Sparse Signature Matrix Optimization for NOMA under Imperfect CSI

Non-orthogonal multiple access (NOMA) can support the rapid development of the Internet of Things (IoT) with its potential to support high spectral eﬃciency and massive connectivity. The low-density superposition modulation (LDSM) scheme is one of the NOMA schemes and uses the sparse signature matrix to reduce multiple access interferences (MAI). In order to improve the NOMA system performance in practice, this paper focuses on designing the sparse signature matrix with a large girth for LDSM under imperfect channel state information (CSI). Based on the orthogonal pilot and linear minimum mean square error (LMMSE) estimation, the LDSM optimized by bare-bone particle swarm optimization (BBPSO) algorithm has a larger girth and can gather more accurate information in the process of iterative decoding convergence. An extrinsic information transfer (EXIT) chart analysis is designed for the LDSM-OFDM system as a theoretical analysis tool. The simulation results show that the optimized LDSM outperforms the reference LDSM system, bringing about a 0.5 dB performance gain.


Introduction
With the rapid development of the mobile Internet and mobile Internet of Things (IoT) [1][2][3], the Sixth generation (6G) mobile communication systems require a higher spectrum efficiency, connection number, and reliability [4]. The traditional Orthogonal Multiple Access (OMA) scheme can not satisfy the rapid demand of the mobile communication systems for the number of access users because of the limitation of time-frequency resources. Non-orthogonal multiple access (NOMA) is a potential technology that can improve users' access or devices in an IoT scenario. In the NOMA system, multiple users' signals are assigned to the same time-frequency resource for superimposed transmission. It can increase the number of access users or IoT devices through overload transmission. Interference is introduced in multiuser superimposed transmission, which requires an advanced detection algorithm to separate multi-user information.
To combat the multi-user superimposed interference, many NOMA schemes based sparse signature matrix have been proposed, such as low-density spreading (LDS) [5,6], pattern division multiple access (PDMA) [7], sparse code multiple access (SCMA) [8], low density superposition modulation (LDSM) [9] and so on. The signals of users are mapped to the resource element (RE) by the sparse signature matrix. Sparse signature matrix reflects the collision of multiple users on the same resources. The signals superimposed on different resources are not uniform, relying on non-zero entries in the sparse signature matrix. Due to the sparsity of the signature matrix, the message passing algorithm (MPA) is used to separate multi-user superimposed information. The MPA detection algorithm is an iterative decoding algorithm based on the factor graph between variable node (VND) and check node (CND) of the signature matrix. It is similar to the belief propagation (BP) decoding algorithm of low-density parity-check (LDPC) code. The sparse structure of the signature matrix affects the receiver's complexity and affects the performance of the NOMA system.
Various research works have been done to improve the performance of the NOMA system or reduce the computational complexity of the receiver for designed the sparse signature matrix. Most of the NOMA schemes are based on designing a low dimension signature matrix, such as LDS, SCMA, PDMA [10][11][12]. The NOMA sparse signature matrix dimension is low. The user's signals are mapped by the signature matrix and allocated to the closer resources, improving the transmission error rate. Moreover, there may be a short cycle in the low dimension NOMA sparse signature matrix, which affects the accuracy of iterative detection. [9] proposes a new LDSM scheme with a large dimension sparse signature matrix. The large dimension sparse signature matrix generally has a larger girth, so its iterative detection accuracy is higher than the low dimension matrix. The large dimension LDSM's user data symbols are more dispersed in time-frequency resources, so it has higher diversity gain. In [13], a method to optimize the degree distribution of the LDSM sparse signature matrix is proposed, which structures a larger girth for LDSM. Most of the research on NOMA technology is carried out in the perfect channel state information (CSI) scenario. To improve the practicability of NOMA technology, this paper studies the design of the NOMA sparse signature matrix under an imperfect CSI scenario.
Recently, some scholars have studied the NOMA system under realistic channel estimation. The least mean squares (LMS) algorithm is proposed to estimate the channel information of the power domain NOMA system [14]. In [15], the channel estimation error of linear minimum mean square error (LMMSE) algorithm is smaller than that of weighted least square (WLS) algorithm for uplink power domain NOMA systems. The error caused by imperfect CSI decreases the performance of the NOMA system. The sparse structure of the NOMA signature matrix also influences the performance of the NOMA system due to its dramatic impact on the multiple user interference. This paper focuses on the optimization sparse signature matrix of NOMA under imperfect CSI conditions with bare-bone particle swarm optimization (BBPSO) algorithm [16]. The main contributions of this paper can be summarized as follows: • We propose an uplink LDSM-OFDM system under imperfect CSI conditions, and the orthogonal pilot design of the NOMA system is given.
• We formulate the optimization problem for the sparse signature matrix of LDSM. Based on the EXIT chart, we utilize the BBPSO algorithm to optimize the degree distribution of the LDSM signature matrix. EXIT chart analysis is provided to illustrate the optimum solutions.
• Compare the performance of the optimized LDSM and reference LDSM [9]. The bit block error rate (BLER) simulations show that optimized LDSM outperforms the corresponding LDSM scheme. The reason is that the optimized matrix has a larger girth. The rest of the paper is organized as follows. We present the proposed LDSM-OFDM system model and related optimization problems in Section 2. Simulation results and discussion are provided in Section 3. Finally, Section 4 includes the main conclusions of the paper.

Method
In this Section, the LDSM-OFDM system model under imperfect CSI conditions is firstly proposed. Then, the sparse signature matrix optimization method for LDSM is presented. Fig.1 describes an uplink LDSM-OFDM communication system with J users under imperfect CSI conditions. At the transmitter side, user j encodes the information sequence b j . The coded binary information data sequence for user j is modulated. The modulated symbols c j are encoded by a S × l sparse signature matrix F j which is part of LDSM sparse signature matrix F = [F 1 , F 2 , · · · F J ] S×Jl . The encoding symbols vector for user j is z j , and the OFDM time-frequency resources are allocated to encoding data symbols. The transmitter generates a pilot signal, inserts a pilot sequence into the OFDM symbol. The symbols of each user are sent after OFDM modulation. In wireless communication, the signals sent by users arrive at the base station through different paths, including data symbols and pilot signals.

System model
At the receiving, the received multi-user superimposed information is demodulated by OFDM first. The channel information of each user is estimated by the received signal at the pilot position. Then the data of each user is separated by the multi-user detection algorithm. The received symbol can be expressed as: where diag(ĥ j ) is a diagonal matrix,ĥ j represents the j-th user's channel coefficient, and n is the additive white Gaussian noise (AWGN) vector with zero mean and variance σ 2 .

LDSM coding
A LDSM coding process for user j is showed in Fig.2. F j is the sparse signature matrix for user j. The column of the F j is a sparse vector consisting of S REs which has many zero elements. Each modulated symbol is spread to S resources through sparse vector. The parallel data streams c j = c j 1 , c j 2 , · · · c j l T are coding by F j . The superposition coding vector z j can be expressed as: The LDSM system's signature matrix F S×Jl has S rows and Jl columns. LDSM sparse signature matrix is constructed by progressive edge-growth (PEG) algorithm [17]. LDSM scheme can obtain the diversity gain in large dimensions. To improve the accuracy of the MPA decoding algorithm, we need to construct a larger girth for LDSM .   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64

Pilot design
The pilot sequence position of the NOMA scheme is designed as shown in Fig. 3. It offers the pilot sequence design for six users. As shown in the figure, the pilot symbols for user j are separated by 6 REs on OFDM symbols, and the pilots of each user are distributed on the first and the seventh OFDM symbols. The remaining OFDM symbols are taken to send the signals to all users. The reference signal for user j denotes R j . The channel information of the data symbols can be obtained using the channel information estimated at the pilots. Suppose that the received signal at the pilot position of the user j is expressed as: Where N j is an AWGN with variance σ 2 and mean 0. H j is the channel frequency response at pilot position for user j. LMMSE algorithm is used to estimate CSI. LMMSE algorithm is a modification of the least square (LS) channel estimation algorithm. The LS channel estimation algorithm calculates the channel response of the pilot position without considering the influence of Gaussian white noise. It then obtains the channel information of all users by interpolation. The H LS channel estimation of each pilot symbol position obtained by the LS algorithm is expressed as:H TheH LM M SE channel estimation by LMMSE algorithm is obtained by: where R HHj = E{HH j H }, R HHj represents the covariance matrices. β = E{|X k | 2 }E{|X k | −2 } is a constellation factor depending on the signal constellation.

The sparse signature matrix optimization method
In this section, the sparse signature matrix optimization method is accomplished in two parts. First, the sparse signature matrix is represented as an optimization problem according to the extrinsic information transfer (EXIT) chart analysis. Second, the BBPSO search algorithm is used to solve the optimization problem.

Optimization problem based on EXIT Chart
The tool of EXIT chart is used for analyzing the iterative decoder process [18].
To get the EXIT charts for the LDSM system, the receiver of the LDSM system contains two parts. As shown in Fig. 4, EXIT function of decoder A and decoder B is T A (·) and T B (·). Decoder A includes the NOMA multi-user detector (MUD) demodulation, and Decoder B represents the FEC decoder. EXIT chart is obtained by EXIT curves of decoder A and decoder B. The information exchange between decoder A and decoder B with iterations is analyzed on multi-path fading channels. We divide the decoder's input information into two parts, which are defined as priority log-likelihood ratio (LLR) and extrinsic LLR. L a denotes priori LLR and 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 L e denotes its output extrinsic LLR. A priori LLR L a is modeled as an independent Gaussian random variable. We can write a priori LLR L a as follows: where u A = σ 2 A 2 and x ∈ ±1. n A is a Gaussian random variable with variance σ 2 A and mean 0. The probability density function is written as: The mutual information I A = I(X; L a ) is calculated by [19]: We define with lim σ→0 J(σ) = 0, lim σ→∞ J(σ) = 1, σ ≥ 0.
The input information of MPA MUD include the information y from channel and a priori information I A A from the FEC decoder. The MPA MUD outputs mutual information I A E and I A E as a priori information is sent to FEC decoder. The extrinsic information I B E of the FEC decoder is passed to the MPA MUD as a priori information I A A . The EXIT curves for MPA MUD and FEC decoders reflect the performance of the iterative system. Therefore, it is useful to design an efficient algorithm to match decoder A chart and decoder B chart. The design of LDSM signature matrix has great influence on system performance. Therefore, it is very important to find an optimization method to design the signature matrix of LDSM. The degree distribution of LDSM VND detector denotes m = [m 1 , m 2 , · · · , m Dv ]. The degree distribution of LDSM CND detector denotes n = [n 1 , n 2 , · · · , n Dc ]. Therefore, the objective function problem is represented as: Where m i is the VND degree distribution of degree i. n j is the CND degree distribution of degree j. The minimum threshold channel is SN R(dB).

BBPSO optimization method
The BBPSO algorithm is to search for the personal optimization solution and the global optimization solution through iterative updating. A feasible solution is regarded as a particle in a swarm. In BBPSO algorithm, the set of particles is represented by X = {x 1 , x 2 , · · · , x W }, which each particle x w = [x w,1 , x w,2 , · · · , x w,D ] T denotes the potential solution, the total number of particles is W and the dimensions of problem is D. The fitness to evaluate a particle is equation (10). The position of particle i at iteration (t + 1) is updated as follows: where u i,j (t) = 0.5(P i,j (t) + G i,j (t)) (14) Where x ij denotes the j-th dimension for the particle i. u i,j is a mean of Gaussian distribution. N (0, 1) represents a standard Gaussian random variable with variance of 1 and mean of 0. P i is the personal best position for the particle i. G g denotes the global optimal position of particle in a particle swarm. The proposed BBPSO-EXIT algorithm can be summarized as follows: According to Algorithm 1, The optimal solution is searched through iteration. The uplink LDSM-OFDM system is considered. LTE turbo code as channel coding and the degree distribution of LDSM signature matrix is optimized based on the BBPSO algorithm. The overload is 150%, QPSK modulation, the bit rate is 1/2, the number of users is 6, and the number of bits sent by each user is 144 bits. Therefore, the corresponding dimension of the LDSM matrix is 576 * 864. The detailed simulation parameters are shown in Table 1. The optimal m 2 is 0.925 and the optimal m 3 is 0.075, respectively. The optimal n 3 is 0.887 and the optimal n 4 is 0.113, respectively. The girth of the large dimensional optimized matrix is 18. The girth of the reference LDSM in [9] is 10. The degree distribution of reference LDSM variable nodes is m 2 =m 4 =0.5. The degree distribution of reference LDSM check nodes is n 4 =n 5 =0.5. The corresponding EXITs analysis is shown in Fig.5. Observe from Fig.5a that the MPA MUD's mutual information I A E for optimized LDSM intersects with the decoder curve later than that for reference LDSM. The simulated SNR threshold for optimized LDSM is 5.0 dB. As shown in Fig.5b, The reference   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 Algorithm 1 :BBPSO-EXIT Algorithm.
1: Initialize: the dimensions S × N L of LDSM signature matrix F , overloading factor L = N L /S, pilot design w, channel estimation algorithm ψ, the number of particle W , the number of iterations T , the dimensions D of particle (m, n) . Randomly initialize each particle (m i , n i ). 2: for t = 1 : Tmax do 3: for j = 1 : W do 4: for k = 1 : D do 5: update the position of the particle by the formula (13), evaluate the objective function value SN R by the formula (10). 6: end for 7: end for 8: 9: for t = 1 : W do 10: (m, n) = arg min [f (Gg(t + 1))] 13: end for 14: end for 15: return the best degree distribution (m, n) of LDSM LDSM can achieve the same I A E at 5.5 dB. According to the predictions of the EXIT, the optimized LDSM scheme can obtain 0.5dB gains than the reference LDSM. The advantage of the optimized LDSM is that it has a larger girth. Consequently, it is possible to gather more accurate information in the process of iterative decoding convergence.

Results and discussion
In this section, uplink level simulation results are provided to evaluate the performance of the optimized LDSM scheme and the reference LDSM scheme. The average total received power and the spectrum efficiency (SE) per UE are kept the same. The signal to noise ratio (SNR) at the receiver is defined as the average total received power of the RE over noise power in the given bandwidth. Assuming 6 UEs with overloading factor of 150% on tapped delay line (TDL)-A-30 channel [20]. The total SE of LDSM system is 1.5 bps/Hz. BLER results for LDSM systems under imperfect CSI is shown in Fig. 6. The optimized LDSM achieves 0.5 dB SNR gain as compared with the reference LDSM [9] at target BLER 10 −2 . Therefore, our EXIT chart predictions are consistent with the simulation results. The optimized LDSM has a smaller average row to reduce MUI and a larger girth to improve decoding accuracy.

Figure 4
Block diagram of the MPA MUD and FEC detector iteratively decoded system. EXIT chart for turbo MUD with 150% at same SNR. 6-user BLER performance comparison of LDSM schemes with load 150% under TDL-A-30 channel.