A Novel Blind Channel Estimation for a 2x2 MIMO System

Transcription

1 Int. J. Communcatons Network and System Scences do:0.436/jcns Publsed Onlne ugust 009 (tt:// Novel Blnd Cannel Estmaton for a x MIMO System Xa LIU Marek E. BILKOWSKI Feng WNG Student Member IEEE Scool of ITEE Te Unversty of Queensland Brsbane ustrala Fellow IEEE Scool of ITEE Te Unversty of Queensland Brsbane ustrala Emal: {xalu meb fwang}@tee.uq.edu.au Receved May 8 009; revsed June 0 009; acceted July BSTRCT novel blnd cannel estmaton metod based on a smle codng sceme for a by multle nut multle outut (MIMO) system s descrbed. Te roosed algortm s easy to mlement n comarson wt conventonal blnd estmaton algortms as t s able to recover te cannel matrx wtout erformng sngular value decomoston (SVD) or egenvalue decomoston (EVD). Te block codng sceme accomanyng te roosed estmaton aroac requres only a block encoder at te transmtter wtout te need of usng te decoder at te recever. Te roosed block codng sceme offers te full codng rate and reduces te nose ower to alf of ts orgnal value. It elmnates te ase ambguty usng only one addtonal lot sequence. Keywords: MIMO Cannel Estmaton Sem-Blnd Cannel Estmaton Pase mbguty. Introducton Multle Inut Multle Outut (MIMO) sgnal transmsson scemes are attractve for g-seed data transmsson n wreless communcaton systems because tey offer an ncreased data trougut (caacty) wtout ncreasng oeratonal bandwdt []. lso tey are caable to enance te qualty of sgnal transmsson troug te use of transmtter or recever dversty. Tese advantages are ossble under te condton tat te MIMO cannel state nformaton (CSI) s avalable at te recever. Tradtonally CSI can be acqured by sendng tranng sequences (also known as lot sgnals) evenly saced along a block of transmt symbols. Te dsadvantage of ts aroac s tat te tranng sequences take u te recous bandwdt. In order to save te bandwdt and ncrease sectral effcency blnd and semblnd cannel estmaton metods can be aled to obtan te CSI. Several blnd cannel estmaton metods ave been descrbed n [34]. Tese metods are based on te subsace algortm [5] wc utlzes te ortogonalty between te cannel matrx and te Sylvester matrxformed nose subsace. Tere are several drawbacks of subsace-based MIMO cannel estmaton metods. One s tat tey suffer from so-called mult-dmensonal ambguty. s a result several lot sequences are needed to elmnate ts ambguty. Two n order to comensate for extra degrees of freedom n te nose subsace wen te number of transmt antennas s smaller tan te number of antennas at te recever te re-codng s requred [34]. lso EVD s an nerent art of te algortm wc leads to g mlementaton comlextes. In [67] a sem-blnd cannel estmaton metod emloyng ortogonal lot maxmum lkelood (OPML) estmator as been roosed. Te metod erforms sngular value decomoston (SVD) to te receved sgnal correlaton matrx to estmate te wtenng matrx of cannel. By usng te wtenng matrx te OPML estmator sows a db mrovement of bt error rate (BER) comared to te conventonal least squares (LS) tranng sceme f te same lengt of tranng sequence s used. owever t stll requres a large number of tranng symbols to aceve te same erformance as LS. Furtermore SVD as to be aled twce to obtan te wtenng matrx and te rotaton matrx. Tese oeratons lead to te ncreased comutatonal comlexty. Te work n [8] resents a new SVD-based blnd cannel estmaton sceme wc uses a smle block re-codng structure. Te advantage of ts aroac s tat CSI can be recovered wtout ambguty f te Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences

2 NOVEL BLIND CNNEL ESTIMTION FOR X MIMO SYSTEM 345 roer modulaton s aled. noter advantage of ts sceme s tat no block decoder s needed at te recever. Tese advantages are ganed at te exense of te codng rate. Te codng rate decreases as more transmttng antennas are used. In artcular for a x MIMO system te code rate s / wc results n wastng of te recous sectrum. In ts aer we roose a novel blnd cannel estmaton algortm wc s of muc lesser comlexty tan tose based on SVD or EVD. Its mortant feature s tat t reserves te advantages of te codng sceme descrbed n [8] wtout sacrfcng te codng rate. Te new sceme offers a full codng rate (codng rate s equal to ) wen te number of transmttng antennas s equal to te number of recevng antennas. In te case of a x MIMO system ts codng sceme reduces te nose ower to te alf of te orgnal nose ower. Ts sceme exbts te ase ambguty. owever t can be elmnated usng only one extra lot sequence. Te rest of te aer s organzed as follows. In Secton a model of MIMO system emloyng a block codng sceme s ntroduced. new blnd cannel estmaton metod s descrbed n Secton 3. Te soluton of elmnatng te ase ambguty s gven n Secton 4. Smulaton results are resented n Secton 5. Secton 6 concludes te aer.. System Descrton & Codng Sceme In ts aer a narrow band block fadng cannel s assumed. Te number of transmttng and recevng antennas s denoted as N t and N r resectvely. Tus te cannel s te N r x N t dmenson cannel matrx wt j reresentng comlex resonse between te -t recevng antenna and te j-t transmttng antenna. In furter consderatons N t s assumed to be equal to N r. Te nut symbols at transmtter can be reresented by X x x...} () { x3 were X stands for ndeendent dentcally dstrbuted (..d) Gaussan random sgnals wt zero mean and te varance matrx gven by s n m E{ x n xm } 0 n m were E{} mles te exectaton and σ s s te ower of one symbol. Te symbols are encoded usng a block encoder structure before beng transmtted. s a result te -t symbol r r t 4 block s an element of matrx grou N N N C C. Te data receved at te oter end of te communcaton cannel s affected by te cannel roertes and an addtve nose. Terefore te relatons between te transmtted encoded symbols and receved data s gven as: Y N () were Y s te N r x N r N t receved sgnal matrx and N s te N r x N r N t (..d) Gaussan random nose matrx wt zero mean. Te coded outut of te transmtter can be wrtten as: were (3) dag( U) X T T 4 dag( U ) X T 4 4 dag( U) X (4) 0... n and X [ x4 x4] and X [ x43 x44]. [ T U U U ] U [ ] U [ ] reresent te encoder structure. Terefore te transmtted coded sgnals are or 4 43 x 4 x 4 x 4 x x x x x (5) 44 x x x x x x x x From exresson (6) one can observe tat 4 symbols are sent n 4 symbol erods durng one block. Terefore te code rate s. Te receved sgnal blocks can be wrtten as (7): (6) Y y y y y y y y y x4 x4 x4 x4 x43 x44 x43 x 44 N x4 x4 x4 x4 x43 x44 x43 x44 N (7) Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences

3 346 n wc N X. LIU ET L. n n n n random Gaussan nose matrx. n equvalent reresentaton of (7) s gven by (8): s te n n n n y x x n 4 4 y x x n 4 4 y x x n y x x n y x x n and y x x n y x x n and y x x n (8) By lnkng (8) drectly to ndvdual cannel matrx elements one obtans: x 4 x 4 x 4 x 4 x 43 x 44 x 43 x 44 y y n n y y n n y y n n y y n n y y n n y y n n y y n n y y n n (9) (0) () () s a result te relatons between te raw (transmtted) data and te receved data s gven by (3): Y XN (3) n wc te ndvdual terms are dentfed by (4) were y y y 4 4 y 0 0 0x n x 4 n 4 y 0 0 y 0x 43 n (4) x4 4 n 44 y y n n n n n n n n n n 4 n4 43 n44 (5) Due to te fact tat te elements n N reresent a Gaussan random nose also te elements n N obey te Gaussan dstrbuton. owever te average ower of eac element n N s alf of tat n N. In ts case te nose ower s suressed by te codng sceme. 3. Blnd Cannel Estmaton Te blnd cannel estmaton requres te knowledge of te correlaton matrx of Y wc s gven as: R E{ YY } E{ X X } E{ N N } (6) Because te ower of eac element n N s alf of tat n te nose matrx N ten Equaton (6) can be rewrtten as: R E{ YY } E{ X X } E{ NN } (7) If te nformaton symbol sequence s of unt ower ten (7) becomes: were R X E{ NN } (8) X E{ X X } I (9) Tus (8) can be converted to (9) N N N N { R E NN } (0) By ntroducng vec( ) te followng olds: r t r t Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences

4 NOVEL BLIND CNNEL ESTIMTION FOR X MIMO SYSTEM 347 dag ( ) () were vec (.) means vector oeraton n wc columns of are stacked on to of eac oter and. denotes te absolute value. s a result NN r t () R dag( ) E{ NN } (3) Te estmaton of s equvalent to fndng te roots of te dagonal elements n R. To obtan te soluton te square-root algortm can be aled. One roblem tat s faced usng ts aroac s tat t ntroduces te ase ambguty n te estmated. Ts s because te square roots are obtaned for te norms of te elements of te cannel matrx. In te next secton a metod for te ase ambguty elmnaton s descrbed. 4. Pase mbguty Elmnaton It s aarent tat te roosed blnd cannel estmaton algortm rovdes te nformaton about te estmated norm of eac element n cannel matrx of te x MIMO system as sown by te followng. ˆ ˆ ˆ ˆ ˆ (4) Now te task s to obtan te ases of tese elements. By sendng one lot sequence P were one obtans P P P (5) Y P N (6) n wc Y and P are known. More secfcally we ave Y X X jy jy e e N By relacng j j j j e e N NN r t te followng s obtaned (7) n (5) by te estmated values X ˆ cos ˆ cos (8) Y ˆ sn ˆ sn (9) X ˆ cos ˆ cos (30) Y ˆ sn ˆ sn (3) To determne θ θ θ and θ Equatons (9)(0) and ()() are used n wc (9)/(0) and ()/() are formed. Ts oeraton results n te followng y y n n j( ) j e e y y n n y y n n j( ) j e e y y n n y y n n j( ) j e e y y n n y y n n j( ) j e e y y n n (3) (33) Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences

5 348 X. LIU ET L. were σ =θ -θ and σ =θ -θ. Usng Equatons (3) and (33) σ and σ can be estmated by ncludng nose mact. Ten θ θ and θ θ can be exressed usng te estmated σ and σ as ˆ (34) ˆ (35) By substtutng (34) nto Equatons (3) and (35) nto Equatons (33) te ases θ θ θ and θ can be determned. Terefore te ase ambguty can be resolved. 5. Smulaton Results Te valdty of te roosed blnd cannel estmaton algortm for a x MIMO s nvestgated va comuter smulatons. For reference uroses comarsons are made wt a tranng-based cannel estmaton usng te least square (LS) metod and a sem-blnd cannel estmaton usng an ortogonal lot maxmum lkelood (OPML) algortm. Te followng s te necessary nformaton tat s used to erform comarsons wt te alternatve cannel estmaton algortms. Te erformance of te roosed cannel estmaton s assessed n terms of mean square error (MSE) as gven by MSE E{ ˆ } (36) n wc. F stands for te Frobenus norm. In te LS metod te estmated cannel matrx s gven as [0] ˆ LS F YP (37) were {.} stands for te seudo-nverse oeraton. Te MSE of LS metod s gven as { ˆ LS LS } F MSE E (38) ccordng to [] and [] te mnmum value of MSE for te LS metod s gven as MSE M M (39) LS t r mn n wc ρ stands for transmtted ower to nose rato n te tranng mode. ere we assume tat te SNR n te roosed estmaton scenaro s equal to ρ. In [6] and [7] a WR-based sem-blnd cannel estmaton metod was ntroduced. Followng tat metod te MIMO cannel matrx can be decomosed by alyng te sngular value (SV) decomoston SVD PQ (40) were P and Q are two sngular vectors corresondng to egenvalues. Σ reresents egenvalues of by te dagonal matrx. For bot P and Q te followng roertes old: PP = P P = I and QQ = Q Q = I. Te wtenng matrx W s gven by W = PΣ and Q s te rotaton matrx. Te wtenng matrx W can be obtaned blndly by comutng te second-order statstcs of a receved sgnal. Detals are gven n Secton.3 of [3] and tus are not reeated ere. Te matrx W s assumed to be erfectly known at te recever. Tranng sequences are used for estmatng te untary matrx Q. It as already been roved n [6] and [7] tat suc a untary matrx Q els ncreasng estmaton gans because t uses a smaller number of arameters. To estmate te rotaton matrx Q several algortms can be aled. Te ortogonal lot maxmum lkelood (OPML) algortm offers te best erformance as demonstrated n [6]. In ts algortm te tranng matrx X s set to ave ortogonal roertes wt unt ower and lengt equal to N t X X = X X = I. Te OPML estmator s exressed as (36) arg mn Y WQ XP subj. to QQ I P (4) were Q s obtaned by mnmzng te lkelood. Let Mˆ W Y X ten by alyng SVD to Mˆ we ave Û M ˆ ˆ SVD( Mˆ ) (4) MVM Qˆ of Q tat mn- It can be sown tat te estmated mzes te lkelood s gven as [6]: ˆ ˆ ˆ (43) Q V M U M Te cannel matrx s ten estmated as ˆ WQˆ (44) In Fgure te erformance of LS metod OPML algortm and te roosed blnd estmaton metod for a x MIMO system are resented. Te lengt of tranng sequences used by LS metod and OPML algortm s set to. From Fgure one can see tat wen te number of receved symbols ncreases te blnd estmaton accuracy s mroved. Te erformance of blnd cannel estmaton s always better tan offered by LS. Ts s manly because te roosed algortm reduces te nose ower to alf of te orgnal one. Wen makng comarson wt OPML algortm one fnds tat te er- Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences

6 NOVEL BLIND CNNEL ESTIMTION FOR X MIMO SYSTEM Blnd Cannel Estmaton for a x MIMO System Receved Symbols=0 Receved Symbols=50 Receved Symbols=00 Receved Symbols=000 Receved Symbols=0000 LS Cannel Estmaton OPML Cannel Estmaton MSE SNR (ρ) Fgure. Performance of LS OPML and newly roosed blnd estmaton algortm for a x MIMO system for dfferent values of SNR. 0 0 Blnd Cannel Estmaton for a x MIMO System SNR=5dB SNR=0dB SNR=5dB MSE Number of receved symbols Fgure. MSE as a functon of number of receved symbols for a newly roosed blnd cannel estmaton metod for a x MIMO system for dfferent values of SNR. formance of te roosed estmaton algortm s always better wen te number of receved symbols s equal or more tan 00. Fgure llustrates te convergence of te blnd cannel estmaton for a x MIMO system. It can be seen tat for eac of te tree assumed values of SNR te convergence occurs wtn te frst 00 receved symbols. Te use of a larger number of symbols (00 to 500) rovdes only a slgt mrovement. Te full convergence occurs at about 700 symbols. One can see tat wt 00 Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences

7 350 X. LIU ET L. receved symbols te estmaton s already very accurate rresectvely of te assumed value of SNR. 6. Conclusons In ts aer we ave resented a novel blnd algortm for estmatng a cannel of a x MIMO. Te roosed algortm oerates n conjuncton wt a sutable codng sceme and elmnates ase ambguty for te estmated cannel matrx coeffcents. Te codng sceme exbts g sectral effcency and reduces te nose ower to te alf of te orgnal one tat s resent n a x MIMO system. Te roosed algortm nvolves te square-root oeraton wc sows a low level of rocessng comlexty. Te smulaton results rove a fast convergence rate for estmatng te cannel. Te erformance of te roosed algortm s better tan of te tranng-based Least Squares (LS) algortm. lso t sows suerorty over te Ortogonal Plot Maxmum Lkelood (OPML) sem-blnd estmaton algortm wen te number of receved sgnal symbols exceeds References [] Telatar Caacty of multle antenna Gaussan cannels Euroean Transactons on Telecommuncatons Vol. 0 No November/December 999. [] G. J. Foscn and M. J. Gans On lmts of wreless communcatons n a fadng envronment wen usng multle antennas Wreless Personal Communcatons Vol [3] S. Zou B. Muquet and G. B. Gannaks Subsace-based (sem-) blnd cannel estmaton for block recoded sace-tme OFDM IEEE Transactons on Sgnal Processng Vol. 50 No May 00. [4] R. Zang Blnd OFDM cannel estmaton troug lnear Precodng: subsace aroac n Proceedngs slomar 0 Pacfc Grove C November 00. [5] E. Moulnes P. Duamel J. F. Cardoso and S. Mayrargue Subsace metods for te blnd dentfcaton of multcannel FIR flters IEEE Transactons on Sgnal Processng Vol February 995. [6]. K. Jagannatam and B. D. Rao Wtenng-rotatonbased sem-blnd MIMO cannel estmaton IEEE Transactons on Sgnal Processng Vol. 54 No. 3 Marc 006. [7]. Jagannatam and B. D. Rao Constraned ML algortms for sem-blnd MIMO cannel estmaton Proceedngs of IEEE Communcaton Socety Globecom 004. [8] X. Lu and M. E. Balkowsk SVD-Based blnd cannel estmaton for a MIMO OFDM system emloyng a smle block re-codng sceme Proceedngs of IEEE Eurocon Poland 007. [9] S. Sabazana. B. Gersman and J.. Manton Closed-form blnd MIMO cannel estmaton for ortogonal sace-tme block codes IEEE Transactons on Sgnal Processng Vol. 53 No. December 005. [0] S. M. Kay Fundamentals of statstc sgnal rocessng: Estmaton teory Prentce-all Incororaton 993. [] M. Bgues and. B. Gersman MIMO cannel estmaton: Otmal tranng and tradeoffs between estmaton tecnques Proceedngs ICC 04 Pars France June 004. [] M. Bgues and. B. Gersman Tranng-based MIMO cannel estmaton: study of estmator tradeoffs and otmal tranng sgnals IEEE Transactons on Sgnal Processng Vol. 54 No. 3 Marc 000. [3]. K. Jagannatam and B. D. Rao sem-blnd tecnque for MIMO cannel matrx estmaton Proceedngs IEEE Workso on SPWC Roma Italy 003. Coyrgt 009 ScRes. Int. J. Communcatons Network and System Scences