Joint uplink SCMA codebook design for correlated fading channels

Journal Pre-proof Joint uplink SCMA codebook design for correlated fading channels Shah Mahdi Hasan, Kaushik Mahata, Md. Mashud Hyder

PII: DOI: Reference:

S1874-4907(19)30646-9 https://doi.org/10.1016/j.phycom.2020.101041 PHYCOM 101041

To appear in:

Physical Communication

Received date : 29 August 2019 Revised date : 27 December 2019 Accepted date : 31 January 2020 Please cite this article as: S.M. Hasan, K. Mahata and Md.M. Hyder, Joint uplink SCMA codebook design for correlated fading channels, Physical Communication (2020), doi: https://doi.org/10.1016/j.phycom.2020.101041. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier B.V.

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Joint Uplink SCMA Codebook Design for

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Correlated Fading Channels

Shah Mahdi Hasan, Kaushik Mahata and Md Mashud Hyder

School of Electrical Engineering and Computation, University of Newcastle

Abstract

Sparse Code Multiple Access (SCMA) is currently one of the most intensively investigated Multicarrier Non-Orthogonal Multiple Access (NOMA) technology for 5G wireless communication system. Careful design of SCMA codebook is one of the major performance criteria of uplink SCMA systems. We took the challenge of tackling SCMA Codebook design problem in correlated Rayleigh fading channels. We propose a novel joint codebook design approach followed by a new signal model, where the design problem is posed as a Quadratically Constrained Quadratic Program (QCQP) which is dealt using Matrix-lifting Semidefinite Relaxation (MSDR). Extensive numerical simulations are presented to provide the validity of proposed codebook comparing with some other well known designs.

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Index Terms

SCMA. Codebook design, Semidefinite relaxation, 5G, mMTC

I. I NTRODUCTION

With the development of wireless communication having the goal of meeting the skyrocketing demand of throughput, spectral efficiency, network security and privacy; one key component

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has seen consistent evolution involving pinnacle of research effort. That is, multiple access (MA) schemes. Starting from Frequency Division Multiple Access (FDMA) for first generation wireless communication (1G), MA schemes underwent fundamental changes in each generation of wireless communication. All of those variants of MA share a common feature among them: the orthogonality of resource (Frequency, Time, orthogonal Walsh-Hadamard codes) allocation. Although, the orthogonality of resource allocation is of paramount importance for attaining interference-free communication and physical layer security, it is a potential hindrance toward

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achieving higher spectral efficiency in a limited resource scenario whereas the ever growing demand for mobile internet and Internet of Things (IoT) has yet to be addressed. A staggering

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17% growth in global number of connected devices by 2025 is predicted whereas 7 billion IoT connection prevailed in 2018 [1].

To enhance spectral efficiency and achievable capacity, researchers are developing new techniques like small-cell densification [2], [3] and Non-Orthogonal Multiple Access (NOMA). In terms of sum capacity, it is established that OMA system cannot approach the Shannon Limit [4]. Unlike OMA schemes, NOMA allows multiple user sharing single REs, thus higher spectral efficiency can be obtained by relaxing the orthogonality requirements [4]–[8]. Due to extensive research effort, NOMA schemes have been integrated with other enabling technologies i.e: massive MIMO beamforming and mmWave communication [9], [10]. In continuation of that, many variants of NOMA have emerged [8]. In this work, we particularly focus on a multi-carrier NOMA known as Sparse Code Multiple Access (SCMA). One of the most promising applications of SCMA systems is in uplink scenarios, especially to support uplink mMTC (Massive Machine Type Communication) services [6], [7], [11]. As a matter of fact, this work is primarily focused on uplink SCMA system.

A. Background and Related Work

Sparse Code Multiple Access (SCMA) is a Code Domain NOMA (CD-NOMA), a subclass of

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NOMA schemes which was first introduced in [12]. In an SCMA system, the users are assigned with codebooks consisting of sparse codewords. As shown in later section, these codebooks are stemmed from multidimensional constellations spread by user specific sparse mapping matrices over resource elements (REs) such as OFDMA tones. The performance of SCMA system largely depends on codebook design alogrithms, which have generated significant effort among research

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community since the inception of SCMA. Numerous design algorithms have been proposed in literature [7], [13]–[18]. In [12], [13], [19], the design problem is turned into multistage optimization problem. First, one finds a mother multidimensional constellation with desired Eucledian distance profile. Then the sets of user specific unitary rotations to generate user specific codebooks is determined. Among recent research works, [14] proposed constellation for SCMA system based on the rotation of Quadrature Amplitude Modulation (QAM) constellation so as to maximize the cutoff rate of equivalent Multiple Input Multiple Output (MIMO) system by an exhaustive computer search approach. These works are primarily on the optimization of

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mother constellations. The joint SCMA codebook and mapping matrix design approach was first introduced in [18]. [17] introduced permutation based SCMA codebook where the non-zero row

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indices of codebooks are not user specific, rather incoming bit stream specific. An important survey on prevailing SCMA codebooks has been conducted in [7], where it was shown that SCMA codebooks performing better under certain channel condition can be outperformed by other codebooks in a different channel condition. As a matter of fact, it is crucial to include operating channel properties in codebook design policy. B. Contributions and Organization

In recent studies, it was shown that for uplink channels, the user specific unitary rotations of multidimensional constellation do not impact the performance of SCMA system [18],[7] because of random Rayleigh fading, hence same multidimensional constellation can be assigned among users. On the other hand, it was argued in [18] that for uplink radio transmission, minimum Eucledian distance among multiplexed signals governs Bit Error Rate (BER) performance and communication reliability of the uplink SCMA system. In this work, we argued that assignment of same constellation among users in SCMA system with correlated Rayleigh fading channel may face performance degradation. Channel correlation may arise as a result of spatial proximity of users or allocation of adjacent REs [7]. This is one of the prime motivations behind this work. To address this issue, we suggest joint optimization of SCMA codebook for all users. Specifically,

•

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we propose the following:

We encapsulate the aforementioned arguments as design criteria to build a high performance SCMA codebook for correlated fading channels. Upon this, we first characterize the set of all possible multiplexed signals. The problem of jointly finding SCMA codebooks that best map users’ transmitted codewords under average power constraint to a predefined set of all

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possible multiplexed signals is modeled as a Quadratically Constrained Quadratic Program (QCQP)

•

The predefined set of all possible multiplexed signals is designed for a given Eucledian Distance Matrix (EDM) using an alternating co-ordinate descent method. The quadratic non-convex constraints which arise from average symbol energy constraints are tackled using matrix-lifting semidefinite relaxation (MSDR)

•

Extensive numerical simulations demonstrate the improved performance of proposed codebook over several well known designs in terms of Bit Error Rate (BER) and Symbol Error

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Rate (SER). We also study the proposed codebook’s performance on different loading (user to resource element ratio) scenario

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Rest of the paper is organized as following: Section II demonstrates uplink SCMA system model. In Section III we delineate the problem statement with examples. The joint codebook optimization procedure is spelled out in steps in Section IVA-IVB. In Section V, we include extensive numerical simulations to conduct comparative performance evaluation and validate the proposed design at different Signal to Noise Ratio (SNR) and overloading. Concluding remark and future research directions are delegated in VI. C. Notation

In the sequel, B, R,Z and C represent the fields of binary, real, natural and complex numbers, respectively. In addition, x, x, X are used to represent a scalar, a vector, and a matrix, xj denotes the j th column of matrix X, whereas X(m, n) is used to denote the element in mth row and nth column. X(a : b, p : q) is used to denote submatrix of X consisting rows a to b and columns p to q. || · ||F is used as Frobenious norm operator. Tr(·) denotes the trace operator. I is the identity

matrix. CN (µ, A) is used to present complex gaussian distribution with mean µ and covariance

matrix A. Finally, diag(A) is used to denote a matrix having diagonal elements identical with A, whereas the off-diagonal elements are set to zero.

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II. U PLINK SCMA S YSTEM M ODEL

Let us consider an uplink SCMA system with J users and K orthogonal resource elements (OFDMA tones), where K < J. Each user is assigned to N < K resource elements (RE) under the constraint that no two users are assigned with the same set of REs. In other words, each
user spreads its information stream over N REs and a system is fully loaded if J = K . N

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At the receiver of an uplink SCMA system, the received signal vector y ∈ CK for all user

j ∈ { 1, 2, · · · , J } is given by

where sj ∈ C

K

y=

J X

diag(hj )sj + w

(1)

j=1

is the transmitted sparse codeword by j th user, w ∈ CK ∼ CN (0, σw2 I)

is complex Gaussian additive noise, σw2 being the noise variance. The Rayleigh fading channel

coefficient vector for j th user is denoted by hj = {h1,j , h2,j , . . . hK,j }T , where hk,j ∼ CN (0, 1),

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Base Station

Uplink fading channel with AWGN

user-RE mapping {01}

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Channel Encoder

User 2

User 3

User 4

User 5

User 6

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User 1

{...0100101100101...} incoming bit stream

Fig. 1. An uplink SCMA system with 4 orthogonal REs and 6 users where each user is assigned with user-specific sparse codebooks

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k ∈ {1, 2, . . . K}, that is the channel coefficients are modeled using complex Gaussian distribu-

tion with zero mean and unit variance. For adjacent RE allocation, a user observe same channel (1) may reduce to

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coefficients over their N REs, i.e: hj = ehj , where e is column vector of all ones. In that case,

y=

J X

hj sj + w

(2)

j=1

A. Non-orthogonal RE Allocation

To enforce the aforementioned constraint of limiting the allocation of a particular set of REs to a user, user specific binary user-to-RE mapping matrices are employed. User-RE mapping matrix for j th user is noted as Fj ∈ BK×N , where j ∈ {1, 2, . . . J}. For example, for an SCMA system with J = 6, K = 4 and N = 2, one can have the following for some user j  1  0  Fj =  0  0

0



 0   0  1

(3)

The non-zero rows of Fj determine the allocation of REs for j th user. In above example, the

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j th user is assigned with 1st and 4th RE. An SCMA system can be entirely represented by the factor graph matrix T = [t1 t2 . . . tJ ] where tj = diag(Fj FTj ) [12]. Since, by design N  K,

factor graph matrix T ∈ BK×J of this system is a sparse matrix. For an SCMA system with

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N = 2, K = 4, J = 6, the factor graph matrix is following  0  1  T= 0  1

 1 1 0 1 0  0 1 0 0 1   1 0 1 0 1  0 0 1 1 0

(4)

B. SCMA Encoding

In an SCMA system, each user is assigned with user specific complex multidimensional constellation. Considering Mj -ary transmission for j th user, the mapping of input bits to a

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complex constellation point can be defined as f : Blog2 Mj −→ Xj where Xj ⊂ CN with

cardinality |Xj | = Mj .

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Denoting Xj ∈ CN ×Mj as the user specific Mj point complex constellation, the set of all

sparse codewords available for j th user can be written with the aid of user-RE mapping matrix

as Vj = Fj Xj . Note that Vj ∈ CK×Mj . This set of sparse codewords Vj = [v 1,j v 2,j · · · v Mj ,J ],

j ∈ {1, 2, · · · J} is known as SCMA codebook for j th user. The sparse codeword transmitted

by j th user can be written as v m,j = Fj xm,j . In other words, as a whole, SCMA encoding can be thought of incoming bits being mapped into sparse complex codewords directly, i.e.: f : Blog2 Mj −→ Vj where Vj ⊂ CK with cardinality |Vj | = Mj .

Without loss of generality, we assume the dimension of codebooks for all user in the SCMA

system are same, i.e: Mj = M ∀ j ∈ {1, 2, . . . J}. Each j th user independently transmits its

codeword v m,j , m ∈ {1, 2, · · · M } over Rayleigh fading channel. SCMA codewords from all users are multiplexed over K orthogonal resources, e.g: OFDMA tones or MIMO spatial layers

[12]. In light of the above discussion, the received multiplexed codewords (1) in an uplink SCMA system can be rearranged as below

y=

J X

diag(hj )v m,j + w

j=1

diag(hj )Fj xm,j + w

(5)

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y=

J X j=1

where v m,j = sj . In this work, we assume that complete Channel State Information (CSI) H = [h1 h2 . . . hJ ] ∈ CK×J is available at the receiver side. Given the CSI and received

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signal y, the set of transmitted codewords S = [s1 s2 . . . sJ ] can be estimated by using joint Maximum a Posteriori (MAP) detection. In this optimal solution, S is estimated as

b = arg max p(S|y, H) S

(6)

S∈V1 ×···×VJ

here, p(S|y, H) is the posterior probability of a given set of transmitted codewords S = [s1 s2 . . . sJ ] given CSI H and received signal y. Taking the advantage of sparsity of factor graph matrix

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T, low-complexity near-optimal Log-MPA (Message Passing Algorithm) can be used to detect O(M J ) to O(M df ) where

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the codewords [12], [20], [21]. In this way, detection complexity is significantly reduced from

df = max k∈K

J X

tj

(7)

j=1

III. P ROBLEM S TATEMENT

In this section we spell out the problem statement thoroughly by characterizing all multiplexed SCMA codebwords then providing examples of the possibility of ambiguous multiplexed codewords in case of when same codebook is assigned for all users in channels with correlated fading.

A. Characterizing all Multiplexed SCMA Codewords

As mentioned in Section II-B, in SCMA encoding, each user maps its incoming bits to sparse complex codewords. For j th user, this mapping of bits to sparse codewords from SCMA codebook can be expressed as following

s j = V j pj

(8)

where pj ∈ BM is a column of I ∈ BM ×M . This is the binary indicator vector representing the

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codeword to be transmitted by j th user by its only non-zero element. Multiplexing of codewords from all J users over K REs can be expressed as following J X z= sj

(9)

j=1

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z=

J X

V j pj

(10)

j=1

Here z ∈ CK is the multiplexed signal for a given set of pj , j ∈ {1, 2, · · · J}. We introduce

b by concatenating indicator joint codebook in this model using combined indicator vector p b = [pT1 pT2 . . . pTJ ]T ∈ BM J×1 . Using this, we can rewrite (10) as vectors of all users i.e. p z = [ V1 V2 . . . VJ ]b p

(11)

z = Vb p

(12)

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where V ∈ CK×M J is the joint SCMA codebook of all J users. Using this approach, we can J

characterize all possible multiplexed signals Z ∈ CK×M by extending the combined indicator J

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vector to the combined indicator matrix P ∈ BM J×M which holds all possible combination of binary indicator vectors. This can be expressed as following

Z = VP

(13)

b1 p b2 . . . p bM J ]. Next, we formulate the optimization problem solving for the where P = [ p

joint SCMA codebook V ∈ CK×M J under average symbol energy constraint. B. Ambiguous Multplexed Codeword Scenario

In [7], it was shown that user specific unitary rotation of multidimensional constellations doesn’t impact the performance of uplink SCMA system as they do in downlink systems because different users experience different fading channels. Hence, same constellation can be utilized over all users. But this policy leads path to an ambiguous multiplexed codeword scenario. This can be best demonstrated using an example. For the purpose of demonstration, let us b using their non-zero indices. For demonstration, we use 4-Bao[14] represent indicator vectors p constellation . Now, joint 4-Bao codebook is expressed as following:

 −0.5019 − 0.4981i −0.5019 + 0.4981i  = 0.5019 − 0.4981i −0.5019 − 0.4981i 0.5019 + 0.4981i −0.5019 + 0.4981i 0.5019 − 0.4981i

0.5019 + 0.4981i

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Xbao



Vbao = [ F1 Xbao F2 Xbao · · · FJ Xbao ]

(14)

bi = [ 1 4 3 1 2 2 ]T It can be shown that for 4-Bao codebooks Vbao , two different indicator vectors p

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bj = [ 1 3 2 2 1 2 ]T yield same multiplexed codeword, i.e: and p bi = Vbao p bj Vbao p

where i 6= j. As a matter of fact, pair of distinct indicator vectors result into multiplexed

codewords having zero Eucledian distance between them. This may cause serious performance degradation in ML detector in terms of Symbol Error Rate (SER) when the channels of uplink SCMA users are strongly correlated, also if uplink sparse structure based uplink equalization is employed [22].

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IV. J OINT SCMA C ODEBOOK D ESIGN In this section, we present our proposed novel SCMA codebook design algorithm. As stated in

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previous section, unlike optimizing user-specific optimal rotations and mother constellation, we propose a joint optimization of sparse SCMA codebooks for all users. For user-to-RE mapping matrices Fj , j ∈ {1, 2, · · · J}, we use the solution proposed in [12]. In following sections, we

formulate the codebook design problem and find relevant heuristics. A. Design Objective Formulation

˜ ∈ CK×M J with a In this section, we first design a set of all possible multiplexed signal Z

desired Eucledian distance matrix D ∈ RM 

J ×M J

0

  d2  2,1 D= .  ..  d2M J ,1

where

d21,2 . . . d21,M J 0 .. .



 . . . d22,M J   ..  .. . .  

... ...

(15)

0

˜ Here di,j denotes Eucledian distance between any two arbitrary multiplexed signal z˜i , z˜j ∈ Z. ˜ from desired Eucledian distance profile D is known as MultidiThe problem of constructing Z mensional Scaling (MDS). An alternating co-ordinate descent algorithm is used to minimize the

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s-stress objective function. An elaboration of this algorithm is beyond the scope of this paper, ˜ our goal is to find hence interested readers are suggested to see [23]. Now, having estimated Z, the joint SCMA codebook V = [V1 V2 · · · VJ ] ∈ CK×M J which provides the best mapping of J ˜ ∈ CK×M J . This can be expressed the combined binary indicator matrix P ∈ BM J×M onto Z

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as following Frobenious norm minimization problem

˜ 2 arg min ||VP − Z|| F

V∈CK×M J

This is similar to the well studied Procrustes problem [24].

(16)

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B. Joint Codebook Optimization The inclusion of average symbol energy constraint turns (16) into a QCQP (Quadratically

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Constrained Quadratic Problem). This can be done by reformulating (16) using the identities of trace including the average power constraint Es in the optimization problem (16) ˜ 2 = Tr{(VP − Z) ˜ H (VP − Z)} ˜ ||VP − Z|| F

˜ H VP) − Tr(PT VH ZV) ˜ = Tr(PT QP) − Tr(Z

˜ H Z) ˜ + Tr(Z

(17)

where Q ∈ CM J×M J is the gram matrix of joint SCMA codebook V ∈ CK×M J which is ˜ H Z) ˜ term in the optimization problem (17) is constant, hence Q = VH V. Note that, the Tr(Z can be ignored. Now, (16) can be written as following

˜ ˜ H VP) − Tr(PT VH Z) arg min Tr(PT QP) − Tr(Z

V∈CK×M J

s.t. Q = VH V

(18)

diag(Q) = Es

To enforce sparsity the joint codebook design, we introduce another linear constraint as following ˜ H VP) − Tr(PT VH Z) ˜ arg min Tr(PT QP) − Tr(Z

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V∈CK×M J

s.t. Q = VH V

(19)

diag(Q) = Es

Vj (Jj , 1 : M ) = 0, j ∈ {1, 2, · · · J}

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In (19), Jj , j ∈ {1, 2, · · · J} is the set of row indices of all-zero rows of corresponding user-

RE binary mapping matrices Fj . Note that, the quadratic equality constraint Q = VH V in the optimization problem (19) is a non-convex constraint which makes the optimization very difficult to solve. This non-convex constraint can be relaxed by incorporating Matrix-lifting Semidefinite

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Relaxation (MSDR) based on Schur complement argument [25], [26] ˜ ˜ H VP) − Tr(PT VH Z) arg min Tr(PT QP) − Tr(Z

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V∈CK×M J



s.t. 

Q VH



<0 V Es I

(20)

diag(Q) = Es

Vj (Jj , 1 : M ) = 0, j ∈ {1, 2, · · · J}

The objective function in (20) is now convex and can be effectively solved by interior-point based methods using reliable software packages like CVX [27]. To carry out numerical investigation in this paper, we used YALMIP [28] along with well known robust solver MOSEK [29] to optimize (20). In next section, extensive numerical simulations are presented and the performance of proposed codebook is assessed and compared with several benchmark design algorithms [14]–[16].

V. N UMERICAL R ESULTS

A. Simulation Setup

In this section, we assess the performance of proposed SCMA codebook in terms of Bit Error Rate (BER) and Symbol Error Rate (SER) in additive white Gaussian noise channels under

•

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following channel conditions:

Case 1: Uncoded fast fading where each user experience independent channel coeffiecients over REs with correlated fading channel

•

Case 2: Uncoded fading where each user experience same channel coefficients over REs with correlated fading channels among user

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Unless otherwise specified, the system parameters are configured as follows: numbers of RE assigned per user N = 2, number of REs available for the SCMA system K = 4, number of users in fully overloaded SCMA system J = 6, size of codebook M = 4, average symbol energy Es = 1. We define SNR as the the average energy per bit of the constellation divided by noise variance, that is

Es LM No Eb = No

SN R =

(21)

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where LM is bits per symbol, Eb is the average energy per bit. Out of 100 trials, best suited

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codebook was picked for performance evaluation. This is worth mentioning that codebook design and optimization are done off-grid, so this is not a serious drawback for practical systems. For comparison, we consider the codebooks proposed in [14]–[16] as benchmark which are denoted as 4-Bao, 4-Beko and 4-LDS respectively. 4-Bao in [14] was proposed for SCMA system based on the rotation of Quadrature Amplitude Modulation (QAM) constellation so as to maximize the cutoff rate of equivalent Multiple Input Multiple Output (MIMO) system by an exhaustive computer search approach. 4-Beko is a well known multidimensional constellation which can be found by maximizing minimum Eucledian distance among symbols whereas 4-LDS is found by repeating 4-QAM constellation over 2 REs [15], [16]. Neither 4-Beko nor 4-LDS was proposed for SCMA systems but are shown to have good performance in uncoded uplink SCMA systems [7]. B. Results and Discussions

As illustrated in Fig.2, the proposed codebook outperforms the codebooks existing in [14]– [16], particularly at SNRs above 0 dB in Case 1. In low SNR, the proposed codebook performs similarly with benchmark codebooks. This is because, in low SNR, the BER performance is primarily dictated by channel noise. The assignment of same codebooks (4-LDS, 4-Beko, 4-

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Bao) among J users in uplink produces multiplexed codewords with small Eucledian distances in correlated fading channel as shown in Section III-B which is not the case for proposed codebook design. The received signals at receiver multiplexed by proposed SCMA codebook has better Eucledian distance profile. So the deviation from the original transmitted codewords caused by both gaussian noise and correlated channel coefficients are less likely to cause erroneous

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detection at receiver. As a matter of fact, proposed SCMA performs better than other codebooks in terms of BER performance in correlated uplink channels, particularly at higher SNR. Similar explanation holds for Case 2 in Fig. 3, but an overall degradation of performance is notable in this scenario. This loss of performance caused by reduced channel diversity gain, which is also illustrated in [6] for 4-LDS, 4-Beko and 4-Bao. To investigate the proposed algorithm’s performance in different loading scenario, we considered three cases: K = 4 and J = 4, K = 4 and J = 5 and K = 4 and J = 6. The first case is the non-overloaded scenario with λ = 1.0 whereas the following cases are overloaded

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10 0

BER

10 -1

10 -2

10 -3 -5

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Proposed 4-Bao 4-Beko 4-LDS

0

5

10

15

Eb /N o (in dB) (a)

Proposed 4-Bao 4-Beko 4-LDS

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10 -1

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SER

10 0

10 -2

-5

0

5

10

15

Eb /N 0 (in dB) (b) Fig. 2. Performance comparison among proposed SCMA codebook design and existing codebook in [14]–[16] for Case 1 using ML detector(a) BER performance comparison (b) SER performance comparison

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BER

10 -1

10 -2 -5

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Proposed 4-Bao 4-Beko 4-LDS

0

5

10

15

Eb /N o (in dB) (a)

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10 -1

-5

Proposed 4-Bao 4-Beko 4-LDS

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SER

10 0

0

5

10

15

Eb /N o (in dB) (b) Fig. 3. Performance comparison among proposed SCMA codebook design and existing codebook in [14]–[16] for Case 2 using ML detector(a) BER performance comparison (b) SER performance comparison

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BER 10

-2

5

7

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= 1.5 = 1.25 = 1.0

10-1

9

11

13

15

Eb/N0 (in dB)

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Fig. 4. BER performance of proposed codebook for overloaded (λ = 1.5) and non-overloaded (λ = 1.0) scenario

scenario with λ = 1.25 and λ = 1.5 respectively. It can be seen that, in overloading scenarios, the codebook performs comparatively worse than non-overloading scenario, especially in high SNR (> 7 dB). This is because, while operating with higher number of users, the Eucledian distance between multiplexed codewords are reduced which eventually led to a performance

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degradation. Importantly, in low SNR, the codebook performs similarly in all of the cases under consideration. This observation turns the proposed codebook into a promising candidate to be used in environments with low SNR and varying loading scenario. VI. C ONCLUSION

In this work, we presented a joint SCMA codebook design approach for uplink systems with correlated channels. Our proposed design encapsulates the introduction of new channel model which is further incorporated in a QCQP problem. We provided relevant relaxation techniques to

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17

solve the codebook optimization problem. Extensive numerical simulation illustrates the promise of the proposed design in correlated channels over a wide range of SNR. Further research

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directions include develop computationally tractable methods of large, high dimension SCMA codebooks for uplink fading channels.

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Declaration of interests

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☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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There are no interest to declare.

Journal Pre-proof Shah Mahdi Hasan received the B.S. degree in electrical and electronic engineering from the Bangladesh University of Engineering and Technology, Dhaka, in 2015. Currently, he is pursuing the Ph.D. degree in the University of Newcastle, Australia. His research interests include sparse signal representation, convex optimization, Non-Orthogonal Multiple Access and Physical Layer Security.

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Kaushik Mahata received the Ph.D. degree in signal processing from Uppsala University, Uppsala, Sweden, in 2003. He is currently an Associate Professor in the University of Newcastle, Callaghan NSW, Australia. His research interests include estimation identification, spectrum analysis, and machine learning.

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Md. Mashud Hyder received the Ph.D. degree in electrical engineering from the University of Newcastle, Callaghan NSW, Australia, in 2012. He is currently a Research Academic in the University of Newcastle. His research interests include sparse signal representation, statistical signal and image processing, sensor array processing, and convex optimization.

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Journal Pre-proof AUTHOR STATEMENT Author SHAH MAHDI HASAN

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KAUSHIK MAHATA

Contribution Conceptualization, Methodology, Software, Data Curation, Writing-Original Draft, Visualization Validation, Writing- Review and Editing, Supervision, Project Administration Validation, Writing- Review and Editing, Supervision, Methodology, Resources

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MD. MASHUD HYDER