為了完全利用多天線系統(multiple-input multiple-output system)所能提供的增益，可以在通訊系統的傳送端加入預編碼(precoding)技術，預偏碼技術可以針對不同的通道響應設計出相對應的編碼方式，當傳送端擁有完整的通道資訊時，便能作出最好的預先偏碼。但在正交分頻多工系統(Orthogonal frequency-division multiplexing system)下的所使用的多天線系統，傳送端無法擁有完整的通道資訊，所以接收端需要額外回饋通道資訊給傳送端。由於頻寬是固定的，所以當回傳頻帶變大時，傳送資料的頻寬就會被相對應的壓縮。
在這篇論文中，我們討論了目前最常被使用來應用在有限回饋預先編碼(limited-feedback precoding)的碼書(codebook)技術。並藉由利用通道的時域關係性以及頻域關係性，設計了差分預先編碼技術；差分編碼技術突破了原本在使用碼書技術時，一組碼書只能擁有有限預編碼的性質，使得效能得以提升。在頻域上，使用了分群回饋的內差技術，使得整個統可以再更有效的利用回傳頻帶。
結合差分預先編碼以及在頻帶上的分群回饋，我們設計出一個低複雜度並應用在有限回饋預先編碼系統的一個編碼技術，藉由利用了時域及頻域的相關性，這個編碼技術可以達到比現存的編碼系統更好的效能。

1Introduction1
2SystemModel4
2.1MIMOTimeCorrelatedChannelModel....................4
2.1.1SISOChannelModel...........................5
2.1.2MIMOChannelModel..........................6
2.1.3TimeCorrelatedChannelModel.....................7
2.2OFDMSystem..................................8
2.2.1ComparisonofOFDMSystemandSingleCarrierSystem.......8
2.2.2OFDMTranseiver.............................9
2.2.3CorrelationofOFDMSubcarriers....................11
2.3TimeCorrelatedMIMOChannelinOFDMSystem..............12
3LinearMIMOPrecoding14
ii

CONTENTS
3.1IntroductionofLinearPrecoding........................14
3.2SVDAlgorithm..................................15
3.3SVDPrecoding..................................17
3.3.1DiversityGainofUsingSVDPrecodingAlgorithm..........17
3.3.2MultiplexingGainofUsingSVDPrecodingAlgorithm........19
4LimitedFeedbackPrecodingwithClusteringFeedback22
4.1IntroductionofLimitedFeedbackPrecodingwithCodebook.........22
4.2GrassmannianBeamformingforMIMOChannel...............24
4.2.1GrassmannianSubspacePacking....................24
4.2.2CodebookDesignCriterion.......................26
4.2.3GrassmannianPrecodinginTxandRx.................28
4.3ClusteringFeedbackwithGrassmannianCodebook..............29
4.3.1ClusteringFeedbackwithPrecoderReuse...............29
4.3.2InterpolationBasedTransmitterBeamforming.............30
4.3.3ReducedInterpolationBasedFeedbackwithGrassmannianCodebook31
5ProposedLowcomplexityDierentialPrecodingwithClusteringFeed-
back 33
5.1DierentialBeamformingonCorrelatedChannel................33
5.1.1IntroductionofDierentialPrecoding..................34
5.1.2DierentialCodebookDesignCriterion.................34
5.1.3DierentialPrecodinginTxandRx..................36
iii NTHUCOM

CONTENTS
5.2ModiedInterpolationExploitTimeandFrequencyCorrelation.......37
5.3ComparisonBetweenGrassmannianPrecodingandDierentialPrecoding..38
6SimulationResults40
6.1TheComparisonofSingleCarrierPrecoding..................40
6.2PerformanceComparisonofInterpolators....................44
6.3PerformanceofProposedAlgorithmandComparisoninOFDMSystem...46
7Conclusion50

L.ZhengandD.Tse,ΣDiversityandmultiplexing:afundamentaltradeoinmultiple
[1]
antennachannels, IEEETrans.Inform.Theory,vol.49,pp.1073V1096,May2003
[2] A.Nrula,M.Lopez,M.Trott,andG.Wornell,ΣEcientuseofsideinformationi
multiple-antennadatatransmissionoverfadingchannels, IEEEJ.Select.AreasCom-
mun.,vol.16,pp.1423V1436,Oct.1998
[3]
K.K.Mukkavilli,A.Sabharwal,E.Erkip,andB.Aazhang,ΣOnbeamfomingwithnit
ratefeedbackinmultipleantennasystem, IEEETrans.Inform.Theory,vol.49,pp.
2562V2579,Oct.2003
[4] W.SpantipachandM.Honig,ΣAsymptoticeperformanceofMIMOwirelesschanne
withlimitedfeedback, IEEETrans.Mil.Electron.,vol.1,pp.141V146,Oct.2003
[5]
L.QiangandC.Georghiades,ΣPrecoderquantizationformimo-ofdmsystemsove
frequencyselectivechannels,i MobileandWirelessCommunicationssummit.16th
IST,2007,pp.1V5
[6]
D.J.LoveandR.W.HealthJr.,ΣGrassmannianbeaformingformultple-inputmultiple
outputwirelesssystem, IEEETrans.Inform.Theory,vol.49,pp.2735V2747,Oct
2003.
[7] D.J.LoveandRobertW.HealthJr.,ΣLimitedfeedbackunitaryprecodingforspatia
multiplexing, IEEETrans.Inform.Theory,vol.51,pp.2967V2976,Aug.2005
[8] D.J.LoveandR.W.J.Health,ΣGrassmannianbeamformingoncorrelatedMIM
channels,i GlobalTelecommunicationsConference(GLOBECOM),2004.
[9] D.SacristanandJ.Pascual-Iserte,ΣDierentialfeedbackofMIMOchannelcorrelatio
matricesbasedongeodesiccurves,i IEEEinternationalConferenceonAcoustics,
SpeechandSignalProcessing(ICASSP),May2009,pp.2553V2556
10]
J.ChoiandR.W.H.Jr.,ΣInterpolationbasedtransmitbeamformingforMIMO-OFD
withlimitedfeedback,i Proc.IEEEInt.Conf.Commun.,Nov.2005,pp.4125V4135
11] J.ChoiandR.W.HealthJr.,ΣInterpolationbasedtransmitbeamformingforMIMO
OFDMwithlimitedfeedback, IEEETrans.SignalProcessing,vol.53,pp.4125V4135
Nov.2005.
12] J.Choi,B.Mondal,andR.W.H.Jr.,ΣInterpolationbasedunitaryprecodingforspatia
multiplexingMIMO-OFDMwithlimitedfeedback, IEEETrans.SignalProcessing,
vol.54,pp.4730V4740,Dec.2006
13] T.Pande,D.J.Love,andJ.Krogeier,ΣAweightedleastsquaresapproachtoprecodin
withpilotsforMIMO-OFDM, IEEETrans.SignalProcessing,vol.54,pp.4067V4073
Oct.2006.
14] L.L.Winger,ΣLinearlyconstrainedgeneralizedLloydalgorithmforreducedcodeboo
vectorquantization, IEEETrans.SignalProcessing,vol.49,pp.1501V1509,2001
L.L.Winger.,ΣLinearlyconstrainedgeneralizedLloydalgorithmforvirtualcodeboo
15]
vectorquantization,i IEEEInternationalConferenceonImageProcessing.,2000,pp.
171V174
[16] S.ZhouandB.Li,ΣBERcriterionandcodebookconstructionfornite-rateprecode
spatialmultiplexingwithlinearreceivers, IEEETrans.SignalProcessing,vol.54,pp.
1653V1665,May2006
[17] C.Li,C.Zhong,L.Yang,andZ.He,ΣCapacitycriterionforcodebookdesigninMIM
systemswithlimitedfeedback,i IEEEint.ConferenceNeuralNetworksandSingal
Processing,Jun.2008,pp.168V172
[18] T.Pande,D.J.Love,andJ.V.Krogemier,ΣReducedfeedbackMIMO-OFDMprecodin
andantennaselection, IEEETrans.SignalProcessing,vol.55,pp.2284V2293,Ma
2007.
[19] W.FerzaliandJ.G.Proakis,ΣAdpativeSVDalgorithmforcovariancematrixeigen
structurecomputation,i 1990InternationalConferenceonAcoustics,Speech,andSig-
nalProcessing.(ICASSP)1990.,,Apr.1990,pp.2615V2618
[20] D.J.Love,R.W.H.Jr.,andJ.G.Andrewsd,ΣMarkovmodelsofmulti-antennalimite
feedbacksystems, IEEETrans.SignalProcessing,vol.4.
[21] J.H.Conway,R.H.Hardin,andN.J.A.Sloane,ΣPackinglines,planes,etc.,packing
ingrassmannianspaces,i ExperimentalMathematics,vol.5.
[22] A.Edelman,T.A.Arias,andS.T.Smith,ΣPackinglines,planes,etc.,packingsi
grassmannianspaces,i SIAMJ.MatrixAnal.Applic,vol.20.
[23] A.BargandD.Nogin,ΣBoundsonpackingsofsphereintheGrassmannmanifold,
vol.48,pp.2450V2454,Sep.2002
K.A.Gallivan,A.Srivastava,X.Liu,andP.VanDoorpen,ΣEcientalgorithmsfo
[24]
inferencesonGrassmannianmanifolds,i IEEEWorkshoponStatisticalSignalPro-
cessing,2003,pp.315V318
[25] H.Y.Sung,ΣWiFivsWiMax, http://www.slideshare.net/hysung/wi-vs-wimax.
[26] T.R.RamyaandS.Bhashya,ΣOnusingchannelpredictioninadaptivebeamform
ingsystem,i IEEEinternatonalconferenceonCommucicationSystemSoftwareand
Middeleware(COMSWARE),Jan.2007,pp.1V6