Downlink SCMA Codebook Design with Low Error Rate by Maximizing Minimum Euclidean Distance of Superimposed Codewords
11 Downlink SCMA Codebook Design with Low ErrorRate by Maximizing Minimum Euclidean Distance ofSuperimposed Codewords
Chinwei Huang,
Student Member, IEEE,
BorchingSu,
Member, IEEE,
Tingyi Lin, and Yenming Huang
Member, IEEE
Abstract —Sparse code multiple access (SCMA), as a codebook-based non-orthogonal multiple access (NOMA) technique, hasreceived research attention in recent years. The codebook designproblem for SCMA has also been studied to some extent sincecodebook choices are highly related to the system’s error rateperformance. In this paper, we approach the downlink SCMAcodebook design problem by formulating an optimization problemto maximize the minimum Euclidean distance (MED) of superim-posed codewords under power constraints. While SCMA codebookswith a larger minimum Euclidean distance (MED) are expected toobtain a better BER performance, no optimal SCMA codebookin terms of MED maximization, to the authors’ best knowledge,has been reported in the SCMA literature yet. In this paper, anew iterative algorithm based on alternating maximization withexact penalty is proposed for the MED maximization problem. Theproposed algorithm, when supplied with appropriate initial pointsand parameters, achieves a set of codebooks of all users whoseMED is larger than any previously reported results. A Lagrangedual problem is derived which provides an upper bound of MEDof any set of codebooks. Even though there is still a nonzero gapbetween the achieved MED and the upper bound given by thedual problem, simulation results demonstrate clear advantages inerror rate performances of the proposed set of codebooks over allexisting ones. The correctness and accuracy of error curves in thesimulation results are further confirmed by the coincidences withthe theoretical upper bounds of error rates derived for any givenset of codebooks.
Index Terms —5G, mMTC, non-orthogonal multiple access(NOMA), sparse code multiple access (SCMA), optimization, min-imum Euclidean distance (MED), semidefinite relaxation (SDR),alternating maximization, exact penalty.
I. I
NTRODUCTION
In the fifth generation (5G) wireless communications andbeyond, to enable the massive connectivity and high spectralefficiency for the Internet of Things (IoT) and the Factoriesof the Future (FoF), non-orthogonal multiple access (NOMA)[1] is considered an important multiple access scheme dueto its extended spatial efficiency as opposed to the conven-tional orthogonal multiple access (OMA), such as orthogonalfrequency-division multiplexing (OFDM). Among the many ex-isting schemes in NOMA [2], SCMA is regarded as a promising multiple access scheme [3]–[5]. Sparse code multiple access(SCMA) [6] is one kind of code-domain NOMA, which distin-guishes multiple users with the aid of codewords [7]. Incomingbits are directly mapped to multi-dimensional codewords of someset of SCMA codebooks, so the codebook design dominates theperformance of the SCMA-based NOMA system [8].Codebook design for SCMA has been studied extensively inthe past few years [9]–[27]. The overall design goal is to finda set of codebooks that result in a good performance, in termsof low error rate or large spectral efficiency, in the scenarios ofAWGN, uplink, and downlink fading channels. One of the majorapproaches for this goal is to find codebooks that have a largeminimum Euclidean distance (MED) [10]–[12], [15], [17], [26].The basic rationale behind the approaches of maximizing MEDis that a codebook with a larger MED usually results in bettererror rate performance and we choose the approach of MEDmaximization in this paper. In this regard, pioneering worksincluding Yu et al [25], and many others [13], [16], [17], [22]considered multi-stage design approaches by first constructinga mother constellation (MC) and then letting every user applythe mother constellation with different rotation and permutationoperations and occupy different resources. Under this multi-stage design approach, the maximization of MED of motherconstellation has been considered as an important issue andhas been studied to some extents [10], [12], [15]. While thisapproach has rather a simple complexity in the optimizationproblem, the fact that codebooks of all users are tied to afixed mother constellation implicitly impose extra and probablyunnecessary constraints to the choice of codebooks, and maylead to a suboptimal codebook design solution.More recently, the idea of MED maximization is studied witha newer definition of MED, namely, the MED of superimposedcodewords [11], [17], [26]. Many previously reported codebookdesign methods [11], [13], [17], [25]–[27] have used MEDof superimposed codewords as one design KPI. However, fewof the previous works have directly maximized the MED ofsuperimposed codewords with only power constraints and thereason may be the overwhelming complexity while dealing a r X i v : . [ c s . I T ] J a n with this non-convex optimization problem. Although an MEDmaximization problem has been formulated in [11], the algorithmproposed therein does not guarantee to obtain the optimal point.In fact, obtaining the set of codebooks with the maximal MEDis still an open question today, nor has an upper bound of themaximal MED been known yet. For convenience, hereafter weuse the term “MED” as the MED of superimposed codewords,rather than of the mother constellation, throughout this paper.In this work, we propose to approach the downlink SCMAcodebook design problem by maximizing the MED of thedesigned set of codebooks. The major contributions of the paperare summarized as follows: 1) A new method is proposed to dealwith the non-convex optimization problem based on the exactpenalty technique [36] with an alternating maximization [35]approach. 2) The aforementioned method achieves a codebookdesign that has a larger MED than any previously reporteddesign, which also shows the best error rate performances amongall existing codebooks. 3) A theoretical upper bound for MEDof any possible codebook designs, that was not known before,is obtained by deriving a Lagrange dual problem of the mainproblem.The rest of the paper is organized as follows. Section IIdescribes the downlink SCMA system model. In Section III, theMED maximization problem is formulated and a correspondingalgorithm is proposed. In Section IV, the dual problem of theMED maximization problem is derived. The numerical resultsare given in Section V. Finally, conclusions are made in SectionVI. A. Notations
Boldfaced lower case letters such as x represent column vec-tors, boldfaced upper case letters such as X represent matrices,and italic letters such as x, X represent scalars. Superscripts asin X T , X H , and X − denote the transpose, transpose-conjugate,and inverse operators, respectively, of a matrix. The binary set { , } is denoted by B . Given any positive integer N , Z N standsfor the set { , , . . . , N } . The N -dimensional complex, binary,and integer vector spaces are expressed as C N , B N , and Z N ,respectively. The ( N × M ) -dimensional complex, binary, andinteger matrix spaces are expressed as C N × M , B N × M , and Z N × M , respectively. The set of all n × n Hermitian matrices isdenoted by H n and the set of all positive semidefinite matricesis denoted by H n + . The notation A (cid:22) B means B − A ∈ H n + .Let N , N , e ( N ) n be the all-ones vector, all-zeros vector, and n -th standard unit vector, respectively, of dimension N . Let O M × N and I N be the M × N all-zeros matrix and N × N identity matrix, respectively. Operators (cid:107)·(cid:107) p , tr( · ) , vec( · ) , ◦ , ⊗ ,and × denote (cid:96) p -norm, trace, vectorization, Hadamard product,Kronecker product, and Cartesian, respectively. For some events A, B , the probability of A and the conditional probability of A given B are denoted by Pr { A } and Pr { A | B } , respectively. Throughout the paper, we adopt one-based indexing. For somevector x and matrix X , the i -th entry of x and the ( i, j ) -th entryof X are denoted by [ x ] i and [ X ] i,j , respectively.II. S YSTEM M ODEL
A. Downlink SCMA System
We consider downlink SCMA transmission on top of an under-lying orthogonal frequency division multiple access (OFDMA)system since SCMA multiplexed symbols need to be transmittedover orthogonal resources [28]. The block diagram of such adownlink SCMA system is shown in Figure 1 where J , K , and N B represent the number of users, the number of resources, andthe FFT size in the underlying OFDMA systems, respectively.The message carrying the data bits of the b -th block transmission(i.e., the b -th OFDM symbol) for the j -th user is encodedand mapped into K -dimensional symbols s j [ b ] by the j -thspecific SCMA encoder, and then the sum of these codewords, s [ b ] , is transmitted over K orthogonal resources, which are K consecutive subcarriers starting from the i sub -th subcarrier of the b -th OFDM block. Other subcarriers of the OFDM block maycontain data from other users. The result of the IFFT operation, x N B [ b ] , is further added a cyclic prefix of length N CP to obtain x [ b ] before being sent to the channel with a finite impulseresponse (FIR), characterized by h [ n ] , whose order is upperbounded by L − . Following the standard OFDMA receiver, atthe output of the FFT operation, excluding subcarriers containingdata from other users, the receiver observes the signal of the b -thblock transmission as r [ b ] = diag( h fsub ) s [ b ] + n (1)where h fsub = [[ h f ] i sub [ h f ] i sub +1 · · · (cid:2) h f ] i sub + K − (cid:3) T ∈ C K is the subvector of the frequency-domain channel gain vector h f ∈ C N B , s [ b ] = (cid:80) Jj =1 s j [ b ] ∈ C K is the transmittedsuperimposed codeword of the b -th block transmission, and n ∼ CN ( K , N I K ) is the additive white Gaussian noise. Thechannel gain vector h f is the discrete Fourier transform of vector (cid:2) h T TN B − L (cid:3) T ∈ C N B , where h = [ h [0] h [1] · · · h [ L − T ∈ C L is the channel vector which represents the channel impulseresponse as FIR filter of length L . B. SCMA Encoder
An SCMA encoder for the j -th user can be regarded as afunction defined as f j : B log M → S j (2)where S j ⊂ C K is the codebook of the j -th user with cardinality |S j | = M , i.e., S j contains M codewords. We require M to be apower of two so that each codeword in S j represents log M bitsof information. For notational convenience, we say that m ( j ) ∈Z M is an SCMA symbol according to the vector of data bits
Fig. 1. Downlink SCMA system model based on OFDMA. b j ∈ B log M from the j -th user: m ( j ) = 1 + (cid:80) log Mi =1 i − [ b j ] i . Each SCMA symbol m ( j ) maps to a K -dimensional complexcodeword s j ∈ S j , which is a sparse vector with N non-zero entries, and K > N . Following [11], [29], the j -th user’scodebook is chosen as S j = { V j C j e ( M ) m | m ∈ Z M } (3)where C j ∈ C N × M and V j ∈ B K × N are the constellationmatrix and the mapping matrix of the j -th user, respectively. Themapping matrix V j is obtained by removing K − N columnsfrom I K . Now, the codeword in S j selected by the j -th user forthe b -th block transmission can be expressed as s j [ b ] = f j ( b j [ b ]) = V j C j e ( M ) m ( j ) [ b ] ∈ C K (4)where b j [ b ] ∈ B log M is the vector of the given data bitsand m ( j ) [ b ] ∈ Z M is the corresponding SCMA symbol. Allcodewords chosen by the J users will be summed up togetherto form a superimposed codeword before being assigned toorthogonal resources: s [ b ] = J (cid:88) j =1 s j [ b ] . A superimposed codeword is determined by all SCMA symbolsfrom J users, collectively a vector in Z JM which we refer toas the multiplexed symbol . There are in total M J multiplexedsymbols, and we denote the k -th multiplexed symbol, k ∈ Z M J ,by m k defined by m k = [ k , k , ..., k J ] ∈ Z JM where k , ..., k J ∈ Z M are the unique integers that satisfy k = 1 + J (cid:88) j =1 ( k j − M j − ∈ Z M J . (5) Note that the set Z JM ⊂ Z J stands for the Cartesian productof J identical sets, Z M . The set of all SCMA superimposedcodewords can be expressed as S = J (cid:88) j =1 V j C j e ( M ) k j (cid:12)(cid:12)(cid:12)(cid:12) k ∈ Z M J , (6)where k ∈ Z M J and k j ∈ Z M follow the relationship definedin (5).Since each of the J users has a distinct mapping matrix V j ,it is obvious that J ≤ (cid:0) KN (cid:1) must hold for any given K and N .Furthermore, a loading factor is defined as λ = J/K [13] thatdirectly determines the spectral efficiency of SCMA. It mustsatisfy λ > for SCMA to be more spectral efficient thanconventional OMA, and it is well known that this implies K ≥ , ≤ N ≤ K − , and J > K [13]. In order to achieve themaximum sparsity, N = 2 is often chosen. In this paper, wechoose to study the simplest case by taking K = 4 , N = 2 , and M = 4 , which is also the most studied case in the literature.Since J ≤ (cid:0) KN (cid:1) , we take J = 6 and set the mapping matrices V , V , . . . , V J as follows: V = V = V = V = V = V = . (7)Moreover, we refer to {S j } Jj =1 , the set of all codebooks, by the collection of codebooks [18], or simply the codebook collection throughout the paper. A codebook collection is completelydetermined by the J constellation matrices C j , j = 1 , ..., J sincethe mapping matrices V j are fixed as in (7). C. SCMA Decoder1) MAP Detection:
Supposing that the channel estimationis perfect and the codebook collection are available for thereceiver, the detection of SCMA can be regarded as a problemof traditional multi-user detection, which can be solved by jointmaximum a posterior (MAP) [30]. Then, given some receivedsignal r of some block transmission (i.e., r [ b ] = r for some b ),the detected multiplexed symbol, denoted by (cid:98) m , will be (cid:98) m = arg max m ∈Z JM Pr { m | r } . (8)The j -th user’s detected symbol is the j -th component of (cid:98) m :[11], [24] (cid:98) m ( j ) = [ (cid:98) m ] j , (9)and the vector of the corresponding detected data bits is denotedby (cid:98) b j . An equivalent way to express the j -th user’s symbol is[30], [41]: (cid:98) m ( j ) = arg max m ∈Z M (cid:88) m ∈Z JM , [ m ] j = m Pr { m | r } (10) = arg max m ∈Z M (cid:88) m ∈Z JM , [ m ] j = m Pr { m } K (cid:89) k =1 Pr { [ r ] k | m } . (11) A proof for the equivalence of (9) and (10) is given in AppendixA. The forms in (10) or (11) are more favorable in some worksbecause it is easier to apply a message passing algorithm basedon (11) than on (9) [31].Given the result of the detection above, we have (cid:98) m ( j ) [ b ] = (cid:98) m ( j ) and (cid:98) b j [ b ] = (cid:98) b j for all j ∈ Z J in Figure 1 if r [ b ] = r .
2) Message Passing Algorithm:
Solving problems (8), (10),or (11) with brute-force has exponential complexity. Thanks tothe sparsity of the codewords, the solution of this problem can beapproximated by an iterative decoding algorithm, message pass-ing algorithm (MPA), which updates the extrinsic information offunction nodes (FNs) and variable nodes (VNs) along the edgesin the factor graph and has moderate complexity [30], [31].III. M
AXIMIZATION OF M INIMUM E UCLIDEAN D ISTANCE
A codebook collection with a large MED tends to have asmaller detection error when an MAP detector (10) is applied[11], [27]. And since the popular MPA detector usually performsvery close to MAP detectors, it makes sense to choose acodebook collection with an MED that is as large as possible.In this section, we formulate the SCMA codebook designproblem as an optimization problem that maximizes the mini-mum Euclidean distance (MED) under some power constraints.We follow the definitions of MED from [11], [26], that is,the MED of any two superimposed codewords of a codebookcollection.Note that in some references, MED is taken as a reasonableKPI only for the AWGN channel [17]. However, the considered system uses OFDMA as the provision of orthogonal resources.It is believed that the adjacent OFDM tones tend to be nearlyidentical over fading channels [28] and thus the received signalover some fading channel will be similar to the one overAWGN channel because putting identical gain on each resourceis playing the same role as amplifying the noise on each resourceby the reciprocal of the gain. Therefore we believe that MEDcan be a reasonable design criterion even when Rayleigh fadingchannel is considered.
A. Problem Formulation1) Minimum Euclidean Distance:
For any given codebookcollection determined by the matrices V j , C j , j ∈ Z J , we firstdefine the square of Euclidean distance of the k -th and l -thpossible superimposed codewords as d kl = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J (cid:88) j =1 V j C j (cid:16) e ( M ) k j − e ( M ) l j (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (12) where k, l ∈ Z M J and k j , l j ∈ Z M , ∀ j ∈ Z J are definedaccording to the same convention as in (5). Then the minimumEuclidean distance (MED) d min is defined as d min = min k,l ∈Z MJ k (cid:54) = l (cid:112) d kl . (13) Note that there are totally (cid:0) M J (cid:1) possible pairs of superimposedcodewords.
2) MED Maximization Problem:
We aim to maximize theMED (13) subject to the power constraint. Therefore the problemis formulated as maximize C ∈ C N × MJ ,t ∈ R t (14a) subject to d kl ≥ t, ∀ k, l ∈ Z M J , k (cid:54) = l (14b) M tr( C Hj C j ) = P, ∀ j ∈ Z J (14c)where C (cid:44) [ C C · · · C J ] contains the codewords for all users, t is an extra real-valued variable representing the square of MED,and P is the limit of each user’s average power of transmittedcodewords. Here we choose equality power constraints so thateach user is ensured to achieve the same power limit. Forconvenience, and without loss of generality, we set P = 1 throughout the paper.Problem (14) can be transformed into an equivalent problemin a QCQP form. Specifically, we define x = vec( C ) ∈ C n x with n x = N M J , and reformulate Problem (14) as [11] maximize x ∈ C nx ,t ∈ R t (15a) subject to x H A i x ≥ t, ∀ i ∈ Z ( MJ ) (15b) x H B j x = M P, ∀ j ∈ Z J (15c) where A i and B j are distance constraint and power constraintmatrix, respectively. The matrices A i and B j have closed-formexpressions as below. Note that they are real, symmetric, verysparse, and with nonzero entries limited to only values − and . The matrix A i is A i = A k,l = J (cid:88) j =1 J (cid:88) q =1 (cid:16) ( e ( J ) q ) T ⊗ I MN (cid:17) T K ( q,j ) k,l (cid:16) ( e ( J ) j ) T ⊗ I MN (cid:17) (16) where K ( q,j ) k,l = (cid:16) ( e ( M ) k q − e ( M ) l q )( e ( M ) k j − e ( M ) l j ) T (cid:17) ⊗ (cid:0) V Tq V j (cid:1) , and the matrix B j is B j = diag( e ( J ) j ) ⊗ I NM . (17)The derivation of (16) is provided in Appendix B.Problem (15) is not convex, so we apply the technique ofsemidefinite relaxation: let X = xx H ∈ H n x + and reformulatethe problem as maximize X ∈ H nx + ,t ∈ R t (18a) subject to tr( A i X ) ≥ t , ∀ i ∈ Z ( MJ ) (18b) tr( B j X ) = M P, ∀ j ∈ Z J (18c) rank( X ) = 1 (18d)Note that problems (15) and (18) are equivalent since for any X ∈ H n x + that satisfies the rank constraint (18d), there alwaysexists some x ∈ C n x (subject to a unit-norm complex ambiguity)such that X = xx H . B. Exact Penalty Approach and Biconvex Problem Formulation
Since the rank constraint (18d) is not a convex constraint,we can not solve it directly by the tools for solving convexoptimization problems. Therefore, we propose a method basedon the concept of alternating maximization and exact penaltyapproach mentioned in [35], [36] to obtain a rank-one solutionof Problem (18). We first formulate a new problem based onProblem (18) as follows maximize X , X ∈ H nx + ,t ,t ∈ R t + t (19a) subject to tr( A i X ) ≥ t , ∀ i ∈ Z ( MJ ) (19b) tr( A i X ) ≥ t , ∀ i ∈ Z ( MJ ) (19c) tr( B j X ) = M P, ∀ j ∈ Z J (19d) tr( B j X ) = M P, ∀ j ∈ Z J (19e) tr( X X ) = tr( X )tr( X ) . (19f)The following theorem shows that Problem (19) is equivalent toProblem (18). Theorem 1. If { X (cid:63) , X (cid:63) , t (cid:63) , t (cid:63) } is an optimal point for (19) , then X (cid:63) = X (cid:63) and t (cid:63) = t (cid:63) and { X , t } = { X (cid:63) , t (cid:63) } is an optimalpoint of (18) . Conversely, if { X (cid:63) , t (cid:63) } is an optimal point of (18) ,then, { X , X , t , t } = { X (cid:63) , X (cid:63) , t (cid:63) , t (cid:63) } is an optimal point of (19) .Proof: By Theorem 1 in [37], if there are any X , X ∈ H n x + satisfying constraint (19f), the necessary and sufficientconditions will be both of rank one and X = α X where α is apositive scalar. Suppose { X (cid:63) , X (cid:63) , t (cid:63) , t (cid:63) } is an optimal point ofProblem (19). Then, (19f) implies X (cid:63) = α X (cid:63) for some α > .Since X (cid:63) and X (cid:63) satisfy constraints (19d) and (19e), we have M P = tr( B j X (cid:63) ) = tr( B j α X (cid:63) ) = α · tr( B j X (cid:63) ) = αM P, which implies α = 1 and X (cid:63) = X (cid:63) . It can also be shownthat t (cid:63) = t (cid:63) = min i tr( A i X (cid:63) ) using (19b), (19c), and (19a).Then, we can show that { X (cid:63) , t (cid:63) } is an optimal point of (18)by contradiction: if { ˜ X , ˜ t } is some optimal point for (18) with ˜ t > t ∗ , then setting { X , X , t , t } = { ˜ X , ˜ X , ˜ t, ˜ t } in (19) willresult in a larger value in (19a) ( t + t = 2˜ t > t (cid:63) = t (cid:63) + t (cid:63) ).Conversely, if { X (cid:63) , t (cid:63) } is an optimal point of Problem (18),then { X , X , t , t } = { X (cid:63) , X (cid:63) , t (cid:63) , t (cid:63) } can be shown to be anoptimal point in (19) by contradiction as follows. If there is anyother feasible point of Problem (19), say, { X (cid:48) , X (cid:48) , t (cid:48) , t (cid:48) } , suchthat t (cid:48) + t (cid:48) > t + t , then the point { X (cid:48) , t (cid:48) } = { X (cid:48) , t (cid:48) } = { X (cid:48) , t (cid:48) } , will results in a larger value in (18a) ( t (cid:48) = ( t (cid:48) + t (cid:48) ) / > ( t + t ) / t (cid:63) ).To deal with constraint (19f), we apply the exact penaltyapproach and introduce the non-positive penalty function tr( X X ) − tr( X )tr( X ) whose value is zero if and only if (19f) holds [37], [40].By the equality constraints (19d), (19e) and using (17), wehave tr( X ) = tr( X ) = JM P , implying that tr( X )tr( X ) is a constant. Therefore, the penalty function can be furthersimplified as tr( X X ) and then we formulate another problemas maximize X , X ∈ H nx + ,t ,t ∈ R t + t + w · tr( X X ) (20a) subject to tr( A i X ) ≥ t , ∀ i ∈ Z ( MJ ) (20b) tr( A i X ) ≥ t , ∀ i ∈ Z ( MJ ) (20c) tr( B j X ) = M P, ∀ j ∈ Z J (20d) tr( B j X ) = M P, ∀ j ∈ Z J . (20e)where w > is a positive weight of the simplified penaltyfunction tr( X X ) , which makes the optimization problem tendto meet constraint (19f).Problem (20) is a biconvex optimization problem [36], mean-ing that it is a convex problem in X and t when X and t are given constants, and vice versa. An iterative algorithm Algorithm 1
Proposed Algorithm based on Alternating Maxi-mization and Exact Penalty Approach
Input: J , C , C , · · · , C J , V , V , · · · , V J , ϕ max , { w ,ϕ } ϕ max ϕ =1 , { w ,ϕ } ϕ max ϕ =1 . Output: ˜ x Initialization : Create A i , ∀ i ∈ Z ( MJ ) and B j , ∀ j ∈ Z J according to (16) and (17). Initialize X (0)2 = xx H , where x = vec([ C C · · · C J ]) . Set t (0)2 = min i tr ( A i X ) and ϕ = 0 . repeat Solve Problem (20) for { X ( ϕ +1)1 , t ( ϕ +1)1 } while fixing { X , t } as { X ( ϕ )2 , t ( ϕ )2 } with the weight being chosen as w = w ,ϕ . Solve Problem (20) for { X ( ϕ +1)2 , t ( ϕ +1)2 } while fixing { X , t } as { X ( ϕ +1)1 , t ( ϕ +1)1 } with the weight being chosenas w = w ,ϕ . ϕ ← ϕ + 1 until | tr( X ( ϕ )1 )tr( X ( ϕ )2 ) − tr( X ( ϕ )1 X ( ϕ )2 ) | < − or ϕ >ϕ max Perform singular value decomposition (SVD) on X ( ϕ )2 : X ( ϕ )2 = UΣV H , with the singular values along the maindiagonal of Σ in non-ascending order. Set σ = (cid:2) Σ (cid:3) , and σ = (cid:2) Σ (cid:3) , . if σ /σ ≤ − then Obtain ˜ x = √ σ · Ue ( n x )1 . else Declare failure of convergence. end if exploiting alternative maximization is presented in Algorithm 1,where ϕ max is the maximum allowable number of iterations,and, noting that the weighting w can be chosen different amongvarious iterations, w ,ϕ and w ,ϕ represent the positive weightsfor the ϕ -th iteration.Unfortunately, Algorithm 1 is not guaranteed to always con-verge within a satisfactory number of iterations. As will be elab-orated in Section V, we observe that if the weight is kept constantduring all iterations, the algorithm is more prone not to converge.A possible solution to this problem, accordingly to our empiricalexperiments, is to manually change the weights { w ,ϕ } , { w ,ϕ } .We found that if the weights are set as a sequence that graduallyincreases, the algorithm tends to converge to a solution with alarge MED.IV. D UAL PROBLEM OF
MED M
AXIMIZATION P ROBLEM
In this section, we derive the Lagrange dual problem associ-ated with the primal problem (15). Unlike the primal problem,which is non-convex, the dual problem is always a convexproblem that is easier to solve. And the optimal value of a dual problem serves as an upper bound of the optimal value of theprimal problem. Noting that Problems (14) and (15) are the twoequivalent forms of the primal problem, one can derive the dualproblem for each of the forms. Here, we choose to derive theLagrange dual of Problem (15) since its QCQP form makes thedual problem derivation much easier than Problem (14). First ofall, the Lagrangian of Problem (15) is L ( { λ i } , { µ j } , x , t )= t + ( MJ ) (cid:88) i =1 λ i ( x H A i x − t ) + J (cid:88) j =1 µ j ( MP − x H B j x )= x H ( MJ ) (cid:88) i =1 λ i A i − J (cid:88) j =1 µ j B j x + − ( MJ ) (cid:88) i =1 λ i t + J (cid:88) j =1 µ j MP , (21) where we introduced Lagrange dual variables λ i and µ j asso-ciated with constraints (15b) and (15c), respectively. Then, theLagrange dual function g ( { λ i } , { µ j } ) , defined as g ( { λ i } , { µ j } ) = sup x ,t L ( { λ i } , { µ j } , x , t ) , (22)is unbounded above if any eigenvalue of (cid:80) ( MJ ) i =1 λ i A i − (cid:80) Jj =1 µ j B j is greater than zero or (cid:80) ( MJ ) i =1 λ i (cid:54) = 1 . Otherwise,the dual function is g ( { λ i } , { µ j } ) = J (cid:88) j =1 µ j M P. (23)Therefore, the Lagrange dual problem is found to be minimize { λ i } , { µ j } ,ν J (cid:88) j =1 µ j M P (24a) subject to ( MJ ) (cid:88) i =1 λ i A i (cid:22) J (cid:88) j =1 µ j B j (24b) ( MJ ) (cid:88) i =1 λ i = 1 (24c) λ i ≥ , ∀ i ∈ Z ( MJ ) . (24d)It is well known that weak duality [32] dictates that the optimalvalue of the dual problem (24) is an upper bound of the optimalvalue of the primal problem (15), as also that of (14) sinceproblems (14) and (15) are equivalent. In fact, as we will findlater in Section V, the optimal values of primal problem (15) anddual problem (24) will coincide, at least for the case of J = 3 ,suggesting that strong duality holds for this case. V. S
IMULATION R ESULTS
In this section, we conduct numerical simulations to verifythe proposed methods presented in Section III and compare theirperformances with existing methods [9], [10], [16]–[18], and alsothe MED upper bound derived in Section IV. Throughout allsimulations, we set the number of resources as K = 4 , thecardinality of codebooks as M = 4 , the constellation sizes as N = 4 , and 15 times message passing iterations. The definitionsof SER and BER are shown as follows. Definition. The symbol error rate (SER) is defined as P e,s = 1 J J (cid:88) j =1 M (cid:88) m =1 Pr { m ( j ) = m } Pr { (cid:98) m ( j ) (cid:54) = m | m ( j ) = m } (25) where (cid:98) m ( j ) was defined in (9) , and Pr { m ( j ) = m } is assumedto be /M for all m ∈ Z M .Definition. The bit error rate (BER) is defined as P e,b = 1 J log M J (cid:88) j =1 log M (cid:88) i =1 (cid:16) Pr { [ b j ] i = 0 } Pr { [ (cid:98) b j ] i = 1 | [ b j ] i = 0 } (26) + Pr { [ b j ] i = 1 } Pr { [ (cid:98) b j ] i = 0 | [ b j ] i = 1 } (cid:17) (27) where, for all j ∈ Z J , b j , (cid:98) b j follow the definitions in SectionII, and Pr { [ b j ] i = 0 } and Pr { [ b j ] i = 1 } are assumed to be / for all i ∈ Z log M . Notations E s and E b , representing the average energy of allusers’ symbols and data bits, respectively, are defined as E s = 1 JM J (cid:88) j =1 M (cid:88) m =1 (cid:107) V j C j e m (cid:107) = 1 JM J (cid:88) j =1 tr( C Hj C j ) (28)and E b = 1log M E s . (29)The normalized minimum Euclidean distance is defined as ˆ d min = d min √ E s (30)where E s is also the average power of each codeword of eachuser by the definition in (28). A. The Case with Three Users ( J = 3 ) We first consider the simple case where only three users areallowed to share the K = 4 orthogonal resources, i.e., J = 3 .Mapping matrices V , V , V in (7) are chosen and this choicemakes sure the number of collisions is as small as 2 [11].Algorithm 1 is applied to solve Problem (20) with the initialpoint X = xx H where all elements of x (i.e., all entries of { C j } j =1 ) are independently generated by a random variableuniformly distributed between 0 and 1. In this relatively simple case, Algorithm 1 is found to be converging and leading to anoptimal codebook collection whose optimal value coincides withthat of the dual problem (24). We used CVX , a MATLAB-basedmodeling system for convex optimization [33], for Steps 5 and 6of Algorithm 1 in each iteration. As a result, we observed that,in about 10 iterations, the algorithm converged. We also used
CVX to solve the dual problem (24) to get the upper bound ofoptimal value. Note that the dual problem (24) is always convex,so it can be solved with just a single
CVX instance.Figure 2 shows the MED comparison of the proposed al-gorithms and various previously reported methods in terms ofMED, along with the bound given by the dual problem (24). Asindicated in the figure, the proposed method achieves the largestvalue among all methods, including the randomization method[11] with L rand = 10 , the starQAM codebook collection [10],and the Top-Down codebook collection [15]. We notice that theMED of the proposed method is the same as the optimal valueof the dual problem (24). As stated in Section IV, it is sufficientto show that the codebook collection we proposed is a set ofoptimal codebooks that achieve the maximum MED. S t a r Q A M [ ] R a n d o m i z a t i o n [ ] T o p - D o w n [ ] P r o p o s e d D u a l P r o b l e m Fig. 2. Minimum Euclidean distance comparison for J = 3 . B. The Case with Six Users ( J = 6 ) We now consider the case when the number of users is J = 6 . As mentioned in Section III, a large number of distanceconstraints greatly increase the consumption of memory andcomputational complexity. It is extremely inefficient for the The codebook collection obtained here do not necessarily coincide withthe one reported in [11], due to the random nature of Gaussian randomizationalgorithm.
CVX tool to handle a large number of more than 8 millionconstraints in constructing the problem settings alone, not tomention solving it. As an alternative approach, we resort todirectly using SDPT3 [34], the default solver of the
CVX tool.We observed that the SDPT3 solver, without the extra burdencaused by the problem-constructing routines of
CVX , is capableof returning correct results within an acceptable time duration.Moreover, it is worthy to note that the sparse properties of thematrices A i , B j with nonzero entries being ± , may have alsoexpedited the computation of the SDPT3 solver [34]. We usedthis new approach to execute Algorithm 1 and tried to solveProblem (15). Although it may not always converge within ϕ max iterations, we usually can obtain an X with the largest eigenvaluedominating all the other eigenvalues (i.e., the ratio of the secondlargest eigenvalue to the largest one σ σ ≤ − ) and thus it isfair enough to consider X as xx H for retrieving x by Step 12in Algorithm 1.Since Algorithm 1 needs the constellation matrices { C j } j =1 of some codebook collection for initialization, we chose twocodebook collections for AWGN channel proposed in [17] and[18] since they have relatively good error rate performancescomparing to other codebook collections over AWGN chan-nels. We first test Algorithm 1 using the codebook collectionfor AWGN channel proposed in [17] (referred to as "Chen’sAWGN codebook collection") for initialization, and we choseweight w = 0 . . The algorithm converged in iterationswith a total computation time of hours. The MED of theresultant codebook collection is 1.17, which is already greaterthan the MEDs of all previously reported codebook collections(The detailed codebook collection is shown in Appendix C).Then, we tried Algorithm 1 using the codebook collection forAWGN channel proposed in [18] (referred to as "Deka’s AWGNcodebook collection") . We at first tried the fixed weight w = 0 . but it ran for over iterations, which takes almost a whole weekof computing with the CPU being AMD Ryzen™ 9 3900X, anddid not converge. Hoping for the convergence of Algorithm 1, wemanually chose weight w between . and . in the process ofthe iterations in Algorithm 1 and it converged with MED being1.30 (The detailed codebook collection is shown in AppendixD). More specifically, we chose: w ,k = . ≤ k ≤ .
15 7 ≤ k ≤ . ≤ k ≤ .
25 36 ≤ k ≤ . ≤ k ≤ .
35 72 , w ,k = . ≤ k ≤ .
15 7 ≤ k ≤ . ≤ k ≤ .
25 35 ≤ k ≤ . ≤ k ≤ .
35 72 . It took iterations and hours to obtain this codebookcollection. Note that the "Chen’s AWGN codebook collection" [17] and "Deka’s AWGNcodebook collection" [18] for initialization we used here are scaled to meet powerconstraint (14c).
The dual problem (24) is also solved via SDPT3 with J = 6 and the resulting upper bound is 1.63. The comparison of thenormalized MEDs of different codebook collections is shownin Figure 3 . Although Deka’s AWGN codebook collection(labeled as “Deka 2020 (AWGN) [18]”) have a smaller MEDthan Chen’s AWGN codebook collection (labeled as “Chen 2020(AWGN) [17]”), the former results in a codebook collection withan even larger MED. We found that this may have been due tothat only a very small number of pairs of superimposed code-words achieve the MED for Deka’s AWGN codebook collection.As the codebook collection obtained by Algorithm 1 withDeka’s AWGN codebook collection [18] for initialization hasthe largest MED, 1.30, it is expected that it will attain relativelybetter error rate performances than other codebook collectionsand therefore we use it as the proposed codebook collection inthe following numerical results. We adopted the MPA algorithmdescribed in Section III-B in reference [30] for the detection.The results, corresponding to the AWGN channel and Rayleighfading channel, are discussed in the following parts respectively.
1) AWGN Channel:
The results for SCMA systems over theAWGN channel, as shown in Figures 4, and 5, demonstrate thatthe proposed codebook collection obtained by Algorithm 1 withDeka’s AWGN codebook collection [18] for initialization indeedachieves the best SER and BER performances since it has thelargest MED. Specifically, there are gains of both about 0.7dB atSER = 10 − and BER = 10 − over the best existing codebookcollection [18].
2) Downlink Rayleigh Fading Channel:
For the simulation ofthe downlink SCMA system based on OFDMA over the Rayleighfading channel, the parameter setting is shown as follows. TheFFT size N B is 256. The channel length L is 18 and cyclic-prefix length N CP is 17. The subcarrier indices of the K -dimensional SCMA signal is set to 127, 128, 129, 130 withinthe 256 subcarriers of an OFDM symbol (i.e. i sub = 127 in(1)). The other signals loaded on the remaining subcarriers areset as independent and identically distributed (i.i.d.) signals withdistribution being CN (0 , E s /K ) . The Rayleigh fading channelis set as h ∼ CN (0 , diag( σ h [0] , σ h [1] , . . . , σ h [ L − )) (31)where (cid:104) σ h [0] σ h [1] · · · σ h [ L − (cid:105) T = h var (cid:107) h var (cid:107) , and h var ∈ R L isthe vector whose elements are linearly spaced between dB and − dB. With the setting above, the simulation results are shownin Figure 6. We notice that most of the codebook collectionshave similar performances on bit error rate, but the codebookcollection proposed by Chen et al. [17] for downlink Rayleighfading channel (labeled as “Chen 2020 (downlink) [17]”) and Note that all codebook collections we used for comparison hereafter arescaled such that the power of the user with maximum average power meet powerconstraint (14c) (i.e., max j ∈Z J M tr (cid:16) C Hj C j (cid:17) = P for all codebook collections). H u a w e i [ ] S t a r Q A M [ ] Z h a n g [ ] C h e n ( A W G N ) [ ] D e k a ( A W G N ) [ ] P r o p o s e d ( C h e n i n i t i a l ) P r o p o s e d ( D e k a i n i t i a l ) D u a l p r o b l e m Fig. 3. Minimum Euclidean distance comparison for J = 6 . the proposed codebook collection have worse performances forSNR lower than 45dB and larger than 33 dB, respectively. Theseconsequences, which probably differ from the ones in some otherworks [17], [18], may result from the differences of the settingof the Rayleigh fading channel in this paper and the one in[17], [19], which just considered the channel gain vector witheach element being i.i.d. complex normal random variable, i.e., h fsub ∼ CN ( K , σ I K ) . Our choice of channel, instead, makes h fsub a colored Gaussian random vector with high correlationsbetween adjacent subcarriers, which is, in our opinion, a morerealistic assumption for the downlink SCMA system based onOFDMA.Although the proposed codebook collection does not outper-form other ones in BER for the case mentioned above, wenotice that the SNR needed for good BER is unrealistically high(about 40dB for BER = 10 − ). The bad performance curves forall methods may have been mostly contributed by deep fadechannels in some Monte Carlo trials. In a real-world SCMAapplication, however, we believe it is reasonable to assume allusers that share the SCMA resources would possess sufficientlygood channel quality on these subcarriers, for otherwise, the basestation would not have assigned the users to these resources atthe first place. With this assumption in mind, we conduct theperformance analysis again by excluding the percent poorestchannels h fsub . The results, as shown in Figure 7, demonstratethat the proposed codebook collection outperforms all the othercodebook collections on bit error rate. Specifically, there is again of about . dB at BER = 10 − over the best existingcodebook collection [18]. -7 -6 -5 -4 -3 -2 -1 Fig. 4. SER performance comparison for J = 6 over AWGN channel. -7 -6 -5 -4 -3 -2 -1 Fig. 5. BER performance comparison for J = 6 over AWGN channel. C. Comparison of Theoretical Results and Simulation Results
To certify the error rate performances reported in the previoussimulation plots, we compare all error rate curves with theoreti-cal bounds in this subsection. For the SER, an upper bound canbe found by simply taking the average over J users of the upperbound of the SER of each single user derived by Bao et al . (eq.(38) in [42]). P e,s ≤ M J · J TM J ( Q ◦ D s ) M J (32) -6 -5 -4 -3 -2 -1 Fig. 6. BER performance comparison for J = 6 over Rayleigh fading channel. -7 -6 -5 -4 -3 -2 -1 Fig. 7. BER performance comparison for J = 6 over Rayleigh fading channel(excluding cases with poor channel). where the matrix Q ∈ R M J × M J is defined with [ Q ] k,l = Q (cid:16)(cid:113) d kl N (cid:17) , d kl was defined in (12), the matrix D s ∈ Z M J × M J is defined with [ D s ] k,l = d s,kl , and d s,kl is the Hammingdistance between the multiplexed symbols m k , m l , i.e., d s,kl = (cid:80) Jj =1 | k j − l j | with |·| being the indicator function of nonzero values. For the BER, an upper bound is found to be P e,b ≤ M J · J log M TM J ( Q ◦ D b ) M J (33)where the matrix D b ∈ Z M J × M J is defined with [ D b ] k,l = d b,kl , and d b,kl is the Hamming distance between the bit pat-terns loaded on the multiplexed symbols m k , m l , i.e., d b,kl = (cid:80) Jj =1 (cid:13)(cid:13) b [ k j ] − b [ l j ] (cid:13)(cid:13) with (cid:107)·(cid:107) being the (cid:96) norm in [39], and b [ k j ] , b [ l j ] ∈ B log M denote the bits corresponding to symbol k j , l j , respectively, according to the convention in Section II-B.The derivation of (33) can be done by the definitions of SER,BER, and (32).In the following comparisons shown in Figures 8, and 9, wewill find that all simulation curves are matching the theoreticalupper bounds (32), (33) in high SNR regions within acceptablemargins. These results not only double-checked the correctnessof the bounds, but also secured all performance advantages ofthe proposed codebook collection that we have seen in Figures4, 5, and 7. The bounds are rather loose in low SNR regions, butthey go tighter as SNR goes higher. Sometimes it is observedthat simulation results even have slightly larger error rates thanthe bounds, and we believe this is because the MPA is still worsethan MAP. Specifically, we notice MPA detection may not alwaystake the closest superimposed codeword to the received signal r [ b ] as the detected one as MAP does, and this results in theslightly larger error rates than the bounds derived based on MLdetection [41], [42], which is equivalent to MAP detection dueto the equally likely input symbols.Based on the tightness of the upper bounds and the simulationresults discussed above, we can further predict the SER, and BERof some codebook collections in the high SNR region as shownin Figures 10. It can be observed that the proposed codebookcollection still has the best error rate performances.Moreover, some remarks about the relation of Euclideandistances and error rates are made. It is observed that the upperbounds, (32), (33), are both proportional to some weighted sumsof all elements in Q , so it is desirable to minimize the entries of Q , which depend on the Euclidean distances, for a low errorrate. Since the largest entry in Q is Q (cid:16) d min √ N (cid:17) , which willdominate the contributions to error rate upper bound formulas(i.e., Q ( d min √ N ) (cid:29) Q ( (cid:113) d kl N ) for most k, l ∈ Z M J , k (cid:54) = l )when the SNR goes to infinity (i.e., N → ), our approachof maximizing d min for the codebook collection design problemis thus justified based on the tightness of these bounds. Atleast, it suggests that the codebook collection designed will haveclear advantages in high-SNR regions, as we have shown in thissection. This can be seen by the fact that lim x →∞ Q ( ax ) /Q ( x ) = 0 for all a > . -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 Fig. 8. Comparison of theoretical and simulation results of SER over AWGNchannel.
VI. C
ONCLUSIONS
In this paper, a new method for downlink SCMA codebookdesign, based on maximizing the minimum Euclidean distance(MED) of superimposed codewords, is proposed. An iterativealgorithm based on alternating maximization is applied by re-formulating the MED-maximization problem into a biconvexform with an exact penalty function. With appropriate choicesof the initial codebook and weighting coefficients, the proposedalgorithm has successfully produced a codebook with an MEDgreater than any existing codebook collection with a largemargin, for the most popular six-user four-resource case. ALagrange dual problem of the MED-maximizing problem wasderived and solved, resulting in a theoretical MED upper boundof any SCMA codebook collections that were unknown before.Although the codebook reported in this article has achievedan MED that is much closer to the upper bound than anypreviously reported codebook collections, the fact that there isstill a nonzero gap between the upper bound and the largestMED suggests there is still room for codebook improvement inthe future.Simulation results demonstrate clear advantages of the ob-tained largest-MED codebook in terms of error-rate performanceover all available reported codebook collections. The perfor- -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 Fig. 9. Comparison of theoretical and simulation results of BER over AWGNchannel.
12 14 16 18 20 2210 -40 -20
10 15 2010 -40 -20 Fig. 10. Performance prediction comparisons over AWGN channel mance advantages are not only seen in AWGN channels butalso in fading channels assuming downlink SCMA resourcesare allocated from an underlying OFDMA system with resourceelements with sufficiently good channel quality. The validity ofall simulation curves is further verified by comparing them withtheoretical error rate bounds.In the future, it is still desirable to further reduce the dualitygap between the attained MED and the upper bound obtainedfrom the dual problem: either a codebook with an even larger MED is to be found, or a smaller upper bound of MED isto be derived, or both. The choices of the initial codebookand weighting coefficients appear to play important roles in theproposed biconvex algorithm. It is therefore desirable to try outother possible combinations of these parameters to find even abetter codebook collection.A
PPENDIX AP ROOF OF THE EQUIVALENCE OF (9)
AND (10)Assume that some multiplexed symbol is transmitted and thereceived signal is r . The possible superimposed codeword whichhas the shortest Euclidean distance from r is (cid:80) Jj =1 V j C j e ( M ) k j .With the assumption of equal probability of each kind of multi-plexed symbol being sent and the probability distribution of theadditive noise, we can view MAP detection as MED detection.Therefore, the concatenated vector of codewords correspondingto the multiplexed symbol, say, m k , detected by (8) will be (cid:98) s = (cid:20)(cid:16) V C e ( M ) k (cid:17) T (cid:16) V C e ( M ) k (cid:17) T · · · (cid:16) V J C J e ( M ) k J (cid:17) T (cid:21) T Moreover, for any j ∈ Z J , we also have Pr (cid:98) s = V C e ( M ) l ... V j − C j − e ( M ) l j − V j C j e ( M ) k j V j +1 C j +1 e ( M ) l j +1 ... V J C J e ( M ) l J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ≥ Pr (cid:98) s = V C e ( M ) l ... V j − C j − e ( M ) l j − V j C j e ( M ) i V j +1 C j +1 e ( M ) l j +1 ... V J C J e ( M ) l J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r , ∀ l ∈ Z M J , ∀ i ∈ Z M . (34) Let P ( i ) j = (cid:80) s ∈ ( S ×···×S J ) , s j = V j C j e ( M ) i Pr { (cid:98) s = s | r [ b ] } ,where s = [ s T s T . . . s TJ ] T ∈ C KJ . Then by (34), we have P ( k j ) j ≥ P ( i ) j , ∀ i ∈ Z M . Therefore, the j -th user’s codewordcorresponding to the symbol detected by (10) will be (cid:98) s j = V j C j e ( M ) k j . Hence, we have (cid:98) s j = (cid:16) ( e ( J ) j ) T ⊗ I K (cid:17)(cid:98) s , which is exactly thecodeword corresponding to the symbol detected by (9).A PPENDIX BD ERIVATION OF THE DISTANCE MATRICES IN (16)First note that the difference of some pair of superimposedcodewords corresponding to some two multiplexed symbols m k , m l is J (cid:88) j =1 V j C j (cid:16) e ( M ) k j − e ( M ) l j (cid:17) = J (cid:88) j =1 vec (cid:16) V j C j (cid:16) e ( M ) k j − e ( M ) l j (cid:17)(cid:17) = J (cid:88) j =1 (cid:20)(cid:16) e ( M ) k j − e ( M ) l j (cid:17) T ⊗ V j (cid:21) x j where x j = vec( C j ) is actually the j -th M N -block of x .Therefore, x j = (cid:16) ( e ( J ) j ) T ⊗ I MN (cid:17) x . (35)Then the square of the Euclidean distance of this pair will be (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J (cid:88) j =1 (cid:20)(cid:16) e ( M ) k j − e ( M ) l j (cid:17) T ⊗ V j (cid:21) x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = J (cid:88) q =1 x Hq (cid:20)(cid:16) e ( M ) k q − e ( M ) l q (cid:17) T ⊗ V q (cid:21) H J (cid:88) j =1 (cid:20)(cid:16) e ( M ) k j − e ( M ) l j (cid:17) T ⊗ V j (cid:21) x j = J (cid:88) q =1 J (cid:88) j =1 x Hq (cid:16)(cid:16) e ( M ) k q − e ( M ) l q (cid:17) ⊗ V Tq (cid:17) (cid:18)(cid:16) e ( M ) k j − e ( M ) l j (cid:17) T ⊗ V j (cid:19) x j = J (cid:88) q =1 J (cid:88) j =1 (cid:20) x Hq (cid:18)(cid:16) e ( M ) k q − e ( M ) l q (cid:17) (cid:16) e ( M ) k j − e ( M ) l j (cid:17) T (cid:19) ⊗ (cid:16) V Tq V j (cid:17) x j (cid:21) . In the end, by substituting (35) for x j for all j ∈ Z J , we havethe expression of x H A i x as shown in Section III-A2. Note thatthe index i ∈ Z ( MJ ) here can be mapped freely to the (cid:0) M J (cid:1) pairsof multiplexed symbols as long as the mapping is confirmed tobe one-to-one. A PPENDIX CT HE CODEBOOK COLLECTION OBTAINED BY A LGORITHM WITH C HEN ’ S AWGN
CODEBOOK COLLECTION [17]
FORINITIALIZATION S − S , shown as follows. S = − . − . i − . − . i − . − . i . . i . . i − . − . i . − . i . . i S = − . . i − . − . i − . . i . . i . − . i − . − . i . − . i . . i S = − . . i − . . i − . . i . . i . − . i − . − . i . − . i . − . i S = − . . i − . . i − . . i . − . i . − . i − . . i . − . i . − . i S = − . − . i − . − . i − . − . i . − . i . . i − . . i . . i . . i S = − . . i − . − . i − . . i . . i . − . i − . − . i . − . i . . i A PPENDIX DT HE PROPOSED CODEBOOK COLLECTION S − S , shown as follows. S = − . − . i . . i − . − . i − . − . i . . i . . i . . i − . − . i S = . − . i . . i . − . i − . − . i − . . i . . i − . . i − . − . i S = . − . i − . − . i . − . i . − . i − . . i − . . i − . . i . . i S = − . . i . − . i . . i . − . i − . − . i − . . i . − . i − . . i S = . . i − . − . i − . − . i − . − . i . . i . . i − . − . i . . i S = . − . i . . i . − . i − . − . i − . . i . . i − . . i − . − . i A CKNOWLEDGMENT
This work was supported by the Ministry of Science andTechnology of Taiwan under Grant MOST 109-2221-E-002-155.R
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