Low-complexity detection for uplink massive MIMO SCMA systems

This paper presents a sparse code multiple access (SCMA) system with massive antennas at the base station. This system is referred to as M-SCMA system. A spectrally-efﬁcient and massive access next-generation wireless network is realized through massive antennas and non-orthogonal SCMA techniques. Two detection algorithms, namely, modiﬁed message passing algorithm (MMPA) and extended message passing algorithm (EMPA) are proposed to detect multiple users’ symbols in M-SCMA. A deep learning (DL)-based detection scheme is also proposed for M-SCMA so as to avoid channel estimation and to lower the detection complexity. Numerical results show that the DL-based detection has similar performance as MMPA even when the channel information is not estimated explicitly. Furthermore, authors also establish the sum rate trade-off between SCMA and orthogonal multiple access in a massive antenna system. The impact of various M-SCMA parameters such as the number of antennas and the overloading factor, on the proposed DL, MMPA, and EMPA-based detection are also investigated.


INTRODUCTION
Contributions: This paper proposes and analyses uplink massive MIMO SCMA system (referred to as M-SCMA system) Both SCMA and massive MIMO give higher throughput and improved system performance. The main contributions in the paper are summarized as follows: • We propose DL-based detection for M-SCMA system. Since DL methods do not require channel information explicitly for data symbol detection and also has lower complexity as compared to conventional MPA-based detection. • The deep neural network (DNN) model is trained offline from the simulated data, then it is used for online signal detection. The number of hidden layers and number of neurons at each layer are fixed in DNN by considering the multiple training to get a good SCMA system's performance. The min-sum approximation is used in lieu of the standard sum-product principle, and (2) the messages are damped in order to ensure correct convergence. The complexity of EMPA increases as the number of antennas increases at the BS as compared to MMPA. However, due to the minsum approximation, the complexity of the detector reduces significantly. This point is further elaborated later in this paper. • Analysis of the proposed M-SCMA system design in terms of bit error rate and the sum rate are carried out. • The impact of M-SCMA system's parameters such as number of antennas and overloading factor are also analysed. Effect of imperfect channel information (ICI) is also studied in MMPA and EMPA-based detection since accurate channel information in M-SCMA is not practically viable [24]. 1 Finally, we show numerical results to demonstrate the efficacy of MMPA, EMPA, and DL detection in the proposed M-SCMA system.
The rest of the paper is organized as follows. Section 2 introduces the proposed M-SCMA system model. Furthermore, DL, MMPA, and EMPA-based detection is also proposed in Section 2. Detection complexity is highlighted in Section 3. Numerical results and discussions are presented in Section 4. Finally, Section 5 concludes the paper.
Notations:  (0, N 0 I K ) denotes the complex Gaussian distribution of zero mean and variance N 0 and I K represents the identity matrix of size K × K . || and ‖(⋅)‖ denote the cardinality of the set  and the Euclidean norm of a signal (⋅), respectively. ℝ and ℂ denote the real and complex numbers, respectively. The diag(h) represents the diagonal matrix corresponding vector h when elements of h are arranged in diagonal of the matrix.

PROPOSED M-SCMA SYSTEM MODEL
In this section, M-SCMA-based communication system is proposed to enhance the system's capacity and detection performance. Uplink M-SCMA system has J users and K orthogonal resources with J > K and referred as J × K M-SCMA system. In M-SCMA, each user has single antenna and the BS is equipped with Z antennas, as shown in Figure 1. Let j th user information symbol and codebook are denoted as d j  and  j , j = 1, 2, … , J , respectively. Symbol d j  can take one value from the M possible values at a time, i.e. d j [4,5]. Therefore, a codebook of each user has M complex valued columns and is denoted as  j ∈ ℂ K ×M . Furthermore, each column of  j has only N (N < K ) non-zero elements out of K . Therefore, codebooks { j }s are sparse in SCMA system. The channel impulse response (CIR) between j th users and zth antenna at BS is denoted as h j,z ∈ ℂ K , z = 1, 2, … , Z. Received signal y z = (y z1 , … , y zK ) ∈ ℂ K at the zth antennas can be expressed as where P j and x j ∈  j represent the transmit power and codeword of j th user, respectively. The j th user symbol detection using maximum a posteriori (MAP) probability is written aŝ where p(x l |{y 1 , y 2 , … , y Z }) denotes the a posteriori probability of the l th user given all the antennas' received signal {y 1 , y 2 , … , y Z }. Computational complexity of the MAP-based detection in (2) is high and it increases exponentially as number of users and antennas increase in the system [4,5]. Therefore, we propose a DNN-based receiver for M-SCMA system below.

DNN-based detection
In general, MPA-based detection is used for data symbol detection in SCMA. MPA-based detection requires exact CIR of each user at the BS. However, the exact CIR of users is not available in practice or complex to estimate in NOMA uplink systems. Therefore, we proposed a DL-based symbols detection in the uplink M-SCMA system. DL-based symbol detection does not require the explicit channel information [25][26][27][28][29], and it learns the joint effect of CIR and signal detection. A neural network (NN) maps the received signal to the data symbols and expressed as [30] f ( , y t ) : where y t is the received signal at BS in t th time frame, and d j ,t is the j th user symbol in t th time frame. is a DNN parameter, which needs to learn during the offline training of DNN. Therefore, DNN function f ( , y t ) should be learned from the data set so that error probability of symbol detection is minimized.

DNN offline training
DNN consists of input, output, and multiple hidden layers, where each layer has multiple neurons [28][29][30], as shown in Figure 2. Each neuron maps weighted sum of inputs form the previous layer to the output using non-linear activation function. The mostly used non-linear activation functions can be Relu, sigmoid, hyperbolic tangent etc., and are defined as , The layers in the DNN are cascaded to generate the final output as shown in Figure 2. Hence, decoder output is expressed aŝ (5) where L is the total number of layers in the DNN. The y 1 is the input vector, and W l and b l are weight and bias vector at the l th layer, l = 1, 2, … , L. The parameter = {W L , … , W 1 , b L , … , b 1 } consist of weight and bias variance. Furthermore, DNN parameter is optimized by minimizing the  2 loss function and it is expressed as where (k) and(k) are the transmit (during training) and estimated data symbols, respectively. Therefore, opt = min { 2 }. Furthermore, a priori domain knowledge (if available) can prescribe the choice of a specific hypothesis class for use in the training process, which results in an efficient and effective DNN training in the system. Let training set  consists of  1 frames with training points (y n 1 ,  n 1 ), n 1 = 1, … ,  1 , where y n 1 and  n 1 the received signal and corresponding output data symbols (of all users) in the n 1 th frame. In the DL-based decoder, aim to derive a predictor (y) from the training data set  that generalize the input-output mapping to the inputs y that is not present in  . Therefore, DL-based decoder predict output  from the posterior distribution by minimizing the loss function  2 . In (7), P (y, ) and  (y) denote the joint and marginal distribution, respectively. The optimal predictor (y) for any received signal y is expressed as Therefore, an optimal predictor (y) is obtained by minimizing the average loss  2 for output value , where the average is evaluated with respect to the posterior distribution  (|y) [28].

MPA-based detection
In this subsection, we propose MPA (MMPA)-based detection for the M-SCMA system. The factor graph of M-SCMA system is modified to use MPA for symbol detection and is described below.

MMPA-based detection
We propose MMPA-based detection for the M-SCMA. Consider the uplink system model shown in Figure 1 where there are J users with single antenna each and one BS with Z antennas. The signal y z = [y z1 , y z2 , … y zK ] T received by the zth antenna at the BS is given by where w z is the additive white Gaussian noise (AWGN) at the zth antenna and is distributed as The message passing procedure in MMPA can be carried out either in probability domain log-likelihood ratio (LLR) domain. However, LLR-domain implementation is preferable in hardware due to numerical stability. Suppose the channel LLR computed from y zk is denoted by L zk ch . The effective channel LLR L k ch received by the kth resource node is given by The procedure of MMPA is illustrated in Figure 3. Here, the factor graph remains the same as in the single-input singleoutput (SISO) case The input channel LLR to a resource node combines the channel LLRs from all the antennas in an equal- gain-combining fashion. Then the normal steps for MPA are carried out to extract the multiple users' data.

EMPA-based detection
Simple receive beamforming based M-SCMA system does not utilize the all antennas' signal effectively to improve performance since conventional J × K SCMA detection is used at the BS. In this subsection, we propose an EMPA to get both beamforming and detection gain in M-SCMA system. The received signal by concatenating all the antennas's signal in a vector form is written as and j th user channel matrix is given as of size ZK × ZK . Therefore, the received signal is expressed as where The modified factor graph by combining all antennas' signal in EMPA is expressed in Figure 4 for Z = 2. Observe that the extended factor graph contains six user nodes as each user is having only one antenna. The number of resource nodes in the extended factor graph is equal to 2 × 4 = 8 as there are Z = 2 antennas in the BS. Observe that the degree of the resource node is the same as d f = 3. However, the degree of a user node increases from 2 to 4. This implies that a user gets higher number of opinions regarding its value. This results in the increase in the detection diversity. Due to this, the EMPA can yield impressive performance.
In EMPA, we offer two contributions which reduce the overall complexity of the detection process. First, the calculation of the messages from the factor nodes are simplified by considering the min-sum approximation. The decoding is represented in the LLR domain. A message is proportional to the negative logarithm of the likelihood. The min-sum approximation is similar to the one considered in the context of the decoding for non-binary low-density parity-check (LDPC) codes [31] Suppose {V ji } j =d f j =1 denote the messages coming to the ith factor node from the d f neighbouring user nodes. The message U i j from the ith factor node to the j th user node is updated as per the min-sum approximation: where the minimum value is obtained among all combinations of the symbols for all the neighbouring user nodes except for the j th one. In (14), the symbol for the j th user node is fixed at x jm . The second contribution in EMPA is the damped update of the messages. The message U (l ) i→ j (x jm ) at the l th iteration is updated as per the following equation: where ∈ (0, 1) is the damping factor. The damped update in (15) is carried out for every message in the EMPA. The damping ensures that the EMPA does not diverge and converges to a correct solution.

DETECTION COMPLEXITY
EMPA operates over the extended factor graph. Observe from Figure 4 that the degree d f of the resource node is the same as that for the MMPA. The complexity depends on the number of bits per symbol, i.e. M and overlapping users at each resource node, i.e. d f , and is expressed as O(M d f +  ), where  depends on the M, K , and d f . However, factor M d f dominates the complexity. Therefore, for I iterations, the complexity is expressed as I × O(M d f ). The min-sum approximation is considered in EMPA. It offers a nice advantage that the signalto-noise ratio (SNR) estimation is no longer required Moreover, observe from (14) that we need to find only the minimum value among M d f terms instead of summing all these terms. This approximation reduces the implementation overhead significantly. DNN is trained offline using a training data set, hence it does not include the training complexity while it uses for online data symbols detection. Considering a well-trained fully connected DNN model, for each data transmission/reception, the online training complexity depends on the number of vector-matrix multiplication. DNN's complexity is L 1 N + L 1 L 2 + L 2 L 3 vector-matrix multiplication, where L i denote the active parameters in all neurons in each hidden layer and N be the number of DNN inputs. Furthermore, modulation order and d f have almost zero effect on the detection time in DNN-based decoder. Therefore, DNN-based decoder has lower complexity as compared to the MPA-based decoder.

SIMULATION RESULTS AND DISCUSSION
In this section, we present the performance of the proposed M-SCMA system using the MMPA, EMPA, and DL-based detection algorithms. The uncoded transmission over Rayleigh fading channels are used. The average SER and sum rate performance metrics are considered for analysis.

System parameters
The M-SCMA system, with K = 4 orthogonal resources and J = 6, 8, 12 users, is considered. These parameters result in overloading factors of = K ∕J = 1.5, 2, 3. Number of antennas Z at the BS are considered in the range of 1-100. The codebooks of users are generated using the methods in [16,32,33]. Furthermore, we also assume each user's codebook is normalized to unity and uses two bits per symbol for transmission, i.e. || = M = 4. DNN at the BS is trained offline generating stochastic data, while online detection is used for considering the independent data set in the M-SCMA system. All the users have same transmitted power and one unit power per user is used in numerical results.

DNN parameters
This paper considered input, output, and three hidden layers in the DNN for data symbol detection. The  1 = 380,  2 = 10, and  3 = 10 neurons are considered at the first, second, and third hidden layers, respectively. The Sigmoid non-linear activation functions is used on hidden layers in DNN with the linear function on the output layer. The 1000 epochs and learning rate = 0.01 are used with the steepest descent algorithm optimization [22] method in the DNN training. The DL decoder is trained by considering randomly generated data symbols over flat Rayleigh fading wireless channel by considering as a black-box. Furthermore, the received signal corresponding the  1 = 10, 000 transmitted data symbols are used in DNN training at SNR=20 dB 2 since it leads to the optimal SER performance over the entire SNR range.

Simulations results
The average SER of 6 × 4 M-SCMA system is shown in Figure 5 for different number of antennas at the BS. As the number of antennas Z increases at BS, array gain of the proposed M-SCMA system also increases, as observed in Figure 5. Therefore, M-SCMA can achieve higher energy and spectral efficiency due to multiple antennas at BS and non-orthogonal SCMA, respectively.
Average SER performance of M-SCMA using the proposed MMPA and EMAP is shown in Figure 6. The MMPA has only SNR gain since it uses only beamforming at the BS as observed in Figure 6. The EMPA has both SNR and decoding gain as shown in Figure 6 due to the large factor graph is used in detection. Furthermore, decoding gain is more prominent for large number of receiver antennas over SNR gain in Figure 6. However, EMPA has higher complexity (marginally) FIGURE 7 Impact of imperfect CIR on MMPA and EMPA detection as compared to the MMPA. Furthermore, conventional MPA [18] has improved performance as compared the proposed MMPA and EMPA, as observed in Figure 6 with increased complexity.

Impact of imperfect CIR
The proposed MMPA and EMPA-based symbol detection require perfect channel state information of each user. The received signal is the superposition of each users' data symbol and their channel information in the M-SCMA system. Therefore, accurate CIR estimation at the BS is either complex or impossible due to non-orthogonal pilots. Thus, MMPA and EMPA detection is analysed for ICI scenario in Figure 7. The actual CIR h j,z and estimated CIRh j,z are related for zth received antenna and j th user as where e is the estimation error between actual CIR h j,z and estimated CIRh j,z . Error e can be modelled as a complex Gaussian random vector with zero mean and the covariance matrix 2 e I K , where I K is the K × K identity matrix. Furthermore, 2 e linearly decreases with increasing SNR [26]. In Figure 7, e = 0.3∕SNR is used by considering the CIR norm is unity. Impact of ICI increases as the number of antennas increases at the BS, as observed in Figure 7 for both MMPA and EMPA detection. We observe that SER performance degrades around 7dB at SER = 10 −4 for both MMPA (Z = 10) and EMPA (Z = 5) in ICI scenario (denoted as 'I') as compared to the perfect CIR (denoted as 'P'), as observed in Figure 7. Hence, a large number of antennas at BS only benefit M-SCMA system in a perfect CIR scenario for both MMPA and EMPA-based detection.

DNN detection
Furthermore, the proposed M-SCMA's SER performance using DNN detector is shown in Figure 8. DNN has close performance to the MMPA-based detection (with perfect CIR), as observed Figure 8. However, DNN outperforms than the MMPA in ICI scenarios. Robustness test against channel statistics mismatches: We consider the robustness of the DNN detection by considering the CIR variation from offline training to online testing, as illustrated in Figure 9. We consider random variation in CIR at each receiver antenna, where 2 var denotes the percentage of the change of channel statistics from offline training to online testing, and 0% means no mismatch. We observe from Figure 9 that CIR variation does not degrade BER performance significantly.
Furthermore, in Figure 10, we highlight the importance of number of antennas at the BS to achieve a higher overloading factor ( ) in M-SCMA system with MMPA detection. Array gain of M-SCMA allows to enhance the spectral efficiency of a wireless system with fixed SER, as observed in Figure 10. Therefore, massive antennas at the BS can significantly reduce the SER in an M-SCMA system.

4.3.3
When does M-SCMA have better sum rate than OMA?
The sum rate R M-SCMA of multiplexed users in the M-SCMA system is analysed and is shown in Figure 11. The R M-SCMA is expressed as where Ω is the set of overlapping users with j th user on the kth resource. For example, second and third users share the same resource node 1 with user one (in Figure 3). Therefore, Ω = {2, 3} in (17) for the first resource node. The sum rate of the proposed M-SCMA system is shown in Figure 11. At high SNR (SNR=10 dB), OMA system sum rate increases exponentially as shown in Figure 11 (left) and crosses the M-SCMA's sum rate. Since M-SCMA system is interference limited. 3 However, M-SCMA can have a higher sum rate for a lower number of antennas than the OMA as observed in Figure 11 (left). Furthermore, M-SCMA's sum rate can also be enhanced by increasing the overloading factor in the system as shown in Figure 11 (left). Since large Z allows the user separation at the BS. Therefore, one should decide the overloading factor and number of antennas in M-SCMA to get a better sum rate than the OMA. Furthermore, at low SNR (SNR = −10 dB), M-SCMA has better sum rate than the OMA for all ranges of antennas, as observed in Figure 11 (right). Hence, in M-SCMA-based system, SNR level, number of antennas (Z ), and overloading factor are essential parameters to decide the system sum rate, unlike the conventional OMA-based system.

CONCLUSIONS
This paper proposed the uplink SCMA system with massive antennas at BS, and it is referred to as M-SCMA system. Furthermore, EMPA, MMMP, and DNN-based detection is proposed for M-SCMA system. Results show that array gain due to massive antennas can be used to improve the spectral and transmit power efficiency of the M-SCMA system. It is also shown that EMPA has better performance than MMPA due to detection gain with marginal increment in complexity. Performance and complexity of MMPA (with and without perfect channel information) and DNN is compared for the proposed M-SCMA. Results show that DNN-based detection is preferable over MMPA to avoid explicit channel estimation and reduces the detection complexity in the proposed M-SCMA system. The trade-off between SCMA and OMA in massive antenna system is investigated by analysing the sum rate for different values of overloading factor.