Precoding and Spatial Modulation in the Downlink of MU-MIMO Systems

Precoding and Spatial Modulation in the Do wnlink of MU-MIMO Systems Azucena Duarte and Raimundo Sampaio Neto Abstract —This work f ocuses on the downlink communication of a multiuser MIMO system where the base station antennas and the users’ recei ving antennas are all activ e, but at each transmission, only a subset of the recei ve antennas is selected by the base station to r eceive the information symbols, and the par- ticular chosen subset (pattern) represents part of the information con veyed to the user . In this paper we present a mathematical model f or the system and develop expr essions that are fairly general and adequate for its analysis. Based on these expressions we propose a procedure to optimize the choice by the ERB of the sets of antenna patterns to be used in the transmissions to the different users, aiming at the maximization of the detection signal-to-noise ratio. Perf ormance results, with and without the optimization pr ocedure, ar e presented f or different scenarios. Index T erms —Multiuser MIMO system, Pr ecoding, Spatial Modulation, optimization I . I N T R O DU C T I O N Spatial Modulation (SM) and Generalized Spatial Modu- lation (GSM) [1]–[3] are recent proposals of communication schemes in MIMO systems. In GSM systems, only a subset of the transmit antennas are activ ated simultaneously at each transmission timeslot and they are used to send symbols belonging to the symbol constellation of the digital modulation employed, and the particular acti ve antenna combination rep- resents part of the transmitted information. This transmission scheme carries advantages o ver con ventional MIMO systems, once it allows the reduction of the RF chains used by the transmitter , and subsequent increase of the energy efficiency , without significant sacrifice of spectral ef ficiency . Recent works [4]–[6] focus on systems called PSM (“Pre- processing aided Spatial Modulation”) and GPSM (“General- ized Pre-coding aided Spatial Modulation”). These systems, differently from the previous ones, activ ates all transmit an- tennas simultaneously to transmit data only to a subset of the receiv e antennas, which are selected by the transmitter , and the particular chosen subset represents part of the information con veyed to the recei ver . Naturally , the implementation of this schemes requires the use of precoders. The abov e cited works consider communication between one transmitter and one receiv er . The w ork herein considers the downlink communication of a multiuser GPSM system, where base station and user antennas are all activ e, but in each transmission, only subsets of the receiv e antennas of each user receiv e information symbols. This paper presents a model for this system and develops fairly general e xpressions suitable for its analysis, and includes the relation between the total transmit energy and energy E k av ailable for detection of the signals destined to user k . This relation depends on channel matrices of all users and on the pattern set used by the base station in the transmissions. Based on this relation, a pattern set selection procedure is done by the base station, aiming at the maximization of E k , and subsequent maximization of the detection signal-to-noise ratio and minimization of the error probability . Performance results, with and without optimization procedure, are presented for different scenarios, in volving number of system users, number of antennas positioned at the users and number of antennas destined to recei ve information. I I . S Y S T E M A N D S I G N A L S Consider the downlink of a MU-MIMO system with N t antennas at the base station and K users, each equipped with N r receiving antennas, where N t ≥ K N r . In this system, all N r antennas belonging to the same user are activ e, but at each transmission only a subset of N iba antennas are selected by the transmitter to receive information symbols, and the particular selected subset represents part of the information transmitted by the base station to the user . Let N iba ( N iba ≤ N r ) the total number of combinations containing N iba out of N r antennas is gi ven by C t =  N r N iba  , (1) and the number of information bits that can be represented by different selections (different patterns) is k ssk = b log 2 ( C t ) c , (2) where b x c denotes the greatest integer less than or equal to x . If M is the modulation order, then the total number of bits transmitted by the base station is R = K ( k ssk + N iba log 2 ( M )) , bits/channel use (3) and N c = 2 k ssk is the number of valid patterns that can be used by the transmitter . For instance, let N r = 4 and N iba = 2 , resulting in C t = 6 , k ssk = 2 and N c = 4 . A possible set of 4 receive antenna combination patterns can be used by the base station to code 2 bits of information destined to user k , and can be represented by: Q k =  q k 1 , q k 2 , q k 3 , q k 4  =     1 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1     , (4) where q k 1 indicates that at each transmission slot the 2 in- formation symbols are con veyed to antennas 1 and 2 of user k , q k 2 indicates that antennas 1 and 3 will be the recipients of this information, and so forth. Note that as N c ≤ C t , L =  C t N c  possible choices exist for the set Q k . In the considered example L = 15 choices exist. As will be shown, the appropriate choice of Q k can impact system performance. A. Signal Model Let s ∈ C K N r × 1 the vector that contains the K information vectors con veyed to the users: s = h s 1 T , s 2 T , . . . , s K T i T , (5) where s k , k = 1 , 2 . . . , K , contains the information destined to user k . The nonzero entries are determined by the position vectors belonging to Q k , ex emplified in (4), which are occu- pied by complex symbols, statistically independent, belonging the signal constellation C of the modulation employed in the system. Statistically independent s k vectors are assumed. For analysis con venience, the vectors s k are represented by s k = p E k D ( q k ) ˙ s k , (6) where E k is the energy of the information symbols destined to user k , D ( z ) is the diagonal matrix that contains in its diagonal vector z and q k is the random vector , statistically independent of ˙ s k , with values drawn from the set Q k =  q k 1 , q k 2 , . . . , q k N c  , with equal probability . In the representa- tion of (6), ˙ s k contains symbols belonging to C in all its N r entries, all zero mean and unit variance. Thus, E [ ˙ s k ] = 0 and E h ˙ s k ˙ s k H i = I N r . V ector x ∈ C N t × 1 containing the elements transmitted by the base station antennas is gi ven by x = K X m =1 P m s m = K X m =1 p E m P m D ( q m ) ˙ s m , (7) where P m , m = 1 , 2 , . . . , K , denotes a N t × N r matrix that precodes the data destined to user m . B. Relation of ener gies The energy spent by the base station for signal transmission is calculated as E T = E [ k x k 2 ] = T r  E [ xx H ]  = T r ( K X m =1 K X l =1 P m E h s m s l H i P l H ] ) = T r ( K X m =1 P m E h s m s m H i P m H ) , (8) where T r { A } denotes the trace of matrix A . From (6), we hav e E h s m s m H i = E m E h D ( q m ) ˙ s m ˙ s m H D H ( q m ) i = E m E [ D ( q m )] = E m D ( q m ) , (9) with q m = E [ q m ] and the equiv alence D H ( q m ) = D ( q m ) = D 2 ( q m ) was employed. Combining (8) and (9), we get E T = K X m =1 E m T r n P m D ( q m ) P m H o = K X m =1 E m T r n D ( q m ) P m H P m o . (10) As D ( q m ) is a diagonal matrix, results that T r n D ( q m ) P m H P m o = q m T g m , (11) with g m = d  P m H P m  =  k p m 1 k 2 , k p m 2 k 2 , . . . , k p m N r k 2  T , (12) where d ( A ) denotes the vector whose components are the ele- ments of the diagonal matrix of A and p m i , m = 1 , 2 , . . . , K , represents the i th column of matrix P m . Combining (10) e (11) results that E T = K X m =1 E m g T m q m = E s K X m =1 ε m g T m q m = E s γ , (13) where E s = 1 /K P K m =1 E m is the av erage energy of the symbols destined to the users and ε m = E m /E s . The relation between the energy E T spent in transmission and the symbol energy destined to user k can be expressed as E k = E s ε k = E T ε k γ , (14) with γ = K X m =1 ε m g T m q m . (15) From (14) and (15) becomes e vident that for a giv en energy E T av ailable at the transmitter, the ener gy E k av ailable for user k depends on the precoding matrices of all K users, via g m , m = 1 , 2 , . . . , K , gi ven by (12), and on the K sets of patterns selected for transmission con veyed to all K user , via q m = 1 N c N c X i =1 q m i , m = 1 , 2 , . . . , K . (16) C. Optimized choice of the set of patterns Note from (14) and (15) that the minimization of γ by means of the K choices of Q m results in the maximization of the energy con veyed to each user . As the parcel of the summation in (15) are all positi ve and each one is a function of the characteristics associated to a single user, the minimization of γ can be carried out by the minimization of the parcels independently . In other words, among all L possible choices for the set Q , the one that results in minimal g T m q must be selected for user m , with q = 1 N c P N c i =1 q i and g m obtained from (12). This optimization procedure will be exemplified in Sec. IV. I I I . R E C E I V E R S V ector x in (7) can be re written as x = Ps , (17) where P ∈ C N t × K N r is giv en by P =  P 1 P 2 . . . P K  (18) and s is defined as in (5). Considering the structure of P giv en by (18), results that P H P contains in its main diagonal K submatrices P m H P m , m = 1 , 2 , . . . , K . Then, taking (12) into consideration, results that the vectors g m , that appear (15) are giv en by  g T 1 g T 2 . . . g T K  T = d  P H P  . (19) Considering (17), the vector containing the signal receiv ed by the k -th user can be expressed by y k = H k x + n k = H k Ps + n k ; (20) where n k is the Gaussian noise vector with components circularly symmetric, zero mean and co variance matrix K n k = σ 2 n I N r . The vector that represents the signals received by all users has the form y =      y 1 y 2 . . . y K      = Hx + n = HPs + n , (21) where n =  n T 1 , n T 2 , . . . , n T K  T and matrix H ∈ C K N r × N t is composed by H =  H T 1 , H T 2 , . . . , H T K  T , where H k ∈ C N r × N t is the matrix that connects the base station antennas to the antennas of user k . A. Systems with ZF pr ecoding The ZF precoding matrix is implemented by the left pseu- doin verse of H P = H H ( HH H ) − 1 , (22) resulting in y = HPs + n = s + n . (23) Then, considering (6), the vector receiv ed by user k becomes y k = s k + n k = p E k D  q k  ˙ s k + n k . (24) B. Detection In this work, maximum likelihood (ML) detection of vector s k was considered. A conv enient alternative representation of this vector is giv en by D  q k  ˙ s k = U k b k , (25) where b k ∈ C N iba × 1 is formed by the symbols belonging to constellation C , all zero mean and unit variance. The N r × N iba matrix U k is a submatrix of identity matrix I N r , obtained from q k according to: if the l th component of q k is zero, then the l th column of I N r is suppressed ( l = 1 , 2 , . . . , N r ) . Then, the set of patterns Q k corresponds to a set of position matrices U k =  U k 1 U k 2 . . . U k N c  (26) and ML detector , that decides ov er the information symbols and corresponding positions in the receiv ed vector , can be expressed as  ˆ U k , ˆ b k  = arg min U ∈ U k b ∈ C N iba k y k − p E k Ub k 2 , (27) where E k = E T ε k γ , γ is giv en by (15), and the vectors g m are obtained from      g 1 g 2 . . . g K      = d  P H P  = d   HH H  − 1  . (28) I V . N U M E R I C A L R E S U LT S In this section performance results, obtained by numerical simulation, are presented and expressed in terms of bit error rate (BER) of system users. The elements of all K channel matrices H k , k = 1 , 2 , . . . , K are modeled as statistically independent complex Gaussian random variables, circularly symmetric with zero mean and unit variance entries. Thus, it is admitted that users experience the same path loss. The influence of the channels in the signal detection of the dif ferent users is made explicit by (14), (15) e (28). Performance results are expressed in terms of the ratio SNR = E T σ 2 n , (29) where E T is the total energy spent in transmission, referred to the reception, and σ 2 n is the variance of the noise components in the reception. Then, results from (14) that the signal-to- noise ratio per recei ved bit av ailable at the detector in (27) is E k log 2 ( M ) σ 2 n = SNR log 2 ( M ) ε k γ . (30) The modulation employed in transmission is QPSK ( M = 4 ) and the transmitter destines equal energy to all users ( ε k = 1 ). Figures 1a and 1b illustrate, for N r = 4 e N r = 5 , respectiv ely , and K = 1 , system performance for different number of information bearing antennas, N iba . Note the perfor- mance improvement in terms of BER for lower N iba values, if N iba < N r is adopted. This performance adv antage is achiev ed at the cost of spectral efficiency reduction, as indicated in T ables I and II. For the results presented in Figures 1a and 1b, in the cases that L > 1 , the set of patterns Q is fixed and was chosen at random among all possible L choices. The simulation consisted of 1,000 channel matrices realizations with the transmission of 19,200 bits in each realization. The following results consider the optimized choice of pat- tern sets used by the transmitter , by means of the minimization of γ , according to the procedure proposed in Sec. II-C. (a) (b) Fig. 1: BER of GPSM and MIMO systems using ZF precoder . (a) N t = 8 , K = 1 and N r = 4 . (b) N t = 10 , K = 1 and N r = 5 . T ABLE I S Y S T EM C H A R AC T E R I S T IC S N t = 8 , N r = 4 , K = 1 . N iba C t N c R L 1 4 4 4 1 2 6 4 6 15 3 4 4 8 1 4 1 1 8 1 Figure 2 presents the results for the scenario used in Figure 1a ( K = 1 , N t = 8 , N r = 4 ) with N iba = 2 , , once this is the only v alue of N iba that permits more than one choice for the set Q . For this example, the C t = 6 possible patterns are giv en by the set C t = { (1 , 2) , (1 , 3) , (1 , 4) , (2 , 3) , (2 , 4) , (3 , 4) } , where the pair ( i, j ) indicates a pattern with 1’ s at the i th and j th positions, respectiv ely , and 0’ s at the remaining two and the L = 15 possible choices for the set Q can be represented by the ordered sets Q 1 = { (1 , 2) , (1 , 3) , (1 , 4) , (2 , 3) } , Q 2 = { (1 , 2) , (1 , 3) , (1 , 4) , (2 , 4) } , . . . , Q 15 = { (1 , 4) , (2 , 3) , (2 , 4) , (3 , 4) } , with corresponding mean vectors q given by q 1 = 1 / 4 [3 , 2 , 2 , 1] T , q 2 = 1 / 4 [3 , 2 , 1 , 2] T , . . . , q 15 = 1 / 4 [1 , 2 , 2 , 3] T , and the vectors g m , m = 1 , 2 . . . , K , obtained by (28). Results in Figure 3 illustrate a scenario with K = 2 users ( L = 15 ). The results of Figures 2 and 3 indicate a gain of approximately 1 dB obtained with the optimization procedure in the considered scenarios. It is noteworthy that once the choice of the sets Q m is done by the transmitter and can vary according to the channel matrix H , the transmitter must inform periodically the users T ABLE II S Y S T EM C H A R AC T E R I S T IC S N t = 10 , N r = 5 , K = 1 . N iba C t N c R L 1 5 4 4 5 2 10 8 7 45 3 10 8 9 45 4 5 4 10 5 5 1 1 10 1 Fig. 2: BER of ZF precoded GPSM system, with randomly chosen Q ( Q ar ), and optimized Q ( Q ot ). N t = 8 , K = 1 , N r = 4 e N iba = 2 . receiv ers which of the L sets is currently in use, in order to enable the correct signal detection. Results in Figures 2 and 3 consider this notification is receiv ed free of errors. For comparison purposes, the figures also present the performance obtained with the adoption of a fixed choice for the set Q , known by the users. A possible way to ex ecute the notification scheme of the set Q in use, is by means of a frame basis transmission scheme, where at the end of each frame the procedure of choosing the sets Q m , m = 1 , 2 , . . . , K is redone by the transmitter and signals informing the choice made are sent to each user during the notification interval of the following frame. In the case N r = 4 , N iba = 2 and QPSK modulation, for example, the information of the index of the L = 15 possible sets can be transmitted using 2 antennas at the receiver ( 4 bits). In order to reduce uncertainty and the possibility of detection error of the notification signal, the antenna pattern used during the periods of notification is known a priori by the receiv ers. A strategy to further reduce the error probability is to send the same notification information multiple times. The receiv er accumulates the received vectors in the notification period and performs detection using the resultant summation vector . In this procedure, if F is the number of repetitions adopted, a signal-to-noise ratio gain of 10 log 10 ( F ) dB is obtained. Figure 4 illustrates the results obtained with the notifi- cation strategy described above for the same scenario used Fig. 3: BER of ZF precoded GPSM system, with randomly chosen Q ( Q ar ), and optimized Q ( Q ot ). N t = 8 , K = 2 , N r = 4 e N iba = 2 . Fig. 4: BER of ZF precoded GPSM system, with optimized Q kno wn by the recei ver ( Q ot ) and notified to the recei ver ( Q not ). N t = 8 , K = 2 , N r = 4 e N iba = 2 . in Figure 3. Frames containing 3 , 200 information signal vectors ( 19 , 200 bits) to each user were adopted, and F = 10 repetitions (40 bits ) in the notification interval was adopted. In the simulation, a new sample of the channel matrix was generated at the end of each frame, totaling 1 , 000 samples of channel matrices. The accordance between the performance results presented in Figure 4 and those obtained with error- free notification evidences the effecti veness of the proposed notification method. V . C O N C L U S I O N This article considered the downlink of GPSM multiuser MIMO systems and developed expressions suitable for the analysis of these systems. Moreover , optimal procedures to determine recei ve antenna combinations were proposed, along with an effecti ve method of periodic notification of these choices to the users’ recei vers. Even more pronounced perfor- mance gains may be achie ved in the optimization procedure from the study of ne w scenarios and channel models that include, for example, path fading, shadowing and correlation among transmission and reception antennas. These studies are underway . N OT E S This paper was published in the Proceedings of the XXXV Brazilian Communications and Signal Processing Sympo- sium [7]. R E F E R E N C E S [1] M. D. Renzo, H. Haas, and P . M. 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