A Novel Scheme for the Construction of the SCMA Codebook

As a code-domain non-orthogonal multiple access technique, sparse code multiple access (SCMA) is considered as a promising technique for future wireless Internet of Things (IoT) networks. The minimum Euclidean distance (MED) and minimum product distance (MPD) have been highlighted as the key performance indicators of the codebooks in additive gaussian white noise (AWGN) and downlink Rayleigh channels respectively. In this paper, based on the mother codebook, a novel codebook design scheme is proposed to achieve better error performance in both AWGN and downlink Rayleigh channel. The problem of constructing the mother codebook is considered as the quadratic assignment problem (QAP), where the Tabu searching algorithm is employed to reduce the complexity of searching for the best permutation result. Then, the rotation matrix is adopted to find the best degrees for the generation of the constellation group, and two algorithms are proposed to assign the obtained constellation in factor graph matrix. Taking the degree optimization and the constellation assignment into joint consideration, an improved unified optimization is further explored to maximize the MED of each user. Besides, a novel polarized modulation scheme is proposed, which places the symbols in the three dimensional (3D) stokes parameters to improve the performance of the system. Finally, simulation results are provided to show the performance of the proposed codebooks, and the comparisons of symbol error performance (SER) in different codebooks are also discussed in detail.

the rapidly growing number of intelligent devices and ser-23 vices, massive devices inevitably cause profound inference, 24 which leads to sharp degradation of system performance [2], 25 [3]. Therefore, traditional orthogonal multiple access (OMA) 26 schemes are not applicable for future communication [3]. 27 In order to meet the demand of IoT communication with a 28 higher spectrum efficiency, non-orthogonal multiple access 29 (NOMA) scheme is viewed as one of the most promising 30 strategies in embracing the future multiple access technology. 31 Existing NOMA schemes can be divided into power-domain 32 The associate editor coordinating the review of this manuscript and approving it for publication was Zilong Liu. that in downlink Rayleigh channel, the codebook with better  For uplink Rayleigh fading channel, the construction of the 82 mother codebook is the key procedure [17], [34], and a good 83 mother codebook can obtain much diversity gains [35]. So far, 84 a lot researches have been conducted to design mother code-85 book with higher symbol mapping diversity. For example, 86 the interleaving method reported in [10] and [25]. Though 87 the complexity of this approach is low, it cannot guarantee the 88 best diversity gains. Besides, the binary switching algorithm 89 (BSA) [36], symbol exchange algorithm (SEA) [17] and 90 dimensional permutation switching algorithm (DPSA) [29] 91 are also proposed to improve the mapping diversity of the 92 mother codebook. While, the symbol mapping problem in 93 the mother codebook can also be viewed as the quadratic 94 assignment problem (QAP) to maximize the diversity gain 95 [35], which provides another aspect to solve the problem, 96 such as Lagrangian relaxation algorithm [37], genetic algo- plexity than metaheuristics algorithms [39]. Besides, similar 101 to SEA algorithm, Tabu searching algorithm further adopted 102 Tabu table to avoid falling into local optimal solutions. 103 Therefore, Tabu algorithm will be adopted to construct the 104 mother codebook. 105 Different from the uplink Rayleigh channel, the MED of 106 the superimposed codewords (MED-SC) should be maxi-107 mized to reduce interference between users. Since the rotation 108 procedure does not change the Euclidean distance between 109 the codewords, which has been widely adopted to generate the 110 multiuser codebooks [10], [27], [32], [40]. However, search-111 ing the optimal rotation angles is time-consuming. Although 112 some researchers directly obtained the rotation angle based 113 on the colliding users over each frequency [10], [40], the 114 results show that the optimized rotation angles can achieve 115 better error rate performance. Thus, the rotation matrix is 116 adopted in this paper to reduce the interference between users 117 and increase the MED-SC. 118 Besides, the limited freedom degree in 2D space blocks 119 the Euclidean distance improvements. Thus, constructing 120 SCMA codebooks with high dimensional freedom can further 121 improve the system's performance. Therefore, based on the 122 proposed scheme, constructing the 3D SCMA codebook is 123 also a promising topic.

125
As mentioned in above part, SCMA codebook design has 126 been investigated for a long time, and the mother codebook 127 based approaches have been widely deployed in the construc-128 tion of SCMA codebooks. In this paper, a modified mother 129 codebook based approach is proposed to construct SCMA 130 codebooks. And the proposed design scheme can also be 131 applied in the construction of the 3D SCMA codebook to 132 further improve the error performance without requiring addi-133 tional energy. However, building 3D constellation requires 134 additional orthogonal parameters, such as frequency [41], 135 polarization [42] et al. In this paper, the polarization dimen-136 sion is firstly adopted to increase the available dimensional 137 information. The main contribution of this work can be sum-138 marized as follows.

139
• Based on the mother codebook, a novel codebook design 140 approach is introduced in this paper. Rather than the 141 permutation method, the symbol mapping diversity in 142 the mother codebook is considered as QAP, where the 143 Tabu searching algorithm is employed. Similar to the 144 SEA algorithm, but with a list of forbidden moves, 145 Tabu searching algorithm can prevent being trapped into 146 local optimal. Then, the rotation matrix is introduced to 147 reduce interference among users. And two assignment 148 algorithms are proposed to assign the optimized con-149 stellation sets. Furthermore, a modified optimization is 150 proposed to improve the performance of the codebook 151 in both AWGN and downlink Rayleigh channels. And 152 the detailed reasons are also discussed.

153
• In order to further improve the system performance of 154 SCMA, the 3D SCMA modulation scheme is firstly pro-155 posed where the 3D symbols are mapped into the stokes 156 parameters to increase the MED of each user. And the 157 In this paper, we consider downlink SCMA system in (1)

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In factor graph matrix F, the row denotes the frequency   where P ( y| x,H) denotes the conditional probability density 210 function (PDF) of the received signal. p (x l ) is the probability 211 of the symbol distribution for the l-th user, Nl l and Nk k 212 represents the neighboring node of user l and frequency k 213 respectively. In this subsection, the pairwise error probability (PEP) of 216 each user is analyzed to introduce the codebook design 217 criteria.

218
Using the Chernoff boundary, the conditional PEP of the 219 received signal is closer to e l than x l under the channel 220 coefficient h l is given by where h l = h 1,l , · · · , h k,l T is the channel vector for the 224 l-th user, E s is the power of the transmitted power, and the 225 Euclidean distance of each user can be written as where x k,l = x k,l −x k,l denotes the Euclidean distance of 228 the wrongly decoded symbols. Since the channel coefficient 229 h k,l is a circular complex gaussian random variable (RV), 230 the probability density function (PDF) of the magnitude is 231 Rayleigh distribution [44], which is given by where σ 2 is the variance of the Rayleigh fading channel. 234 Thus, we can obtain the unconditional PEP of (6) through a 235 simple mathematical integral It can be observed that maximizing the product distance of 239 the superimposed codewords dominates the PEP under high 240 signal to noise ratio (SNR). While in low SNR, maximizing 241 the MED in each frequency resource is an important factor for 242 VOLUME 10, 2022 improving the error performance. Furthermore, in high SNR, 243 according to the average inequality [25] It can be obtained that the MED-SC is the upper bound   Tabu table  6: if symbol switching is allowed (m, j) = 0 7: exchange the m-th symbol with j-th symbol and calculate the cost function (11). 8: else 9: skip the symbol switching. 10: end if 11: if the cost function (11)  As the labelling result is obtained, the mother codebook 288 can be written as When discarding the null dimensional sparse codebooks in 293 each d v frequency resources, the remained codebook formed 294 into a constellation group with d f sub-constellation sets in 295 each frequency resource. And the constellation group can also 296 be used to calculate the MED of users (MED-U). For exam-297 ple, if the first two frequency resources in (1) are adopted, 298 we can obtain the MED of user 1.

299
Therefore, in this subsection, we aim to generate a constel-300 lation group with maximized minimum Euclidean distance. 301 Then, the constellations will be assigned in the factor graph 302 matrix to generate the SCMA codebook. Here, the rotation 303 matrices are introduced to increase the minimum Euclidean 304 distance of the constellation group.
where R (θ i ) is the rotation matrix, and the corresponding 309 And the best rotation angle can be found by solving graph matrix should be avoided [47]. Referring to the meth-331 ods in LDPC codes, we adopted the progressive edge growth 332 algorithm (PEG) to design the factor graph matrix [48]. imum when assigned in factor graph matrix (16). Therefore, 340 two algorithms are proposed to assign the constellation group 341 in the factor graph matrix in order to maximize the MED-U 342 and diversity gains.

343
In Algorithm 2, the frequency diversity of each user is 344 maximized at the cost of the maximum MED-U. While in 345 Algorithm 3, we intend to maximize the MED-U of the 346 most users while the frequency diversity gains are not the 347 maximum. The generated codebooks are showed in (17) 348 and (18), as shown at the bottom of the next page, respec-349 tively. And we refer the codebook generated by Algorithm 2 350 and Algorithm 3 as ''codebook 2'' and ''codebook 3'' 351 respectively.
It can be observed that from (17), the SCMA codebook 353 generated by Algorithm 2 can guarantee the diversity gains 354 of each user, and for the SCMA codebook generated by 355 Algorithm 3, the diversity gains and the MED-U of the most 356 users are maximized at the expense of the error performance 357 of user 5 and 6 in codebook (18).  problem can be written as

383
where l is the adopted frequency resources for the l-th user.

384
For the first user in (17) are not maximum. Therefore, the improved optimization 396 with Algorithm 2 has better error performance and will be 397 employed to construct the 3D SCMA codebook.

399
Although the improved method can achieve better error 400 performance, the traditional In-phase/quadrature (I/Q) mod-401 ulation scheme only provide limited Euclidean distance 402 under the same modulation power. Therefore, exploring high 403 dimensional modulation is important to improve the perfor-404 mance of SCMA system.

405
The modulation of 3D constellation requires additional 406 orthogonal parameters, such as frequency [41], polarization 407 [42] et al. In this paper, the polarization dimension is firstly 408 adopted to increase the available dimensional information. 409 And the 3D symbols are placed on the stokes parameters to 410 increase the MED-U.

412
The electromagnetic wave can be decomposed into two 413 orthogonal components where E 0x , E 0y represent the amplitude of each component, 416 and ϕ x , ϕ y the corresponding phases. Besides, the electromag-417 netic wave can also be described by stokes parameters where δ = ϕ x − ϕ y , and the degree of the polarization can be 420 calculated (23) 422 The stokes parameters S 1 , S 2 and S 3 constitute a three-423 dimensional polarization space, where the transmitted infor-424 mation will be mapped. However, stokes parameters only 425 measure the intensities of the polarized wave, while the Jones 426 vector involves the magnitude and the phase of the electro-427 magnetic, which can better describe the propagation of the 428 electromagnetic wave. Therefore, we will express the Jones 429 vector from the stokes parameters [42] where E = S 2 1 + S 2 2 + S 2 3 is the transmitted energy of the wave can be expressed as codewords, which can be calculated as 447 FIGURE 2. The adopted 3D constellation in stokes parameters.
The transmitted codewords d k can be expressed in Jones 448 vector, which is written as is more complicated than traditional 2D codebooks since 3D 487 rotation matrix introduces more degrees.
(32) 500 Different from 2D rotation procedures, the best angles in 501 3D SCMA codebook are searched in [0, 2π ). And the rotation 502 procedures consist of three dimensions. Thus, searching the 503 3D best angles are more time-consuming than that in 2D 504 codebooks. Here, we only generate the 3D SCMA codebooks 505 with respect to Algorithm 2.

507
In this section, the simulation results are provided to show the 508 performance of the proposed codebooks in AWGN and down-509 link Rayliegh fading channel respectively, and the iteration 510 number of MPA is set as 10 times. Besides, Table 1 Table 1, it can be observed that the joint opti-  of each node since the MPA also affect the performance of 535 the system through the messages circles in the factor graph 536 matrix. 537 Fig. 4 shows the SER comparison of the proposed code-538 books in downlink Rayleigh fading channel. We can observe 539 that the improved codebook generated by Algorithm 2 has 540 nearly the same error performance with codebook [17]. And 541 compared with previous codebook 2, a gain of 2dB can be 542 observed at a SER of 10 −4 in the improved codebook 2. 543 As for codebook 3, the constellation sets distribution algo-544 rithm ensure the MED in one-dimensional resources, and 545 codebook 3 has a larger MPD than codebook 2 (0.28 com-546 pared to 0). Thus, codebook 3 outperforms codebook 2 in 547 downlink Rayleigh fading channels. Besides, although the 548 average MED-U of all users in improved codebook 3 is 549 decreased by the modified optimization, it further increases 550 the MED in one-dimensional resources (from 0.28 to 0.40). 551 Therefore, a slight error improvement can be noticed in down-552 link Rayleigh channel.

553
Besides, observing from Fig. 3 and Fig. 4, we can find 554 that the improved codebook 2 has better performance in both 555 AWGN and Rayleigh fading channels. Thus, this codebook 556 VOLUME 10, 2022 design scheme and resources assignments method will be 557 further employed in high codebook size. And the improved 558 codebook 2 is provided in Appendix I. 559 In Fig. 5, we provided the performance of each user in the 560 AWGN channel. In Table 1, the MED-U is not maximized 561 for all users in codebook 2, thus, we can find that the fairness 562 of the error performance among users cannot be guaranteed, 563 and the same condition also occurred in Deka's codebook. 564 Thus, improving the MED-U is important for improving the 565 performance of each user. As for Huang's codebook, it can 566 be noticed that the MED-SC is the largest in the provided 567 codebook, while the MED-U varies among different users. 568 And the users with small MED-U influence the nodes' beliefs 569 and error performance of each user. Thus, we can observe that 570 Hunag's codebook does not achieve the best error rate when 571 compared with improved codebook 2. 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0.0000i + 0.0000j 0.0000 + 0. the existing uplink Rayleigh codebooks. In Fig. 6, the SER 579 performance is provided. We can notice that the modified 580 codebooks have better performance than Chen's codebook 581 [17], especially in lower SNR.

582
In addition to applying the proposed scheme in high code-583 book sizes, it can also be introduced in constructing the 584 3D SCMA codebook with the aid of 3D rotation matrix. 585 We firstly provided the performance of the 3D SCMA code-