=TTfnTjdfefBnTx2/12/12)()(π()exp{2}nnBfbjfnTπ∞=−∞=∑∫−−=TTfnTjndfefBTb2/12/12)(π)(nTTxbn−=⎩⎨⎧≠==00,,0nnTbnTTmfXm=+∑)(00,,01)(≠=⎩⎨⎧=nnnTxTfB=)(DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing11SupposeCase1:whenT1/2Wor1/T2W,thereisnochoiceforZ(f)=TinthiscaseandthereisnowaythatwecandesignasystemwithnoISI()0,CffW=()0,XffW=()()()()TRXfGfCfGf=TfTTmfXfZm21)()(≤=+=∑DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing12Case2:whenT=1/2Wor1/T=2W,theNyquistrate,itisclearthatinthiscasethereexistonlyoneZ(f)thatresultsinZ(f)=T.()0TfWXf⎧=⎨⎩其它()()sinctXfxtT⎛⎞⇔=⎜⎟⎝⎠TfTTmfXm21)(≤=+∑DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing13Tworemarkszisnoncausal,and,therefore,nonrealizable.Tomakeitrealizable,usuallyadelayedversionofit,t0ischosensuchthatfort0,wehave,sowecanregarditascausalandrealizable.zThetailsofdecayas1/t,consequently,asmallmistimingerrorinsamplingtheoutputofthematchedfilteratthedemodulatorresultsinaninfiniteseriesofISIcomponents.Suchaseriesisnotabsolutelysummablebecauseofthe1/trateofdecayofthepulse,and,hence,thesumoftheresultingISIdoesnotconverge()sinctxtT⎛⎞=⎜⎟⎝⎠()0sinc0ttT−⎛⎞≈⎜⎟⎝⎠()nnmamnT≠⋅Δ−+Δ∑()sinctxtT⎛⎞=⎜⎟⎝⎠DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing14Case3:whenT1/2Wor1/T2W,thereexistnumerouschoicesforX(f)suchthatZ(f)=T.Fromtheabove,wecanrealizethatSymbolrateRBcannotbegreaterthanthetwiceofchannelbandwidth2W，Themaximumbitrateofthechanneloftransmissionsymbol2BRW=b/HzTfTTmfXm21)(≤=+∑DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing15Aparticularpulsespectrum,thathasdesirablespectralpropertiesandhasbeenwidelyusedinpracticeistheraisedcosinespectrum.ThefrequencycharacteristicWhere0≤α≤1iscalledtheroll-offfactor，thepulsex(t),havingraisedcosinespectrum,is()(),0121111cos,222210,2rcTfTTTXfffTTTfTαπααααα⎧⎪≤≤−⎪⎡−⎤−+⎪⎛⎞=+−≤≤⎨⎜⎟⎢⎥⎝⎠⎣⎦⎪⎪+⎪⎩()()()()222222cossin12cossinc12tTtTxttTtTtTtTtTπαππαπαα=⋅−=⋅−DesignofBand-LimitedSignalsforNoIntersymbolInterferenceDigitalCommunicationsHanYuBing16bandwidth12WTα+=SymbolsperHz()12121TTαα=++Thetailsofthesig