Sparse code multiple access (SCMA) is a promising uplink multiple access technique that can achieve superior spectral efficiency, provided that multidimensional codebooks are carefully designed. In this letter, we investigate the multiuser codebook design for SCMA systems over Rayleigh fading channels. The criterion of the proposed design is derived from the cutoff rate analysis of the equivalent multiple-input multiple-output system. Furthermore, new codebooks with signal-space diversity are suggested, while simulations show that this criterion is efficient in developing codebooks with substantial performance improvement, compared with the existing ones

Bao, J.

Ding, Z.

Karagiannidis, G. K.

Ma, Z.

Zhu, Z.

Lancaster E-Prints

IEEE ProofIEEE COMMUNICATIONS LETTERS 1On the Design of Multiuser Codebooks for Uplink SCMA SystemsJinchen Bao, Zheng Ma, Member, IEEE, Zhiguo Ding, Senior Member, IEEE,George K. Karagiannidis, Fellow, IEEE, and Zhongliang ZhuAbstract— Sparse code multiple access (SCMA) is a promising1uplink multiple access technique that can achieve superior2spectral efficiency, provided that multidimensional codebooks are3carefully designed. In this letter, we investigate the multiuser4codebook design for SCMA systems over Rayleigh fading chan-5nels. The criterion of the proposed design is derived from the6cutoff rate analysis of the equivalent multiple-input multiple-7output system. Furthermore, new codebooks with signal-space8diversity are suggested, while simulations show that this criterion9is efficient in developing codebooks with substantial performance10improvement, compared with the existing ones.11Index Terms— Sparse code multiple access (SCMA), multidi-12mensional constellations, cutoff rate, signal-space diversity.13I. INTRODUCTION14SPARSE code multiple access (SCMA) is a code/frequence15 domain non-orthogonal multiple access technique, pro-16posed for the fifth generation (5G) wireless networks [1]–[6].17It is able to provide good spectrum efficiency due to the18use of multidimensional complex codebooks/constellations.19However, the joint design of multiple multidimensional code-20books is generally challenging. Scanning the open literature,21the design of SCMA codebooks are limited to a few works,22while – to the best of the authors’ knowledge – the optimal23design criterion for multiuser codebooks is unknown. More24specifically, Nikopour and Baligh [1] and Taherzadeh et al. [4]25proposed a unified framework, where a multi-stage suboptimal26approach is used, for the joint design of multiuser codebooks.27Following this direction, several methods were proposed to28increase the coding gain of SCMA codebooks, or to lower the29complexity of multiuser detection. These approaches include30constellation with low projections [4], [5], spherical code-31books [7], dimensional rotation and interleave [8] and star-32QAM based multidimensional signaling [9]. However, they33only provide some rough insights on the multiuser codebooks,34and an efficient design criterion was not introduced.35Manuscript received July 12, 2016; revised July 28, 2016; acceptedJuly 28, 2016. This work is supported by Huawei HIRP project(no.YB201504), National Nature Science Foundation of China underGrant 61571373, Fundamental Research Funds for the Central Universities(no. 2682016ZDPY06). The associate editor coordinating the review of thispaper and approving it for publication was A. Lioumpas.J. Bao, Z. Ma, and Z. Zhu are with the Provincial Key Laboratoryof Information Coding and Transmission, Southwest Jiaotong Uni-versity, Chengdu 611756, China (e-mail: jinchen_bao@my.swjtu.edu.cn;zma@home.swjtu.edu.cn; jczuoyi@163.com).Z. Ding is with the School of Computing and Communications, LancasterUniversity, Lancaster LA1 4YW, U.K. (e-mail: z.ding@lancaster.ac.uk).G. K. Karagiannidis is with the Provincial Key Laboratory of InformationCoding and Transmission, Southwest Jiaotong University, Chengdu 611756,China, and also with the Electrical and Computer Engineering Department,Aristotle University of Thessaloniki, Thessaloniki 54124, Greece (e-mail:geokarag@auth.gr).Digital Object Identifier 10.1109/LCOMM.2016.2596759In this letter, a novel and efficient performance criterion is 36proposed, for the joint design of multiuser codebooks in uplink 37SCMA systems. The basic idea is to optimize the mutual 38information and cutoff rate of the equivalent multiple input 39multiple output (MIMO) system. Based on this criterion, we 40propose a rotated multidimensional constellation with signal- 41space diversity in order to choose the multiuser constellations. 42In addition, we investigate the applications of the proposed 43codebooks in a coded system with iterative multiuser detec- 44tion [6], through an extrinsic information transfer (EXIT) 45chart analysis. Simulation results verify the efficiency of 46the proposed codebook design and the significant achieved 47performance gain, compared to other existing methods. 48II. SYSTEM MODEL 49We consider J users/layers employing K -dimensional con- 50stellations transmitting over a multiple access channel, where 51each user employs only one layer. The message, which carries 52a data stream in bits for each user, is encoded and mapped 53into a K -dimensional symbol, x j . Then, each symbol is 54transmitted over K resources, such as time slots or orthog- 55onal frequency division multiplexing (OFDM) tones, without 56inter-carrier interference. At the receiver, the K -dimensional 57channel output vector, y = [y[1], · · · , y[K ]]t , is expressed as 58y =J∑j=1diag(h j )x j + n = HX + n, (1) 59where the diag(h), which denotes the diagonal matrix with its 60k-th diagonal element, corresponds to the k-th entry of vector 61h, and [·]t denotes transpose. The K × 1 data symbol (x j ), 62the K ×1 additive white Gaussian noise (AWGN) vector with 63independent and identically distributed (i.i.d) components of 64zero mean and variance N02 per dimension, and the K × 1 65channel gain vector, h j , between the j -th user and the receiver 66are given by 67x j =⎡⎢⎣x j [1]...x j [K ]⎤⎥⎦, n =⎡⎢⎣n[1]...n[K ]⎤⎥⎦, h j =⎡⎢⎣h j [1]...h j [K ]⎤⎥⎦. (2) 68The equivalent channel matrix H with size K × K J is given 69by 70H = [diag(h1), diag(h2), · · · , diag(hJ )]K×K J , (3) 71where X = [xt1, xt2, · · · , xtJ]t is the combined transmitted 72signal vector for J users. Therefore, the multiple access 73channel model can be viewed as a MIMO communication 74system with K J inputs and K outputs. The discrete input 75to the equivalent MIMO channel is a combined constellation, 76which can be represented as, X˜ = X1 × · · · × XJ , where X j 771558-2558 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.IEEE Proof2 IEEE COMMUNICATIONS LETTERSis the constellation of the j -th user, and it is assumed that its78norm, |X j | = M , for all j .79The sparseness of codewords can be completely char-80acterized by a factor graph matrix [1]. Denote φk =81 { j : 1 ≤ j ≤ J, x j [k] = 0}, the index of users that contribute82to the k-th resource, then the cardinality of φk is equal to the83row weight of the k-th row in this matrix.84III. MUTUAL INFORMATION ANALYSIS85Assuming that perfect channel knowledge is available at the86receiver. The mutual information I (X; y) between the discrete87input X and the continuous output y, conditioned on the fixed88channel matrix H, is given by89I (X; y|H)90= log C − 1C∑Xa1(π N0)K{∫yexp(− 1N0‖y − HXa‖2)91× log⎡⎣∑Xb,b =aexp(‖y − HXa‖2 − ‖y − HXb‖2N0)⎤⎦ dy},92(4)93where C = M J , is the number of all possible inputs to94the equivalent channel, and Xa, Xb ∈ X˜ are two distinct95symbols from the combined constellation. The average mutual96information can be obtained by averaging over the distribution97of H.98Since it is quite difficult—if not impossible—to deal with99an expression for the mutual information, we consider the100cutoff rate [10], instead. The conditional cutoff rate for a given101channel gain H is defined as102R(X; y|H)103= log C−log⎡⎣1+ 1C∑Xa∑Xb,b =aexp(−‖H(Xa − Xb)‖24N0)⎤⎦.104(5)105Let us define106(Xa, Xb, H) = exp(−‖H(Xa − Xb)‖24N0)107= exp(−∑Kk=1 ‖h[k]†(xa[k] − xb[k])‖24N0),108(6)109where xa[k] = [x1,a[k], · · · , x J,a[k]]t , is the vector of the k-th110component for J users, and h[k]† = [h1[k], · · · , h J [k]], are111the corresponding channel gains, and [·]† denotes conjugate112transpose.113Using the matrix decomposition, it holds that114(xa[k] − xb[k])(xa[k] − xb[k])† = UkkU†k , (7)115where Uk is unitary and k is a diagonal matrix, i.e. k =116diag(λ2k,1, · · · , λ2k,J ), with λ2k, j being the ordered eigenvalues117of the matrix (xa[k] − xb[k])(xa[k] − xb[k])†. Hence,118‖h[k]†(xa[k] − xb[k])‖2 = h[k]†UkkU†kh[k]119= hˆ[k]†k hˆ[k] = λ2k |hˆ1[k]|2, (8)120where we define, hˆ[k]† = h[k]†Uk . Thus, hˆ[k] has the same 121distribution as h[k], and λ2k = ‖xa[k] − xb[k]‖2, is the unique 122nonzero singular value in k (the index 1 is dropped here for 123λ2k,1). Due to the sparseness of the codebooks, 124λ2k =∑j∈φk|x j,a[k] − x j,b[k]|2. (9) 125Therefore, it holds that 126(Xa, Xb, H) =K∏k=1exp(−λ2k |hˆ1[k]|24N0). (10) 127Averaging over the distribution of H and assuming indepen- 128dent Rayleigh fadings, results in 129EH [(Xa, Xb, H)] =K∏k=1(1 + λ2k4N0)−1, (11) 130where E[·] denotes the mean. Thus, the mean cutoff rate is 131given by 132R(X; y) = EH[R(X; y|H)] 133≥ log C−log(1+ 1C∑Xa∑Xb,b =aEH [(Xa, Xb, H)])134= log C − log⎡⎣1+ 1C∑Xa∑Xb,b =aK∏k=1(1 + λ2k4N0)−1⎤⎦ . 135(12) 136The first step above follows from the Jensen inequality. 137It should be noted that the term in the square bracket of (12) 138is the union bound for the joint symbol error probability (SEP) 139for multiple users in i.i.d Rayleigh fading, which means 140that optimizing the mean cutoff rate is equivalent to the 141optimization of the average SEP. Therefore, we can formulate 142the cutoff rate criterion for the multiuser codebooks design, 143by making the union bound on the average SEP as small as 144possible. 145IV. DESIGN MULTIUSER CODEBOOKS 146According to the analysis in the previous section, the goal 147is to design the constellation X˜ , which maximizes the cutoff 148rate of the equivalent MIMO system. From (12), maximizing 149R(X; y) is equivalent to choose the combined constellation 150such that 151X˜ = arg minX˜∑Xa∑Xb,b =aK∏k=1(1 + λ2k4N0)−1152s.t. E[‖X‖2]= Es, (13) 153where Es is the average power. It is expected that the criterion 154is optimal in the sense of designing codebooks with large 155mutual information, since it involves all pairs of possible 156symbols for multiple users. 157Recall that the problem of designing multidimensional con- 158stellations for fading channels has been solved by using signal- 159space diversity, which rotates the QAM constellations with a 160unitary matrix, constructed either from the algebraic number 161theory or by computer search [11], [12]. Therefore, in this 162IEEE ProofBAO et al.: ON THE DESIGN OF MULTIUSER CODEBOOKS FOR UPLINK SCMA SYSTEMS 3work, we use the rotated constellation to build the multiuser163codebooks. In particular, we obtain the rotation matrices164through computer search over compact parameterizations of165unitary matrices. Note, that the null dimensions of codebooks166are discarded before the rotations. The N × N unitary matrix167can be written as [12]168R =N−1∏m=1N∏n=m+1Tmn, (14)169where Tmn is a complex Givens matrix [13], which changes170the identity matrix by replacing its (m, m)th, (n, n)th, (m, n)th171and (n, m)th elements with cos θmn , cos θmn , e−iηmn sin θmn172and −eiηmn sin θmn , respectively. The angles satisfy θmn ∈173[−π, π] and ηmn ∈ [−π/2, π/2]. So, the search for R is174reduced to the search for a sequence of phase of175θ = {θ1,2, · · · , θ1,N , θ2,3, · · · , θ2,N , · · · , θN−1,N },176η = {η1,2, · · · , η1,N , η2,3, · · · , η2,N , · · · , ηN−1,N }. (15)177It seems that optimizing the constellations using the above178criterion is intractable even for a moderate number of users179and codebook size, since we need to search N(N − 1)J180angles, and the summation in the right hand side of (13) will181add up C(C − 1) terms. However, searching results for two-182dimensional constellations with a small number of users show183that the rotation matrices are the same for all codebooks, and184are independent of the number of users. Therefore, we simplify185the optimization process by searching over a single rotation186matrix, even though this is suboptimal. Furthermore, we can187use the approach developed in [14], where all the entries188of the rotation matrix are equal in magnitude. Therefore,189by expanding the product in (14), we get θ = {π4}and190θ = {π4 , 0.6155, π4}for N = 2 and N = 3, respectively.191Exhaustive search is computationally feasible, provided, that192each user occupies a moderate number of resources such that193N ≤ 3.194V. SIMULATIONS AND DISCUSSIONS195We investigate the block error rate (BLER) performance for196an uncoded SCMA system with receive diversity, with two197antennas. The message passing algorithm (MPA) detector is198used with 6 iterations all the time. The SCMA system follows199a factor graph matrix as in (16),200F =⎡⎢⎢⎣1 0 1 0 1 00 1 1 0 0 11 0 0 1 0 10 1 0 1 1 0⎤⎥⎥⎦ , (16)201which occupies four resources and affords up to 6 users and202a maximum overloading factor 150% (J/K ). The simulations203are performed over i.i.d Rayleigh fading channels for 4-ary204and 16-ary codebooks, where the codebook from [4] and205spherical codes [7] are used as the benchmark. We also206provide the results of the star-QAM based codebooks [9] after207optimization using our approach, for which we extend α to208complex numbers and get β = 1, α = −i and α = 0.8 − 0.8i209for 4-ary and 16-ary codebooks, respectively.1210Fig. 1. Mutual information of 4-ary SCMA codebooks for 4 users.Fig. 2. Mutual information of 16-ary SCMA codebooks for 4 users.Figs. 1 and 2 plot the codebook constraint capacities of 2114-ary and 16-ary codebooks for 4 users with one receiving 212antenna, together with the theoretical limit rates of i.i.d 213Gaussian inputs. As it is evident, the proposed scheme outper- 214forms all the other codebooks in the high rate region for both 2154-ary and 16-ary codebooks, while the mutual information 216gain is more clear for 4-ary case. While the rate of the star- 217QAM scheme is quite small, a significant gain is achieved 218after optimization with our criterion, and it becomes as good 219as the proposed one for 4-ary codebook. 220Figs. 3 and 4 compare the BLER performance of the 221proposed scheme with existing ones for 4-ary and 16-ary code- 222boks, respectively. As it is observed, the proposed codebook 223has a gain about 0.8 dB and 0.6 dB over that in [4] for 4-ary 224and 16-ary cases, and a gain about 0.5 dB and 0.3 dB over the 225spherical codes, respectively. Without optimization, the star- 226QAM scheme yields the worst error performance. However, it 227performs much better after the optimization, which coincides 228with the result of mutual information in Figs. 1 and 2. 229The iterative multiuser receiver [6] was introduced for 230SCMA, because of its low-complexity feature and its good per- 231formance. To examine whether the proposed codebook works 2321The star-QAM based codebook targets on downlink channels, while itsperformance deteriorates in the uplink and for large constellation size.IEEE Proof4 IEEE COMMUNICATIONS LETTERSFig. 3. BLER of 4-ary codebooks for uplink SCMA systems with 6 usersover Rayleigh fading channels.Fig. 4. BLER of 16-ary codebooks for uplink SCMA systems with 6 usersover Rayleigh fading channels.Fig. 5. Exit chart curve of MPA detector for 6 users over Rayleigh fadingchannels.well when an error-correcting code is employed, we provide233the results of the EXIT chart [15] analysis of the MPA detector234in Fig. 5. For both 4-ary and 16-ary codebooks, the proposed235scheme has the steepest gradient as a function of the mutual236information of the a priori knowledge, and it ends up with the237largest extrinsic information when ideal a priori information is 238available. If an appropriate outer code is designed to match the 239EXIT curves, more reliable detection is possible and a larger 240iterative gain can be achieved for the proposed codebook, 241compared to other ones. Note that the EXIT curves do not 242necessarily to match the result of uncoded performance, since 243in addition to the constellations, the labeling also plays an 244important role in an iterative decoding scenarios. 245VI. CONCLUSIONS 246A performance criterion for the joint design of multiuser 247codebooks for uplink SCMA systems over Rayleigh fading 248channels was proposed, based on the cutoff rate of the equiv- 249alent MIMO system. The provided analysis showed that the 250used criterion significantly affects the average SEP. Based on 251this criterion, a codebook design with signal-space diversity, 252was proposed. Simulations showed that the proposed scheme 253provides substantial performance improvement compared to 254existing codebooks. 255REFERENCES 256[1] H. 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On the design of multiuser codebooks for uplink SCMA systems

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On the design of multiuser codebooks for uplink SCMA systems

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