Combined ML and QR Detection Algorithm for MIMO-OFDM Systems with Perfect ChanneI State Information

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1 Combined ML and QR Deecion Algorihm for MIMO-OFDM Sysems wih Perfec ChanneI Sae Informaion Weizhi You, Lilin Yi, and Weisheng Hu An effecive signal deecion algorihm wih low complexiy is presened for iple-inpu iple-oupu orhogonal frequency division iplexing sysems The proposed echnique, QR-, combines he convenional maximum likelihood deecion () algorihm and he QR algorihm, resuling in much lower complexiy compared o The proposed echnique is compared wih a similar algorihm, showing ha he complexiy of he proposed echnique wih T= is a 95% improvemen over ha of, a he expense of abou a -db signalo-noise-raio (SR) degradaion for a bi error rae (BER) of 0 3 Addiionally, wih T=, he proposed echnique reduces he complexiy by 73% for iplicaions and 80% for iions and enhances he SR performance abou db for a BER of 0 3 Keywords: MIMO-OFDM, QR decomposiion, QR- algorihm, maximum likelihood deecion () Manuscrip received Sep 6, 0; revised ov 5, 0; acceped Dec 3, 0 This work was suppored by 973 Program (0CB3560 and 00CB3804), aure Science Foundaion China (600704, , and ), 863 Program, Program of Shanghai Subjec Chief Scienis (09XD4000), and Program of Shanghai Chen Guang Scholar (CG) Weizhi You (phone: , ywz758@6com), Lilin Yi (lilinyi@sjueducn), and Weisheng Hu (wshu@sjueducn) are wih he Sae Key Lab of Advanced Opical Communicaion Sysems and eworks, Deparmen of Elecronic Engineering, Shanghai Jiao Tong Universiy, Shanghai, China hp://dxdoiorg/048/erij I Inroducion Wireless communicaion via iple-inpu iple-oupu (MIMO) channels has received exensive aenion over he years Orhogonal frequency division iplexing (OFDM) has become a preferred echnique o ransmi signals over wireless channels OFDM based on MIMO (MIMO-OFDM) is an imporan echnique for he nex-generaion wireless communicaion However, i is difficul o obain a signal deecion algorihm wih low complexiy and high performance for he MIMO-OFDM sysems Many signal deecion algorihms have been proposed []-[], such as QR [3], a dynamic QR-decomposiion-based M-algorihm wih he feedback equalizaion deecion echnique based on QR decomposiion (QRDM-QR) [4], and maximum likelihood deecion () [5] The QR deecion algorihm has very low compuaional complexiy, bu is performance is no suiable Alhough allows for opimum performance, he complexiy of his algorihm is very high when he modulaion consellaion se and he number of ransmier anennas become large Therefore, hese algorihms are hardly used in a pracical deecion process QRDM-QR is a suiable signal deecion algorihm regarding boh performance and complexiy, bu here is room for improvemen in hese wo parameers In his paper, an effecive signal deecion algorihm combining and QR (QR-) is presened for MIMO- OFDM sysems For he proposed deecion algorihm, a parameer T is adoped A he beginning of he proposed algorihm, he channel marix H is sored as he column norm ETRI Journal, Volume 35, umber 3, June Weizhi You e al 37

2 from minimum o maximum Then, he firs T signals are deeced by he QR algorihm Finally, he convenional algorihm is used o deec he las T signals, where represens he number of ransmier anennas If he value of T is small, he sysem performs well and has high compuaional complexiy, which is similar wih he algorihm On he oher hand, when he value of T is large, he performance qualiy diminishes and he complexiy of he deecion scheme is reduced Hence, he performance and he compuaional complexiy of he QR- algorihm can be conrolled by adjusing he value of his new parameer T The proposed echnique is compared wih and QRDM-QR Even hough QRDM-QR [4] is a suiable signal deecion algorihm in boh performance and complexiy, i is much worse han QR- a a given value of T Compared wih, QR- significanly reduces he compuaional complexiy a he expense of minimal degradaion of he bi error rae (BER) performance Therefore, he proposed QR- deecion algorihm demonsraes desirable performance and low complexiy From siaion and compuaional complexiy comparison resuls, he complexiy of he proposed echnique wih T= is reduced by 95% compared wih ha of a he expense of abou a -db signal-o-noise-raio (SR) degradaion for a BER of 0 3 Compared wih he QRDM-QR deecion algorihm, no only is he complexiy of he proposed echnique wih T= decreased by 73% for iplicaions and 80% for iions, bu is performance is enhanced by abou db for a BER of 0 3 Therefore, he proposed echnique can be efficienly used for pracical MIMO-OFDM requiring very low complexiy and desirable performance The res of his paper is organized as follows The model for MIMO-OFDM sysems is inroduced in secion II In secion III, wo convenional MIMO deecion algorihms are briefly described In secion IV, he proposed echnique combining and he QR deecion echnique is presened A compuaional complexiy comparison is described in secion V The siaion resuls and discussion are presened in secion VI, and he conclusion is presened in secion VII II Sysem Model Sysem Model Consider a wireless MIMO-OFDM sysem wih ransmiing anennas and r receiving anennas, supposing r This sysem can be denoed by (, r ) The ransmier and receiver srucures of he MIMO-OFDM sysem are shown in Figs and For he ransmier side, he inpu informaion source is iplexed o symbol sreams by S/P Mapping Mapping IFFT IFFT +CP +CP P/S P/S Fig Transmier srucure of MIMO-OFDM sysem S/P S/P CP CP FFT FFT Fig Receiver srucure of MIMO-OFDM sysem Deecion and demodulae serial-o-parallel (S/P) convering Then, he symbols are passed hrough he consellaion mapping, he inverse fas Fourier ransform (IFFT) modulaors, and he cyclic prefix (CP) inserion Finally, he OFDM signals are ransmied by anennas For he receiver side, afer he cyclic prefix is removed, he r symbols are launched ino he fas Fourier ransform (FFT) for he demodulaion and deecion Finally, he r symbols are oupu by parallel-o-serial (P/S) convering Afer he demodulaion, he receiver signals can be given by P/S Rk ( ) = HSk ( ) + Wk ( ), () where R(k) is he r vecor of received signals, S(k) is he vecor of ransmied signals, W(k) is he r vecor iive whie Gaussian noise (AWG), and H is he r MIMO channel marix We can denoe R(k)=[R (k),, R r (k)] T, S(k)=[S (k),, S (k)] T, and W(k)=[W (k),,w r (k)] T, where S i (k) (k=0,,, K) is he k-h subcarrier of he complex baseband signal ransmied by he i-h anenna and K is he lengh of he OFDM daa sequence H can be represened as æ h h h ö, h h h ç, H = = ( h ) ç èh h h ø r, r, r, i, j r, () where h i,j is he channel impulse response from ransmier anenna j o receiver anenna i Here are some assumpions: (A) W(k) is zero mean, circularly symmeric complex H Gaussian wih EW ( ( kw ) ( k) ) = σ ni ; r H (B) S(k) is zero mean and wih ESkSk ( ( ) ( ) ) = I ; 37 Weizhi You e al ETRI Journal, Volume 35, umber 3, June 03

3 (C) W(k) and S(k) are independen of each oher Channel Model Le H be he perfec channel sae informaion and well known by he deecor Each elemen of H is an asymmeric complex Gaussian disribuion wih zero mean and uni variance, which are independen of each oher We consider he MIMO channel o be a quasisaic fla Rayleigh fading channel From [], we know ha he envelope response a any ime has a Rayleigh probabiliy disribuion and ha he phase is uniformly disribued in he inerval (0, π) Tha is, r σ r PRayleigh () r = e, r 0, (3) σ where r is he envelope of he received signal and σ is he variance of he received signals We knowσ = Var( r) = E( r ), ha is, E(r)=0 (zero mean) From [4], we know ha he Rayleigh fading can be defined as μ j πθl h () = le S ( l) μ ρ τ, (4) l = ρ l = l where S() is he inpu signal, θ l is he phase shif from he scaering of he l-h pah, ρ l is he aenuaion of he l-h pah, τ l is he relaive delay of he l-h pah, and μ is he oal number of ipahs III Convenional and QR Deecion Algorihms In his secion, we give an overview of he convenional opimal algorihm and QR algorihm Algorihm The can be wrien as Sk ( ) = arg min Rk ( ) HSk ( ), (5) S( k) Ω where Ω denoes he modulaion consellaion To obain he, an exhausive search is conduced Ω imes, where Ω represens he oal number of elemens of he consellaion se When boh and Ω are large, he complexiy of he algorihm is exremely high Therefore, i is difficul o use in he pracical deecion process QR Algorihm We apply QR decomposiion o () wih H=QR, where he r marix Q is an orhonormal marix saisfying Q H Q=I and marix R is an upper riangular marix wih R i,j (j i) denoing is non-zero elemens and R j,j (j=,,, ) being a posiive real number We have Therefore, Rk ( ) = QRSk ( ) + Wk ( ) (6) Yk ( ) = RSk ( ) + nk ( ), (7) where Y(k)=Q H R(k), n(k)=q H W(k), and n(k) are sill AWG Equaion (7) can be wrien as æy( k) ö ær, R, R ö, æ S( k) ö æ n( k) ö Y( k) 0 R, R, S( k) n( k) = + ç Y ( k) 0 0 R S ( k) è ø ç è øèç ø è çn ( k) ø, Signal Y ( k ) can be wrien as, (8) Y ( k) = R S ( k) + n ( k) (9) Finally, he decision saisic (9) can be used o esimae S ( k ) as S k Y k R, ( ) = quan( ( )/ ) (0) The process of QR deecing can be described by using he following recursive algorihm Si k = Yi k - Ri, j Sj k Ri, i j=+ i ( ) quan(( ( ) å ( ))/ ), () i= -,, Alhough he QR deecion scheme has low compuaional complexiy, i suffers from error propagaion If he i-h layer is deeced successively and incorrecly, hose fauly decisions will affec he subsequen decisions IV Combined and QR Deecion Algorihm Algorihm Developmen To lower he complexiy of he algorihm and improve he performance of he QR algorihm, a combined and QR deecion echnique is proposed for he MIMO-OFDM sysems We denoe his proposed deecion algorihm as QR- During he deecion process, a new parameer T is adoped o decrease he deecion complexiy The value of parameer T deermines he number of signals deeced by he QR algorihm, and he number T deermines he number of signals deeced by he algorihm Obviously, he value of T is a se Tha is, T={0,,, } The whole process of he QR- deecion algorihm is as follows Sep : We sor he columns of he channel marix H by he column norm from minimum o maximum Tha is, ETRI Journal, Volume 35, umber 3, June 03 Weizhi You e al 373

4 H = sor( H, H,, H ), () new where sor() represens a soring from minimum o maximum Hi is he norm of he i-h column of he channel marix H The soring order can be recorded in a se, denoed by B Sep : The deecion process wih QR is execued for he signals oe ha he channel marix H is represened by H new and ha parameers R(k), S(k), and W(k) are ordered by se B The firs T signals can be obained by (3) æ ö Si ( k) = quan( Yi( k) Ri, j Sj( k) ), R ç -å ii, çè j=+ i ø (3) i=,, - T + where S i ( k ) = 0 if he value of T is zero Reserve he T deeced signals as T QR ( ) [ ( ),, ( )] T+ S k = S k S k (4) Sep 3: The las T signals are deeced by he algorihm S ( k) = T T T (5) arg min Rk ( ) HS [ ( k) + S ( k) ], T SM ( k) Ω where Ω denoes he modulaion consellaion se and an exhausive search will be conduced Ω T imes Sep 4: Finally, he esimae of he ransmied signal is ~ T T T Sk ( ) = [ S ( k), SQR ( k) ] According o he firs soring ~ ~ ~ order recorded in se B in Sep, sor Sk ( ) by Slas = sor( S) ~ and hen he oupu S las is he resul of he QR- deecion algorihm Algorihm Analysis A he beginning of he QR- algorihm, he channel marix H is sored as he column norm from minimum o maximum In he firs deecion sage, when he QR deecion echnique is execued, is reliabiliy has grealy enhanced since he maximum column norm represens he maximum SR The bigger he SR is, he more reliable he deecion signal is In he second sage, he convenional opimal algorihm is execued Is performance is he bes among oher deecion algorihms Due o he firs sage of he QR deecion scheme, he exhausive search will be conduced Ω T imes, raher han Ω When T 0, is complexiy will be reduced significanly Obviously, when T=0, he proposed algorihm works as if i is he algorihm, and is performance is much beer han ha of he QR algorihm, considering ha channel H is sored a he beginning of he QR- algorihm The comparison of he compuaional complexiy among differen algorihms will be given in secion V M QR V Compuaional Complexiy Comparison In his secion, we calculae he compuaional complexiy for he proposed QR- deecion scheme, convenional algorihm, and QRDM-QR algorihm proposed in [4] Real iplicaions and iions are considered for he purpose of compuaional complexiy comparison From [4], we know ha we coun he oal number of iplicaions as four if here is one complex iplicaion operaion and we coun he oal number of iions as wo if here is one complex iion operaion For simpliciy, we suppose = r = and L-QAM modulaion is used From [3], [4], we know ha when M=L, he performance of he QRDM algorihm is similar o ha of he algorihm, where M represens he number of branches kep in every layer Therefore, o compare he complexiy beween QR- and QRDM-QR, we suppose M=L The number of iplicaions,, for he deecion algorihm can be wrien as The number of iions, algorihm can be wrien as = 4 L ( + ) (6), for he deecion L = 4 ( + ) + ( ) + + ( ) (7) The number of iplicaions, algorihm can be wrien as for he QR deecion QR, = L + 4L + 4 Ln (8) QR 3 The number of iions, algorihm can be wrien as QR, n= for he QR deecion QR 3 n= = L + (4Ln + Ln) + 4L + (9) QR- Therefore, he number of he iplicaions,, for he QR- deecion algorihm can be wrien as T QR- 3 = = C T L L L + 4 L ( T + ) + 4 C3 C (0) The number of iions, QR-, for he QR- deecion algorihm can be wrien as T QR- 3 = C 4 = L+ (4L+ L) + 4L Weizhi You e al ETRI Journal, Volume 35, umber 3, June 03

5 T + 4 L ( T + ) + ( T ) + ( T) + ( T ) ( ) C 6 () From [4], we know ha he compuaional complexiy of he QRDM-QR algorihm is as follows: C5 T T QRDM-QR 3 = = = C7 + 4D + D4, = + C8 L D L D L () T QRDM-QR 3 = (4 + 4 ) = C9 T + (4DL + 4 L) + D(4+ ) + 4D +, = = T+ C0 L D L D L (3) where D = ML = L and D = L In (0) hrough (3), C and C4 are he number of iplicaions and iions of he QR process in he Table umber of iplicaions for each algorihm wih 6- QAM modulaion, r = =4 Algorihm T= T= T=3 T=4 5,4,880 QRDM-QR 6,46 5,07,048, QR- 63,680 3,95,368,76 QR QR- QRDM-QR QR- algorihm Similarly, C and C5 represen he number of iplicaions and iions of he process in he proposed algorihm Finally, C3 and C6 are he number of iplicaions and iions in he soring process in he QR- deecion scheme Furhermore, C7 and C9 are he number of operaions o calculae iplicaions and iions of he QRDM process in he QRDM-QR algorihm, and C8 and C0 are he number of operaions o calculae iplicaions and iions of he QR process in he QRDM-QR echnique The compuaional complexiy of he QR- deecion algorihm compared wih and he QRDM-QR is illusraed in Table and Table wih 6-QAM modulaion and r = =4 From Tables and, i is ineresing o noe ha he complexiy of he proposed deecion algorihm is significanly reduced for boh iplicaions and iions compared wih he convenional algorihm whenever T is equal o,, 3, or 4 Furhermore, compared wih he QRDM-QR algorihm proposed in [4], he complexiy of he QR- algorihm is reduced by 73% for iplicaions and 80% for iions when he value of T is When T is equal o 3 or 4, he complexiy of he QR- algorihm is similar wih ha of QRDM-QR The principle o choose he proper value of T for differen and r and he performance siaion of he proposed echnique are presened in secion VI VI Siaion Resuls and Discussion Siaion Resuls In his subsecion, compuer siaions are presened o Table umber of iions for each algorihm wih 6-QAM modulaion, r = =4 Algorihm T= T= T=3 T=4 5,4,95 QRDM-QR 49,480 67,784,376,47 QR- 63,770 4,095,596,473 QR QR- QRDM-QR BER QR- (T=) QR- (T=) QR- (T=3) QR- (T=4) SR (db) Fig 3 BER performance of algorihm and proposed echnique wih = r =4, according o he value of T ETRI Journal, Volume 35, umber 3, June 03 Weizhi You e al 375

6 BER QR- (T=) QR- (T=) QR- (T=3) QR- (T=4) QRDM-QR (T=, M=6) QRDM-QR (T=, M=6) QRDM-QR (T=3, M=6) QRDM-QR (T=4, M=6) SR (db) Fig 4 BER performance of QRDM-QR and ha of QR- algorihm wih = r =4, according o value of T evaluae he performance of he proposed QR- algorihm for MIMO-OFDM sysems Assuming perfec channel esimaion, he receiver deecion is based on a five-pah Rayleigh fading environmen wihou channel coding During he siaion, he lengh of he OFDM daa sequence K is equal o 5, = r =4, and 6-QAM modulaion is used Figure 3 shows he BER performance of he algorihm and he proposed QR- algorihm wih T being equal o,, 3, and 4, respecively Figure 4 plos he BER performance of he proposed QR- algorihm and he QRDM-QR algorihm Discussion Firs, he BER performance of he QR- algorihm is compared wih ha of he algorihm in Fig 3 The lower he value of T is, he beer he BER performance of he QR- algorihm is From Table, Table, and Fig 3, i can be concluded ha in he case of = r =4, wih T=, he performance of he proposed QR- algorihm reflecs only a -db degradaion for a BER of 0 3 and a 95% decrease in complexiy compared wih he algorihm I is reasonable o decrease he complexiy of he algorihm by 95% a he expense of a -db performance degradaion because he convenional algorihm is difficul o use in pracical sysems As shown in Fig 4, alhough he BER performance of he QRDM-QR deecion algorihm is similar o ha of he proposed QR- deecion algorihm, here are noable differences For example, when he value of T is, compared wih ha of he QRDM-QR algorihm, he performance of he QR- algorihm is enhanced abou db for a BER of 0 3 Furhermore, considering he complexiy of he algorihm, i is amazing o find ha he complexiy of he proposed echnique is decreased by 73% for iplicaions and 80% for iions compared wih he QRDM-QR deecion algorihm When he value of T is 3 or 4, alhough he compuaional complexiy of QR- is abou 5% higher han ha of QRDM-QR, he performance of QR- is abou db improved for a BER of 0 3 compared wih he QRDM-QR algorihm Finally, we can conclude ha even hough QRDM-QR is a suiable deecion algorihm regarding performance and complexiy, he level of performance and complexiy of QR- when an appropriae value of T is seleced is beer Compared wih he convenional algorihm, i is reasonable o use he QR- deecion echnique wih much lower complexiy Therefore, he proposed selecive echnique can be efficienly used for pracical MIMO-OFDM sysems requiring very low complexiy by assigning he proper value of T VII Conclusion In his paper, a highly efficien and minimally complex algorihm for MIMO-OFDM sysems was proposed The proposed QR- algorihm enhances he deecion performance by applying he echnique and decreases he compuaional complexiy by using he QR algorihm Siaion resuls show ha he complexiy of he proposed echnique wih T= is reduced by 95% compared wih ha of, a he expense of abou a -db SR degradaion for a BER of 0 3 Is performance and complexiy are much beer han hose of he effecive QRDM-QR algorihm by appropriaely selecing T Considering ha he convenional scheme is difficul o use in pracical MIMO-OFDM sysems because of is high complexiy, he proposed QR- deecion scheme can be effecively used In conclusion, by assigning he proper value of T, he proposed combined and QR deecion echnique can be used for MIMO- OFDM sysems requiring high-level performance and very low complexiy References [] KJ Kim and RA Ilis, Join Deecion and Channel Esimaion Algorihms for QS-CDMA Signals over Time-Varying Channels, IEEE Trans Commun, vol 50, May 00, pp [] KJ Kim e al, A QRD-M/Kalman Filer-Based Deecion and Channel Esimaion Algorihm for MIMO-OFDM Sysems, IEEE Trans Wireless Commun, vol 4, no, Mar 005, pp Weizhi You e al ETRI Journal, Volume 35, umber 3, June 03

7 [3] B Hassibi, An Efficien Square-Roo Algorihm for Blas, Proc IEEE In Conf Acousic, Speech, Signal Process, Isanbul, Turkey, June 000, pp 5-9 [4] M-S Baek, Y-H You, and H-K Song, Combined QRD-M and DFE Deecion Technique for Simple and Efficien Signal Deecion in MIMO-OFDM Sysems, IEEE Trans Wireless Commun, vol 8, no 4, Apr 009 [5] S Sun e al, Pseudo-Inverse MMSE Based QRD-M Algorihm for MIMO-OFDM, IEEE Veh Technol Conf, May 006, pp [6] W You, L Yi, and W Hu, Reduced Complexiy Maximum- Likelihood Deecion for MIMO-OFDM Sysems, IEEE WiCOM, 0 [7] W You, L Yi, and W Hu, A Hybrid Dynamic QRDM and ZF Deecion Algorihm for MIMO-OFDM Sysems, IEEE WiCOM, 0 [8] H Jian e al, A Low Complexiy Sof-Oupu QRD-M Algorihm for MIMO OFDM Sysems, IEEE WiCOM, 00 [9] Y Liu e al, Super-Low-Complexiy QR Decomposiion-M Deecion Scheme for MIMO-OFDM Sysems, IET Commun, vol 5, no 9, 0, pp [0] PA Haris e al, Performance of Combined Low Complexiy QRD-M and DFE Deecion in MIMO-OFDM Sysems, IEEE ICCSP, 0, pp [] ZY Huang e al, High-Throughpu QR Decomposiion for MIMO Deecion in OFDM Sysems, IEEE ISCAS, 00, pp [] JG Proakis, Digial Communicaions, 4h ed, 008, pp 80-8 [3] L Wei, ear Opimum Tree-Search Deecion Schemes for Bi- Synchronous Muliuser CDMA Sysems over Gaussian and Two-Pah Rayleigh Fading Channels, IEEE Trans Commun, vol 45, no 6, June 997, pp [4] TM Aulin, Breadh-Firs Maximum Likelihood Sequence Deecion, IEEE Trans Commun, vol 47, no, Feb 999, pp 08-6 Weizhi You received his BS in elecommunicaion engineering from Hangzhou Dian Zi Universiy, Hangzhou Ciy, Zhejiang Province, China, in 00 Since Sepember 0, he has been pursuing his MS in communicaion and informaion sysems a Shanghai Jiao Tong Universiy, Shanghai, China, and is scheduled o graduae in March 04 He works in he Sae Key Lab of Advanced Opical Communicaion Sysems and eworks, Shanghai Jiao Tong Universiy His major research ineress are in informaion and communicaions echnology, MIMO-OFDM signal processing, opical OFDM-PO resource scheduling, OFDM- PO power consumpion, and image processing He has published several papers abou MIMO-OFDM signal processing and opical OFDM resource scheduling In he 009 Inernaional Compeiion of Mahemaical Cones in Modeling, he was named Meriorious Winner for his ousanding performance Lilin Yi received his BS and MS in physics from Shanghai Jiao Tong Universiy, Shanghai, China, in 00 and 005, respecively He earned his PhD in communicaions and elecronics from École aionale Supérieure des Télécommunicaions (EST), France, and in elecronics engineering from Shanghai Jiao Tong Universiy, Shanghai, China, in 008, as a join-educaed PhD suden Afer graduaion, he worked a Oclaro Incorporaion as an engineering manager Since 00, he has been a member of he Shanghai Jiao Tong Universiy faculy He is currenly an associae professor in he Elecronics Engineering Deparmen His research ineress are mainly focused on novel opical access neworks and phoonic informaion processing He has published more han 50 peerreviewed echnical journal and conference papers Weisheng Hu received his BS from Tsinghua Universiy, Beijing, China, his MS from Beijing Universiy of Science and Technology, Beijing, China, and his PhD from anjing Universiy, anjing, China, in 986, 989, and 997, respecively A Shanghai Jiao Tong Universiy, Shanghai, China, he worked as a posdocoral fellow from 997 o 998 and became a professor in 999 He is he direcor of he Sae Key Laboraory of Advanced Opical Communicaion Sysems and eworks, Shanghai Jiao Tong Universiy He serves on he boards of four journals and conferences, including IEEE JLT, OFC, and ACP His ineress include he opical ranspor nework wih GMPLS conrolled and opical packe swiching He is he co-auhor of over 00 peer-reviewed journal and conference papers ETRI Journal, Volume 35, umber 3, June 03 Weizhi You e al 377