Layered Space-Time Codes for Wireless Communications Using Multiple Transmit Antennas

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1 Layered Spae-Time Codes for Wireless Communiations Using Multiple Transmit Antennas Da-shan Shiu and Joseph M. Kahn University of California at Bereley Abstrat Multiple-antenna systems provide very high apaity ompared to single antenna systems in a Rayleigh fading environment. Spae-time odes are hannel odes designed to exploit this high apaity for multiple-antenna systems without requiring instantaneous hannel nowledge at the transmitter. A pratial onern for high data rate spae-time odes is their deoding omplexity. The deoding omplexity with ML riterion an be prohibitively large. In this paper we fous on layered spae-time (LST) odes. Two types of LST odes the horizontally-layered spae-time (HLST) odes and the diagonally-layered spae-time (DLST) odes are presented. We analyze the performane of both types of LST odes under slow and fast fading onditions. We onlude that in a slow fading environment DLST odes have superior performane over HLST odes. The design riteria for DLST odes are proposed. I. INTRODUCTION Reent studies have explored the ultimate limit of multipleantenna systems from the information-theoreti point of view [1] - [3]. Consider a multiple-antenna system that has n transmitting and m reeiving antennas. It is shown that if the narrowband slow fading hannel an be modeled as an n m matrix with i.i.d. omplex Gaussian random entries the average hannel apaity of suh a system is approximately min(n m)-times higher than that of a single antenna system for the same overall transmitting power. In most appliations a major obstale to utilizing this high throughput is that the transmitter annot have the instantaneous information about the fading hannel. The transmitter thus must employ a hannel ode that an guarantee good performane with the majority of possible hannel realizations. Suh a hannel ode is inherently multi-spatial-dimensional and thus is alled a spae-time ode [4] [5]. Aside from the onsideration of ombatting hannel unertainty another pratial onsideration for spae-time odes is the deoding omplexity. As stated above the hannel apaity of a multiple-antenna system is approximately proportional to min(n m). This means that eah hannel usage on average an onvey proportional to min(n m) bits of information. The omplexity of deoding suh a high data-rate hannel ode using the maximal-lielihood (ML) riterion an be prohibitively high even if min(n m) is just moderately large; thus spae-time odes that admit high performane low omplexity suboptimal deoding algorithms are desirable. The layered spae-time (LST) arhiteture proposed by Foshini in [6] is a framewor of proessing spae-time signals. An LST ode is a hannel ode that is designed and proessed aording to the LST arhiteture. An LST ode is onstruted by assembling 1-D onstituent odes. With the use of interferene suppression and interferene anellation at the reeiver these onstituent odes an be separated and then deoded using onventional deoding algorithm developed for 1-D onstituent odes leading to a muh lower deoding omplexity ompared to ML deoding. Other possible low-omplexity deoding tehniques inlude sequential deoding (SD) [7] and multistage deoding [4] [8]. In this paper we analyze the performane of both horizontally-layered spae-time (HLST) odes and diagonally-layered spae-time (DLST) odes. We will show that DLST odes an ahieve better performane ompared to HLST odes. From our analysis we propose the design riteria i.e. the riteria to hoose the onstituent ode for DLST odes based on the trunated multidimensional effetive ode length (TML) and the trunated multidimensional produt distane (TMPD) of the onstituent ode. The remainder of this paper is organized as follows. In Setion II the baground of multiple-antenna systems and spae-time odes is reviewed. In Setion III we introdue the LST arhiteture. In Setion IV the performanes of HLST and DLST odes are analyzed. The analysis leads to the design riteria for DLST odes. We also present an example DLST ode. Conluding remars an be found in Setion V. II. BACKGROUND In this paper we fous on single-user to single-user ommuniation using multiple antennas at both ends over narrowband flat-fading hannels. We refer to a multiple-antenna system in whih the transmitter has n transmitting antennas and the reeiver has m reeiving antennas as an (n m) system. A general spae-time ode an be desribed as follows. The enoder first applies the spae-time ode to the input information bits to generate an n-row (possibly semi-infinite) matrix C. The matrix C represents the signal that is to be transmitted by the transmitter. Speifially the th row th olumn element of C denoted by represents the signal to be transmitted by antenna at time slot. We emphasize that there is no mehanism suh as time- frequeny- or ode-division multiplexing employed to ensure that the signals transmitted by different transmitting antennas are orthogonal upon reeption by the reeiver. The signal reeived by the reeiving antenna l during the time slot is denoted by r l. This

2 reeived signal r l ontains a superposition of transmitted signals 1 n 1 and an AWGN omponent v l. For a narrowband flat-fading hannel the gain onneting transmitting antenna and reeiving antenna l at time an be denoted by a omplex number h l. Define the vetors 1 ( 1 n )' r ( r 1 r r m )' v ( v 1 v v m )' and the hannel matrix H ( H l ) h l. The disrete-time input-output relation of an (n m) multiple-antenna system over a narrowband flat-fading hannel an be written in the following vetor notation: r H + v. (1) The following terminology is used in this paper. The matrix C a oded matrix output of the transmitter enoder is referred to as a spae-time odeword matrix. A spae-time odeword matrix an be thought of as a serial onatenation of n-tuples and an n-tuple is omposed of n symbols. Note that the first row of the matrix C is indexed as row zero not row one. To failitate the omparison between multiple transmit-antenna systems using spae-time odes and single transmit-antenna systems using onventional 1-D hannel odes at equal average transmit powers (total over all transmit antennas) the average energy of an n-tuple is regardless of the spatial dimensionality n. III. TH LAYRD SPAC-TIM ARCHITCTUR A. noding The enoding proess is illustrated in Fig. 1. In Fig. 1(a) the input information bit sequene is first demultiplexed into n subsequenes and eah subsequene is subsequently enoded by a 1-D enoder. These 1-D hannel odes are referred to as the onstituent odes (CC). The output of the onstituent oder is a sequene of symbols s 1. In the LST arhiteture the multi-spatial dimensional odeword matrix C is onstruted by assigning these symbols to the slots of C in a systemati fashion with the goal of reduing the reeiver omplexity. One intuitive assignment rule is to simply plae s at the th row th olumn slot of C. With this assignment rule the output oded symbol from onstituent enoder is always transmitted using the transmit antenna. This is illustrated in Fig. 1(b). Under this assignment rule the spae-time odeword matrix has an obvious horizontally layered struture. Therefore it is referred to as the horizontally layered spaetime (HLST) arhiteture. HLST was originally proposed by Foshini in [6]. Another assignment rule also proposed in [6] is the diagonally layered spae-time (DLST) arhiteture. In DLST instead of always sending the output symbols from a onstituent oder to a partiular transmit antenna they are fed to the n transmitting antennas in turn. DLST is illustrated in 1. The transpose and onjugate transpose of x are denoted by x' and x respetively. Fig. 1(). Using our odeword matrix notation to onstrut the DLST odeword matrix C the output symbols from onstituent oder are used to fill the leftmost NW-S diagonal of C and the output symbols from onstituent oder 1 are used to fill the next diagonal and so on. The layered struture of both HLST and DLST odeword matries are shown in Fig.. If the data rate of the onstituent ode maintains onstant regardless of n the data rate of an LST ode is obviously proportional to n. B. Deoding To deode an LST ode the reeived signal is proessed along both the spatial and temporal dimensions. Here we introdue the spatial signal proessing in the LST arhiteture. Fous on a given instane in time say. The transmitted n-tuple is and the reeived m-tuple is r H + v. The reeived signal r is a superposition of transmitted oded symbols saled by the hannel gain and orrupted by AWGN. The tas here is to determine the values of the n omponents information bits CC CC 1 CC 1 Fig. 1. LST ode enoding proess. Here n 3. ah square represents a symbol. (a) The inoming information bit sequene is first demultiplexed into n subsequenes. ah subsequene is then enoded using a onstituent ode. (b) In HLST the oded symbols from onstituent enoder are transmitted by antenna. () In DLST the oded symbols from a onstituent enoder are transmitted by the n transmitting antennas in turn. 3 1 to antenna 5 to antenna 1 to antenna 3 to antenna 4 (a) (b) HLST 1 () DLST to antenna 1 to antenna

3 noder α noder β C α β α 1 β 1 α β α 3 β 3 omponent. Beause R is upper triangular y ) + v ' + { onstribution from n T }. (3) noder γ (a) γ γ 1 γ γ 3 Note that in (3) only the interferenes from l l < are suppressed in y. The interferene term in (3) is aneled by using the available deisions ĉ + 1 ĉ + ĉ n 1. Assuming that these deisions are all orret the deision variable z is noder γ z ) + v ' 1 n 1. (4) noder β noder α The omplexity of the spatial proessing desribed above is O( n + nm ) per transmitted n-tuple. Other interferene suppression riteria suh as the linear least square riterion an also be employed instead [9]. of i.e. 1 n 1 with the only available information being r and H. Under the LST arhiteture the deisions on the values of these n omponents are made sequentially aording to a predetermined order. In this paper without loss of generality given a fixed is deided in desending order of. Consider the symbol n 1 whih is to be deteted first. An interferene suppression operation is employed to extrat a deision variable about n 1 from the reeived signal r. This deision variable denoted by z n 1 ontains a low level of interferene from other transmitted symbols. The deision on n 1 is then made based on z n 1. Maing use of the deision on n 1 the reeiver an modify the reeived signal r by removing the ontribution of n 1 to it. This modifying operation is referred to as interferene anellation. The proess of interferene suppression symbol value deision and interferene anellation is repeated for the remaining symbols n n 3. Suppose that the interferene from other transmitted symbols is to be ompletely suppressed using linear operation. Let the QR deomposition of H be H ( U R ) R where ( U R ) is a unitary matrix and R is an upper triangular matrix. Left-multiply the reeived signal r by ( U R ) ( ) r R + v ' () where C α β α1 (b) Fig.. The layered struture of an LST odeword matrix. (a) HLST. (b) DLST. v ' y ( U R ) v U R γ β1 α α 3 γ1 β β 3 α4 γ γ 3 β4 α 5 γ4 β 5 γ 5 is an m-tuple of i.i.d. AWGN noise IV. LAYRD SPAC-TIM CODS A. HLST To deode an HLST odeword matrix C the reeiver first extrats the deision variables for the symbols of the bottommost row of C. The resulting deision variable sequene { z n 1 } 1 is then passed to a onventional 1-D deoder of the orresponding onstituent ode to produe the deisions on the symbols of this row. The reeiver uses these deisions to modify the reeived signal sequene {r } and then proeeds to deode row n n 3 and so on. In short the HLST odeword matrix C is deoded row by row or layer by layer from bottom to top. Consider the th row of an HLST odeword matrix C. Let { } denote the atual transmitted symbol sequene on this row and { e } denote a distint possible transmitted symbol sequene. Conditioned on the hannel realization H {H H 1 } the probability that the lielihood of transmitting { e } is higher than { } is Prob e ( H ) Q N ) e N ) exp e where the matrix R omes from the QR deomposition of H i.e. H U R and Qx ( ) ( π) 1 / exp( x ) dx. In x (5) we mae use of the Chernoff bound for the Q funtion Qx ( ) exp( x ). Prob( e H ) is the onditional pairwise error probability between { } and { e }. The average pairwise error probability an thus be upper-bounded by taing the expeted value of the right side of (5) over the distribution of ) whih is a hi-squared distribution with (m ) degrees of freedom [1]. In a fast fading environment the ) are i.i.d. for distint. The average pairwise error probability Prob( e ) an be upper bounded (5)

4 by: Prob( e ) η( e ) exp N ) e 1 e ( m ) N η( e ) where η( e ) { e }. In a slow fading environment ) R for all and Prob( e ) an be upper bounded by: Prob( e ) R 4N e e 4N ( m ) +. B. DLST A DLST odeword matrix is deoded diagonal by diagonal. To illustrate this onsider the example DLST odeword matrix in Fig.. The reeiver first generates the deision variables for the symbols of the first diagonal of C namely α α 1 and α. Based on the deision variables this diagonal is deoded and the deision is then fed ba to remove the ontribution of this diagonal to the reeived signal. The reeiver then ontinues to deode the next diagonal and so on. Here we onsider the probability of a diagonal deision error. Consider the leftmost NW-S diagonal of a DLST odeword matrix. On this diagonal the transmitted symbols are 1 n 1. The probability that under the DLST deoding algorithm the lielihood of a distint diagonal e { e e 11 e n 1 n 1 } is higher than that of the transmitted diagonal { 11 n 1 n 1 } onditioned on the hannel realization H {H H 1 } an be derived by applying (3) noting that the interferene term in (3) is zero for this diagonal: Prob( e H ) Q n 1 quation (8) applies in both fast and slow fading environments beause the ) are i.i.d. for 1 n 1. The upper bound of the average pairwise error probability is again obtained by taing the expeted value of the righthand side of (8). When the SNR is high ( m ) N ) e n N ) exp e. (6) (7) (8) Prob( e) 1 + e N η( e ) ( e )( m ) η( e ) 4N where η( e ) { e }. When the SNR is low i.e. e 4N «1 for all using the approximation ( 1 + mx) 1 ( 1 + x) m for small mx (9) an be approximated by Prob( e) 1 + ε ε 1 4N η( e ) 1 + ε ε 1 4N (9) (1) where ε is an arbitrarily small positive number. C. Comparison By omparing (7) and (9) we onlude that the performane of HLST odes in slow fading environments is inferior to that of DLST odes. For an HLST ode the average pairwise error probability of the bottommost row is inversely proportional to the ( m n+ 1)th power of SNR. In ontrast in DLST the average pairwise diagonal error probability between two diagonals and e is inversely proportional to the ( m )th power of SNR. Therefore if onstituent η( e ) odes of equivalent data rate and omplexity are deployed the error probability of a DLST ode in a slow fading environment an be muh lower than that of an HLST ode. D. Design Criteria for DLST Codes Define the trunated multi-dimensional effetive length (TML) and the trunated multi-dimensional produt distane (TMPD) between two distint diagonals and e as TML m η( e ) TMPD e η( e ) and ( m ) m η( e ) (11a). (11b) At high SNR the pairwise error probability between and e is approximated by Prob( e) ( TMPD) 1 ( 4N ) TML. The ode design riterion is to maximize the minimum value of Prob( e) ( TMPD) 1 ( 4N ) TML over all pairs of distint diagonals. If the exat operating SNR is not nown but an be assumed to be reasonably high an approximate design riterion is to maximize the minimum two-tuple (TML TMPD) in ditionary order. At low SNR the pairwise error probability is approximated by (1). We define the exponent e ( m ) η( e ) to be the trunated multi-dimensional ulidean distane { e ( m ) } e ( m ) η( e) ( m )

5 (TMD) between and e. The ode design riterion at low SNR is to maximize the minimum TMD between any pair of distint diagonals.. xample In this example a (7 3) RS ode is used as a onstituent ode of a 7-D DLST ode. The (7 3) RS enoder maps three 8-ary input digits into seven 8-ary output digits. ah 8-ary digit selets a point (symbol) on the 8-PSK onstellation aording to the Gray ode mapping and these seven onstellation points are used to fill the slots of a diagonal of the 7-D DLST ode. The minimum TML of this ode given n m 7 is 15. Monte-Carlo simulations are performed to obtain the performane of this DLST ode in slow fading environments given n m 7. Fig. 3 shows the average error probability assuming perfet deision feedba. Fig. 3 shows that with an average SNR of 1 db it is possible to use this simple DLST ode to ahieve a data rate of 9 bits/s/hz with an average diagonal detetion error probability lower than 1-4. V. SUMMARY In this paper we onsidered layered spae-time odes. We showed that if the wireless hannel is i.i.d. Rayleigh fading an (n m) multiple-antenna system employing an LST ode an ahieve a throughput min(n m) times higher than that of a single-antenna system for a given overall transmit power limit. LST odes have two important harateristis. First the transmitter is not required to have the instantaneous hannel information to employ an LST ode. Seondly LST odeword matries are onstruted from one-spatial-dimensional onstituent odewords. The deoding omplexity of an LST ode 1-1 is only quadrati in the number of antennas maing it suitable for systems that have a large number of antennas. Furthermore the existing tehnology of 1-D ode an be effetively leveraged. We analyzed the performane of both HLST and DLST odes. Our result indiates that in a slow fading environment DLST odes have superior performane ompared to HLST odes. We proposed the design riteria for DLST odes based on the following parameters of the onstituent ode: TML TMPD and TMD. VI. RFRNCS [1]. Telatar Capaity of Multi-antenna Gaussian Channels AT&T- Bell Labs Internal Teh. Memo. June [] G. J. Foshini and M. J. Gans On Limits of Wireless Communiation in a Fading nvironment When Using Multiple Antennas aepted for publiation in Wireless Personal Communiations. [3] G. J. Foshini R. A. Valenzuela Initial stimation of Communiation ffiieny of Indoor Wireless Channel Wireless Networs 3 pp [4] V. Taroh N. Seshadri A.R. Calderban Spae-Time Codes for High Data Rate Wireless Communiation: Performane Criterion and Code Constrution I Trans. Inform. Theory vol. 44 no. Marh [5] J. Guey M.P. Fitz M.R. Bell and W. Kuo Signal Design for transmitter diversity wireless ommuniation systems over Rayleigh fading hannels in Pro. I VTC 96 pp [6] G. J. Foshini Layered Spae-Time Arhiteture For Wireless Communiation in a Fading nvironment When Using Multi-lement Antennas Bell Labs Tehnial Journal Autumn [7] S. Lin D. J. Costello rror Control Coding: Fundamentals and Appliations Prentie Hall New Jersey [8] A.J. Calderban Multilevel Codes and Multistage Deoding I Trans. Commun. vol. 37 no. 3 pp.-9 Marh [9] D. Tse Capaity Saling in Multi-Antenna Systems in preparation. [1] R. J. Muirhead Aspets of Multivariate Statistial Theory John Wiley & Sons New Yor 198. Average error probability SNR /N (db) Fig. 3. Simulation results of the error probability of a 7-D DLST ode using the (7 3) RS as its onstituent ode in slow fading environments. Here n m 7.