# Adaptive Modulation SCMA Codebook Design Based on Constellation Rotation

Authors: Meiming Fu; Yumiao Shang; Qingyang Liu; Shuming Xu; Gang Xie
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Sparse Code Multiple Access (SCMA) is one of the competitive nonorthogonal multiple access (NOMA) techniques for the fifth-generation (5G) communication system. The SCMA codebook design is particularly important for SCMA systems. However, according to the results of our investigation of as many literature as possible, there is no research on adaptive modulation SCMA at present. Also, there is no systematic study concerning adaptive modulation SCMA codebook optimization. In this paper, an adaptive modulation SCMA transmission mechanism in different fading channels is proposed. Then, we optimize the adaptive modulation SCMA codebook design based on constellation rotation (AMSCR) for each adaptive mode including different modulation methods. The codebook optimization criterion is to maximize the minimum Euclidean distance between joint codewords. Since the space of the set of joint codewords is too large, the calculation of the minimum Euclidean distance has become an NP-hard problem. We use a genetic algorithm to solve this problem. As compared with the traditional SCMA codebook, the proposed AMSCR has better bit error rate performance in Gaussian channel and Rayleigh channel. The adaptive modulation strategy can maximize the throughput and has the best throughput performance compared with the traditional SCMA system and adaptive modulation SCMA system using traditional codebook.


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## Body

## 1. Introduction

Multiple access [1] is one of the important technologies of wireless communications. Many authors have studied the orthogonal multiple access technology [2]. The authors of [3] present and compare the main OFDM scheduling techniques. In [3], the influences of bandwidth granularity on the resource allocation performances are studied in particular. A multiorder orthogonal frequency division multiplexing (OFDM) frequency diversity approach using properties of order theory and Hamming distance is proposed in [4]. Also, the proposed approach in [4] outperforms other OFDM diversity techniques, such as Maximal Ratio Combination (MRC) and Space Frequency Block Coding (SFBC). However, with the development of data transmission and applications, the existing orthogonal multiple access technology cannot support a large number of users and devices. To meet the challenges of future wireless communication such as low latency, massive connections, and high spectrum efficiency (SE), the nonorthogonal multiple access (NOMA) becomes a hot topic [5].NOMA [6] achieves high spectrum efficiency by serving multiple users on the same frequency or time resource. It is a promising fifth-generation wireless communication technology. The authors of [7] make a comprehensive overview about the promising NOMA schemes. The grant-free NOMA incorporates NOMA technique with uplink uncoordinated access and is expected to address the massive connectivity requirement of 5G [7]. A new NOMA technology, sparse code multiple access (SCMA), is proposed in [8]. It is a multicarrier NOMA [9], which can support multiple users to serve at the same time with resource blocks less than the number of users. Therefore, it is a very attractive multiple access technology for 5G.The detection algorithm and SCMA codebook design are two of the main research directions of SCMA [10–27].First, to reduce the complexity of the detection algorithm, SCMA uses a message-passing algorithm (MPA) based multiuser detector at the receiver to eliminate interuser interference in SCMA, while keeping the maximum likelihood (ML) performance. This method takes advantage of the sparsity of the codebook and has lower complexity than the joint maximum likelihood detector. However, the complexity of the receiver is still very high when a large number of users communicate. Hence, all users are divided into several groups, and in each group, each user has its own dedicated codebook. In [19], a serial strategy message-passing algorithm is proposed, which can reduce the complexity of detection while reducing the detection performance slightly. In [20], the complexity of the overall detection is reduced by accelerating the iterative convergence. In [21], the Log-MPA algorithm is proposed, which saves more than 90% multiplication and completely eliminates the exponent calculations with negligible performance. In this paper, we use Log-MPA algorithm as the detection algorithm of receiver.Second, the SCMA codebook design is also particularly important for SCMA systems. The currently known SCMA codebooks are all designed based on the traditional SCMA, and the codebooks of all users are specific codebooks generated from the same mother constellation through different degrees of rotation. Codebook optimization mainly includes several aspects such as mother constellation, matrix, and phase rotation factor. The authors of [22] present a framework to design the codebooks by taking into account the entire system including the SCMA encoder and the MPA-based detector, and the minimization of symbol-error rate is the design criterion. The codebook in [23, 24] achieved the pursuit of lower bit error rate (BER) through the optimization design of the mother constellation. In [24], a new method of SCMA codebook design based on the principle of lattice constellation design is proposed, and it is shown that the performance of SCMA is better than LDS due to the shaping gain of multidimensional codebook. In [23], the SCMA codebook based on star quadrature amplitude modulation (star-QAM) constellation is proposed, and its BER performance was proved to be better than low-density signature (LDS) [25] and the constellation design based on the lattice principle in [24].The design of the mapping matrix in [26] significantly reduced the BER. A reasonable calculation method for the phase rotation factor is given in [25, 27]. According to the modulation order of the user, the appropriate phase rotation factor is calculated to achieve the even distribution of the constellation points on the two-dimensional plane as much as possible. Adaptive modulation technology is a way to improve the spectrum efficiency (SE) of SCMA systems. However, in adaptive modulation SCMA systems, due to the different modulation orders, it is impossible to obtain the appropriate phase rotation factor by simple calculation according to the traditional method. Therefore, it is necessary to redesign the phase rotation factor to generate a new constellation.In an adaptive modulation communication system, users can choose different modulation methods or different transmission powers according to the different signal-to-noise ratio (SNR) caused by different fading channels to enable the system to achieve the highest spectrum efficiency or achieve the highest throughput. Adaptive modulation has a wide range of applications in various communication systems. The authors of [28] proposed a variable-power rate-adaptive MQAM modulation scheme for high-speed data transmission under fading channels, which maximized the spectrum efficiency. Sampei and Harada discussed the subcarrier-level adaptive modulation of a single-cell multiplex bandwidth cellular system based on orthogonal frequency division multiplexing (OFDM) in [29], while considering the application of transmit power control. The adaptive quadrature amplitude modulation scheme is introduced in [30], and its performance in known and predicted channels is studied. In [31], a MIMO-OFDM system based on superposition-based adaptive modulation is proposed. When the transmitting antennas have different channel conditions, each transmitting antenna selects a different modulation method, and at the same time, the optimal modulation corresponding to each channel condition is selected through superimposed space-time coding and decoding, which improves the spectral efficiency. In addition to changing the modulation mode and power, other parameters in the modulation and channel coding that affect the performance of the adaptive scheme can also be changed. There are also many studies on power allocation. It can be seen that, in various types of communication systems such as OFDM and MIMO, adaptive modulation can be used to improve the system performance. The SCMA system can also support adaptive modulation, and each user adopts different modulation methods according to the SNR.However, in the adaptive modulation SCMA system, the codebook generated based on the traditional phase rotation factor is no longer applicable. The minimum Euclidean distance between the joint codewords no longer meets the maximum minimum Euclidean distance. Hence, the adaptive modulation SCMA codebook must be redesigned. At the same time, since the space of the set of joint codewords is too large, the calculation of the minimum Euclidean distance has become an NP-hard problem. In this paper, we use a genetic algorithm to search for the minimum Euclidean distance to solve this problem.The main contributions of this paper are summarized as follows:(i)
An adaptive modulation SCMA transmission mechanism in different fading channels is proposed to maximize the throughput. In the adaptive modulation SCMA system, we set up different adaptive modes. Then, we choose the corresponding adaptive mode according to the channel condition.(ii)
For each adaptive mode in which different users use different modulation, the adaptive modulation SCMA codebook design based on constellation rotation (AMSCR) is proposed. Also, the optimization criterion is to maximize the minimum Euclidean distance between joint codewords.The main contents of other parts of this paper are organized as follows. In Section2, we introduce the adaptive modulation SCMA downlink system model, including the system structure, the codebook generation of the sender, adaptive modulation scheme, and receiver multiuser detection algorithm. The codebook optimization scheme of the adaptive SCMA system is presented in Section 3. Section 4 evaluates the BER performance and system throughput of the proposed codebook in the adaptive modulation SCMA system through simulation. Section 5 gives some conclusions and prospects.Notations: in this paper, we useB to denote the set of binary numbers and ℂ represents the set of complex numbers. The operations ⋅T and ⋅2 represent the transpose and Euclidean norm, respectively.


## 2. System Model

Figure1 shows the downlink adaptive modulation SCMA system in this paper. The channel is completely known by the transmitter, that is, assuming that the channel state information fed back by the receiver is completely accurate and without delay, a certain adaptive mode p that needs to be selected in the system is determined by the instantaneous signal-to-noise ratio. According to the channel quality, each user can choose its own modulation order. According to the situation of the modulation order selected by each user, there are several adaptive modes p, p=0,1,…,P−1. There are P adaptive modes in the adaptive modulation SCMA system.Figure 1
The downlink adaptive modulation SCMA system model.In each adaptive mode, there is a set of codebooks forJ users with all selectable modulation orders. After the user codebook is determined, it enters the SCMA encoding module and then superimposes the codewords of all users to generate a multiuser joint codeword.After going through the channel, receivers use the existing MPA algorithm to decode.The Adaptive Modulator of the transmitter in Figure1 is the contribution of this paper. An adaptive mode selection strategy is given for different channel states, and a new codebook design method is proposed for each adaptive mode. The receiver uses the traditional low-complexity MPA algorithm [21] to decode; some other low-complexity decoding algorithms such as the MPA algorithms in [19, 20] are also applicable.


### 2.1. Sender and Adaptive Modulation Strategy

In a downlink SCMA system with multiple users and one base station,K resource blocks support access to J users, J>K, and the reload factor is λ=J/K,λ>1, where each user occupies N resource blocks N<K, J=K/N, and the number of users carried by each resource block is df=K−1/N−1. Compared with TDMA, OFDMA, and other orthogonal multiple access methods (J≤K), it has significantly higher spectrum efficiency. For the SCMA system with K=4 and J=6, its structure is represented by a factor graph as follows. Each resource block is shared by df=3 users, and each user is connected to N=2 resource blocks.As shown in Figure2, each layer node represents a user and each resource node represents a carrier resource block. It is expressed as a factor graph as F=f1,f2,…,fJ. fjk=1 means that user j is connected to resource block k.(1)F=111000100110010101001011.Figure 2
Factor graph.At the sender of the adaptive modulation SCMA system, first, the bit stream input by each user is multidimensionally modulated. Taking userj as an example, its modulation order is Mj, the input bit stream is bj, and every log2Mj bit is a symbol. After multidimensional modulation, an N-dimensional complex codeword zj=x1′,…,xN′T is generated. The mapping from bits to N-dimensional codewords is denoted as(2)zi=gbi.Then, aK-dimensional complex codeword is generated through the mapping matrix xi=x1,x2,…,xKT, which contains N nonzero elements, and the mapping matrix for each user is given by(3)V1=10010000,V2=10000100,V3=10000001,V4=00100100,V500100001=00100001,V6=00001001.Each mapping matrix containsN nonzero elements. It expands the N-dimensional complex codeword into the K-dimensional complex codeword of SCMA, where diagVjVjT=fj.The codeword generated during the entire encoding process above is expressed as(4)xj=Vj⋅gbj,where Vj∈BK×N.Userj has its specific codebook, which is generated by the rotation and expansion of the mother constellation A corresponding to the modulation mode selected by user j. Users who choose the same modulation method have the same mother constellation. The rotation matrix and mapping matrix of the user further make the codebook of each user different. The codebook of user j can be given by(5)Cj=VjΔjA,where Cj∈ℂK contains Mj symbols and Δj is a N×N rotation matrix, which contains the phase rotation factor θ. In the adaptive modulation SCMA system, the phase rotation factor θ of the traditional SCMA system will not be the optimal value because the mother constellation of each user is no longer the same. Therefore, the codebook of the adaptive modulation SCMA system needs to be optimized afterwards by optimizing this parameter.In the adaptive modulation method, the sender can obtain the channel information. In its simplest form, the instantaneous SNR is available, but for more complex channels, more channel information can be obtained.Given the maximum error rateBERth, the link error rate should be lower than or equal to the maximum error rate. The transmission power is a fixed value, and the transmission rate changes with the change of the channel quality due to the change of the adaptive mode p. It is assumed that the channel gain is fully known to both the sender and the receiver.When the instantaneous SNRγ satisfies γp<γ<γp+1, the adaptive mode p is selected, where γP=+∞.For allγ>γ0, it is necessary to satisfy BER≤BERth, which is given by(6)BERpγ≤BERth.The average BER under system modep in adaptive modulation SCMA is BERpγ.The spectrum efficiency [30] of the adaptive modulation SCMA system can be given by(7)SEΓ=∑p=06kp∫γpγp+1fγdγ,fγ=1Γexp−γΓ,γ≥0,0,γ<0,,where Γ is the average SNR, kp is the number of bits transmitted per unit symbol on the unit resource block corresponding to the adaptive mode p, and fγ is the probability density function of the instantaneous SNR. To maximize the spectrum efficiency of the system, an appropriate SNR boundary value γp should be selected by(8)maxγp∑p=0P−1kp∫γpγp+1fγdγ,p=0,1,…,P−1,s.t.BERpγp≤BERth.Maximizing the system spectrum efficiency means makingkp as large as possible under the premise of ensuring communication quality. As the SNR increases, the adaptive mode with larger kp should be adopted as soon as possible. Therefore, the spectrum efficiency reaches the maximum when the inequality constraint in equation (8) takes the equal sign.Specifically, the average BER of the adaptive modulation SCMA system is very complicated, and it is difficult to give a closed-form solution. Therefore, after the rotation-based constellation optimization in Section3 is completed, γp will be determined directly by simulation.


### 2.2. Receiver Multiuser Detection Algorithm

In the SCMA downlink communication link, after synchronous multiplexing, the received signal can be expressed as(9)y=H∑j=1Jxj+n=H∑j=1JVjfjbj+n,where H=diagh and h=h1,…,hKT is the channel vector. n=n1,n2,…,nKT is additive white Gaussian noise vector. Each nk is independently and identically distributed with zero mean and N0 variance. The transmitted codewords of J users are directly superimposed and sent, and the receiver receives the signal as y.At the receiver, it uses the MPA algorithm in [21] to decode. In the decoding process, messages are iteratively exchanged between user nodes and resource nodes along the lines of the factor graph representation in Figure 2.The MPA algorithm in [21] is briefly described as follows. Taking the third resource block as an example, the message (10)Ic3⟶v6m6=∑m3=1M3∑m4=1M4φkyk,m2,m4,m6,nk,hkIv2⟶c3m2Iv4⟶c3m4,m6=1,…,M6,Ic3⟶v2m2=∑m4=1M4∑m6=1M6φkyk,m2,m4,m6,nk,hkIv4⟶c3m4Iv6⟶c3m6,m2=1,…,M2,Ic3⟶v4m4=∑m2=1M2∑m6=1M6φkyk,m2,m4,m6,nk,hkIv2⟶c3m2Iv6⟶c3m6,m4=1,…,M4,(11)φkyk,m1,m2,m3,nk,hk=exp1nkyk−hkx1,km1+hkx2,km2hkx3,km32,transmitted from resource block c3 to the three user nodes v6,v2,andv4 is given by equation (10), where m1,m2,andm3 are the transmitted codewords of three users on resource block k. Mj is the modulation order of user j. φk is given by equation (11), which represents the maximum likelihood function that the transmitted signal is m1, m2, and m3 when the receiver receives yk.Taking user nodev6 as an example, the message it transmits to resource nodes c3andc4 can be expressed as(12)Iv6⟶c3m6=Ic4⟶v6m6∑m6Ic4⟶v6m6,m6=1,…,M6,Iv6⟶c4m6=Ic3⟶v6m6∑m6Ic3⟶v6m6,m6=1,…,M6,and at the first iteration, the value of Iv6⟶c4m6 is initialized to 1/M6.When the maximum number of iterations is reached, the decision output is performed. The symbol probability is represented by the message and a priori estimate. It can be obtained as(13)Qvjm=1MjIc1⟶vjIc2⟶vj.According to the bit log-likelihood ratio (LLR), the symbols are mapped to bits. For uservj, the calculation of i-th bit’s LLR is given by(14)LLRi=logPbi=1Pbi=0=log∑m:bm,i=1Qvjm∑m:bm,i=0Qvjm.In order to reduce the complexity of MPA, the authors of [21] applied Jacobian logarithm logea+eb≈maxa,b to transfer the operation to log domain and eliminate the exponential operation in equation (10).


## 2.1. Sender and Adaptive Modulation Strategy

In a downlink SCMA system with multiple users and one base station,K resource blocks support access to J users, J>K, and the reload factor is λ=J/K,λ>1, where each user occupies N resource blocks N<K, J=K/N, and the number of users carried by each resource block is df=K−1/N−1. Compared with TDMA, OFDMA, and other orthogonal multiple access methods (J≤K), it has significantly higher spectrum efficiency. For the SCMA system with K=4 and J=6, its structure is represented by a factor graph as follows. Each resource block is shared by df=3 users, and each user is connected to N=2 resource blocks.As shown in Figure2, each layer node represents a user and each resource node represents a carrier resource block. It is expressed as a factor graph as F=f1,f2,…,fJ. fjk=1 means that user j is connected to resource block k.(1)F=111000100110010101001011.Figure 2
Factor graph.At the sender of the adaptive modulation SCMA system, first, the bit stream input by each user is multidimensionally modulated. Taking userj as an example, its modulation order is Mj, the input bit stream is bj, and every log2Mj bit is a symbol. After multidimensional modulation, an N-dimensional complex codeword zj=x1′,…,xN′T is generated. The mapping from bits to N-dimensional codewords is denoted as(2)zi=gbi.Then, aK-dimensional complex codeword is generated through the mapping matrix xi=x1,x2,…,xKT, which contains N nonzero elements, and the mapping matrix for each user is given by(3)V1=10010000,V2=10000100,V3=10000001,V4=00100100,V500100001=00100001,V6=00001001.Each mapping matrix containsN nonzero elements. It expands the N-dimensional complex codeword into the K-dimensional complex codeword of SCMA, where diagVjVjT=fj.The codeword generated during the entire encoding process above is expressed as(4)xj=Vj⋅gbj,where Vj∈BK×N.Userj has its specific codebook, which is generated by the rotation and expansion of the mother constellation A corresponding to the modulation mode selected by user j. Users who choose the same modulation method have the same mother constellation. The rotation matrix and mapping matrix of the user further make the codebook of each user different. The codebook of user j can be given by(5)Cj=VjΔjA,where Cj∈ℂK contains Mj symbols and Δj is a N×N rotation matrix, which contains the phase rotation factor θ. In the adaptive modulation SCMA system, the phase rotation factor θ of the traditional SCMA system will not be the optimal value because the mother constellation of each user is no longer the same. Therefore, the codebook of the adaptive modulation SCMA system needs to be optimized afterwards by optimizing this parameter.In the adaptive modulation method, the sender can obtain the channel information. In its simplest form, the instantaneous SNR is available, but for more complex channels, more channel information can be obtained.Given the maximum error rateBERth, the link error rate should be lower than or equal to the maximum error rate. The transmission power is a fixed value, and the transmission rate changes with the change of the channel quality due to the change of the adaptive mode p. It is assumed that the channel gain is fully known to both the sender and the receiver.When the instantaneous SNRγ satisfies γp<γ<γp+1, the adaptive mode p is selected, where γP=+∞.For allγ>γ0, it is necessary to satisfy BER≤BERth, which is given by(6)BERpγ≤BERth.The average BER under system modep in adaptive modulation SCMA is BERpγ.The spectrum efficiency [30] of the adaptive modulation SCMA system can be given by(7)SEΓ=∑p=06kp∫γpγp+1fγdγ,fγ=1Γexp−γΓ,γ≥0,0,γ<0,,where Γ is the average SNR, kp is the number of bits transmitted per unit symbol on the unit resource block corresponding to the adaptive mode p, and fγ is the probability density function of the instantaneous SNR. To maximize the spectrum efficiency of the system, an appropriate SNR boundary value γp should be selected by(8)maxγp∑p=0P−1kp∫γpγp+1fγdγ,p=0,1,…,P−1,s.t.BERpγp≤BERth.Maximizing the system spectrum efficiency means makingkp as large as possible under the premise of ensuring communication quality. As the SNR increases, the adaptive mode with larger kp should be adopted as soon as possible. Therefore, the spectrum efficiency reaches the maximum when the inequality constraint in equation (8) takes the equal sign.Specifically, the average BER of the adaptive modulation SCMA system is very complicated, and it is difficult to give a closed-form solution. Therefore, after the rotation-based constellation optimization in Section3 is completed, γp will be determined directly by simulation.


## 2.2. Receiver Multiuser Detection Algorithm

In the SCMA downlink communication link, after synchronous multiplexing, the received signal can be expressed as(9)y=H∑j=1Jxj+n=H∑j=1JVjfjbj+n,where H=diagh and h=h1,…,hKT is the channel vector. n=n1,n2,…,nKT is additive white Gaussian noise vector. Each nk is independently and identically distributed with zero mean and N0 variance. The transmitted codewords of J users are directly superimposed and sent, and the receiver receives the signal as y.At the receiver, it uses the MPA algorithm in [21] to decode. In the decoding process, messages are iteratively exchanged between user nodes and resource nodes along the lines of the factor graph representation in Figure 2.The MPA algorithm in [21] is briefly described as follows. Taking the third resource block as an example, the message (10)Ic3⟶v6m6=∑m3=1M3∑m4=1M4φkyk,m2,m4,m6,nk,hkIv2⟶c3m2Iv4⟶c3m4,m6=1,…,M6,Ic3⟶v2m2=∑m4=1M4∑m6=1M6φkyk,m2,m4,m6,nk,hkIv4⟶c3m4Iv6⟶c3m6,m2=1,…,M2,Ic3⟶v4m4=∑m2=1M2∑m6=1M6φkyk,m2,m4,m6,nk,hkIv2⟶c3m2Iv6⟶c3m6,m4=1,…,M4,(11)φkyk,m1,m2,m3,nk,hk=exp1nkyk−hkx1,km1+hkx2,km2hkx3,km32,transmitted from resource block c3 to the three user nodes v6,v2,andv4 is given by equation (10), where m1,m2,andm3 are the transmitted codewords of three users on resource block k. Mj is the modulation order of user j. φk is given by equation (11), which represents the maximum likelihood function that the transmitted signal is m1, m2, and m3 when the receiver receives yk.Taking user nodev6 as an example, the message it transmits to resource nodes c3andc4 can be expressed as(12)Iv6⟶c3m6=Ic4⟶v6m6∑m6Ic4⟶v6m6,m6=1,…,M6,Iv6⟶c4m6=Ic3⟶v6m6∑m6Ic3⟶v6m6,m6=1,…,M6,and at the first iteration, the value of Iv6⟶c4m6 is initialized to 1/M6.When the maximum number of iterations is reached, the decision output is performed. The symbol probability is represented by the message and a priori estimate. It can be obtained as(13)Qvjm=1MjIc1⟶vjIc2⟶vj.According to the bit log-likelihood ratio (LLR), the symbols are mapped to bits. For uservj, the calculation of i-th bit’s LLR is given by(14)LLRi=logPbi=1Pbi=0=log∑m:bm,i=1Qvjm∑m:bm,i=0Qvjm.In order to reduce the complexity of MPA, the authors of [21] applied Jacobian logarithm logea+eb≈maxa,b to transfer the operation to log domain and eliminate the exponential operation in equation (10).


## 3. Adaptive Modulation SCMA Codebook Design

The codebook design of traditional SCMA considers that different users in the SCMA system adopt the same modulation method, and their respective codebooks are rotated based on the same mother constellation, thereby reducing the interference between users. Because of the combination of adaptive modulation technology and SCMA, when the adaptive SCMA system contains users with different modulation orders, their mother constellations are also different. The phase rotation factor cannot be obtained through the user’s unified parent constellation and unified modulation method according to the traditional method. Hence, it is necessary to redesign the phase rotation factor to optimize the codebook.For simplicity, the adaptive modulation in this paper involves two modulation methods, QPSK and 16QAM. The number of users using 16QAM is recorded asn, and the number of users using QPSK modulation is J−n. The number of users in the adaptive modulation SCMA system is J=6, and the number of resource blocks is K=4. Ignoring the channel quality differences between users, we only consider the overall channel quality changes of the system. Users can choose different modulation methods from QPSK Ma=4 and 16QAM Mb=16. The transmission rate is changed by changing the number of high-order modulation users nn=0,1,…,6.Specifically,p=n, all users in the adaptive mode p=0 adopt QPSK modulation, corresponding to n=0. When the adaptive mode p=6, all users adopt 16QAM modulation, corresponding to n=6.In a word, SCMA with QPSK and SCMA with 16QAM are two special cases of all adaptive modes. The adaptive modesp=0 and p=6 are traditional SCMA with QPSK and the traditional SCMA with 16QAM, respectively. Therefore, these two adaptive modes do not require codebook optimization. The remaining five adaptive modes, p=1,2,3,4,5, corresponding to 5 different groups of constellations need to be optimized.There are∏j=1JMj types of joint codewords sent by the sender. The probability that the joint codeword sa sent to be translated into sb is given by(15)Psa⟶sb|H=QHsa−sb22N0,where si=∑j=1Jxj. Therefore, in the fading channel, the decoding error rate for sending a certain joint codeword is given by(16)Psa|H=∑b=1b≠a∏j=1JMjPsa⟶sb|H=∑b=1b≠a∏j=1JMjQ∑k=1Khk2da,b2k2N0≤∑b=1b≠a∏j=1JMjQ∑k=1Kh2da,b2k2N0=∑b=1b≠a∏j=1JMjQh2∑k=1Kda,b2k2N0≤∑b=1b≠a∏j=1JMjQh2d22N0=∏j=1JMj−1Qh2d22N0,where da,b2k=sak−sbk2, h2=minkhk2, and d2=mina≠ba,b∈1,2,…,∏j=1JMj∑k=1Kda,b2k.In the AWGN channel, equation (16) can be simplified as(17)Psa|H=∑b=1b≠a∏j=1JMjQ∑k=1Kda,b2k2N0≤∑b=1b≠a∏j=1JMjQd22N0=∏j=1JMj−1Qd22N0,(18)d2=mina≠ba,b∈1,2,…,∏j=1JMj∑k=1Ksak−sbk2=mina≠ba,b∈1,2,…,∏j=1JMj∑k=1K∑j=1JVjΔjθpAcpa,jk−∑j=1JVjΔjθpAcpb,jk2,where d2=mina≠ba,b∈1,2,…,∏j=1JMj∑k=1Kda,b2k. Therefore, the criterion of adaptive modulation SCMA constellation optimization is to minimize equation (17), that is, to maximize minimum Euclidean distance d2.In adaptive modep, for any joint codeword index a, its corresponding codeword sequence number mj sent by user j is represented by (19)mj=cpa,j.Then, the minimum Euclidean distance, which is an important factor affecting the decoding error rate, can be further expressed as equation (18).The rotation matrixΔj needs to satisfy the condition that the generator matrix G obtained by Vj and Δj is a Latin generator matrix, that is, the nonzero elements of each row are different and the nonzero elements of each column are also different. According to [25], the generator matrix with Latin characteristics can improve the BER performance.According to [25, 32], this paper also uses the rotation matrix that can make the generator matrix satisfy the Latin characteristics. The rotation matrix Δj is given by(20)Δ1=eiθ00ei⋅2θ,Δ2=100ei⋅3θ,Δ3=ei⋅3θ00eiθ,Δ4=100ei⋅2θ,Δ5=eiθ001,Δ6=100ei⋅3θ.The specific generator matrixG can be expressed as(21)G=g1,g2,…,gJ=eiθ1ei⋅3θ000ei⋅2θ001eiθ00ei⋅3θ0ei⋅2θ0100eiθ01ei⋅3θ,where gj=diagVjΔjVjT. The phase rotation factor θ in equations (20) and (21) is the most critical influencing factor. The meaning of the phase rotation factor θ can be understood as the rotation interval of adjacent users. Derived from [25, 27, 32], the phase rotation factor of traditional SCMA is calculated as(22)θ=8π11M−12.Figures3 and 4 show the constellation of three users carried on a single resource block. It must be mentioned that not all users have the same modulation order in p=1,2,…,5 modes of the adaptive modulation SCMA system. If we use equation (22) to get the phase rotation factor, the distribution of constellation points will be very uneven, as shown in Figure 3.(23)θp=argmaxθpmina≠b∑k=1K∑j=1JVjΔjθpAcpa,jk−∑j=1JVjΔjθpAcpb,jk2,0≤θp≤π2,p=1,…,J−1.Figure 3
The adaptive modep=1.Figure 4
SCMA with QPSK.The rotation factors in Figures3 and 4 are both obtained by equation (22). In Figure 4, we can see that all users adopt the same modulation order and their constellation points are evenly distributed. However, Figure 3 includes users with modulation orders of 4 and 16 at the same time. After each user is rotated by equation (22), it can be seen from Figure 3 that the constellation points are unevenly distributed. The distance between user 1 and user 2 is much less than that between users 2 and 3. This phenomenon is caused by improper rotation. Hence, in the adaptive mode with different modulation orders, if equation (22) is used to calculate the phase rotation factor, the BER and throughput performance will be very poor. Also, its throughput will even be worse than that of SCMA with QPSK and SCMA with 16QAM. These conclusions are confirmed in the simulation.Therefore, we must redesign the adaptive modulation SCMA codebook by optimizing the phase rotation factor.The authors of [33] gave some rules that need to be met when designing the phase rotation factor to minimize the decoding error rate, that is, the user’s constellation is distributed on a two-dimensional plane as uniformly as possible as shown in Figure 4. In traditional SCMA, the appropriate θ is calculated according to the user’s modulation order to achieve the even distribution of constellation points as much as possible. However, in the adaptive modulation SCMA system, due to the different constellation orders among users, it is impossible to obtain θ through simple calculations according to the traditional method. Therefore, to minimize the decoding error rate, that is, to maximize equation (18), it is necessary to optimize the design of the phase rotation factor. The optimization objectives are given by equation (23).Since there are∏j=1JMj different joint codewords sent by the sender, to calculate the minimum Euclidean distance between two of these codewords, ∏j=1JMj⋅∏j=1JMj−1/2 complex number additions and modulo operations are required. In the adaptive mode p=5, the user’s modulation order is five 16QAM and one QPSK J=6. In this case, the number can reach 8,796,090,925,056. Even in the case of p=4, the computational complexity is unacceptable. Therefore, a suboptimal algorithm is needed to calculate the minimum Euclidean distance d2. In this paper, we use genetic algorithm. According to [34], the genetic algorithm simulates the natural selection of Darwin’s biological evolution theory and the biological evolution process of genetic mechanism, so that the initial population (consisting of a certain number of individuals encoded by genes) evolves generation by generation to produce an approximate solution. In each generation, individuals are selected according to their fitness, with the help of natural genetic operators to combine crossover and mutation to generate a new generation of populations. The algorithm stops when the number of generations reaches the upper limit or satisfies the hysteresis generation number (the fitness value does not increase significantly after multiple generations). The optimal individual in the last population is decoded as the approximate optimal solution d2.Through the genetic algorithm, we can locate the two codewordssa and sb corresponding to the minimum Euclidean distance in the current codebook.The phase rotation factor is also searched by genetic algorithm.The overall process is summarized as follows. Under the initial valueθp, we obtain d2 and its corresponding sasa and sb through the genetic algorithm, and then use these two codeword indexes a and b to directly calculate d2 as the target of genetic algorithm search.For different θp, the codeword index corresponding to the minimum Euclidean distance of the generated codebook is not exactly the same. So after getting the current θp, the genetic algorithm is used to search the a and b corresponding to the minimum Euclidean distance again. The algorithm process is shown in Algorithm1.Algorithm 1: The phase rotation factor of adaptive modulation SCMA codebook design algorithm.
Input: Adaptive modulation SCMA system parameters K, V, N, Ma, Mb.Output:θ^p(1)
x=a,b, 1≤a<b≤∏j=1JMj. The range of θp is 0∼π/2.(2)
Initialize the initial population to be emptypopulationx= (randomly generate the first-generation population without specifying). PopulationSize=40,MaxStallGenerations=50.(3)
Initializeθ^p by random. Initialize number of iterations, Iter = 5.(4)
forIter = 1 : 5 do(5)
Generate codebook according toθ^p.(6)
Generate a list of sent joint codewordsS (contains ∏j=1jMj joint codewords).(7)
functionGENETICA (populationx)(8)
The fitness function isda,b2=∑k=1ksak−sbk2(9)
returnx^=a^,b^=argmina,bda,b2(10)
end function(11)
populationx=populationx∪x^(12)
functionGENETICB (populationx)(13)
returnθp′=argminθp−∑k=1K∑j=1JVj⋅Δjθp⋅Acpa^,jk−∑j=1jVj⋅Δjθp⋅AcpB^,jk2(14)
end function(15)
updateθ^p=θp′(16)
end for(17)
returnθ^p


## 4. Simulation Results

This section gives the simulation results and analysis to illustrate the BER performance of the adaptive modulation SCMA codebook design based on constellation rotation (AMSCR) compared with adaptive modulation SCMA using traditional codebook (AM-TRA) and the throughput (the number of correct bits transmitted per second, Mbps) performance of adaptive modulation SCMA system.In the simulation of BER performance, both AWGN channel and Rayleigh fading channel have been considered. The number of symbols is107. The modulation methods included in the simulation are QPSK and 16QAM. The energy of each codeword in the codebook is the same value. The SCMA modulated signal is carried on the OFDM subcarrier for transmission.In Table1, we present the optimized θp value in each adaptive mode p=1,2,3,4,5.Table 1
Searching results ofθp.
p12345θp0.72320.7350.680.70320.6812Figure5 shows the BER performance comparison between AMSCR and AM-TRA in AWGN channel in adaptive modes p=1,2,3,4,5. We can see the overall BER performance has been significantly improved. In the two adaptive modes p=1 and p=2, the BER performance is slightly improved after constellation optimization. The SNR corresponding to AMSCR is about 0.1 dB less than that of AM-TRA for given BERth. In the adaptive mode p=3, the SNR of AMSCR is approximately 0.5 dB smaller than that of AM-TRA. In adaptive modes p=4 and p=5, the corresponding SNR gaps are larger. They are 4 dB and 7 dB, respectively. The BER performance improves more obviously in the high SNR.Figure 5
The BER performance in AWGN channel.We can see that the optimization effect is better when the adaptive mode is higherp=4,5. We suspect it is because the BER performance of AM-TRA with more users using 16QAM is poor that the optimization effect is more obvious in higher adaptive modes.As shown in Figure6, the BER performance of the AMSCR is better than AM-TRA in Rayleigh fading channel in each adaptive mode. For given BERth, the SNR of AMSCR is less than that of AM-TRA by 0.5 dB on average in adaptive modes p=1,2,3. Similarly, in adaptive modes p=4 and p=5, the SNR gaps are larger. The corresponding SNR gaps are approximately 4 dB and 6 dB.Figure 6
The BER performance in Rayleigh fading channel.Similarly, we suspect it is because the BER performance of AM-TRA with more users using 16QAM and fewer users using QPSK is poor that the optimization effect is more obvious in higher adaptive mode.Channel coding is not added to the adaptive modulation SCMA system in the simulation process. In the simulation, the SCMA modulated signal is carried on the OFDM subcarrier for transmission. There are 4096 subcarriers, and the sampling frequency is 60 kHz, the frame length is 20 (the number of OFDM symbols in each frame), andBERth=0.01. We directly drop the frames with a BER greater than BERth.Table2 shows the SNR range corresponding to different adaptive modes in AWGN channel and Rayleigh fading channel. Also, the adaptive modulation SCMA system depends on these SNR ranges.Table 2
γ ranges of 7 adaptive modes.
pγ range (AWGN)γ range (Rayleigh)0γ<23.6γ<27.7123.6≤γ<2427.7≤γ<28224≤γ<24.528≤γ<28.3324.5≤γ<2628.3≤γ<29.3426≤γ<26.729.3≤γ<30526.7≤γ<28.430≤γ<31.76γ≥28.4γ≥31.7Figures7 and 8 present the throughput performance of adaptive modulation SCMA, SCMA with QPSK, and SCMA with 16QAM in AWGN channel and Rayleigh fading channel.Figure 7
The throughput performance in AWGN channel.Figure 8
The throughput performance in Rayleigh channel.From Figure7, we can see that, in AWGN channel, the adaptive modulation SCMA system uses adaptive modes from p=1 to p=5 when the SNR is from 24 dB to 28 dB. Also, the throughput of AMSCR is significantly improved in this SNR range. As compared with SCMA with QPSK and SCMA with 16QAM, the maximum increase is about 13.6 Mbps at 26 dB.The throughput performance of AM-TRA is very poor. When SNR is 27.2 dB, its throughput is about 27 Mbps less than that of SCMA using QPSK and about 42 Mbps less than that of SCMA using 16QAM. This performance proves that the conclusion from Figures3 and 4 is reasonable.From Figure8, we can see that, in Rayleigh fading channel, from 28 dB to 31.5 dB, the adaptive modulation SCMA system uses adaptive modes from p=1 to p=5, and the throughput has been greatly improved. As compared with SCMA with QPSK and SCMA with 16QAM, the maximum increase is about 7.5 Mbps at 29.5 dB.Similarly, the throughput performance of AM-TRA is poor. When SNR is 30.5 dB, its throughput is about 25 Mbps less than that of SCMA using QPSK and about 34 Mbps less than that of SCMA using 16QAM. This performance also proves that the conclusion from Figures3 and 4 is reasonable.


## 5. Conclusions

An adaptive modulation strategy for SCMA system is proposed for both Gaussian and Rayleigh fading channels. SCMA with QPSK and SCMA with 16QAM are two special cases of all adaptive modes. Therefore, these two adaptive modes do not require codebook optimization. In the remaining adaptive modes,p=1,2,3,4,5, the codebook is redesigned by optimizing the phase rotation factor θp. Therefore, the adaptive modulation SCMA codebook design based on constellation rotation (AMSCR) is proposed.The codebook has a lower BER than the traditional SCMA codebook in adaptive modes. The simulation results have shown that the proposed adaptive modulation SCMA codebook can significantly improve the average BER of the system and the adaptive modulation SCMA system can reach the maximum throughput. In the future, we can also try to further improve the BER performance of each adaptive mode by optimizing the mother constellation or add more modulation methods such as 64QAM into the adaptive modulation SCMA system.



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*Source: 5031271-2022-05-24.xml*