MIMO IDENTICAL EIGENMODE TRANSMISSION SYSTEM (IETS) A CHANNEL DECOMPOSITION PERSPECTIVE

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1 MIMO IDENTICAL EIGENMODE TRANSMISSION SYSTEM (IETS) A CANNEL DECOMPOSITION PERSPECTIVE M. Zeesan Sakir, Student member IEEE, and Tariq S. Durrani, Fellow IEEE Department of Electronic and Electrical Engineering, University of Stratclyde 04 George Street, G XW, Glasgow, U pone: + (44) , fax: + (44) , {msakir, tsd }@eee.strat.ac.uk web: ABSTRACT In te past few years considerable attention as been given to te design of Multiple-Input Multiple-Output (MIMO) Eigenmode Transmission Systems (EMTS). Tis paper presents an in-dept analysis of a new MIMO eigenmode transmission strategy. Te non-linear decomposition tecnique called Geometric Mean Decomposition (GMD) is employed for te formation of eigenmodes over MIMO flatfading cannel. Exploiting GMD tecnique, identical, parallel and independent transmission pipes are created for data transmission at iger rate. Te system based on suc decomposition tecnique is referred to as MIMO Identical Eigenmode Transmission System (IETS). Te comparative analysis of te MIMO transceiver design exploiting nonlinear and linear decomposition tecniques for variable constellation is presented in tis paper. Te new transmission strategy is tested in combination wit te Vertical Bell Labs Layered Space Time (V-BLAST) decoding sceme using different number of antennas on bot sides of te communication link. Te analysis is supported by various simulation results.. INTRODUCTION Exploiting multiple antennas at te transmitter and te receiver in wireless communication systems is an extremely promising way to enance te data rate of future wireless communication systems [] and []. Multiple-Input Multiple-Output (MIMO) communication system tat transmit data troug parallel sub-cannels and exploit te Cannel State Information (CSI) at te transmitter, are termed as MIMO Eigenmode Transmission System (EMTS). owever, in suc systems, ric-scattering and flat-fading conditions are a requisite condition, in order to exploit ig MIMO EMTS capacity [3]. By deploying multiple transmitters and receivers in sufficiently ric in scattering environment, te MIMO EMTS as a ig potential to increase te capacity linearly wit te number of spatial cannels [4]. Te same as been illustrated in [4] and [5] for te Bell Labs Layered Space Time (BLAST) arcitecture. Since ten, many transmission strategies ave been proposed [] and []. Among te two most important and well known approaces, one is space time coding metod tat attempts to improve te communication reliabilities by coding and diversity gain acieved by appropriate coding design []. Te oter is spatial multiplexing metod wic transmits data over spatial subcannels, often in conjunction wit an outer cannel code, e.g., te BLAST arcitecture wic focuses on maximizing te cannel trougput [] and [6]. Bot design scemes assume tat te CSI is available only at te receiver. By transmitting troug parallel, spatial sub-cannels and exploiting te CSI at te receiver, spatial multiplexing systems can provide ig data rates [3]. owever, if te communication environment is slowly time varying, suc as indoor communication via Wireless Local Area Networks (WLANs), te availability of te CSI is also possible at te transmitter, using feedback or te reciprocal tecnique wen Time Division Duplex (TDD) is used [3]. Wit te CSI available at te transmitter as well, cannel capacity can be acieved exploiting a linear transformation at te transmitter. Te linear decomposition tecniques can be employed at te receiver to convert te MIMO EMTS cannel into set of parallel and independent scalar subcannels. Te design of MIMO EMTS transceiver includes preco ing at te transmitter and employing an equalizer at te receiver []. Several designs ave been proposed based on conventional decomposition tecniques and using a variety of criteria, including maximum Signal to Noise Ratio (SNR), Minimum Mean Squared Error (MMSE) and Bit Error Rate () based criteria [] and [7]. In [], a transmission strategy as been designed using linear transformation only. Te linear transformation used is Singular Value Decomposition (SVD), wic decomposes te MIMO EMTS flat-fading cannel into multiple parallel sub-cannels. Te water filling algoritm is used to acieve te capacity of eac sub-cannel []. Te non-zero singular values of te diagonal matrix represent te Signal to Noise Ratios SNRs) of te sub-cannels formed by SVD. owever, due to very ig variations in te SNRs of te sub-cannels and ig condition number, tis apparently simple linear decomposition sceme requires a very intelligent and adaptive bit allocation in order to matc te capacity of eac sub-cannel and acieve te prescribed []. Bit allocation among te eigenmodes of te MIMO EMTS not only increases te coding/decoding complexity but also inerently capacity loss because of finite constellation granularity [] and [4]. Alternatively, assignment of equal power to te sub-cannels, results in te same constellation

2 among all eigenmodes. owever, more transmitting power could be allocated to te cannel aving a poorer SNR i.e. te eigenmode aving lowest SNR. A fundamental trade off is always required to be made between capacity and performance wen dealing wit linear decomposition tecnique. In tis paper, we first review linear decomposition tecnique referred to as SVD. Ten we introduce non-linear decomposition tecnique called Geometric Mean Decomposition (GMD). Te main aim of tis paper is to develop a MIMO EMTS transceiver design exploiting GMD in combination wit V-BLAST decoding. Using GMD, te MIMO EMTS cannel can be decomposed into set of identical, parallel and independent sub-cannels. Tis non-linear decomposition tecnique brings about muc more convenience in coding/decoding and modulation/demodulation processes. In order to demonstrate te effectiveness of new transmission strategy, various simulation scenarios are created in Matlab. Te performance curves are obtained for different number of antennas at te transmitter and te receiver wit two different types of modulation scemes. Results of te transmission strategy based on nonlinear decomposition tecnique ensure superior performance as compare to conventional transmission strategy based on SVD []. Te remainder of te paper is organized as follows. Section introduces cannel model and decomposition tecniques. Tis section presents an overview of cannel decomposition perspective of Rayleig flat-fading MIMO cannel and associated callenges. After reviewing te well understood, linear decomposition tecnique SVD, we introduce GMD to construct te eigenmode over MIMO cannel. Tis section also presents matematical concept for te formation of identical, parallel and independent subcannels. Section 3 presents te testing, comparative results and analyses of GMD based V-BLAST detection sceme. Te following notation is used trougout tis paper. Boldface uppercase letters denote matrices, boldface italic lowercase letters denote column vectors, and italics denote M N scalars. R andc M N represent te set of M N matrices wit real and complex valued entries respectively. Te superscripts (.) T and (.) denote transpose and ermitian operation respectively.. CANNEL MODEL AND DECOMPOSITION TECNIQUES In tis section of paper, we present flat-fading cannel model and GMD tecnique to design te precoder at te transmitter and te equalizer at te receiver. We also discuss te formation of identical, parallel and independent transmission pipes over MIMO cannel.. Te Cannel Model Let us consider a MIMO EMTS aving N transmitting antennas and M receiving antennas as sown in Figure.,,N,N,, MN, M,, M, Figure Model of MIMO communication system wit N transmitting and M receiving antennas over Rayleig flat-fading cannel. Te signal model corresponding to a transmission troug a flat-fading MIMO cannel is given by [] and [3] y= s+ n () N M N were s C is te transmitted vector, C is te cannel matrix wit te ( mnt, ) element denoting te fading coefficient between te mt receiving and nt transmitting antennas, y C is te received signal vector, M and N n C is a zero mean circularly symmetric complex Gaussian interference-plus-noise vector wit arbitrary covariance matrix. Trougout tis paper we assumed tat minmn, and ) ) denote te rank of suc tat ( ) perfect CSI at te transmitter and te receiver i.e. is known at bot sides of te communication link.. Geometric Mean Decomposition Algoritm In order to investigate te geometric mean decomposition tecnique, we review linear transformation of te cannel matrix called SVD and is given by = UΣ V. () were U and V are unitary matrices and Σ is a diagonal matrix wit singular values equal to te square root of eigenvalues of te Wisart matrix, given by,, suc tat following condition is satisfied >. (3) 0 By calculating te filtered receive vector, we ave r=u y, (4) By substituting () and apply decomposition defined in (), we ave = U s+ U n= U UΣ V s+ U n, were r= Σ s + n, (5) s =V s and n =U n.

3 Q ˆx N M N M Figure Sceme of MIMO communication system wit non-linear transceiver over Rayleig flat-fading cannel. Te MIMO EMTS defined in () is now decompose into parallel sub-cannels, eac representing single in-put single out put (SISO) system given by r= s + n. (6) i i i i were i=,,, k,, and i denotes te eigenvalues of wisart matrix. It is obvious tat te information can only be transmitted over tose equivalent cannels wit non zero singular values []. It is very important to note tat if te number of transmission layers exceeds te number of strong singular values, te performance of te MIMO system degrades. Te complexity in te design of linear transceiver is due to large variations among te eigenvalues of te cannel matrix [] and [3]. M N For any rank, matrix C wit singular values > 0, tere exist an upper trian- gular matrix R R wit identical diagonal elements given by [7] and [8] / rii== i. i (7) i= suc tat te singular value decomposition of R is given by R= U Σ V, (8) R R wit it diagonal element equal tor ii =. ere Σ is a diagonal matrix wose elements are equal to te singular values of te matrix i.e. { } i= = U V. Since we also know tat Σ and combining wit (8), we ave = UUR RVRV, (9) ence, te decomposition derived in (9), is referred to as te Geometric Mean Decomposition (GMD) [3] and [8] = QRP. (0) were Q and P are semi-unitary matrices denoting te linear operations at te receiver and te transmitter respectively..3 Transmission over Identical and Parallel Pipes Te sceme of a general MIMO EMTS communication system wit non-linear transceiver is sown in Figure. First we calculate te GMD of cannel matrix as= QRP. Next we encode te information symbols s via te linear precoder P, suc tat te transmitted vector is given by Figure 3 Concept of formation of identical, parallel and independent pipes over Rayleig flat-fading MIMO cannel using Geometric Mean Decomposition (GMD). were Decomposed wireless MIMO cannel s=px. () N P C is te transmit matrix (precoder) and x C is te data vector tat contains te symbols to be transmitted (zero mean, normalized and uncorrelated, tat is, E xx = I ) drawn from a set of constellations. At te receiver side, te decomposition algoritm is exploited in combination wit a receiver interface, referred to as V-BLAST decoding. Te decoding algoritm is based on sequential nulling and cancellation in order to decode te transmitted information symbols s [], [6] and [7]. Te nulling step can be implemented by eiter using zero forcing (ZF) or minimum mean squared error (MMSE) criterion []. Te main drawback of te V-BLAST detection algoritm lies in te computational complexity, as it requires multiple calculations of te pseudoinverse of cannel matrix in te ZF case [7]. Tus, we consider GMD sceme in order to design a reduced complexity version of V-BLAST detection sceme. Te estimated data vector at te receiver is given by =Q y, () M were Q C is te receiver matrix (equalizer). Substituting () in (), we ave = Q s+ Q n, (3) we consider te sceme presented in [7]. Restating te successive interference cancellation sceme employing GMD (0), te resulting received vector (3) becomes = Q QRP s+ z, (4) also substituting () in above, we ave = Q QRP Px+ z, (5) knowing QQ = I & PP = I, (5) becomes = Rx+ z. (6)

4 0 - N=M= 0 - N=M= (a) (a) 0 - N=M= 0 - N=M= (b) Figure 4 performance of GMD V-BLAST over Rayleig flatfading MIMO IETS wit (a) QAM-6 (b) QPS. also in component wise notation, (6) becomes r, r, r, x z r, x z = +. (7) r, x z 0 Due to upper triangular structure of R, te it element of ˆx is given by i = r x+ r x + z, (8) i ii, i i+, i i+ i i+ But for GMD, we know tat r ii= for =,,, k,,. Ignoring propagation effects i.e. r x i+, i i+ i+ = 0, we can regard te resulting sub-cannels as identical, parallel and independent sub cannels given by i= xi+ zi. fori=,, (9) Te concept of formation of identical, parallel and (b) Figure 5 performance of SVD V-BLAST over Rayleig flatfading MIMO system wit (a) QAM-6 (b) QPS. independent pipes using GMD is sown in Figure 3. Te main advantage of tis combined strategy comes wit te complexity reduction, as it requires only a fraction of computational effort as compare to te original V-BLAST algoritm [], [] and [7]. Te cannel gain of eac eigenmode is given by. Beam steering tecniques on bot te transmitter and te receiver sides are acieved by multiplying vector p at te transmitter and matrix Q at te receiver, were p m denotes mt column of P. As a result, an equivalent cannel matrix can be expressed as Q p m= 0,,,, 0. (0) In MIMO systems, data transmission over te equivalent cannel given by (0) is referred to as identical eigenmode transmission system (IETS) and te sub-cannels andp m are called as eigenmodes and eigenvectors, respectively. 3. PERFORMANCE ANALYSIS In tis Section, we present te performance analysis based on curves obtained after various simulation scenarios. In all simulation experiments, we assumed tat te T m

5 cannel is Rayleig flat-fading. To determine te effectiveness of te new GMD based V-BALST detection strategy, te performance curves are compared wit SVD based V-BLAST decoding strategy. In eac scenario te curves are obtained after averaging 5000 Monte Carlo trails of. In first case, we consider GMD V-BLAST over Rayleig flat-fading MIMO cannel wit N=,,, 3 4 and M =,,, 3 4 transmitting and receiving antennas respectively. In Figure 4(a), we present a scenario were N independent symbols, modulated as Quadrature Amplitude Modulation (QAM-6) are transmitted. Using GMD over, we ave identical, parallel and independent sub-cannels for symbol transmission. After observing te curves for different number of transmitting and receiving antennas it is demonstrated tat for te curven= M= 4, GMD V- BLAST performs superior, from moderate to ig SNRs. ence, te performance of GMD based V-BLAST sceme is increased wit te increase in te number of transmitting and receiving antennas. Wit te increase in cannel dimension te condition number is usually ig as well and terefore, te performance curves for cannel wit iger condition number outperforms te curves wit lower condition number. In Figure 4(b), te same simulation scenario is repeated for a different constellation. In tis simulation N independent symbols, modulated as Quaternary Pase Sift eying (QPS) are transmitted over identical, parallel and independent sub-cannels. It is observed tat te performance curves for QPS outperform te curves obtained for QAM-6. Next, we present te simulation scenarios for MIMO linear transceiver over Rayleig flat-fading cannel again wit N=,,, 3 4 and M =,, 3, 4 transmitting and receiving antennas respectively. In Figure 5(a), we present a scenario were N independent symbols, modulated as QAM- 6 are transmitted. Exploiting SVD over, we ave parallel and independent sub-cannels for symbol transmission. From te performance curves it is observed tat wit te increase in number of transmitting and receiving antennas te performance of te SVD based V-BLAST detection is degraded. For iger cannel dimension i.e. 4 4 C te condition number of is usually very ig tat results in ill conditioned sub-cannels. Allocating more power to te poor cannel degrades te performance of te system. To avoid te system degradation, SVD sould be used in combination wit water filling algoritm wic suggests tat te binary pase sift keying (BPS) or QPS sould be used to matc te capacity of te worst subcannels and someting like QAM-6 or QAM-64 to te best sub-cannels, te same is observed in following simulation scenario. In Figure 5(b), te sceme is tested for same number of transmitting and receiving antennas. Te N independent symbols, now modulated as QPS i.e. same constellation over all eigenmodes, are transmitted over parallel and independent sub-cannels. Te performance curves sow better performance as compared to te curves obtained wit QAM-6 using SVD based V-BLAST detection sceme. 4. CONCLUSION Efficient and less complex non-linear decomposition tecnique referred to as GMD is used for te formation of identical, parallel and independent sub-cannels over Rayleig flat-fading MIMO cannel. Te comparison of GMD and SVD based V-BLAST detection sceme using performance curves is being done. Te effectiveness of te strategy is observed for various numbers of transmitting and receiving antennas. In SVD based transceiver design we cannot use te same constellation among all parallel sub-cannels. Terefore, it involves te use of adaptive bit allocation to matc te capacity of eac sub-cannel wic increases te complexity. A trade off is always required to be defined between te capacity and te wen dealing wit SVD based strategy. On te oter and, for GMD tecnique, te same constellation wit moderate size e.g. QAM-6 or QAM-64 can be used to matc te capacity of most of te sub-cannels. Results of new MIMO IETS ensure superior performance due to and complexity in comparison wit conventional MIMO transmission system. ACNOWLEDGEMENT Te autors would like to acknowledge te financial support of Picsel Tecnologies Ltd, Glasgow, U. REFERENCES [] T. aiser, A. Bourdoux,. Boce, J. R. Fonollosa, et al., Smart antennas - state of te art, indawi Publising Corporation. New York, USA, 005. [] G. Tsoulos, MIMO system tecnology for wireless communications, CRC Press, 006. [3] Y. Jiang, J. Li, and W. ager, Joint transceiver design for MIMO communications using geometric mean decomposition, in IEEE Trans. on Signal Processing, vol. 53, no. 0, pp , Oct [4] P. W. Wolniansky, G. J. Foscini, G. D. Golden, et al., V-BLAST: an arcitecture for realizing ig data rates over te ric scattering wireless cannels, in Proc. Int. symp. on Signals, Systems and Electronics, pp , Pisa, Italy, September 998. [5] G. J. Foscini Layered space time arcitecture for wireless communication in a fading environment wen using multiple antennas, in Bell Labs. Tec. Jour., vol., no., pp. 4-59, 996. [6] G. J. Foscini, and M. J. Gans, On limits of wireless communications in a fading environment wen using multiple antennas, in Springer Jour. on Wireless Personal Communs., vol. 6, no. 3, pp , 998. [7] D. Wubben, R. Bonke, V. un, and. D. ammeyer, MMSE extention of V-BLAST based on sorter QR decomposition, in Proc. IEEE Conf. on Veicular Tecnology, vol., pp , Orlando, Fla, USA, October 003. [8] Y. Jiang, J. Li, and W. ager, Te geometric mean decomposition, in Elsevier Jour. on Linear Algebra and its Application, vol. 396, pp , Feb [9] D. Wubben, R. Bonke, V. un, and. D. ammeyer, Efficient algoritm for decoding layered space time codes, in IEE Electronic Let., vol. 37, no., pp , 00.