Research ArticleAn Efficient SCMA Codebook Optimization Algorithm Based onMutual Information Maximization

Chao Dong Guili Gao Kai Niu and Jiaru Lin

Key Laboratory of Universal Wireless Communications Ministry of Education Beijing University of Posts and TelecommunicationsBeijing 100876 China

Correspondence should be addressed to Chao Dong dongchaobupteducn

Received 21 November 2017 Revised 2 February 2018 Accepted 26 February 2018 Published 1 April 2018

Academic Editor Imran S Ansari

Copyright copy 2018 Chao Dong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An efficient codebook optimization algorithm is proposed tomaximizemutual information in sparse codemultiple access (SCMA)At first SCMA signal model is given according to superposition modulation structure in which the channel matrix is column-extended The superposition model can well describe the relationship between the codebook matrix and received signal Basedon the above model an iterative codebook optimization algorithm is proposed to maximize mutual information between discreteinput and continuous output This algorithm can efficiently adapt to multiuser channels with arbitrary channel coefficients Thesimulation results show that the proposed algorithm has good performance in both AWGN and non-AWGN channels In additionmessage passing algorithm (MPA) works well with the codebook optimized according to the proposed algorithm

1 Introduction

Nonorthogonal multiple access (NOMA) [1ndash3] has attractedmuch attention from both industry and academia and hasbecome the key technique in 5G NOMA can accommo-date more users than orthogonal multiple access (OMA)and improve the spectral efficiency At the same time theinteruser interference is introduced in NOMA and moreefficient detection is required at the receiver

Code-domain NOMA is considered as an important can-didate technique [3] At first low density spreading CDMA(LDS-CDMA) which belongs to code-domain NOMA isproposed in [4] where the spreading sequence is sparse andmessage passing algorithm (MPA) is introduced to lowermultiuser detection complexity In [5] message passing algo-rithm (MPA) is proven to be optimal when the length oflow density spreading sequence tends to be infinity Later in[6] LDS-OFDM is proposed where the iteration number ofMPA is optimized according to extrinsic information transfer(EXIT) chart [7] In [8] LDPC codes and LDS constitutethree-layer factor graph and the joint belief propagationalgorithm exhibits good performance In [9] the progressiveedge growth (PEG) algorithm is introduced to improve LDSpattern and the multiuser detection performance becomesbetter

Recently sparse code multiple access (SCMA) [10 11]has been proposed to combine low density spreading (LDS)with codebook design In [10] the codebook design is imple-mented by rotating the phases of different usersrsquo constellationpoints At the receiver message passing algorithm (MPA) isapplied and SCMA exhibits better performance than LDS-CDMA [10] Afterwards several methods to improve MPAin SCMA are proposed in [12ndash14] Furthermore in the recentpaper [15] SCMA detection can be seen as a specific treesearch problem and sphere decoding algorithm shows muchlower complexity with negligible performance loss Inspiredby transmit signal optimization in two-user multiple accesschannels [16] SCMA codebook design is mainly focusedon phase rotation optimization in [17 18] In addition theproduct distance and cutoff-rate are introduced to optimizeSCMA codebook in [19] and [20] respectively

In this paper SCMA signal model is given accordingto superposition modulation structure where the channelmatrix is column-extended The proposed model can welldescribe the relationship between the codebook of eachuser and received signal Therefore the codebook optimiza-tion algorithm is derived according to maximizing mutualinformation between the discrete input and continuousoutput Our analysis shows that the proposed algorithm can

HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 8910907 13 pageshttpsdoiorg10115520188910907

2 Wireless Communications and Mobile Computing

efficiently adapt to multiuser channels with random channelcoefficients

This paper is organized as follows In Section 2 SCMAsignal model is represented as superposition modulationstructure where the channel matrix is extended in the col-umn space to explicitly demonstrate the relationship betweenthe codebook and received signal Afterwards the principlesof codebook optimization to maximize mutual informationare described in Section 3 The concrete expressions ofmutual information and its gradient are given SubsequentlyKarush-Kuhn-Tucker (KKT) conditions are introduced torealize codebook optimization In Section 4 the implemen-tation steps of the proposed iterative codebook optimizationalgorithm are elaborated The simulation results are given inSection 5 Section 6 draws the conclusions

In the following parts lower and upper boldface lettersdenote the vector and matrix respectively For the matrixA Aminus1 A119879 and A119867 denote its inverse transpose andHermitian respectively The row vector e119870119895 denotes the 119895throw of the 119870 times 119870 identity matrix I119870 The matrix A =blkdiagA1 A119873 denotes the block diagonal matrix inwhich A119894 is the submatrix on the 119894th diagonal block Theoperator otimes denotes Kronecker product In addition 119864x[sdot]denotes the expectation over the random variable x

2 SCMA Signal Model withSuperposition Modulation

In this section SCMA signal model is given according tosuperposition modulation structure The analysis shows thatthe channel matrix is column-extended in the proposedstructure In addition the relationship between the codebookand received signal is given The analysis in this section laysfoundation for the codebook optimization For clearness thetypical SCMA signal model is detailed in the first subsection

21 Typical SCMA Model A typical SCMA factor graphis given in Figure 1 In this paper 119870 denotes the numberof multiple access users and 119873 denotes the number ofsubchannels In factor graph the user node is usually calledldquovariable noderdquo and the subchannel is usually called ldquofunctionnoderdquo The variable node degree 119889V denotes the number ofsubchannels occupied by one user Meanwhile the functionnode degree 119889119891 denotes the number of users carried by onesubchannel

In the factor graph shown in Figure 1 119889V is equal to 2and 119889119891 is equal to 3 The load is equal to 119870119873 The mappingbetween variable nodes and function nodes in Figure 1 isgiven by

F = [[[[[[

1 1 1 0 0 01 0 0 1 1 00 1 0 1 0 10 0 1 0 1 1]]]]]] (1)

The matrix F has 119870 columns and 119873 rows It can be seenthat the 119895th row of F denotes the mapping from all variablenodes to the 119895th function node Similarly the 119894th column of

V1 V2 V3 V4 V5 V6

F1 F2 F3 F4

Figure 1 Factor graph with 119870 = 6 and119873 = 4

F denotes the mapping from all function nodes to the 119894thvariable node In addition the number of nonzero elementsin each row is equal to 119889119891 and the number of nonzeroelements in each column is equal to 119889V

The matrix F denotes the channel matrix in AWGNscenario The existing codebook designs in [10 11 17 18]are mainly based on F in (1) However in the wirelesscommunication the channel response amplitudes and phasesof 119870 users are usually different from each other In thefollowing ℎ119895119894 denotes the channel response of the 119894th useron the 119895th subchannel Therefore the channel matrix H119878corresponding to the factor graph in Figure 1 can be given by

H119878 = [[[[[[

ℎ11 ℎ12 ℎ13 0 0 0ℎ21 0 0 ℎ24 ℎ25 00 ℎ32 0 ℎ34 0 ℎ360 0 ℎ43 0 ℎ45 ℎ46]]]]]] (2)

Based on the channel matrix in (2) SCMA codebookoptimization proposed in [10] can be denoted by

Glowast = argmaxG

119898(119878 (H119878G 119873119870 119889V119872)) (3)

where 119872 denotes the modulator order of each user 119878(sdot)denotes thematrix function related to the variables in (3) and119898(sdot) gives performance measure in codebook optimizationIn this paper ourmain focus is on the factor graph in Figure 1with119870 = 6119873 = 4 119889119891 = 3 119889V = 2 and119872 = 4

In the next subsection the superposition modulationSCMA signal model is carefully analyzed

22 Superposition Modulation Model It can be seen fromFigure 1 that user 1 is connected to function nodes 1198651 and 1198652In SCMA the signal of user 1mapped to1198651 and1198652 is given by11990911 and 11990921 respectively For clearness the above two signalelements are collected to generate the following signal vector

x1 = [11990911 11990921]119879 (4)

In SCMAmodel with119872 = 4 the signal elements 11990911 and 11990921of user 1 carry the same two information bits In [10 11] 11990911and 11990921 are generated by phase rotation of the chosenmotherconstellation

In this paper the superposition modulation structureis introduced In [21 22] the superposition modulation isproven to be an efficient modulation scheme to approach

Wireless Communications and Mobile Computing 3

the channel capacity Based on the above analysis x1 can berewritten as follows

x1 = [1199091111990921] = [[119892(1)11 119892(1)12119892(1)21 119892(1)22]] sdot [119887(1)1119887(1)2 ] = G1b1 (5)

where the superscript (1) denotes the user index G1 denotesthe codebook matrix of user 1 and the bit vector b1 containsthe two information bits of user 1

By extending the above model to 119870 = 6 users thetransmit signal vector of user 119894 is given by

x119894 = G119894b119894 1 le 119894 le 119870 (6)

Therefore the overall transmitted vector can be obtained bystacking x1 to x6

x = [x1198791 x1198792 x1198793 x1198794 x1198795 x1198796 ]119879 = Gb

= blkdiag G1 G6 times b (7)

where G denotes the block diagonal codebook matrixand b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 b1198794 b1198795 b1198796 ]119879

In order to adapt to the transmit signal expression in (7)the channel expressionH119878 shown in (2) should be extended inthe column space It is noted that the dimension of x is equalto 119889V times 119870 = 12 Therefore the column number of channelmatrix should be extended from 6 to 12

H119878 =[[[[[[[

ℎ11 0 ℎ12 0 ℎ13 0 0 0 0 0 0 00 ℎ21 0 0 0 0 ℎ24 0 ℎ25 0 0 00 0 0 ℎ32 0 0 0 ℎ34 0 0 ℎ36 00 0 0 0 0 ℎ43 0 0 0 ℎ45 0 ℎ46

]]]]]]] (8)

For example in the first column of H119878 the channelcoefficients corresponding to 11990911 and 11990921 are ℎ11 and ℎ21respectively To match the two-dimensional transmissionvector x1 shown in (5) the first column of H119878 should beextended to generate the following two columns

h11 = [ℎ11 0 0 0]119879 h21 = [0 ℎ21 0 0]119879 (9)

Based on the above analysis the column-extended chan-nel matrixH119878 is given by the long expression in (8) It can beseen from (8) thatH119878 is obtained by dividing each column ofH119878 into two columns in order to match the two-dimensionalmodulation symbol vector of each user

Based onH119878 in (8) the received signal is rewritten as

y = H119878x + n = H119878Gb + n (10)

where n is 119873-dimensional additive white Gaussian noisevector with distribution CN(0 1205902119899I119873)

According to block diagonal property of G the receivedsignal y in (10) can be rewritten as the superposition resultsof119870 = 6 usersrsquo signals

y = H119878Gb + n = 119870sum119894=1

H119894G119894b119894 + n (11)

where H119894 is the equivalent channel matrix of user 119894 Forexample the equivalent channel matrixH1 corresponding touser 1 is given by

H1 = [[[[[[

ℎ11 00 ℎ210 00 0]]]]]] (12)

Based on the above analysis the received signal y isconnected to the codebook matrix G119894 of each user with thehelp of the column-extended channel matrix H119894 1 le 119894 le 119870Therefore the codebook optimization can be implementedaccording to various criteria In the next section the code-book matrix G119894 1 le 119894 le 119870 is optimized according tomaximizingmutual information between the bit vector b andthe received signal y shown in (11)

3 Principle of SCMA Codebook Optimization

In this section the codebook optimization to maximizemutual information is carefully analyzedWe assume that thenumber of users and the channel responses are known bythe transmitter In practical wireless communication systemsthis assumption is possible for the downlink transmissionwith channel state information feedback but not possible forthe uplink

At first the concrete expression of mutual informationbetween the bit vector b and the continuous received signal yis given Afterwards KKT conditions based on the gradientof mutual information are introduced to realize codebook

4 Wireless Communications and Mobile Computing

optimization It is shown that the gradient of mutual infor-mation with respect to codebook matrix G119894 1 le 119894 le 119870depends on the mean squared error matrix E119887 For clearnessthe details of calculating mean squared error matrix E119887 aregiven in Appendix A

31 Detailed Expression of Mutual Information Similar tothat in [23 24] mutual information between discrete inputb and continuous output y can be given by

119868 (b y) = 119867 (b) minus 119867 (b | y) = 119870 log2119872minus 119872119870sum119898=1

inty119901 (b119898 y) log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y = 119870

sdot log2119872minus 119872119870sum119898=1

inty119901 (b119898) 119901 (y | b119898)

sdot log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y

(13)

In this paper all possible input vectors are assumed to haveequal probability The input constellation alphabet size isequal to 119872119870 and 119901(b119898) = 1119872119870 When signal-to-noiseratio (SNR) tends to be infinity themutual information is notlarger than the entropy119867(b) which is equal to119870 log2119872 Thesubscript119898 denotes the index of the constellation point from1 to119872119870 With additive white Gaussian noise the conditionalprobability distribution function 119901(y | b119898) is given by

119901 (y | b119898) = 1(1205871205902119899)119873 exp(minus10038171003817100381710038171003817y minusH119878Gb1198981003817100381710038171003817100381721205902119899 ) (14)

In addition the probability distribution function 119901(y) in (13)can be given by

119901 (y) = 119872119870sum119896=1

119901 (b119896) 119901 (y | b119896)

= 119872119870sum119896=1

1119872119870 (1205871205902119899)119873 exp(minus10038171003817100381710038171003817y minusH119878Gb1198961003817100381710038171003817100381721205902119899 )

(15)

where the subscript 119896 denotes the constellation point indexWhen the bit vector b119898 is transmitted the received signal

is given by y = H119878Gb119898 + n In this case the unknowncontained in y is only additive white Gaussian noise vectorn Therefore the integral of y can be expressed as the integralof n Consequently in the 119898th integral of the summation in(13) y is replaced byH119878Gb119898 + n

inty119901 (b119898) 119901 (y | b119898) log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y= int

n119901 (b119898) 119901 (H119878Gb119898 + n | H119878Gb119898)

sdot log 119901 (H119878Gb119898 + n)119901 (b119898) 119901 (H119878Gb119898 + n | H119878Gb119898)119889n

= intn119901 (b119898) 119901 (n) log 119901 (H119878Gb119898 + n)119901 (b119898) 119901 (n) 119889n

= intn119901 (b119898) 119901 (n)

sdot log sum119872119870119896=1 119901 (b119896) 119901 (H119878Gb119898 + n | H119878Gb119896)119901 (b119898) 119901 (n) 119889n(16)

where 119901(n) = 1(1205871205902119899)119873 times exp(minusn21205902119899) For the firstequation we assume that the channel matrix and codebookmatrix are perfectly known With the expression of 119901(y | b119898)in (14) the second equation is achieved In the third equation119901(y | b119896) is replaced by 119901(H119878Gb119898 + n | H119878Gb119896) whoseexpression is given by

119901 (H119878Gb119898 + n | H119878Gb119896)= 1(1205871205902)119873 exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902 ) (17)

Based on the above analysis the integral value in (16) dependson the Euclidean distances betweenH119878Gb119898 and all the otherreceived signal constellations

Under the equal probability input assumption themutualinformation in (13) can be rewritten as

119868 (b y) = 119870 log2119872minus 1119872119870

119872119870sum119898=1

intn119901 (n) log119872119870sum

119896=1

exp (minus119902119898119896) 119889n= 119870 log2119872minus 1119872119870

119872119870sum119898=1

119864n [[log119872119870sum119896=1

exp (minus119902119898119896)]] (18)

where 119902119898119896 is given by

119902119898119896 =10038171003817100381710038171003817H119878G (b119898 minus b119896) + n100381710038171003817100381710038172 minus n21205902119899 (19)

It should be noted that 119898 and 119896 are both constellation pointindexes and they are independent of each other

From the above analysis it can be seen that mutualinformation is the function of codebook matrices In thefollowing subsection the gradient of mutual informationwith respect to codebook matrix of each user is analyzedand the KKT conditions are introduced to maximize mutualinformation

Wireless Communications and Mobile Computing 5

32 KKT Conditions When Maximizing Mutual InformationTo optimize mutual information the gradient of mutualinformation with respect to G119894 1 le 119894 le 119870 is calculated

According to the results in [25 26] the gradient withrespect to the overall block diagonal codebook matrix G isgiven ApplyingTheorem 2 in [26] we have

nablaG119868 (b y) = 120597120597Glowast 119868 (b y) = 1ln 2H119867119878 (1205902119899I119873)minus1H119878GE119887

= 1ln 2 sdot 1205902119899H119867119878 H119878GE119887

(20)

where the factor 1 ln 2 is appended because the naturallogarithm is applied in [26] The matrix Eb denotes the meansquared error matrix In [26] it is proven that the abovegradient expression holds for the linear received signal modelin (10) regardless of the structure of the channel matrix H119878and the codebook matrix G

In SCMA we assume that the codebook matrix of eachuser satisfies individual power constraint This requires thegradient with respect to each userrsquos codebook matrixG119894 1 le119894 le 119870 Based on the fact thatG119894 is 119889Vtimes119889V submatrix on the 119894thdiagonal block of G the gradient with respect to G119894 is givenby

nablaG119894119868 (b y) = 120597120597Glowast119894 119868 (b y)= (e119870119894 otimes I119889V) 120597120597G119868 (b y) (e119870119894 otimes I119889V)119867= 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)H119867119878 H119878GEb (e119870119894 otimes I119889V)119867

(21)

where e119870119894 is the 119894th row of the119870times119870 identity matrix I119873 From(19) it can be seen thatnablaG119894119868(b y) can be easily calculated fromthe result of nablaG119868(b y)

In order to maximize mutual information between b andy the optimization problem is given by

maxG119894 1le119894le119870

119868 (b y)st tr (G119894G119867119894 ) le 119875119894 1 le 119894 le 119870 (22)

Unfortunately 119868(b y) is not a convex function of the code-book matrix G119894 1 le 119894 le 119870 and it is difficult to calculate itsglobally optimal solution An efficient method to solve thiskind of problem is to find locally optimal solution accordingto KKT conditions [22] Therefore we have the followinglemma

Lemma 1 With the power constraint of each user the KKTconditions corresponding to problem (22) are given by

120582119894G119894 = nablaG119894119868 (b y) 120582119894 ge 0

tr (G119894G119867119894 ) le 119875119894120582119894 [tr (G119894G119867119894 ) minus 119875] = 0

(23)

Proof According to the result in [22] the Lagrangian corre-sponding to problem (22) is given by

119871 (120582119894G119894) = minus119868 (b y) + 119870sum119894=1

120582119894 [tr (G119894G119867119894 ) minus 119875119894] (24)

where 120582119894 is the Lagrangian dual variable corresponding to the119894th userrsquos power constraint By making the gradient of (24)with respect to G119894 equal to zero the first equation in (23)is achieved Afterwards by adding the power constraint andnonnegative Lagrangian dual variable constraint the KKTconditions shown in (23) are obtained

Depending on the KKT conditions the line searchmethod shown in [22] can be applied to optimize thecodebookmatrix It should be noted that mutual informationshown in (18) contains rather complex integrals and it isdifficult to achieve its closed-form expression In Section 4the calculation of mutual information is achieved by MonteCarlo simulations and the iterative codebook optimizationalgorithm is proposed

In addition it can be seen that when calculating thegradient with respect to G119894 in (21) the expression of Eb isrequired The details of deriving the expression of Eb aregiven in Appendix A It can be seen that Eb also contains verycomplex integrals and its value is obtained by Monte Carlosimulations

4 Iterative Codebook Optimization Algorithm

In Section 3 the KKT conditions do not give explicit methodto find the optimal codebook matrix In this section inspiredby the line search method in [22] the iterative codebookoptimization algorithm is proposed where the codebookoptimization is implemented by searching the suitable updatestep size along the direction of the gradient

In the first subsection the line search applied in the iter-ative codebook optimization algorithm is described After-wards the steps of the proposed algorithm are elaboratedBecause the mutual information and mean squared errordo not have closed-form expressions the optimization isimplemented based on their Monte Carlo simulation results

41 Line Search Optimization Method Based on the linesearch method in [22] the codebook matrix of each usershould be updated along the direction of the gradient Duringoptimization the update step size should be optimized tomake sure that mutual information after codebook updatingis nondecreasing In this paper the backtracking line searchmethod [22] is introduced to determine the step size

There are twonested loops in the proposed algorithmTheouter-loop index denotes the iteration number and the innerloop index denotes the user index from 1 to119870

In the 119899th outer loop the expression of 119868(b y) after the(119894 minus 1)th userrsquos updating is denoted by

119868(119899119894minus1) (b y)= 119891 (G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119894+1 G(119899)119896 ) (25)

6 Wireless Communications and Mobile Computing

Input Randomly select codebook matrix G(1)119894 1 le 119894 le 119870 tr(G119894G119867119894 ) = 119875119894(1) Initialization G(10) = blkdiag[G(1)1 G(1)119896 ](2) Outer loop for 119899 = 1 1 119873ite(3) Inner loop for 119894 = 1 1 119870

(a) Perform monte-carlo simulations to calculate 119868(119899119894minus1)(b y) and E(119899119894minus1)b(b) Calculate the gradient nablaG(119899)

119894

119868(119899119894minus1)(b y) according to (28)Do

(c) Update nablaG(119899)119894 according to (27)(d) Calculate G(119899)119894 to satisfy the power constraint according to (29)(e) Perform monte-carlo simulations to calculate 119868(119899119894)(b y) according to (30)(f) Update step size 119905 = 119905 times 120573

While 119868(119899119894)(b y) lt 119868(119899119894minus1)(b y) + 120572119905 times nablaG(119899)119894

119868(119899119894minus1)(b y)2119865(g) Generate the updated codebook matrix G(119899119894) = blkdiag[G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119896 ]

(4) End Inner loop(h) Generate G(119899+10) = G(119899119870)

(5) End Outer loop

Algorithm 1 Concrete process of iterative codebook optimization algorithm

where mutual information is considered as the function ofcodebook matrix of each user and the superscript (119899 119894 minus 1) of119868(b y) denotes the outer loop and inner loop index pair Thecodebook matrix corresponding to 119868(119899119894minus1)(b y) is given by

G(119899119894minus1) = blkdiag [G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119896 ] (26)

where the matrices from G(119899)1 to G(119899)119894minus1 denote the codebooksthat have been updated in the 119899th outer loop

In addition G(1198990) = blkdiag[G(119899)1 G(119899)119896 ] denotes theinitial codebook matrix in the 119899th iteration the correspond-ing mutual information is 119868(1198990)(b y) = 119891(G(119899)1 G(119899)119896 )

Based on the gradient expression in (21) the line searchresult is given by

nablaG(119899)119894 = G(119899)119894 + 119905nablaG(119899)119894

119868(119899119894minus1) (b y) (27)

where 119905 is the step size and the expression of nablaG(119899)119894

119868(119899119894minus1)(b y)is given by

nablaG119894119868(119899119894minus1) (b y) = 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)timesH119867119878 H119878G

(119899119894minus1)E(119899119894minus1)b

times (e119870119894 otimes I119889V)119867 (28)

where the mean squared error matrix E(119899119894minus1)b is calculatedbased on the codebook matrix G(119899119894minus1)

In addition the codebook matrix of each user shouldsatisfy the power constraint Assuming that the maximumtransmit power of user 119894 is equal to 119875119894 the normalizedcodebook matrix is given by

G(119899)119894 = radic119875119894 times nablaG(119899)11989410038171003817100381710038171003817nablaG(119899)119894 10038171003817100381710038171003817119865 (29)

Afterwards the 119894th 119889V times119889V diagonal blockG(119899)119894 is replacedby G(119899)119894 and the updated mutual information is calculatedaccording to

119868(119899119894) (b y) = 119891 (G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119894+1 G(119899)119896 ) (30)

Based on the backtracking line search method [22] thefollowing constraint should be satisfied to make sure that theupdated mutual information is nondecreasing

119868(119899119894) (b y) gt 119868(119899119894minus1) (b y) + 120572119905 100381710038171003817100381710038171003817nablaG(119899)119894 119868(119899119894minus1) (b y)1003817100381710038171003817100381710038172119865 (31)

where 120572 is the predetermined parameter and always belongsto the interval (0 03) [22]

If the above constraint is not satisfied the calculations in(27)ndash(30) are repeated to update 119868(119899119894)(b y) and the ldquoback-trackingrdquo is performed with updated size 119905 = 119905 times 120573 where120573 isin (0 08) is the predetermined parameter [22] Afterwardsthe constraint in (31) is retested

In the next subsection the detailed steps of the proposediterative codebook optimization algorithm are given

42 Concrete Steps of Iterative Codebook Optimization Algo-rithm From the analysis in Section 3 and Appendix A itis shown that both 119868(b y) and Eb contain rather complexintegrals and it is difficult to derive their closed-form expres-sions Therefore in the proposed algorithm the calculationof 119868(b y) and Eb is realized according to the Monte Carlosimulations which should cover all the 119872119870 constellationpoints It is believed that the computational complexity isproportional to119872119870

According to the above analysis concrete steps of theproposed algorithm are given in Algorithm 1 The parameter119873ite is the number of outer loops

It should be noted that the performance of backtrackingline search method depends on the initial values of the

Wireless Communications and Mobile Computing 7

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information non-AWGN channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10 12

Figure 2Mutual information performance in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6 and119873 = 4 Thechannel responses are given in Appendix B

codebook matrices Therefore in simulations the iterativeoptimization shown in Algorithm 1 should be repeated mul-tiple times with different initial codebook matrices

In order to evaluate upper bound of the proposed algo-rithm Gaussian channel capacity with the same channelcoefficient matrix should be calculated According to [27]under Gaussian input assumption the iterative water-fillingalgorithm is able to find the globally optimal power allocationresult which achieves Gaussian capacity bound This can beseen as the upper bound of the proposed iterative codebookoptimization algorithm

5 Simulation Results

In this section the simulation results are given With thefactor graph in Figure 1 and 119872 = 4 mutual informationbetween the information bit vector b and received signal y isbounded by119867(b) = 119870 log2119872 = 12 bit The codebook matrixof each user should satisfy the power constraint tr(G119894G119867119894 ) le119875119894 1 le 119894 le 119870 In the following simulations we set 119875119894 =119873119870 = 23 1 le 119894 le 119870 Simulation results in non-AWGNandAWGNchannels are given in Sections 51 and 52respectively

51 Non-AWGN Channel Simulation Results In Figure 2mutual information achieved by the proposed iterativecodebook optimization algorithm in non-AWGN channel isshown The responses of non-AWGN channel are given inAppendix B In addition the channel setting makes sure thatthe channel power satisfies the following constraint

tr (H119878H119867119878 ) = 119873119889119891 = 119870119889V = 12 (32)

1 2 3 4 5 6 7 8Outer loop number

2

3

4

5

6

7

8

9

Mut

ual i

nfor

mat

ion

(bit)

Iterative codebook optimization

Optimized codebook on 4 dBOptimized codebook on 2 dBOptimized codebook on 0 dB

Figure 3 Convergence performance of the proposed iterativeoptimization algorithm innon-AWGNchannelThe SNR is set equalto 0 dB 2 dB and 4 dB respectively

According to the analysis in Section 4 the performanceof the proposed iterative codebook optimization algorithmdepends on values of the initial codebook matrices There-fore the codebook optimization result is chosen from 20realizations with different initial codebook matrices

In Figure 2 the result of the proposed iterative code-book optimization algorithm is denoted by ldquooptimized code-bookrdquo The Gaussian capacity bound with the same channelresponses according to [27] is denoted by ldquogaussian capacityrdquoIn addition we introduce the scheme called ldquoGaussian powerinputrdquo In this setting the codebook matrix G119894 1 le 119894 le 119870is squared root of the power distribution matrix obtainedfrom iterative water-filling algorithm in [27] With above G119894mutual information between discrete input b and continuousoutput y is calculated and denoted by ldquoGaussian powerinputrdquo in Figure 2 From the analysis in [27] iterative water-filling algorithm also requires channel state informationIn addition the result of random codebook satisfying thepower constraint is denoted by ldquorandom codebookrdquo Figure 2demonstrates that the proposed iterative codebook optimiza-tion algorithm can approach Gaussian capacity bound in lowand medium SNR regime Due to the inability to track thechannel responses the performance of ldquorandom codebookrdquois worse than that of ldquooptimized codebookrdquo When SNRis lower than 1 dB the result of ldquoGaussian power inputrdquo isbetter than that of ldquorandom codebookrdquo However when SNRincreases ldquoGaussian power inputrdquo method fails to approachthe performance of ldquooptimized codebookrdquo This indicatesthat iterative water-filling algorithm with Gaussian inputassumption cannot be directly applied in the discrete inputchannel even with perfect channel state information

Furthermore in Figure 3 the convergence of the pro-posed iterative codebook optimization algorithm is shown

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

Wireless Communications and Mobile Computing 3

the channel capacity Based on the above analysis x1 can berewritten as follows

x1 = [1199091111990921] = [[119892(1)11 119892(1)12119892(1)21 119892(1)22]] sdot [119887(1)1119887(1)2 ] = G1b1 (5)

where the superscript (1) denotes the user index G1 denotesthe codebook matrix of user 1 and the bit vector b1 containsthe two information bits of user 1

By extending the above model to 119870 = 6 users thetransmit signal vector of user 119894 is given by

x119894 = G119894b119894 1 le 119894 le 119870 (6)

Therefore the overall transmitted vector can be obtained bystacking x1 to x6

x = [x1198791 x1198792 x1198793 x1198794 x1198795 x1198796 ]119879 = Gb

= blkdiag G1 G6 times b (7)

where G denotes the block diagonal codebook matrixand b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 b1198794 b1198795 b1198796 ]119879

In order to adapt to the transmit signal expression in (7)the channel expressionH119878 shown in (2) should be extended inthe column space It is noted that the dimension of x is equalto 119889V times 119870 = 12 Therefore the column number of channelmatrix should be extended from 6 to 12

H119878 =[[[[[[[

ℎ11 0 ℎ12 0 ℎ13 0 0 0 0 0 0 00 ℎ21 0 0 0 0 ℎ24 0 ℎ25 0 0 00 0 0 ℎ32 0 0 0 ℎ34 0 0 ℎ36 00 0 0 0 0 ℎ43 0 0 0 ℎ45 0 ℎ46

]]]]]]] (8)

For example in the first column of H119878 the channelcoefficients corresponding to 11990911 and 11990921 are ℎ11 and ℎ21respectively To match the two-dimensional transmissionvector x1 shown in (5) the first column of H119878 should beextended to generate the following two columns

h11 = [ℎ11 0 0 0]119879 h21 = [0 ℎ21 0 0]119879 (9)

Based on the above analysis the column-extended chan-nel matrixH119878 is given by the long expression in (8) It can beseen from (8) thatH119878 is obtained by dividing each column ofH119878 into two columns in order to match the two-dimensionalmodulation symbol vector of each user

Based onH119878 in (8) the received signal is rewritten as

y = H119878x + n = H119878Gb + n (10)

where n is 119873-dimensional additive white Gaussian noisevector with distribution CN(0 1205902119899I119873)

According to block diagonal property of G the receivedsignal y in (10) can be rewritten as the superposition resultsof119870 = 6 usersrsquo signals

y = H119878Gb + n = 119870sum119894=1

H119894G119894b119894 + n (11)

where H119894 is the equivalent channel matrix of user 119894 Forexample the equivalent channel matrixH1 corresponding touser 1 is given by

H1 = [[[[[[

ℎ11 00 ℎ210 00 0]]]]]] (12)

Based on the above analysis the received signal y isconnected to the codebook matrix G119894 of each user with thehelp of the column-extended channel matrix H119894 1 le 119894 le 119870Therefore the codebook optimization can be implementedaccording to various criteria In the next section the code-book matrix G119894 1 le 119894 le 119870 is optimized according tomaximizingmutual information between the bit vector b andthe received signal y shown in (11)

3 Principle of SCMA Codebook Optimization

In this section the codebook optimization to maximizemutual information is carefully analyzedWe assume that thenumber of users and the channel responses are known bythe transmitter In practical wireless communication systemsthis assumption is possible for the downlink transmissionwith channel state information feedback but not possible forthe uplink

At first the concrete expression of mutual informationbetween the bit vector b and the continuous received signal yis given Afterwards KKT conditions based on the gradientof mutual information are introduced to realize codebook

4 Wireless Communications and Mobile Computing

optimization It is shown that the gradient of mutual infor-mation with respect to codebook matrix G119894 1 le 119894 le 119870depends on the mean squared error matrix E119887 For clearnessthe details of calculating mean squared error matrix E119887 aregiven in Appendix A

31 Detailed Expression of Mutual Information Similar tothat in [23 24] mutual information between discrete inputb and continuous output y can be given by

119868 (b y) = 119867 (b) minus 119867 (b | y) = 119870 log2119872minus 119872119870sum119898=1

inty119901 (b119898 y) log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y = 119870

sdot log2119872minus 119872119870sum119898=1

inty119901 (b119898) 119901 (y | b119898)

sdot log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y

(13)

In this paper all possible input vectors are assumed to haveequal probability The input constellation alphabet size isequal to 119872119870 and 119901(b119898) = 1119872119870 When signal-to-noiseratio (SNR) tends to be infinity themutual information is notlarger than the entropy119867(b) which is equal to119870 log2119872 Thesubscript119898 denotes the index of the constellation point from1 to119872119870 With additive white Gaussian noise the conditionalprobability distribution function 119901(y | b119898) is given by

119901 (y | b119898) = 1(1205871205902119899)119873 exp(minus10038171003817100381710038171003817y minusH119878Gb1198981003817100381710038171003817100381721205902119899 ) (14)

In addition the probability distribution function 119901(y) in (13)can be given by

119901 (y) = 119872119870sum119896=1

119901 (b119896) 119901 (y | b119896)

= 119872119870sum119896=1

1119872119870 (1205871205902119899)119873 exp(minus10038171003817100381710038171003817y minusH119878Gb1198961003817100381710038171003817100381721205902119899 )

(15)

where the subscript 119896 denotes the constellation point indexWhen the bit vector b119898 is transmitted the received signal

is given by y = H119878Gb119898 + n In this case the unknowncontained in y is only additive white Gaussian noise vectorn Therefore the integral of y can be expressed as the integralof n Consequently in the 119898th integral of the summation in(13) y is replaced byH119878Gb119898 + n

inty119901 (b119898) 119901 (y | b119898) log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y= int

n119901 (b119898) 119901 (H119878Gb119898 + n | H119878Gb119898)

sdot log 119901 (H119878Gb119898 + n)119901 (b119898) 119901 (H119878Gb119898 + n | H119878Gb119898)119889n

= intn119901 (b119898) 119901 (n) log 119901 (H119878Gb119898 + n)119901 (b119898) 119901 (n) 119889n

= intn119901 (b119898) 119901 (n)

sdot log sum119872119870119896=1 119901 (b119896) 119901 (H119878Gb119898 + n | H119878Gb119896)119901 (b119898) 119901 (n) 119889n(16)

where 119901(n) = 1(1205871205902119899)119873 times exp(minusn21205902119899) For the firstequation we assume that the channel matrix and codebookmatrix are perfectly known With the expression of 119901(y | b119898)in (14) the second equation is achieved In the third equation119901(y | b119896) is replaced by 119901(H119878Gb119898 + n | H119878Gb119896) whoseexpression is given by

119901 (H119878Gb119898 + n | H119878Gb119896)= 1(1205871205902)119873 exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902 ) (17)

Based on the above analysis the integral value in (16) dependson the Euclidean distances betweenH119878Gb119898 and all the otherreceived signal constellations

Under the equal probability input assumption themutualinformation in (13) can be rewritten as

119868 (b y) = 119870 log2119872minus 1119872119870

119872119870sum119898=1

intn119901 (n) log119872119870sum

119896=1

exp (minus119902119898119896) 119889n= 119870 log2119872minus 1119872119870

119872119870sum119898=1

119864n [[log119872119870sum119896=1

exp (minus119902119898119896)]] (18)

where 119902119898119896 is given by

119902119898119896 =10038171003817100381710038171003817H119878G (b119898 minus b119896) + n100381710038171003817100381710038172 minus n21205902119899 (19)

It should be noted that 119898 and 119896 are both constellation pointindexes and they are independent of each other

From the above analysis it can be seen that mutualinformation is the function of codebook matrices In thefollowing subsection the gradient of mutual informationwith respect to codebook matrix of each user is analyzedand the KKT conditions are introduced to maximize mutualinformation

Wireless Communications and Mobile Computing 5

32 KKT Conditions When Maximizing Mutual InformationTo optimize mutual information the gradient of mutualinformation with respect to G119894 1 le 119894 le 119870 is calculated

According to the results in [25 26] the gradient withrespect to the overall block diagonal codebook matrix G isgiven ApplyingTheorem 2 in [26] we have

nablaG119868 (b y) = 120597120597Glowast 119868 (b y) = 1ln 2H119867119878 (1205902119899I119873)minus1H119878GE119887

= 1ln 2 sdot 1205902119899H119867119878 H119878GE119887

(20)

where the factor 1 ln 2 is appended because the naturallogarithm is applied in [26] The matrix Eb denotes the meansquared error matrix In [26] it is proven that the abovegradient expression holds for the linear received signal modelin (10) regardless of the structure of the channel matrix H119878and the codebook matrix G

In SCMA we assume that the codebook matrix of eachuser satisfies individual power constraint This requires thegradient with respect to each userrsquos codebook matrixG119894 1 le119894 le 119870 Based on the fact thatG119894 is 119889Vtimes119889V submatrix on the 119894thdiagonal block of G the gradient with respect to G119894 is givenby

nablaG119894119868 (b y) = 120597120597Glowast119894 119868 (b y)= (e119870119894 otimes I119889V) 120597120597G119868 (b y) (e119870119894 otimes I119889V)119867= 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)H119867119878 H119878GEb (e119870119894 otimes I119889V)119867

(21)

where e119870119894 is the 119894th row of the119870times119870 identity matrix I119873 From(19) it can be seen thatnablaG119894119868(b y) can be easily calculated fromthe result of nablaG119868(b y)

In order to maximize mutual information between b andy the optimization problem is given by

maxG119894 1le119894le119870

119868 (b y)st tr (G119894G119867119894 ) le 119875119894 1 le 119894 le 119870 (22)

Unfortunately 119868(b y) is not a convex function of the code-book matrix G119894 1 le 119894 le 119870 and it is difficult to calculate itsglobally optimal solution An efficient method to solve thiskind of problem is to find locally optimal solution accordingto KKT conditions [22] Therefore we have the followinglemma

Lemma 1 With the power constraint of each user the KKTconditions corresponding to problem (22) are given by

120582119894G119894 = nablaG119894119868 (b y) 120582119894 ge 0

tr (G119894G119867119894 ) le 119875119894120582119894 [tr (G119894G119867119894 ) minus 119875] = 0

(23)

Proof According to the result in [22] the Lagrangian corre-sponding to problem (22) is given by

119871 (120582119894G119894) = minus119868 (b y) + 119870sum119894=1

120582119894 [tr (G119894G119867119894 ) minus 119875119894] (24)

where 120582119894 is the Lagrangian dual variable corresponding to the119894th userrsquos power constraint By making the gradient of (24)with respect to G119894 equal to zero the first equation in (23)is achieved Afterwards by adding the power constraint andnonnegative Lagrangian dual variable constraint the KKTconditions shown in (23) are obtained

Depending on the KKT conditions the line searchmethod shown in [22] can be applied to optimize thecodebookmatrix It should be noted that mutual informationshown in (18) contains rather complex integrals and it isdifficult to achieve its closed-form expression In Section 4the calculation of mutual information is achieved by MonteCarlo simulations and the iterative codebook optimizationalgorithm is proposed

In addition it can be seen that when calculating thegradient with respect to G119894 in (21) the expression of Eb isrequired The details of deriving the expression of Eb aregiven in Appendix A It can be seen that Eb also contains verycomplex integrals and its value is obtained by Monte Carlosimulations

4 Iterative Codebook Optimization Algorithm

In Section 3 the KKT conditions do not give explicit methodto find the optimal codebook matrix In this section inspiredby the line search method in [22] the iterative codebookoptimization algorithm is proposed where the codebookoptimization is implemented by searching the suitable updatestep size along the direction of the gradient

In the first subsection the line search applied in the iter-ative codebook optimization algorithm is described After-wards the steps of the proposed algorithm are elaboratedBecause the mutual information and mean squared errordo not have closed-form expressions the optimization isimplemented based on their Monte Carlo simulation results

41 Line Search Optimization Method Based on the linesearch method in [22] the codebook matrix of each usershould be updated along the direction of the gradient Duringoptimization the update step size should be optimized tomake sure that mutual information after codebook updatingis nondecreasing In this paper the backtracking line searchmethod [22] is introduced to determine the step size

There are twonested loops in the proposed algorithmTheouter-loop index denotes the iteration number and the innerloop index denotes the user index from 1 to119870

In the 119899th outer loop the expression of 119868(b y) after the(119894 minus 1)th userrsquos updating is denoted by

119868(119899119894minus1) (b y)= 119891 (G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119894+1 G(119899)119896 ) (25)

6 Wireless Communications and Mobile Computing

Input Randomly select codebook matrix G(1)119894 1 le 119894 le 119870 tr(G119894G119867119894 ) = 119875119894(1) Initialization G(10) = blkdiag[G(1)1 G(1)119896 ](2) Outer loop for 119899 = 1 1 119873ite(3) Inner loop for 119894 = 1 1 119870

(a) Perform monte-carlo simulations to calculate 119868(119899119894minus1)(b y) and E(119899119894minus1)b(b) Calculate the gradient nablaG(119899)

119894

119868(119899119894minus1)(b y) according to (28)Do

(c) Update nablaG(119899)119894 according to (27)(d) Calculate G(119899)119894 to satisfy the power constraint according to (29)(e) Perform monte-carlo simulations to calculate 119868(119899119894)(b y) according to (30)(f) Update step size 119905 = 119905 times 120573

While 119868(119899119894)(b y) lt 119868(119899119894minus1)(b y) + 120572119905 times nablaG(119899)119894

119868(119899119894minus1)(b y)2119865(g) Generate the updated codebook matrix G(119899119894) = blkdiag[G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119896 ]

(4) End Inner loop(h) Generate G(119899+10) = G(119899119870)

(5) End Outer loop

Algorithm 1 Concrete process of iterative codebook optimization algorithm

where mutual information is considered as the function ofcodebook matrix of each user and the superscript (119899 119894 minus 1) of119868(b y) denotes the outer loop and inner loop index pair Thecodebook matrix corresponding to 119868(119899119894minus1)(b y) is given by

G(119899119894minus1) = blkdiag [G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119896 ] (26)

where the matrices from G(119899)1 to G(119899)119894minus1 denote the codebooksthat have been updated in the 119899th outer loop

In addition G(1198990) = blkdiag[G(119899)1 G(119899)119896 ] denotes theinitial codebook matrix in the 119899th iteration the correspond-ing mutual information is 119868(1198990)(b y) = 119891(G(119899)1 G(119899)119896 )

Based on the gradient expression in (21) the line searchresult is given by

nablaG(119899)119894 = G(119899)119894 + 119905nablaG(119899)119894

119868(119899119894minus1) (b y) (27)

where 119905 is the step size and the expression of nablaG(119899)119894

119868(119899119894minus1)(b y)is given by

nablaG119894119868(119899119894minus1) (b y) = 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)timesH119867119878 H119878G

(119899119894minus1)E(119899119894minus1)b

times (e119870119894 otimes I119889V)119867 (28)

where the mean squared error matrix E(119899119894minus1)b is calculatedbased on the codebook matrix G(119899119894minus1)

In addition the codebook matrix of each user shouldsatisfy the power constraint Assuming that the maximumtransmit power of user 119894 is equal to 119875119894 the normalizedcodebook matrix is given by

G(119899)119894 = radic119875119894 times nablaG(119899)11989410038171003817100381710038171003817nablaG(119899)119894 10038171003817100381710038171003817119865 (29)

Afterwards the 119894th 119889V times119889V diagonal blockG(119899)119894 is replacedby G(119899)119894 and the updated mutual information is calculatedaccording to

119868(119899119894) (b y) = 119891 (G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119894+1 G(119899)119896 ) (30)

Based on the backtracking line search method [22] thefollowing constraint should be satisfied to make sure that theupdated mutual information is nondecreasing

119868(119899119894) (b y) gt 119868(119899119894minus1) (b y) + 120572119905 100381710038171003817100381710038171003817nablaG(119899)119894 119868(119899119894minus1) (b y)1003817100381710038171003817100381710038172119865 (31)

where 120572 is the predetermined parameter and always belongsto the interval (0 03) [22]

If the above constraint is not satisfied the calculations in(27)ndash(30) are repeated to update 119868(119899119894)(b y) and the ldquoback-trackingrdquo is performed with updated size 119905 = 119905 times 120573 where120573 isin (0 08) is the predetermined parameter [22] Afterwardsthe constraint in (31) is retested

In the next subsection the detailed steps of the proposediterative codebook optimization algorithm are given

42 Concrete Steps of Iterative Codebook Optimization Algo-rithm From the analysis in Section 3 and Appendix A itis shown that both 119868(b y) and Eb contain rather complexintegrals and it is difficult to derive their closed-form expres-sions Therefore in the proposed algorithm the calculationof 119868(b y) and Eb is realized according to the Monte Carlosimulations which should cover all the 119872119870 constellationpoints It is believed that the computational complexity isproportional to119872119870

According to the above analysis concrete steps of theproposed algorithm are given in Algorithm 1 The parameter119873ite is the number of outer loops

It should be noted that the performance of backtrackingline search method depends on the initial values of the

Wireless Communications and Mobile Computing 7

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information non-AWGN channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10 12

Figure 2Mutual information performance in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6 and119873 = 4 Thechannel responses are given in Appendix B

codebook matrices Therefore in simulations the iterativeoptimization shown in Algorithm 1 should be repeated mul-tiple times with different initial codebook matrices

In order to evaluate upper bound of the proposed algo-rithm Gaussian channel capacity with the same channelcoefficient matrix should be calculated According to [27]under Gaussian input assumption the iterative water-fillingalgorithm is able to find the globally optimal power allocationresult which achieves Gaussian capacity bound This can beseen as the upper bound of the proposed iterative codebookoptimization algorithm

5 Simulation Results

In this section the simulation results are given With thefactor graph in Figure 1 and 119872 = 4 mutual informationbetween the information bit vector b and received signal y isbounded by119867(b) = 119870 log2119872 = 12 bit The codebook matrixof each user should satisfy the power constraint tr(G119894G119867119894 ) le119875119894 1 le 119894 le 119870 In the following simulations we set 119875119894 =119873119870 = 23 1 le 119894 le 119870 Simulation results in non-AWGNandAWGNchannels are given in Sections 51 and 52respectively

51 Non-AWGN Channel Simulation Results In Figure 2mutual information achieved by the proposed iterativecodebook optimization algorithm in non-AWGN channel isshown The responses of non-AWGN channel are given inAppendix B In addition the channel setting makes sure thatthe channel power satisfies the following constraint

tr (H119878H119867119878 ) = 119873119889119891 = 119870119889V = 12 (32)

1 2 3 4 5 6 7 8Outer loop number

2

3

4

5

6

7

8

9

Mut

ual i

nfor

mat

ion

(bit)

Iterative codebook optimization

Optimized codebook on 4 dBOptimized codebook on 2 dBOptimized codebook on 0 dB

Figure 3 Convergence performance of the proposed iterativeoptimization algorithm innon-AWGNchannelThe SNR is set equalto 0 dB 2 dB and 4 dB respectively

According to the analysis in Section 4 the performanceof the proposed iterative codebook optimization algorithmdepends on values of the initial codebook matrices There-fore the codebook optimization result is chosen from 20realizations with different initial codebook matrices

In Figure 2 the result of the proposed iterative code-book optimization algorithm is denoted by ldquooptimized code-bookrdquo The Gaussian capacity bound with the same channelresponses according to [27] is denoted by ldquogaussian capacityrdquoIn addition we introduce the scheme called ldquoGaussian powerinputrdquo In this setting the codebook matrix G119894 1 le 119894 le 119870is squared root of the power distribution matrix obtainedfrom iterative water-filling algorithm in [27] With above G119894mutual information between discrete input b and continuousoutput y is calculated and denoted by ldquoGaussian powerinputrdquo in Figure 2 From the analysis in [27] iterative water-filling algorithm also requires channel state informationIn addition the result of random codebook satisfying thepower constraint is denoted by ldquorandom codebookrdquo Figure 2demonstrates that the proposed iterative codebook optimiza-tion algorithm can approach Gaussian capacity bound in lowand medium SNR regime Due to the inability to track thechannel responses the performance of ldquorandom codebookrdquois worse than that of ldquooptimized codebookrdquo When SNRis lower than 1 dB the result of ldquoGaussian power inputrdquo isbetter than that of ldquorandom codebookrdquo However when SNRincreases ldquoGaussian power inputrdquo method fails to approachthe performance of ldquooptimized codebookrdquo This indicatesthat iterative water-filling algorithm with Gaussian inputassumption cannot be directly applied in the discrete inputchannel even with perfect channel state information

Furthermore in Figure 3 the convergence of the pro-posed iterative codebook optimization algorithm is shown

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

4 Wireless Communications and Mobile Computing

optimization It is shown that the gradient of mutual infor-mation with respect to codebook matrix G119894 1 le 119894 le 119870depends on the mean squared error matrix E119887 For clearnessthe details of calculating mean squared error matrix E119887 aregiven in Appendix A

31 Detailed Expression of Mutual Information Similar tothat in [23 24] mutual information between discrete inputb and continuous output y can be given by

119868 (b y) = 119867 (b) minus 119867 (b | y) = 119870 log2119872minus 119872119870sum119898=1

inty119901 (b119898 y) log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y = 119870

sdot log2119872minus 119872119870sum119898=1

inty119901 (b119898) 119901 (y | b119898)

sdot log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y

(13)

In this paper all possible input vectors are assumed to haveequal probability The input constellation alphabet size isequal to 119872119870 and 119901(b119898) = 1119872119870 When signal-to-noiseratio (SNR) tends to be infinity themutual information is notlarger than the entropy119867(b) which is equal to119870 log2119872 Thesubscript119898 denotes the index of the constellation point from1 to119872119870 With additive white Gaussian noise the conditionalprobability distribution function 119901(y | b119898) is given by

119901 (y | b119898) = 1(1205871205902119899)119873 exp(minus10038171003817100381710038171003817y minusH119878Gb1198981003817100381710038171003817100381721205902119899 ) (14)

In addition the probability distribution function 119901(y) in (13)can be given by

119901 (y) = 119872119870sum119896=1

119901 (b119896) 119901 (y | b119896)

= 119872119870sum119896=1

1119872119870 (1205871205902119899)119873 exp(minus10038171003817100381710038171003817y minusH119878Gb1198961003817100381710038171003817100381721205902119899 )

(15)

where the subscript 119896 denotes the constellation point indexWhen the bit vector b119898 is transmitted the received signal

is given by y = H119878Gb119898 + n In this case the unknowncontained in y is only additive white Gaussian noise vectorn Therefore the integral of y can be expressed as the integralof n Consequently in the 119898th integral of the summation in(13) y is replaced byH119878Gb119898 + n

inty119901 (b119898) 119901 (y | b119898) log 119901 (y)119901 (b119898) 119901 (y | b119898)119889y= int

n119901 (b119898) 119901 (H119878Gb119898 + n | H119878Gb119898)

sdot log 119901 (H119878Gb119898 + n)119901 (b119898) 119901 (H119878Gb119898 + n | H119878Gb119898)119889n

= intn119901 (b119898) 119901 (n) log 119901 (H119878Gb119898 + n)119901 (b119898) 119901 (n) 119889n

= intn119901 (b119898) 119901 (n)

sdot log sum119872119870119896=1 119901 (b119896) 119901 (H119878Gb119898 + n | H119878Gb119896)119901 (b119898) 119901 (n) 119889n(16)

where 119901(n) = 1(1205871205902119899)119873 times exp(minusn21205902119899) For the firstequation we assume that the channel matrix and codebookmatrix are perfectly known With the expression of 119901(y | b119898)in (14) the second equation is achieved In the third equation119901(y | b119896) is replaced by 119901(H119878Gb119898 + n | H119878Gb119896) whoseexpression is given by

119901 (H119878Gb119898 + n | H119878Gb119896)= 1(1205871205902)119873 exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902 ) (17)

Based on the above analysis the integral value in (16) dependson the Euclidean distances betweenH119878Gb119898 and all the otherreceived signal constellations

Under the equal probability input assumption themutualinformation in (13) can be rewritten as

119868 (b y) = 119870 log2119872minus 1119872119870

119872119870sum119898=1

intn119901 (n) log119872119870sum

119896=1

exp (minus119902119898119896) 119889n= 119870 log2119872minus 1119872119870

119872119870sum119898=1

119864n [[log119872119870sum119896=1

exp (minus119902119898119896)]] (18)

where 119902119898119896 is given by

119902119898119896 =10038171003817100381710038171003817H119878G (b119898 minus b119896) + n100381710038171003817100381710038172 minus n21205902119899 (19)

It should be noted that 119898 and 119896 are both constellation pointindexes and they are independent of each other

From the above analysis it can be seen that mutualinformation is the function of codebook matrices In thefollowing subsection the gradient of mutual informationwith respect to codebook matrix of each user is analyzedand the KKT conditions are introduced to maximize mutualinformation

Wireless Communications and Mobile Computing 5

32 KKT Conditions When Maximizing Mutual InformationTo optimize mutual information the gradient of mutualinformation with respect to G119894 1 le 119894 le 119870 is calculated

According to the results in [25 26] the gradient withrespect to the overall block diagonal codebook matrix G isgiven ApplyingTheorem 2 in [26] we have

nablaG119868 (b y) = 120597120597Glowast 119868 (b y) = 1ln 2H119867119878 (1205902119899I119873)minus1H119878GE119887

= 1ln 2 sdot 1205902119899H119867119878 H119878GE119887

(20)

where the factor 1 ln 2 is appended because the naturallogarithm is applied in [26] The matrix Eb denotes the meansquared error matrix In [26] it is proven that the abovegradient expression holds for the linear received signal modelin (10) regardless of the structure of the channel matrix H119878and the codebook matrix G

In SCMA we assume that the codebook matrix of eachuser satisfies individual power constraint This requires thegradient with respect to each userrsquos codebook matrixG119894 1 le119894 le 119870 Based on the fact thatG119894 is 119889Vtimes119889V submatrix on the 119894thdiagonal block of G the gradient with respect to G119894 is givenby

nablaG119894119868 (b y) = 120597120597Glowast119894 119868 (b y)= (e119870119894 otimes I119889V) 120597120597G119868 (b y) (e119870119894 otimes I119889V)119867= 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)H119867119878 H119878GEb (e119870119894 otimes I119889V)119867

(21)

where e119870119894 is the 119894th row of the119870times119870 identity matrix I119873 From(19) it can be seen thatnablaG119894119868(b y) can be easily calculated fromthe result of nablaG119868(b y)

In order to maximize mutual information between b andy the optimization problem is given by

maxG119894 1le119894le119870

119868 (b y)st tr (G119894G119867119894 ) le 119875119894 1 le 119894 le 119870 (22)

Unfortunately 119868(b y) is not a convex function of the code-book matrix G119894 1 le 119894 le 119870 and it is difficult to calculate itsglobally optimal solution An efficient method to solve thiskind of problem is to find locally optimal solution accordingto KKT conditions [22] Therefore we have the followinglemma

Lemma 1 With the power constraint of each user the KKTconditions corresponding to problem (22) are given by

120582119894G119894 = nablaG119894119868 (b y) 120582119894 ge 0

tr (G119894G119867119894 ) le 119875119894120582119894 [tr (G119894G119867119894 ) minus 119875] = 0

(23)

Proof According to the result in [22] the Lagrangian corre-sponding to problem (22) is given by

119871 (120582119894G119894) = minus119868 (b y) + 119870sum119894=1

120582119894 [tr (G119894G119867119894 ) minus 119875119894] (24)

where 120582119894 is the Lagrangian dual variable corresponding to the119894th userrsquos power constraint By making the gradient of (24)with respect to G119894 equal to zero the first equation in (23)is achieved Afterwards by adding the power constraint andnonnegative Lagrangian dual variable constraint the KKTconditions shown in (23) are obtained

Depending on the KKT conditions the line searchmethod shown in [22] can be applied to optimize thecodebookmatrix It should be noted that mutual informationshown in (18) contains rather complex integrals and it isdifficult to achieve its closed-form expression In Section 4the calculation of mutual information is achieved by MonteCarlo simulations and the iterative codebook optimizationalgorithm is proposed

In addition it can be seen that when calculating thegradient with respect to G119894 in (21) the expression of Eb isrequired The details of deriving the expression of Eb aregiven in Appendix A It can be seen that Eb also contains verycomplex integrals and its value is obtained by Monte Carlosimulations

4 Iterative Codebook Optimization Algorithm

In Section 3 the KKT conditions do not give explicit methodto find the optimal codebook matrix In this section inspiredby the line search method in [22] the iterative codebookoptimization algorithm is proposed where the codebookoptimization is implemented by searching the suitable updatestep size along the direction of the gradient

In the first subsection the line search applied in the iter-ative codebook optimization algorithm is described After-wards the steps of the proposed algorithm are elaboratedBecause the mutual information and mean squared errordo not have closed-form expressions the optimization isimplemented based on their Monte Carlo simulation results

41 Line Search Optimization Method Based on the linesearch method in [22] the codebook matrix of each usershould be updated along the direction of the gradient Duringoptimization the update step size should be optimized tomake sure that mutual information after codebook updatingis nondecreasing In this paper the backtracking line searchmethod [22] is introduced to determine the step size

There are twonested loops in the proposed algorithmTheouter-loop index denotes the iteration number and the innerloop index denotes the user index from 1 to119870

In the 119899th outer loop the expression of 119868(b y) after the(119894 minus 1)th userrsquos updating is denoted by

119868(119899119894minus1) (b y)= 119891 (G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119894+1 G(119899)119896 ) (25)

6 Wireless Communications and Mobile Computing

Input Randomly select codebook matrix G(1)119894 1 le 119894 le 119870 tr(G119894G119867119894 ) = 119875119894(1) Initialization G(10) = blkdiag[G(1)1 G(1)119896 ](2) Outer loop for 119899 = 1 1 119873ite(3) Inner loop for 119894 = 1 1 119870

(a) Perform monte-carlo simulations to calculate 119868(119899119894minus1)(b y) and E(119899119894minus1)b(b) Calculate the gradient nablaG(119899)

119894

119868(119899119894minus1)(b y) according to (28)Do

(c) Update nablaG(119899)119894 according to (27)(d) Calculate G(119899)119894 to satisfy the power constraint according to (29)(e) Perform monte-carlo simulations to calculate 119868(119899119894)(b y) according to (30)(f) Update step size 119905 = 119905 times 120573

While 119868(119899119894)(b y) lt 119868(119899119894minus1)(b y) + 120572119905 times nablaG(119899)119894

119868(119899119894minus1)(b y)2119865(g) Generate the updated codebook matrix G(119899119894) = blkdiag[G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119896 ]

(4) End Inner loop(h) Generate G(119899+10) = G(119899119870)

(5) End Outer loop

Algorithm 1 Concrete process of iterative codebook optimization algorithm

where mutual information is considered as the function ofcodebook matrix of each user and the superscript (119899 119894 minus 1) of119868(b y) denotes the outer loop and inner loop index pair Thecodebook matrix corresponding to 119868(119899119894minus1)(b y) is given by

G(119899119894minus1) = blkdiag [G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119896 ] (26)

where the matrices from G(119899)1 to G(119899)119894minus1 denote the codebooksthat have been updated in the 119899th outer loop

In addition G(1198990) = blkdiag[G(119899)1 G(119899)119896 ] denotes theinitial codebook matrix in the 119899th iteration the correspond-ing mutual information is 119868(1198990)(b y) = 119891(G(119899)1 G(119899)119896 )

Based on the gradient expression in (21) the line searchresult is given by

nablaG(119899)119894 = G(119899)119894 + 119905nablaG(119899)119894

119868(119899119894minus1) (b y) (27)

where 119905 is the step size and the expression of nablaG(119899)119894

119868(119899119894minus1)(b y)is given by

nablaG119894119868(119899119894minus1) (b y) = 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)timesH119867119878 H119878G

(119899119894minus1)E(119899119894minus1)b

times (e119870119894 otimes I119889V)119867 (28)

where the mean squared error matrix E(119899119894minus1)b is calculatedbased on the codebook matrix G(119899119894minus1)

In addition the codebook matrix of each user shouldsatisfy the power constraint Assuming that the maximumtransmit power of user 119894 is equal to 119875119894 the normalizedcodebook matrix is given by

G(119899)119894 = radic119875119894 times nablaG(119899)11989410038171003817100381710038171003817nablaG(119899)119894 10038171003817100381710038171003817119865 (29)

Afterwards the 119894th 119889V times119889V diagonal blockG(119899)119894 is replacedby G(119899)119894 and the updated mutual information is calculatedaccording to

119868(119899119894) (b y) = 119891 (G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119894+1 G(119899)119896 ) (30)

Based on the backtracking line search method [22] thefollowing constraint should be satisfied to make sure that theupdated mutual information is nondecreasing

119868(119899119894) (b y) gt 119868(119899119894minus1) (b y) + 120572119905 100381710038171003817100381710038171003817nablaG(119899)119894 119868(119899119894minus1) (b y)1003817100381710038171003817100381710038172119865 (31)

where 120572 is the predetermined parameter and always belongsto the interval (0 03) [22]

If the above constraint is not satisfied the calculations in(27)ndash(30) are repeated to update 119868(119899119894)(b y) and the ldquoback-trackingrdquo is performed with updated size 119905 = 119905 times 120573 where120573 isin (0 08) is the predetermined parameter [22] Afterwardsthe constraint in (31) is retested

In the next subsection the detailed steps of the proposediterative codebook optimization algorithm are given

42 Concrete Steps of Iterative Codebook Optimization Algo-rithm From the analysis in Section 3 and Appendix A itis shown that both 119868(b y) and Eb contain rather complexintegrals and it is difficult to derive their closed-form expres-sions Therefore in the proposed algorithm the calculationof 119868(b y) and Eb is realized according to the Monte Carlosimulations which should cover all the 119872119870 constellationpoints It is believed that the computational complexity isproportional to119872119870

According to the above analysis concrete steps of theproposed algorithm are given in Algorithm 1 The parameter119873ite is the number of outer loops

It should be noted that the performance of backtrackingline search method depends on the initial values of the

Wireless Communications and Mobile Computing 7

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information non-AWGN channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10 12

Figure 2Mutual information performance in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6 and119873 = 4 Thechannel responses are given in Appendix B

codebook matrices Therefore in simulations the iterativeoptimization shown in Algorithm 1 should be repeated mul-tiple times with different initial codebook matrices

In order to evaluate upper bound of the proposed algo-rithm Gaussian channel capacity with the same channelcoefficient matrix should be calculated According to [27]under Gaussian input assumption the iterative water-fillingalgorithm is able to find the globally optimal power allocationresult which achieves Gaussian capacity bound This can beseen as the upper bound of the proposed iterative codebookoptimization algorithm

5 Simulation Results

In this section the simulation results are given With thefactor graph in Figure 1 and 119872 = 4 mutual informationbetween the information bit vector b and received signal y isbounded by119867(b) = 119870 log2119872 = 12 bit The codebook matrixof each user should satisfy the power constraint tr(G119894G119867119894 ) le119875119894 1 le 119894 le 119870 In the following simulations we set 119875119894 =119873119870 = 23 1 le 119894 le 119870 Simulation results in non-AWGNandAWGNchannels are given in Sections 51 and 52respectively

51 Non-AWGN Channel Simulation Results In Figure 2mutual information achieved by the proposed iterativecodebook optimization algorithm in non-AWGN channel isshown The responses of non-AWGN channel are given inAppendix B In addition the channel setting makes sure thatthe channel power satisfies the following constraint

tr (H119878H119867119878 ) = 119873119889119891 = 119870119889V = 12 (32)

1 2 3 4 5 6 7 8Outer loop number

2

3

4

5

6

7

8

9

Mut

ual i

nfor

mat

ion

(bit)

Iterative codebook optimization

Optimized codebook on 4 dBOptimized codebook on 2 dBOptimized codebook on 0 dB

Figure 3 Convergence performance of the proposed iterativeoptimization algorithm innon-AWGNchannelThe SNR is set equalto 0 dB 2 dB and 4 dB respectively

According to the analysis in Section 4 the performanceof the proposed iterative codebook optimization algorithmdepends on values of the initial codebook matrices There-fore the codebook optimization result is chosen from 20realizations with different initial codebook matrices

In Figure 2 the result of the proposed iterative code-book optimization algorithm is denoted by ldquooptimized code-bookrdquo The Gaussian capacity bound with the same channelresponses according to [27] is denoted by ldquogaussian capacityrdquoIn addition we introduce the scheme called ldquoGaussian powerinputrdquo In this setting the codebook matrix G119894 1 le 119894 le 119870is squared root of the power distribution matrix obtainedfrom iterative water-filling algorithm in [27] With above G119894mutual information between discrete input b and continuousoutput y is calculated and denoted by ldquoGaussian powerinputrdquo in Figure 2 From the analysis in [27] iterative water-filling algorithm also requires channel state informationIn addition the result of random codebook satisfying thepower constraint is denoted by ldquorandom codebookrdquo Figure 2demonstrates that the proposed iterative codebook optimiza-tion algorithm can approach Gaussian capacity bound in lowand medium SNR regime Due to the inability to track thechannel responses the performance of ldquorandom codebookrdquois worse than that of ldquooptimized codebookrdquo When SNRis lower than 1 dB the result of ldquoGaussian power inputrdquo isbetter than that of ldquorandom codebookrdquo However when SNRincreases ldquoGaussian power inputrdquo method fails to approachthe performance of ldquooptimized codebookrdquo This indicatesthat iterative water-filling algorithm with Gaussian inputassumption cannot be directly applied in the discrete inputchannel even with perfect channel state information

Furthermore in Figure 3 the convergence of the pro-posed iterative codebook optimization algorithm is shown

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

Wireless Communications and Mobile Computing 5

32 KKT Conditions When Maximizing Mutual InformationTo optimize mutual information the gradient of mutualinformation with respect to G119894 1 le 119894 le 119870 is calculated

According to the results in [25 26] the gradient withrespect to the overall block diagonal codebook matrix G isgiven ApplyingTheorem 2 in [26] we have

nablaG119868 (b y) = 120597120597Glowast 119868 (b y) = 1ln 2H119867119878 (1205902119899I119873)minus1H119878GE119887

= 1ln 2 sdot 1205902119899H119867119878 H119878GE119887

(20)

where the factor 1 ln 2 is appended because the naturallogarithm is applied in [26] The matrix Eb denotes the meansquared error matrix In [26] it is proven that the abovegradient expression holds for the linear received signal modelin (10) regardless of the structure of the channel matrix H119878and the codebook matrix G

In SCMA we assume that the codebook matrix of eachuser satisfies individual power constraint This requires thegradient with respect to each userrsquos codebook matrixG119894 1 le119894 le 119870 Based on the fact thatG119894 is 119889Vtimes119889V submatrix on the 119894thdiagonal block of G the gradient with respect to G119894 is givenby

nablaG119894119868 (b y) = 120597120597Glowast119894 119868 (b y)= (e119870119894 otimes I119889V) 120597120597G119868 (b y) (e119870119894 otimes I119889V)119867= 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)H119867119878 H119878GEb (e119870119894 otimes I119889V)119867

(21)

where e119870119894 is the 119894th row of the119870times119870 identity matrix I119873 From(19) it can be seen thatnablaG119894119868(b y) can be easily calculated fromthe result of nablaG119868(b y)

In order to maximize mutual information between b andy the optimization problem is given by

maxG119894 1le119894le119870

119868 (b y)st tr (G119894G119867119894 ) le 119875119894 1 le 119894 le 119870 (22)

Unfortunately 119868(b y) is not a convex function of the code-book matrix G119894 1 le 119894 le 119870 and it is difficult to calculate itsglobally optimal solution An efficient method to solve thiskind of problem is to find locally optimal solution accordingto KKT conditions [22] Therefore we have the followinglemma

Lemma 1 With the power constraint of each user the KKTconditions corresponding to problem (22) are given by

120582119894G119894 = nablaG119894119868 (b y) 120582119894 ge 0

tr (G119894G119867119894 ) le 119875119894120582119894 [tr (G119894G119867119894 ) minus 119875] = 0

(23)

Proof According to the result in [22] the Lagrangian corre-sponding to problem (22) is given by

119871 (120582119894G119894) = minus119868 (b y) + 119870sum119894=1

120582119894 [tr (G119894G119867119894 ) minus 119875119894] (24)

where 120582119894 is the Lagrangian dual variable corresponding to the119894th userrsquos power constraint By making the gradient of (24)with respect to G119894 equal to zero the first equation in (23)is achieved Afterwards by adding the power constraint andnonnegative Lagrangian dual variable constraint the KKTconditions shown in (23) are obtained

Depending on the KKT conditions the line searchmethod shown in [22] can be applied to optimize thecodebookmatrix It should be noted that mutual informationshown in (18) contains rather complex integrals and it isdifficult to achieve its closed-form expression In Section 4the calculation of mutual information is achieved by MonteCarlo simulations and the iterative codebook optimizationalgorithm is proposed

In addition it can be seen that when calculating thegradient with respect to G119894 in (21) the expression of Eb isrequired The details of deriving the expression of Eb aregiven in Appendix A It can be seen that Eb also contains verycomplex integrals and its value is obtained by Monte Carlosimulations

4 Iterative Codebook Optimization Algorithm

In Section 3 the KKT conditions do not give explicit methodto find the optimal codebook matrix In this section inspiredby the line search method in [22] the iterative codebookoptimization algorithm is proposed where the codebookoptimization is implemented by searching the suitable updatestep size along the direction of the gradient

In the first subsection the line search applied in the iter-ative codebook optimization algorithm is described After-wards the steps of the proposed algorithm are elaboratedBecause the mutual information and mean squared errordo not have closed-form expressions the optimization isimplemented based on their Monte Carlo simulation results

41 Line Search Optimization Method Based on the linesearch method in [22] the codebook matrix of each usershould be updated along the direction of the gradient Duringoptimization the update step size should be optimized tomake sure that mutual information after codebook updatingis nondecreasing In this paper the backtracking line searchmethod [22] is introduced to determine the step size

There are twonested loops in the proposed algorithmTheouter-loop index denotes the iteration number and the innerloop index denotes the user index from 1 to119870

In the 119899th outer loop the expression of 119868(b y) after the(119894 minus 1)th userrsquos updating is denoted by

119868(119899119894minus1) (b y)= 119891 (G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119894+1 G(119899)119896 ) (25)

6 Wireless Communications and Mobile Computing

Input Randomly select codebook matrix G(1)119894 1 le 119894 le 119870 tr(G119894G119867119894 ) = 119875119894(1) Initialization G(10) = blkdiag[G(1)1 G(1)119896 ](2) Outer loop for 119899 = 1 1 119873ite(3) Inner loop for 119894 = 1 1 119870

(a) Perform monte-carlo simulations to calculate 119868(119899119894minus1)(b y) and E(119899119894minus1)b(b) Calculate the gradient nablaG(119899)

119894

119868(119899119894minus1)(b y) according to (28)Do

(c) Update nablaG(119899)119894 according to (27)(d) Calculate G(119899)119894 to satisfy the power constraint according to (29)(e) Perform monte-carlo simulations to calculate 119868(119899119894)(b y) according to (30)(f) Update step size 119905 = 119905 times 120573

While 119868(119899119894)(b y) lt 119868(119899119894minus1)(b y) + 120572119905 times nablaG(119899)119894

119868(119899119894minus1)(b y)2119865(g) Generate the updated codebook matrix G(119899119894) = blkdiag[G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119896 ]

(4) End Inner loop(h) Generate G(119899+10) = G(119899119870)

(5) End Outer loop

Algorithm 1 Concrete process of iterative codebook optimization algorithm

where mutual information is considered as the function ofcodebook matrix of each user and the superscript (119899 119894 minus 1) of119868(b y) denotes the outer loop and inner loop index pair Thecodebook matrix corresponding to 119868(119899119894minus1)(b y) is given by

G(119899119894minus1) = blkdiag [G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119896 ] (26)

where the matrices from G(119899)1 to G(119899)119894minus1 denote the codebooksthat have been updated in the 119899th outer loop

In addition G(1198990) = blkdiag[G(119899)1 G(119899)119896 ] denotes theinitial codebook matrix in the 119899th iteration the correspond-ing mutual information is 119868(1198990)(b y) = 119891(G(119899)1 G(119899)119896 )

Based on the gradient expression in (21) the line searchresult is given by

nablaG(119899)119894 = G(119899)119894 + 119905nablaG(119899)119894

119868(119899119894minus1) (b y) (27)

where 119905 is the step size and the expression of nablaG(119899)119894

119868(119899119894minus1)(b y)is given by

nablaG119894119868(119899119894minus1) (b y) = 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)timesH119867119878 H119878G

(119899119894minus1)E(119899119894minus1)b

times (e119870119894 otimes I119889V)119867 (28)

where the mean squared error matrix E(119899119894minus1)b is calculatedbased on the codebook matrix G(119899119894minus1)

In addition the codebook matrix of each user shouldsatisfy the power constraint Assuming that the maximumtransmit power of user 119894 is equal to 119875119894 the normalizedcodebook matrix is given by

G(119899)119894 = radic119875119894 times nablaG(119899)11989410038171003817100381710038171003817nablaG(119899)119894 10038171003817100381710038171003817119865 (29)

Afterwards the 119894th 119889V times119889V diagonal blockG(119899)119894 is replacedby G(119899)119894 and the updated mutual information is calculatedaccording to

119868(119899119894) (b y) = 119891 (G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119894+1 G(119899)119896 ) (30)

Based on the backtracking line search method [22] thefollowing constraint should be satisfied to make sure that theupdated mutual information is nondecreasing

119868(119899119894) (b y) gt 119868(119899119894minus1) (b y) + 120572119905 100381710038171003817100381710038171003817nablaG(119899)119894 119868(119899119894minus1) (b y)1003817100381710038171003817100381710038172119865 (31)

where 120572 is the predetermined parameter and always belongsto the interval (0 03) [22]

If the above constraint is not satisfied the calculations in(27)ndash(30) are repeated to update 119868(119899119894)(b y) and the ldquoback-trackingrdquo is performed with updated size 119905 = 119905 times 120573 where120573 isin (0 08) is the predetermined parameter [22] Afterwardsthe constraint in (31) is retested

In the next subsection the detailed steps of the proposediterative codebook optimization algorithm are given

42 Concrete Steps of Iterative Codebook Optimization Algo-rithm From the analysis in Section 3 and Appendix A itis shown that both 119868(b y) and Eb contain rather complexintegrals and it is difficult to derive their closed-form expres-sions Therefore in the proposed algorithm the calculationof 119868(b y) and Eb is realized according to the Monte Carlosimulations which should cover all the 119872119870 constellationpoints It is believed that the computational complexity isproportional to119872119870

According to the above analysis concrete steps of theproposed algorithm are given in Algorithm 1 The parameter119873ite is the number of outer loops

It should be noted that the performance of backtrackingline search method depends on the initial values of the

Wireless Communications and Mobile Computing 7

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information non-AWGN channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10 12

Figure 2Mutual information performance in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6 and119873 = 4 Thechannel responses are given in Appendix B

codebook matrices Therefore in simulations the iterativeoptimization shown in Algorithm 1 should be repeated mul-tiple times with different initial codebook matrices

In order to evaluate upper bound of the proposed algo-rithm Gaussian channel capacity with the same channelcoefficient matrix should be calculated According to [27]under Gaussian input assumption the iterative water-fillingalgorithm is able to find the globally optimal power allocationresult which achieves Gaussian capacity bound This can beseen as the upper bound of the proposed iterative codebookoptimization algorithm

5 Simulation Results

In this section the simulation results are given With thefactor graph in Figure 1 and 119872 = 4 mutual informationbetween the information bit vector b and received signal y isbounded by119867(b) = 119870 log2119872 = 12 bit The codebook matrixof each user should satisfy the power constraint tr(G119894G119867119894 ) le119875119894 1 le 119894 le 119870 In the following simulations we set 119875119894 =119873119870 = 23 1 le 119894 le 119870 Simulation results in non-AWGNandAWGNchannels are given in Sections 51 and 52respectively

51 Non-AWGN Channel Simulation Results In Figure 2mutual information achieved by the proposed iterativecodebook optimization algorithm in non-AWGN channel isshown The responses of non-AWGN channel are given inAppendix B In addition the channel setting makes sure thatthe channel power satisfies the following constraint

tr (H119878H119867119878 ) = 119873119889119891 = 119870119889V = 12 (32)

1 2 3 4 5 6 7 8Outer loop number

2

3

4

5

6

7

8

9

Mut

ual i

nfor

mat

ion

(bit)

Iterative codebook optimization

Optimized codebook on 4 dBOptimized codebook on 2 dBOptimized codebook on 0 dB

Figure 3 Convergence performance of the proposed iterativeoptimization algorithm innon-AWGNchannelThe SNR is set equalto 0 dB 2 dB and 4 dB respectively

According to the analysis in Section 4 the performanceof the proposed iterative codebook optimization algorithmdepends on values of the initial codebook matrices There-fore the codebook optimization result is chosen from 20realizations with different initial codebook matrices

In Figure 2 the result of the proposed iterative code-book optimization algorithm is denoted by ldquooptimized code-bookrdquo The Gaussian capacity bound with the same channelresponses according to [27] is denoted by ldquogaussian capacityrdquoIn addition we introduce the scheme called ldquoGaussian powerinputrdquo In this setting the codebook matrix G119894 1 le 119894 le 119870is squared root of the power distribution matrix obtainedfrom iterative water-filling algorithm in [27] With above G119894mutual information between discrete input b and continuousoutput y is calculated and denoted by ldquoGaussian powerinputrdquo in Figure 2 From the analysis in [27] iterative water-filling algorithm also requires channel state informationIn addition the result of random codebook satisfying thepower constraint is denoted by ldquorandom codebookrdquo Figure 2demonstrates that the proposed iterative codebook optimiza-tion algorithm can approach Gaussian capacity bound in lowand medium SNR regime Due to the inability to track thechannel responses the performance of ldquorandom codebookrdquois worse than that of ldquooptimized codebookrdquo When SNRis lower than 1 dB the result of ldquoGaussian power inputrdquo isbetter than that of ldquorandom codebookrdquo However when SNRincreases ldquoGaussian power inputrdquo method fails to approachthe performance of ldquooptimized codebookrdquo This indicatesthat iterative water-filling algorithm with Gaussian inputassumption cannot be directly applied in the discrete inputchannel even with perfect channel state information

Furthermore in Figure 3 the convergence of the pro-posed iterative codebook optimization algorithm is shown

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

6 Wireless Communications and Mobile Computing

Input Randomly select codebook matrix G(1)119894 1 le 119894 le 119870 tr(G119894G119867119894 ) = 119875119894(1) Initialization G(10) = blkdiag[G(1)1 G(1)119896 ](2) Outer loop for 119899 = 1 1 119873ite(3) Inner loop for 119894 = 1 1 119870

(a) Perform monte-carlo simulations to calculate 119868(119899119894minus1)(b y) and E(119899119894minus1)b(b) Calculate the gradient nablaG(119899)

119894

119868(119899119894minus1)(b y) according to (28)Do

(c) Update nablaG(119899)119894 according to (27)(d) Calculate G(119899)119894 to satisfy the power constraint according to (29)(e) Perform monte-carlo simulations to calculate 119868(119899119894)(b y) according to (30)(f) Update step size 119905 = 119905 times 120573

While 119868(119899119894)(b y) lt 119868(119899119894minus1)(b y) + 120572119905 times nablaG(119899)119894

119868(119899119894minus1)(b y)2119865(g) Generate the updated codebook matrix G(119899119894) = blkdiag[G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119896 ]

(4) End Inner loop(h) Generate G(119899+10) = G(119899119870)

(5) End Outer loop

Algorithm 1 Concrete process of iterative codebook optimization algorithm

where mutual information is considered as the function ofcodebook matrix of each user and the superscript (119899 119894 minus 1) of119868(b y) denotes the outer loop and inner loop index pair Thecodebook matrix corresponding to 119868(119899119894minus1)(b y) is given by

G(119899119894minus1) = blkdiag [G(119899)1 G(119899)119894minus1G(119899)119894 G(119899)119896 ] (26)

where the matrices from G(119899)1 to G(119899)119894minus1 denote the codebooksthat have been updated in the 119899th outer loop

In addition G(1198990) = blkdiag[G(119899)1 G(119899)119896 ] denotes theinitial codebook matrix in the 119899th iteration the correspond-ing mutual information is 119868(1198990)(b y) = 119891(G(119899)1 G(119899)119896 )

Based on the gradient expression in (21) the line searchresult is given by

nablaG(119899)119894 = G(119899)119894 + 119905nablaG(119899)119894

119868(119899119894minus1) (b y) (27)

where 119905 is the step size and the expression of nablaG(119899)119894

119868(119899119894minus1)(b y)is given by

nablaG119894119868(119899119894minus1) (b y) = 1ln 2 sdot 1205902119899 (e119870119894 otimes I119889V)timesH119867119878 H119878G

(119899119894minus1)E(119899119894minus1)b

times (e119870119894 otimes I119889V)119867 (28)

where the mean squared error matrix E(119899119894minus1)b is calculatedbased on the codebook matrix G(119899119894minus1)

In addition the codebook matrix of each user shouldsatisfy the power constraint Assuming that the maximumtransmit power of user 119894 is equal to 119875119894 the normalizedcodebook matrix is given by

G(119899)119894 = radic119875119894 times nablaG(119899)11989410038171003817100381710038171003817nablaG(119899)119894 10038171003817100381710038171003817119865 (29)

Afterwards the 119894th 119889V times119889V diagonal blockG(119899)119894 is replacedby G(119899)119894 and the updated mutual information is calculatedaccording to

119868(119899119894) (b y) = 119891 (G(119899)1 G(119899)119894minus1 G(119899)119894 G(119899)119894+1 G(119899)119896 ) (30)

Based on the backtracking line search method [22] thefollowing constraint should be satisfied to make sure that theupdated mutual information is nondecreasing

119868(119899119894) (b y) gt 119868(119899119894minus1) (b y) + 120572119905 100381710038171003817100381710038171003817nablaG(119899)119894 119868(119899119894minus1) (b y)1003817100381710038171003817100381710038172119865 (31)

where 120572 is the predetermined parameter and always belongsto the interval (0 03) [22]

If the above constraint is not satisfied the calculations in(27)ndash(30) are repeated to update 119868(119899119894)(b y) and the ldquoback-trackingrdquo is performed with updated size 119905 = 119905 times 120573 where120573 isin (0 08) is the predetermined parameter [22] Afterwardsthe constraint in (31) is retested

In the next subsection the detailed steps of the proposediterative codebook optimization algorithm are given

42 Concrete Steps of Iterative Codebook Optimization Algo-rithm From the analysis in Section 3 and Appendix A itis shown that both 119868(b y) and Eb contain rather complexintegrals and it is difficult to derive their closed-form expres-sions Therefore in the proposed algorithm the calculationof 119868(b y) and Eb is realized according to the Monte Carlosimulations which should cover all the 119872119870 constellationpoints It is believed that the computational complexity isproportional to119872119870

According to the above analysis concrete steps of theproposed algorithm are given in Algorithm 1 The parameter119873ite is the number of outer loops

It should be noted that the performance of backtrackingline search method depends on the initial values of the

Wireless Communications and Mobile Computing 7

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information non-AWGN channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10 12

Figure 2Mutual information performance in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6 and119873 = 4 Thechannel responses are given in Appendix B

codebook matrices Therefore in simulations the iterativeoptimization shown in Algorithm 1 should be repeated mul-tiple times with different initial codebook matrices

In order to evaluate upper bound of the proposed algo-rithm Gaussian channel capacity with the same channelcoefficient matrix should be calculated According to [27]under Gaussian input assumption the iterative water-fillingalgorithm is able to find the globally optimal power allocationresult which achieves Gaussian capacity bound This can beseen as the upper bound of the proposed iterative codebookoptimization algorithm

5 Simulation Results

In this section the simulation results are given With thefactor graph in Figure 1 and 119872 = 4 mutual informationbetween the information bit vector b and received signal y isbounded by119867(b) = 119870 log2119872 = 12 bit The codebook matrixof each user should satisfy the power constraint tr(G119894G119867119894 ) le119875119894 1 le 119894 le 119870 In the following simulations we set 119875119894 =119873119870 = 23 1 le 119894 le 119870 Simulation results in non-AWGNandAWGNchannels are given in Sections 51 and 52respectively

51 Non-AWGN Channel Simulation Results In Figure 2mutual information achieved by the proposed iterativecodebook optimization algorithm in non-AWGN channel isshown The responses of non-AWGN channel are given inAppendix B In addition the channel setting makes sure thatthe channel power satisfies the following constraint

tr (H119878H119867119878 ) = 119873119889119891 = 119870119889V = 12 (32)

1 2 3 4 5 6 7 8Outer loop number

2

3

4

5

6

7

8

9

Mut

ual i

nfor

mat

ion

(bit)

Iterative codebook optimization

Optimized codebook on 4 dBOptimized codebook on 2 dBOptimized codebook on 0 dB

Figure 3 Convergence performance of the proposed iterativeoptimization algorithm innon-AWGNchannelThe SNR is set equalto 0 dB 2 dB and 4 dB respectively

According to the analysis in Section 4 the performanceof the proposed iterative codebook optimization algorithmdepends on values of the initial codebook matrices There-fore the codebook optimization result is chosen from 20realizations with different initial codebook matrices

In Figure 2 the result of the proposed iterative code-book optimization algorithm is denoted by ldquooptimized code-bookrdquo The Gaussian capacity bound with the same channelresponses according to [27] is denoted by ldquogaussian capacityrdquoIn addition we introduce the scheme called ldquoGaussian powerinputrdquo In this setting the codebook matrix G119894 1 le 119894 le 119870is squared root of the power distribution matrix obtainedfrom iterative water-filling algorithm in [27] With above G119894mutual information between discrete input b and continuousoutput y is calculated and denoted by ldquoGaussian powerinputrdquo in Figure 2 From the analysis in [27] iterative water-filling algorithm also requires channel state informationIn addition the result of random codebook satisfying thepower constraint is denoted by ldquorandom codebookrdquo Figure 2demonstrates that the proposed iterative codebook optimiza-tion algorithm can approach Gaussian capacity bound in lowand medium SNR regime Due to the inability to track thechannel responses the performance of ldquorandom codebookrdquois worse than that of ldquooptimized codebookrdquo When SNRis lower than 1 dB the result of ldquoGaussian power inputrdquo isbetter than that of ldquorandom codebookrdquo However when SNRincreases ldquoGaussian power inputrdquo method fails to approachthe performance of ldquooptimized codebookrdquo This indicatesthat iterative water-filling algorithm with Gaussian inputassumption cannot be directly applied in the discrete inputchannel even with perfect channel state information

Furthermore in Figure 3 the convergence of the pro-posed iterative codebook optimization algorithm is shown

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

Wireless Communications and Mobile Computing 7

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information non-AWGN channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10 12

Figure 2Mutual information performance in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6 and119873 = 4 Thechannel responses are given in Appendix B

codebook matrices Therefore in simulations the iterativeoptimization shown in Algorithm 1 should be repeated mul-tiple times with different initial codebook matrices

In order to evaluate upper bound of the proposed algo-rithm Gaussian channel capacity with the same channelcoefficient matrix should be calculated According to [27]under Gaussian input assumption the iterative water-fillingalgorithm is able to find the globally optimal power allocationresult which achieves Gaussian capacity bound This can beseen as the upper bound of the proposed iterative codebookoptimization algorithm

5 Simulation Results

In this section the simulation results are given With thefactor graph in Figure 1 and 119872 = 4 mutual informationbetween the information bit vector b and received signal y isbounded by119867(b) = 119870 log2119872 = 12 bit The codebook matrixof each user should satisfy the power constraint tr(G119894G119867119894 ) le119875119894 1 le 119894 le 119870 In the following simulations we set 119875119894 =119873119870 = 23 1 le 119894 le 119870 Simulation results in non-AWGNandAWGNchannels are given in Sections 51 and 52respectively

51 Non-AWGN Channel Simulation Results In Figure 2mutual information achieved by the proposed iterativecodebook optimization algorithm in non-AWGN channel isshown The responses of non-AWGN channel are given inAppendix B In addition the channel setting makes sure thatthe channel power satisfies the following constraint

tr (H119878H119867119878 ) = 119873119889119891 = 119870119889V = 12 (32)

1 2 3 4 5 6 7 8Outer loop number

2

3

4

5

6

7

8

9

Mut

ual i

nfor

mat

ion

(bit)

Iterative codebook optimization

Optimized codebook on 4 dBOptimized codebook on 2 dBOptimized codebook on 0 dB

Figure 3 Convergence performance of the proposed iterativeoptimization algorithm innon-AWGNchannelThe SNR is set equalto 0 dB 2 dB and 4 dB respectively

According to the analysis in Section 4 the performanceof the proposed iterative codebook optimization algorithmdepends on values of the initial codebook matrices There-fore the codebook optimization result is chosen from 20realizations with different initial codebook matrices

In Figure 2 the result of the proposed iterative code-book optimization algorithm is denoted by ldquooptimized code-bookrdquo The Gaussian capacity bound with the same channelresponses according to [27] is denoted by ldquogaussian capacityrdquoIn addition we introduce the scheme called ldquoGaussian powerinputrdquo In this setting the codebook matrix G119894 1 le 119894 le 119870is squared root of the power distribution matrix obtainedfrom iterative water-filling algorithm in [27] With above G119894mutual information between discrete input b and continuousoutput y is calculated and denoted by ldquoGaussian powerinputrdquo in Figure 2 From the analysis in [27] iterative water-filling algorithm also requires channel state informationIn addition the result of random codebook satisfying thepower constraint is denoted by ldquorandom codebookrdquo Figure 2demonstrates that the proposed iterative codebook optimiza-tion algorithm can approach Gaussian capacity bound in lowand medium SNR regime Due to the inability to track thechannel responses the performance of ldquorandom codebookrdquois worse than that of ldquooptimized codebookrdquo When SNRis lower than 1 dB the result of ldquoGaussian power inputrdquo isbetter than that of ldquorandom codebookrdquo However when SNRincreases ldquoGaussian power inputrdquo method fails to approachthe performance of ldquooptimized codebookrdquo This indicatesthat iterative water-filling algorithm with Gaussian inputassumption cannot be directly applied in the discrete inputchannel even with perfect channel state information

Furthermore in Figure 3 the convergence of the pro-posed iterative codebook optimization algorithm is shown

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

8 Wireless Communications and Mobile Computing

2 4 6 8 10 12 14SNR

BER

Uncoded BER non-AWGN channel

MLMPA 1 iteMPA 2 ite

MPA 4 ite

100

10minus1

10minus2

10minus3

10minus4

Figure 4 Uncoded BER performance of maximum likelihooddetection algorithm (ML) and message passing algorithm (MPA) innon-AWGN channel

The maximum number of outer loops in the proposedalgorithm is set as 8 In addition the initial value of stepsize parameter 119905 is set as 1 During iterative codebookoptimization the parameter 120572 is set as 01 and 120573 is set as 05It can be seen that after 6 iterations the increment of mutualinformation becomesmarginalThismeans that the proposedalgorithm tends to converge after limited outer loops

In the following the optimized codebook with mutualinformation equal to 6 bits is applied The concrete code-book expressions are given in Appendix B In Figure 4 theuncoded bit error rate (uncoded BER) results of maximumlikelihood algorithm (ML) and message passing algorithm(MPA) are given It can be seen that MPA can approach theperformance of ML detection after 4 iterations When BER isequal to 10minus3 the loss of MPA with 4 iterations is only about06 dBThis indicates thatMPAworkswell with the optimizedcodebook

In Figure 5 the coded bit error rate (coded BER) withthe optimized codebook matrix is given Turbo code in LTE[28] is applied and the information bit length is equal to1024 Because the codebook in Appendix B is optimizationresult whenmutual information is equal to 6 bits the channelcode rate is set as 05 The inner iteration number of Turbodecoding is equal to 7 In multiuser detection the iterationnumber of MPA is equal to 4 Two channel coding schemesare involved in Figure 5 In scheme 1 each user in SCMA hasits own channel coding block Figure 5 shows that the bestuser is about 3 dB better than the worst user In addition inscheme 1 the average bit error rate is limited by the worstuser In scheme 2 the channel coding across all119870 = 6 users isintroduced According to the statement in [29] coding acrosschannels with different reliabilities can achieve better coded

1 2 3 4 5 6 7SNR

BER

Coded BER with code rate = 05

Average BER in scheme 1BER of worst user in scheme 1BER of best user in scheme 1Average BER in scheme 2

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 5 Coded BER performance with the optimized codebookmatrix in non-AWGN channel Two channel coding schemes areinvolved

SNR

BER

Coded BER with outer iteration

Ourter iteration 1Outer iteration 2Outer iteration 4

Outer iteration 10

2 25 3 35 4 45 5 55 6 65 7

10minus1

10minus2

10minus3

10minus5

10minus4

Figure 6 Coded BER performance with outer iteration betweenchannel decoding and message passing algorithm (MPA) in non-AWGNchannelThe SCMA structure is given in Figure 1 with119870 = 6and119873 = 4

BER performance In Figure 5 it is shown that the average biterror rate of scheme 2 is about 1 dB better than scheme 1

In addition the performance of outer-loop iterationbetween channel decoder and message passing algorithm(MPA) with scheme 1 is given in Figure 6 In scheme 1 eachuser has its own channel coding block Similar to that in

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

Wireless Communications and Mobile Computing 9

SNR

2

3

4

5

6

7

8

9

10

11

12

Mut

ual i

nfor

mat

ion

(bit)

Mutual information Rayleigh fading channel

Gaussian capacityGaussian power inputRandom codebook

Optimized codebook

minus2 0 2 4 6 8 10

Figure 7 Mutual information performance averaging over 1000Rayleigh fading channels The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4Figure 5 the information bit length is equal to 1024 andthe channel coding rate is equal to 05 The inner iterationnumber of Turbo decoding is equal to 7 and the iterationnumber ofMPA is equal to 4 Because channel decoding feed-back provides high-reliability extrinsic information forMPAthe outer-loop iteration can greatly improve the receiverperformance After 10 outer-loop iterations the performanceimprovement is about 3 dB when BER is equal to 10minus5

In order to improve the credibility we further givethe simulation results averaging over 1000 Rayleigh fadingchannels in Figure 7 The curve legends in Figure 7 are thesame as that in Figure 2 The simulation results show thatthe performance of ldquooptimized codebookrdquo is better thanthat of ldquorandom codebookrdquo and ldquoGaussian power inputrdquoCompared with Gaussian capacity upper bound the loss ofldquooptimized codebookrdquo is not very large in low and mediumSNR regime When SNR is lower than 0 dB the resultof ldquoGaussian power inputrdquo is better than that of ldquorandomcodebookrdquo With the increase of SNR ldquoGaussian powerinputrdquo is unable to approach the performance of ldquooptimizedcodebookrdquo The above analysis shows that when averagingover many Rayleigh channels the proposed optimizationalgorithm still has better performance

52 AWGN Channel Simulation Results In this subsectionsimulation results in AWGN channel are given Figure 8demonstrates mutual information for the factor graph inFigure 1 in AWGN channel The result of the proposediterative codebook optimization algorithm is denoted byldquooptimized codebookrdquo The Gaussian capacity bound isdenoted by ldquoGaussian capacityrdquo In addition the result ofthe existing codebook proposed by Huawei Corporationin [30] is denoted by ldquoHuawei codebookrdquo It can be seen

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Mutual information AWGN channel

Gaussian capacityHuawei codebookOptimized codebook

minus2 0 2 4 6 8 1210

Figure 8 Mutual information performance in AWGN channelTheSCMA structure is given in Figure 1 with 119870 = 6 and119873 = 4

SNR

BER

Uncoded BER AWGN channel

dB codebook 8 iteOpt 10Opt 10 dB codebook 4 iteHuawei codebook 8 iteHuawei codebook 4 ite

10minus1

10minus2

10minus3

10minus4

10 11 12 13 14 15 16 17 18

Figure 9 Uncoded BER performance of message passing algorithm(MPA) in AWGN channel The SCMA structure is given in Figure 1with 119870 = 6 and119873 = 4

that the proposed algorithm can achieve the same mutualinformation performance as ldquoHuawei codebookrdquo In low andmedium SNR regime the proposed algorithm can approxi-mate ldquoGaussian capacityrdquo boundwith small performance loss

Furthermore the uncoded bit error rate (uncoded BER)of the optimized codebook in AWGN channel is given inFigure 9Themessage passing algorithm (MPA) is performedat the receiver The codebook matrices are optimization

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

10 Wireless Communications and Mobile Computing

V1 V2 V3

F1 F2

Figure 10 Factor graph with119870 = 3 and119873 = 2

Es_N0

2

3

4

5

6

7

8

9

10

11

12

13

Mut

ual i

nfor

mat

ion

(bit)

Gaussian capacityOptimized codebook

Mutual information (3 2) graph M = 16 AWGN channel

0 2 4 6 8 10 12 14 16 18 20

Figure 11Mutual information performancewith119872 = 16 in AWGNchannel The SCMA structure is given in Figure 10 with 119870 = 3 and119873 = 2

results of the proposed algorithm when SNR = 10 dB whoseexpressions are detailed in Appendix C Compared withldquoHuawei codebookrdquo in [30] ldquooptimized codebookrdquo accordingto the proposed algorithm has better performance With 8iterations of MPA ldquooptimized codebookrdquo has 1 dB perfor-mance gain over ldquoHuawei codebookrdquo For clearness Huaweicodebook in [30] is rewritten according to superpositionmodulation matrices and its concrete expressions are givenin Appendix C

The above simulationsrsquo results are all based on the factorgraph in Figure 1 with 119872 = 4 In the following simulationthe codebook design is extended to the case with 119872 =16 Considering the codebook optimization complexity ourfocus is on the factor graph with 2 subchannels and 3 userswhose structure is shown in Figure 10

The proposed column-extended channel model can welldescribe the codebook optimization problem with 119872 =16 The detailed signal model analysis with 119872 = 16 isgiven in Appendix D Figure 11 demonstrates the simulationresult of (3 2) factor graph with 119872 = 16 in AWGNchannel The optimized codebook can efficiently approachGaussian capacity upper bound When SNR is lower than

6 dB the performance loss between the optimized codebookand upper bound is negligible

6 Conclusion

In this paper an efficient SCMA codebook optimizationalgorithm is proposed according tomaximizingmutual infor-mation between the discrete input and continuous outputFirstly SCMA signal model is given based on the super-position modulation structure which can well representthe relationship between the codebook matrix and receivedsignal Based on the superposition model the iterative code-book optimization algorithm is proposed where the linesearch method is applied to find locally optimal codebooksIt is shown that the superposition model can be appliedin multiuser channel with random channel coefficients InAWGN channel the proposed optimization codebook canapproachGaussian capacity upper bound in low andmediumSNR regime In non-AWGN channel the performance losscompared with upper bound is not very large In additionwith the optimized codebook message passing algorithm(MPA) at the receiver exhibits good performance

Appendix

A Details of Mean Squared Error

Based on the result in [31]mean squared errormatrix denotesthe error correlation between the transmit bit vectorb and thedetection result b(y) Therefore we have

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] (A1)

where b(y) is achieved by calculating the conditional mean ofthe transmit bit vector based on the received signal y and it isdenoted by

b (y) = 119872119870sum119898=1

b119898119901 (b119898 | y)= sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)

(A2)

Furthermore expression (A1) can be rewritten as follows

Eb = 119864by [(b minus b (y)) (b minus b (y))119867] = 119872119870sum119898=1

119901 (b119898)sdot int

y(b119898 minus b (y)) times (b119898 minus b (y))119867 119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) inty(b119898b119867119898 minus b119898b

119867 (y)minus b (y) b119867119898 + b (y) b119867 (y)) 119901 (y | b119898) 119889y

(A3)

There are four parts included in the integral of the aboveexpression and the derivation details of each part are givenas follows

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

Wireless Communications and Mobile Computing 11

For the first part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867119898119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 inty119901 (y | b119898) 119889y

= 119872119870sum119898=1

119901 (b119898) b119898b119867119898 = I119870times119889V (A4)

In the above expression the second equation holds because119901(y | b119898) shown in (14) is Gaussian distributed probabilitydensity function with inty 119901(y | b119898)119889y = 1

For the second part we have

119872119870sum119898=1

119901 (b119898) intyb119898b119867 (y) 119901 (y | b119898) 119889y

= inty

119872119870sum119898=1

b119898119901 (b119898) 119901 (y | b119898) b119867 (y) 119889y= int

y

sum119872119870119898=1 b119898119901 (b119898) 119901 (y | b119898)sum119872119870119898=1 119901 (b119898) 119901 (y | b119898)times 119872119870sum119898=1

119901 (b119898) 119901 (y | b119898) times b119867 (y) 119889y= 119872119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y

(A5)

In the above expression the third equation is achieved basedon the expression of b(y) shown in (A2)

It can be seen that the third part and the fourth part havethe same result as (A5) Therefore the mean squared errormatrix in (A3) is rewritten as

Eb = I119870times119889V

minus 119873119870sum119898=1

119901 (b119898) intyb (y) b119867 (y) 119901 (y | b119898) 119889y (A6)

With equal probability input assumption the expressionof Eb can be further denoted by

Eb = I119870times119889V minus 1119872119870sdot 119872119870sum119898=1

119864n[[[(sum119872119870119896=1

b119896119906119898119896) (sum119872119870119896=1 b119896119906119898119896)119867(sum119872119870119896=1

119906119898119896)2]]] (A7)

where the variable 119906119898119896 is given by

119906119898119896 = exp(minus10038171003817100381710038171003817H119878G (b119898 minus b119896) + n1003817100381710038171003817100381721205902119899 ) (A8)

The above analysis shows that it is difficult to derive theclosed-form expression of Eb During the implementation ofiterative codebook optimization algorithm in Section 4 Eb isachieved fromMonte Carlo simulations

B Details of Non-AWGN ChannelResponse and Codebook Expressions

The channel responses applied in non-AWGN scenario aregiven by

ℎ11 = 04843 minus 11249119894ℎ21 = 05868 minus 03945119894ℎ12 = 05700 + 05846119894ℎ32 = 09879 minus 05978119894ℎ13 = minus06148 minus 06748119894ℎ43 = 08837 + 06211119894ℎ24 = minus01626 + 08983119894ℎ34 = minus10336 minus 03137119894ℎ25 = 11138 minus 03047119894ℎ45 = 07967 minus 01786119894ℎ36 = 03878 + 05912119894ℎ46 = 12039 minus 02250119894

(B1)

The optimized codebook matrices from G1 to G6 withmutual information equal to 6 bits are given by

G(opt)1 = [02570 + 05092119894 04398 minus 0349511989400385 + 01257119894 minus00789 + 00480119894] G(opt)2 = [minus02314 + 00851119894 01009 minus 0035911989401741 minus 04712119894 minus03640 minus 04568119894] G(opt)3 = [minus01860 minus 00235119894 01222 + 0509511989400277 + 05366119894 minus01556 + 02099119894] G(opt)4 = [ 03295 minus 01191119894 03341 minus 02843119894minus02860 + 03745119894 00675 minus 03533119894] G(opt)5 = [minus05955 minus 01326119894 minus03274 + 0369411989401337 + 00159119894 minus01658 + 00717119894] G(opt)6 = [minus01840 minus 00873119894 minus01115 minus 0012611989400684 + 04833119894 minus05262 minus 03122119894]

(B2)

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

12 Wireless Communications and Mobile Computing

C Details of AWGN ChannelCodebook Expressions

In AWGN channel the optimized codebook matrices forfactor graph in Figure 1 when SNR =10 dB are given by

G(10 dB)1 = [minus04537 minus 02942119894 minus01114 minus 0250311989402563 + 00679119894 minus02026 minus 04334119894] G(10 dB)2 = [minus04935 + 01406119894 minus01302 minus 0191011989400727 minus 01944119894 minus04370 minus 03404119894] G(10 dB)3 = [ 01123 minus 03513119894 minus01685 + 04982119894minus04298 + 00382119894 minus02537 minus 00589119894] G(10 dB)4 = [minus01376 + 01682119894 03903 minus 03857119894minus04810 + 01877119894 minus02117 + 00831119894] G(10 dB)5 = [minus00548 minus 02603119894 minus05657 minus 0054111989403334 + 02855119894 minus02423 + 01469119894] G(10 dB)6 = [minus00130 + 03272119894 minus00089 minus 0391411989400410 minus 04939119894 minus00347 minus 03992119894]

(C1)

In addition Huawei codebook proposed in [30] can begiven by the following superposition modulation matrices

G(HW)1 = [02269 minus 01648119894 04083 minus 0296511989403132 minus 03958119894 minus01740 + 02199119894]

G(HW)2 = [ minus02804 minus05047minus04083 minus 02965119894 02269 + 01648119894]

G(HW)3 = [minus00122 minus 05045119894 00068 + 0280311989402269 minus 01648119894 04083 minus 02965119894]

G(HW)4 = [ minus02804 minus0504703132 minus 03958119894 minus01740 + 02199119894]

G(HW)5 = [minus04083 minus 02965119894 02269 + 01648119894minus02804 minus05047 ]

G(HW)6 = [ minus02804 minus05047minus00122 minus 05045119894 00068 + 02803119894]

(C2)

D Signal Model of Figure 9 with119872 = 16Based on the factor graph in Figure 9 the mapping matrixbetween the user nodes and subchannels is given by

F = [1 1 11 1 1] (D1)

In AWGN scenario the channel matrix H119878 is equal to aboveF After column extension the followingHS is achieved

H119878 = [1 0 1 0 1 00 1 0 1 0 1] (D2)

With119872 = 16 the bit vector of the 119896th user 1 le 119896 le 3 isgiven by

b119896 = [119887(119896)1 119887(119896)2 119887(119896)3 119887(119896)4 ]119879 (D3)

The corresponding codebookG119896 1 le 119896 le 3 is a 2times4matrixConsequently the overall block diagonal codebook matrix isgiven by

G = blkdiag G1G2G3 (D4)

Based on the above analysis the received signal is given by

y = H119878Gb + n (D5)

where b collects all the usersrsquo information bits with b =[b1198791 b1198792 b1198793 ]119879In addition the multiuser access model can be further

denoted by

y = H119878Gb + n = 3sum119894=1

H119894G119894b119894 + n (D6)

where H119894 is column-extended result of the 119894th column of H119878and it is given by

H119894 = [1 00 1] (D7)

According to the above expression the proposed iterativecodebook optimization algorithm can be implemented

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (61601047 61671080 and 61771066)

References

[1] L Dai B Wang Y Yuan S Han C-L I and Z Wang ldquoNon-orthogonalmultiple access for 5G Solutions challenges oppor-tunities and future research trendsrdquo IEEE CommunicationsMagazine vol 53 no 9 pp 74ndash81 2015

[2] Z DIng X Lei G K Karagiannidis R Schober J Yuan andV K Bhargava ldquoA Survey on Non-Orthogonal Multiple Accessfor 5GNetworks ResearchChallenges andFutureTrendsrdquo IEEEJournal on Selected Areas in Communications vol 35 no 10 pp2181ndash2195 2017

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

Wireless Communications and Mobile Computing 13

[3] Y Cai Z Qin F Cui G Y Li and J A McCann ldquoModulationand Multiple Access for 5G Networksrdquo IEEE CommunicationsSurveys amp Tutorials vol 20 no 1 pp 629ndash646 2018

[4] R Hoshyar F P Wathan and R Tafazolli ldquoNovel low-densitysignature for synchronous CDMA systems over AWGN chan-nelrdquo IEEE Transactions on Signal Processing vol 56 no 4 pp1616ndash1626 2008

[5] D Guo andC-CWang ldquoMultiuser detection of sparsely spreadCDMArdquo IEEE Journal on SelectedAreas inCommunications vol26 no 3 pp 421ndash431 2008

[6] R Hoshyar R Razavi and M Al-Imari ldquoLDS-OFDM anefficient multiple access techniquerdquo in Proceedings of the 2010IEEE 71st Vehicular Technology Conference VTC 2010-SpringTaiwan May 2010

[7] R Razavi M Al-Imari M A Imran R Hoshyar and D ChenldquoOn receiver design for uplink low density signature OFDM(LDS-OFDM)rdquo IEEE Transactions on Communications vol 60no 11 pp 3409ndash3508 2012

[8] L Wen R Razavi M A Imran and P Xiao ldquoDesign of JointSparseGraph forOFDMSystemrdquo IEEETransactions onWirelessCommunications vol 14 no 4 pp 1823ndash1836 2015

[9] M-C Chang and Y T Su ldquoOverloadedmultiple access systemsA generalized model and a low-complexity multiuser decoderrdquoin Proceedings of the 9th International Symposium on TurboCodes and Iterative Information Processing ISTC 2016 pp 231ndash235 France September 2016

[10] H Nikopour and H Baligh ldquoSparse code multiple accessrdquo inProceedings of the IEEE 24th Annual International SymposiumonPersonal Indoor andMobile Radio Communications (PIMRCrsquo13) pp 332ndash336 IEEE London UK September 2013

[11] M Taherzadeh H Nikopour A Bayesteh and H BalighldquoSCMA codebook designrdquo in Proceedings of the 80th IEEEVehicular Technology Conference VTC 2014-Fall CanadaSeptember 2014

[12] Y Wu S Zhang and Y Chen ldquoIterative multiuser receiverin sparse code multiple access systemsrdquo in Proceedings of theIEEE International Conference on Communications ICC 2015pp 2918ndash2923 UK June 2015

[13] HMu Z MaM Alhaji P Fan and D Chen ldquoA fixed low com-plexity message pass algorithm detector for Up-Link SCMAsystemrdquo IEEEWireless Communications Letters vol 4 no 6 pp585ndash588 2015

[14] B Xiao K Xiao S Zhang Z Chen B Xia andH Liu ldquoIterativedetection and decoding for SCMA systems with LDPC codesrdquoin Proceedings of the International Conference on Wireless Com-munications and Signal Processing WCSP 2015 China October2015

[15] F Wei and W Chen ldquoLow Complexity Iterative ReceiverDesign for Sparse Code Multiple Accessrdquo IEEE Transactions onCommunications vol 65 no 2 pp 621ndash634 2017

[16] J Harshan and B S Rajan ldquoOn two-user Gaussian multipleaccess channels with finite input constellationsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 57 no 3 pp 1299ndash1327 2011

[17] M Cheng Y Wu and Y Chen ldquoCapacity analysis for non-orthogonal overloading transmissions under constellation con-straintsrdquo in Proceedings of the International Conference onWireless Communications and Signal Processing WCSP 2015China October 2015

[18] S Zhang K Xiao B Xiao et al ldquoA capacity-based codebookdesign method for sparse code multiple access systemsrdquo in

Proceedings of the 8th International Conference on WirelessCommunications and Signal Processing WCSP 2016 ChinaOctober 2016

[19] J Bao Z Ma G K Karagiannidis M Xiao and Z Zhu ldquoJointMultiuser Detection of Multidimensional Constellations overFading Channelsrdquo IEEE Transactions on Communications vol65 no 1 pp 161ndash172 2017

[20] J Bao Z Ma Z Ding G K Karagiannidis and Z Zhu ldquoOnthe design of multiuser codebooks for uplink SCMA SystemsrdquoIEEECommunications Letters vol 20 no 10 article no A42 pp1920ndash1923 2016

[21] X Ma and L Ping ldquoCoded modulation using superimposedbinary codesrdquo Institute of Electrical and Electronics EngineersTransactions on Information Theory vol 50 no 12 pp 3331ndash3343 2004

[22] L Ping J Tong X Yuan and Q Guo ldquoSuperposition codedmodulation and iterative linearMMSE detectionrdquo IEEE Journalon Selected Areas in Communications vol 27 no 6 pp 995ndash1004 2009

[23] C Xiao Y R Zheng and Z Ding ldquoGlobally optimal linear pre-coders for finite alphabet signals over complex vector Gaussianchannelsrdquo IEEE Transactions on Signal Processing vol 59 no 7pp 3301ndash3314 2011

[24] M Wang W Zeng and C Xiao ldquoLinear precoding for MIMOmultiple access channels with finite discrete inputsrdquo IEEETransactions on Wireless Communications vol 10 no 11 pp3934ndash3942 2011

[25] D Guo S Shamai and S Verdu ldquoMutual information andminimummean-square error in Gaussian channelsrdquo Institute ofElectrical and Electronics Engineers Transactions on InformationTheory vol 51 no 4 pp 1261ndash1282 2005

[26] D P Palomar and S Verdu ldquoGradient of mutual informationin linear vector Gaussian channelsrdquo Institute of Electrical andElectronics Engineers Transactions on Information Theory vol52 no 1 pp 141ndash154 2006

[27] W YuW Rhee S Boyd and J Cioffi ldquoIterative water-filling forGaussian vector multiple-access channelsrdquo Institute of Electricaland Electronics Engineers Transactions on Information Theoryvol 50 no 1 pp 145ndash152 2004

[28] Multiplexing and channel coding Release 8 2009 3GPP TS36212

[29] D Tse and P Viswanath Fundamentals ofWireless Communica-tion Cambridge University Press Cambridge UK 2005

[30] httpwwwinnovateasiacom5gengp2html SCMA Code-books (Jun 2015)

[31] A Lozano A M Tulino and S Verdu ldquoOptimum power allo-cation for parallel Gaussian channels with arbitrary inputdistributionsrdquo Institute of Electrical and Electronics EngineersTransactions on InformationTheory vol 52 no 7 pp 3033ndash30512006

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