A New Space-Repetition Code Based on One Bit Feedback Compared to Alamouti Space-Time Code

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1 Proceedigs of the 4th WSEAS It. Coferece o Electromagetics, Wireless ad Optical Commuicatios, Veice, Italy, November 0-, A New Space-Repetitio Code Based o Oe Bit Feedback Compared to Alamouti Space-Time Code BITA SOBHANI Electrical Egieerig Departmet Isfaha Uiversity of Techology Isfaha, Ira IRAN Abstract: - Alamouti space-time code for two trasmit ateas is most attractive due to simplicity of its ecodig ad decodig algorithms. However, it does ot provide ay codig gai. This paper proposes a ew space-repetitio code for two trasmit ateas ad oe receive atea based o oe bit feedback. This ca provide a relatively sigificat codig gai while it s ecodig ad decodig complexity is less tha Alamouti code. Although the orthogoal desig theory does ot satisfy, the ew code achieves maximum trasmit diversity. Eve if the reverse chael is ot reliable, the performace of the ew code still remais better tha that of the Alamouti code. All the above advatages are achieved while preservig spectral efficiecy with respect to Alamouti code. The oly price paid for these advatages is oe bit feedback from the receiver to the trasmitter which ca be easily retured with the ACK sigal to the trasmitter. Eergy computatios are also preseted ad it is show that the ew code ca save eergy compared to Alamouti code. Key-Words: - MIMO chaels, space-time codig, atea diversity, multipath propagatio, chael modelig, chael estimatio. 1 Itroductio Multiple-Iput Multiple-Output (MIMO) systems ca cosiderably icrease wireless commuicatio capacity by usig multiple trasmit ad receive ateas. MIMO chael capacity grows approximately liearly with the miimum umber of trasmit ad receive ateas [1],[]. A effective ad practical way to approach the capacity of MIMO chaels is to employ space-time codig. Space-time codig is a codig techique desiged for multiple atea trasmissio. There are several space-time codig techiques that itroduce correlatio betwee sigals trasmitted from various ateas at various time periods [3]. Space-time block codes are attractive due to simplicity of their ecodig ad decodig algorithms [4]. The codeword matrix of these codes is costructed by orthogoal desig theory [5] so that each two rows of the codeword matrix are orthogoal. The orthogoal property makes the code achieve maximum trasmit diversity. It also simplifies the maximum likelihood receiver by separately decodig sequeces trasmitted from ateas ad employig oly liear operatios o the received sigals. Alamouti code for two trasmit ateas is the oly space-time block code that is full rate for complex modulatios. For more tha two trasmit ateas ad complex modulatios, there is o full rate space-time block code. However, Alamouti code does ot provide ay codig gai. That is, its performace is the same as that of the ucoded system with the same diversity gai. This paper proposes a ew space-repetitio code for two trasmit ateas ad oe receive atea based o oe bit feedback that provides a relatively sigificat codig gai while it s ecodig ad decodig complexity is less tha Alamouti code. Although the orthogoal desig theory does ot satisfy, the ew code achieves maximum trasmit diversity. All the above advatages are achieved while preservig spectral efficiecy with respect to Alamouti code. The oly price paid for these advatages is oe bit feedback from the receiver to the trasmitter. The feedback bit provides partial chael state iformatio (CSI) at the trasmitter. This paper is orgaized as follows. Sectio gives a overview of space-time codig ad well kow Alamouti code. The it presets basic ideas of the ew code. Sectio 3 cofirms theory results by simulatios. Eergy computatios are also preseted i this sectio ad it is show that the ew code ca save eergy compared to Alamouti code. It is show that eve with ureliable reverse chael, the performace of the ew code still remais better tha that of the Alamouti code. Effect of imperfect

2 Proceedigs of the 4th WSEAS It. Coferece o Electromagetics, Wireless ad Optical Commuicatios, Veice, Italy, November 0-, chael estimatio o performace is also cosidered. Sectio 4 cocludes with a brief summary of results. New Code Basics Space-time codes are defied by a codeword matrix. Each of the rows determies sequeces trasmitted from a specific atea at various time periods ad each of the colums determies sequeces trasmitted from various ateas at a specific time period. It ca be deoted as: x 1 K xl = x1 K xl X (1) M M O M T T T x1 K xl i Where x t is the modulated symbol trasmitted from atea i at time period t, L is the frame legth ad T is the umber of trasmit ateas. For a space-time code, coditioal pairwise error probability is give by [3],[6]: ˆ E s P( X, X H) = Q ˆ H( X X) () N 0 where E s is the symbol eergy, N 0 is the oise power spectral desity, H is the R T chael matrix, R is the umber of receive ateas, X ad Xˆ are two T L codeword matrices ad H( X Xˆ ) is defied as: L R T i i H ( X Xˆ ) = h ( x xˆ ) (3) t= 1 j= 1 i= 1 j, i where h j, i is the chael coefficiet betwee trasmit atea i ad receive atea j..1 A review of Alamouti space-time code Alamouti space-time code for two trasmit ateas is the most famous space-time block code. Its codeword matrix is defied as: * x 1 X = * (4) x1 where x 1 ad x are two symbols modulated with m some -level real or complex modulatio techique. Usig (), coditioal pairwise error probability of Alamouti code for oe receive atea ca be computed as: t t P( X, Xˆ H) = E (5) s Q ( h1 + h )( d1 + d ) N 0 where d1 = x1 ˆx 1 ad d = ˆx, h 1 ad h are chael coefficiets betwee trasmit ateas 1 ad ad the oe receive atea, respectively.. New space-repetitio code Cosider a space-repetitio code for two trasmit ateas with the codeword matrix defied by: x1 (6) ax1 a where a is a real umber. a is chose equal to 1 so that the eergy trasmitted from each trasmit atea at a time period be the same as the eergy of x 1 or x. Hece, the trasmitter trasmits either: x1 (7.a) x1 or: x1 (7.b) x1 Usig (), pairwise error probability of this code, for oe receive atea is obtaied as: P( X, Xˆ H) = E (8) s * Q ( h + + Re{ })( + ) 1 h a h1h d1 d N 0 It is clear that if the term a * Re{ h 1 h } always remais positive, the the argumet of Q (.) for the ew code will always be more tha the argumet of Q (.) i (5) for Alamouti code. Therefore, its error probability will always be less tha that of the Alamouti code due to descedig property of Q (.). * Hece, if Re{ h 1 h } is positive, a is chose equal to 1 ad if egative, a is chose equal to -1. Therefore the receiver should compute the term Re{ h * 1 h } ad feedback oe bit determiig it s sig to the * trasmitter. If Re{ h 1 h } is positive the trasmitter trasmits (7.a) ad if egative, it trasmits (7.b). It is clear that fadig should be slow eough so that the chael coefficiets remai costat at least over oe block of data. From the special form of the codeword matrix, it is clear that we ca trasmit the codeword matrix: x (9) ax

3 Proceedigs of the 4th WSEAS It. Coferece o Electromagetics, Wireless ad Optical Commuicatios, Veice, Italy, November 0-, for every symbol x istead of (6). So the trasmitter trasmits either: x (10.a) x or x (10.b) x I this case the chael coefficiets should be costat over at least two blocks. It should be oted that sice oe symbol of m bits is trasmitted durig oe time period, spectral efficiecy of the ew code remais equal to m, which is the same as that of the Alamouti code. 3 Simulatio Results This sectio compares the ew code to Alamouti space-time code by simulatios. I the simulatios, complex basebad liear system model described i discrete time is cosidered. Idividual chaels betwee trasmit ad receive ateas are modeled by idepedet Rayleigh, flat ad slow fadig processes. It is assumed that the chael coefficiets chage from frame to frame ad the receiver ca estimate perfect chael state iformatio. Fig.1 plots i terms of SNR for the ew code, each of the compoet codes (10.a) ad (10.b), as well as Alamouti code usig complex QPSK modulatio. Fig. shows the same plots cosiderig real BPSK modulatio. The frame legth is assumed to be 30 symbols. Because chael coefficiets vary from frame to frame, the feedback bit must be computed at every frame. Space repetitio code with a= 1 or -1 depedig o CSI Alamouti code Space repetitio code with a= 1 Space repetitio Code with a= Fig. Performace compariso of the ew code ad Alamouti code for BPSK modulatio ad frame legth of 30 symbols. From the above simulatios, it ca be see that the ew code provides a codig gai of about 1.5dB at a of 0.4. While each of the compoet codes (10.a) or (10.b) achieves trasmit diversity gai of 1, the ew code achieves full trasmit diversity gai of. If the complexity of ecodig ad decodig algorithm is measured as the time elapsed i simulatios by MATLAB, it is see that ecodig ad decodig complexity of the ew code decreases by a amout of about 0% compared to Alamouti code because of the simple form of the codeword matrix. It should be oted that the above simulatios give a compariso of the ew space-repetitio code give that partial CSI is retured to the trasmitter ad Alamouti code which its performace is the same as that of the ucoded system whe full CSI is available at the trasmitter (trasmit beamformig). Space repetitio code with a= 1 or -1 depedig o CSI Alamouti code Space repetitio code with a= 1 Space repetitio code with a= Fig.1 Performace compariso of the ew code ad Alamouti code for QPSK modulatio ad frame legth of 30 symbols. 3.1 Eergy discussio It is clear that by icreasig the symbol eergy i Alamouti code, error probability ca be decreased util it reaches the error probability of the ew code. O the other had, for correct operatio of the ew code, the trasmitter must receive the feedback bit with a very low error probability. This requires that the feedback bit is set with high eergy. Now oe may ask whether the eergy cosumed for accurately estimatig the feedback bit be more that the extra eergy for achievig the same error probability i Alamouti code ad the ew code. This questio is aswered by followig computatios. From Figs.1 ad, it is see that for achievig

4 Proceedigs of the 4th WSEAS It. Coferece o Electromagetics, Wireless ad Optical Commuicatios, Veice, Italy, November 0-, the same error rate i Alamouti code ad ew code, Alamouti code eeds about 1.5dB more SNR compared to the ew code. So symbol eergy should be icreased from 1 uit (assumed for modulated symbol eergy) to 1.4 uits accordig to relatioship: E SNR = s T (11) N 0 (where the factor is iserted for expressig symbol eergy i basebad form ad T is i this case). Sice two symbols are trasmitted at every time period, extra eergy i each time period is 0.4=0.8 uits. If the forward chael is used as the reverse chael, the reverse chael becomes a 1 chael. Fig.3 plots BER i terms of SNR for a 1 ucoded system. For this chael, if BER of 10-4 is acceptable for estimatig the feedback bit, it requires a SNR equal to about 17dB accordig to this figure I digital commuicatio system desig, symbol period is typically chose so that the chael coherece time is betwee 10 to 100 symbol periods. For example if the chael coherece time is 100 symbol times, the usig the ew code will save eergy of about 17dB. Clearly, with slower fadig rate, more eergy is saved. By employig more equipmet such as icreasig the umber of receive ateas i the reverse chael, more eergy will be saved. For example if a 1 4 chael is used as the reverse chael istead of a 1 chael, the ew code will save eergy after 4 symbol times (istead of 40 symbol times). If a SISO chael is used as the reverse chael, the ew code will save eergy after 1978 symbol times. The feedback bit ca be easily trasmitted with the ACK sigal from the receiver to the trasmitter. Sice it may ot be possible to trasmit the feedback bit with high eergy due to, for example, iterferig with adjacet statios, it ca be trasmitted repeatedly istead of icreasig the eergy. I this case the above eergy discussio ad slow fadig costrait do ot apply. BER Fig.3 BER i terms of SNR for ucoded system with oe trasmit atea ad two receive ateas. The above eergy discussio does ot apply to high SNRs, sice at high SNRs, the chael oise is low ad the receiver does ot eed to trasmit the feedback bit with high eergy. Therefore, if for example SNR is assumed to be 5dB for the ew code (which is equivalet to a oise power spectral desity of about 1.6 uits accordig to (11) where T is ad Es is 1 i this case), the feedback bit trasmit eergy should be about 3 uits accordig to (11) where E s is replaced by bit eergy, SNR is 17dB, N 0 is 1.6 uits ad T is 1 i this case. Thus up to about 3/0.8=40 symbol times, icreasig symbol eergy i Alamouti code is preferred. After 40 symbol times, the ew code will save eergy of about 0.8 uits i every symbol time. 3. Effect of o ideal reverse chael Simulatios of Figs.1 ad are based o the assumptio that the reverse chael is ideal. That is, the feedback bit is estimated accurately at the trasmitter. However, i practice the reverse chael is ot ideal. Therefore oe may ask whether estimatio error of the feedback bit at the trasmitter may deteriorate the performace of the ew code so that it becomes eve worse tha that of the Alamouti code. I order to itroduce the effect of estimatio error of the feedback bit i the above simulatios, it is assumed that the trasmitter estimates the feedback bit with a error probability of about For example if the forward chael is used as the reverse chael, SNR i the reverse chael ca be cosidered to be about 1dB accordig to Fig.3. Based o this assumptio, the performace of the ew code compared to Alamouti code is show i Fig.4. As see i Fig.4, despite the feedback bit estimatio error of about 0.04, the performace of the ew code still remais better tha that of the Alamouti code. For example at a of 0.4, the performace of the ew code deteriorates about 0.3dB compared to the ideal reverse chael ad it is still about 1.dB better tha Alamouti code. At a of 0., the performace of the ew code

5 Proceedigs of the 4th WSEAS It. Coferece o Electromagetics, Wireless ad Optical Commuicatios, Veice, Italy, November 0-, deteriorates about 0.6dB compared to the ideal reverse chael ad it is still about 0.9dB better tha Alamouti code. slightly more tha Alamouti code. However its performace still remais better tha that of the Alamouti code. For example at of 0.3, the performace of the ew code is still about 1.4dB better tha Alamouti code. Space repetitio code (ideal reverse chael) Alamouti code Space repetitio code (o ideal reverse chael) Fig.4 Performace compariso of the ew code assumig o-ideal reverse chael ad Alamouti code for BPSK modulatio ad frame legth of 30 symbols. 3.3 Effect of imperfect chael estimatio So far, we have assumed that the receiver ca estimate perfect chael state iformatio. Now we cosider imperfect chael state iformatio. The chael fadig coefficiets are estimated by isertig pilot sequeces at the begiig of each trasmitted frame. It is assumed that the chael is costat over the duratio of a frame ad idepedet betwee the frames. With T trasmit ateas we eed to have T orthogoal pilot sequeces P 1, P, K, P T. The receiver estimates the chael fadig coefficiets h j, i by usig the j observed sequeces r. The miimum mea square error estimate of h, is give by [7] : j i j r. Pi j, i =, i = 1,, K P i ~ h, T (1) Fig.5 plots i terms of SNR of the ew code ad Alamouti code for real BPSK modulatio ad imperfect chael state iformatio. It is assumed that the chael coefficiets are costat over a frame of 30 symbols. The pilot sequeces iserted i each frame have a legth of 0 symbols. The simulatio results show that due to imperfect chael estimatio, both codes have degradatio i performace compared to the case of perfect chael estimatio. The ew code performace is degraded Space repetitio code with perfect chael estimatio Alamouti code with perfect chael estimatio Space repetitio code with imperfect chael estimatio Alamouti code with imperfect chael estimatio Fig.5 Performace compariso of the ew code ad Alamouti code for BPSK modulatio ad frame legth of 30 symbols assumig imperfect chael estimatio. 4 Coclusio The ew space-repetitio code based o oe-bit feedback preseted i this paper was show to be better tha Alamouti space-time code i both performace ad complexity eve if the feedback is ureliable. The feedback bit ca be easily retured with the ACK sigal to the trasmitter. All these advatages are achieved while preservig spectral efficiecy compared to Alamouti code. Refereces: [1] E. Telatar, Capacity of Multi-Atea Gaussia Chaels, Techical Report, October [] G.J. Foschii ad M.J. Gas, O Limits of Wireless Commuicatios i a Fadig Eviromet whe Usig Multiple Ateas, Wireless Persoal Commuicatios, Vol.6, No.3, 1998, pp [3] B. Vucetic ad J. Yua, Space-Time Codig, Wiley, 003. [4] S. Sadhu, R. Heath ad A. Paulraj, Space-Time Block Codes versus Space-Time Trellis Codes, Stadford Uiversity, IEEE Iteratioal Coferece o Commuicatios, Vol.4, 001, pp [5] V. Tarokh, H. Jafarkhai ad A.R. Calderbak, Space-Time Block Codes from Orthogoal

6 Proceedigs of the 4th WSEAS It. Coferece o Electromagetics, Wireless ad Optical Commuicatios, Veice, Italy, November 0-, Desigs, IEEE Trasactios o Iformatio Theory, Vol.45, No.5, 1999, pp [6] D. Tse ad P. Viswaath, Fudametals of Wireless Commuicatio, Cambridge Uiversity Press, 005. [7] V. Tarokh, A. Naguib, N. Seshadri ad A.R. Calderbak, Space-Time Codes for High Data Rate Wireless Commuicatio: Performace Criteria i the Presece of Chael Estimatio Errors, Mobility, ad Multiple Paths, IEEE Trasactios o Commuicatios, Vol.47, No., 1999, pp

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