Large Scale MIMO Analysis Using Enhanced LAMA

In this research work, we propose a enhanced large scale multi-input multi-output (MIMO) approximate message passing (LAMA) based optimal data detector for large scale MIMO systems. Existing LAMA and sub-optimal detection techniques suffer from iteration complexity and performance loss in finite dimensional systems due to large scale user fading. To overcome these, Gram matrix and message damping techniques are incorporated in the traditional LAMA. The effectiveness of the proposed enhanced LAMA and existing techniques are analyzed with 64, 32 and 16 user antennas, 256, 128, 64 and 16 base station elements with 64QAM, 16QAM, QPSK and BPSK. The simulation results show that the proposed enhanced LAMA gives better performance when compared to existing matrix inversion methods such as Gauss-Seidel and Neumann, box techniques such as optimal co-ordinate descent and alternating direction method of multipliers based on the infinity norm, minimum mean square error and LAMA.


Introduction
Large scale multi-input multi-output (MIMO) (also known as massive multi user MIMO) is an extension of small scale MIMO systems plays a vital role in the fifth generation and beyond of communication systems [1][2][3]. In comparison to current small scale MIMO systems, large scale MIMO technology promises substantial improvements in spectral efficiency [4,5]. The basic concept behind the large scale MIMO system is the base station (BS) equipped with hundreds of antenna elements and the user terminals consists of tens of antenna elements and the communication between user terminals and BS take place within the same radio frequency element [6]. On the other hand, finite dimensional (practical) antenna setups necessitate more complex data detection algorithms with high computational complexity, as shown in [7].

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The benefit of large scale MIMO comes with the disadvantage of computational complexity at the BS [8]. Maximum likelihood (ML) algorithm [9] is the safest bet for optimum detection criteria, but computational complexity increases significantly with the number of antennas. In order to achieve the low complexity, minimum mean square error and zero forcing linear detectors are used in [10]. A set of approximate inverse detectors such as Neumann Series Approximation [11], Gauss-Seidel Method [12] and Conjugate Gradient Method [13] are significantly used to reduce complexity. These detectors [14,15] provide good performance only when the system ratio (i.e. the ratio between the number of BS antennas to user terminals) is greater than 2 otherwise it provides unsatisfactory response. Later, the authors of [16] have proposed approximate semidefinite relaxation based data detectors for massive MU-MIMO systems. It gives good performance to the square size matrix only. However, this is limited to BPSK and QPSK. In [17], coordinate descent algorithm is implemented for the large scale MIMO-OFDM for large system ratio. The alternating direction method of multipliers based on the infinity norm (ADMIN) equalization is described in [18]. ADMIN outperforms linear detectors, even though the system ratio is small.
Initially, approximate message passing (AMP) was proposed for compressive sensing and sparse signal recovery [19]. Later, it is extended to different applications, including phase retrieval [20], signal detection [21], channel estimation [22] and denoising [23]. Further AMP related algorithms are being explored for data detection in various communication systems [24,25]. Large scale multi-input multi-output (MIMO) approximate message passing (LAMA) was developed on the basis of the AMP [26] algorithm. According to [27], LAMA attains better error rate efficiency for AWGN channels with a large antenna limit (where the BS to User antenna ratio is greater than two). Unlike minimum mean square error (MMSE), matrix inversion based detectors, and box based detectors, LAMA improves performance in similar user and BS configurations by using knowledge about the constellation set. However, LAMA [27] is unable to provide the best performance to practical channels. Furthermore, as the number of receive antennas increases, so does the complexity. In order to support the practical (finite dimensional) channels and to reduce the complexity at the receiver, few modifications are done to the traditional LAMA at the algorithmic level. This is referred to as Enhanced LAMA.
Notation: Bold lower case alphabet denotes the vector, bold upper case alphabet denotes the matrix, (⋅) H denotes the Hermition transpose (complex conjugate transpose), || ⋅ || denotes the frobenius norm, (⋅) −1 denotes the inverse operation, I N denotes the identity matrix of size N, Λ denotes the LLR, and (⋅) indicates the Dirac delta function.
Outline: The remainder of the paper is as follows: Sect. 2 describes the large scale MIMO system model. Section 3 includes mathematical derivation of the proposed Enhanced LAMA method along with its algorithm. Section 4 highlights the simulation results of proposed work and compares with the existing techniques. It also includes simulations of the Enhanced LAMA in different scenarios w.r.t to varying number of BS antennas and modulation techniques. Finally, the work is concluded in Sect. 5.

System Model
Consider an uplink large scale MIMO system consisting of N t transmit and N r receive antennas. Let be the transmit data vector having the entries l , l = 1, … , N t are first encoded by the channel encoder and then mapped to constellation points a j , j = 1, … , | | . These points are chosen from the QAM or PSK are taken from the energy normalised constellation set at , where = 2 Q . Assume p(s) = ∏ N t l=1 p(s l ) be prior distribution for each transmit symbol s l is where p a is the prior probability of each constellation point a ∈ . Let ∈ ℂ N t ×1 be the transmit vector and ∈ ℂ N r ×1 be the receive vector and they are related by [27] where ∈ ℂ N r ×N t is the known system matrix to the receiver and the entries follow i.i.d. circularly symmetric complex Gaussian with zero mean and variance of N r is noise vector which is additive in nature with variance of N 0 . The signal to noise ratio (SNR) is defined on an average basis is where = N r ∕N t denotes the system ratio. The individually optimal (IO) data detection problem in [28] by keeping the above assumptions are given by where ̃ l indicates the exclusion of lth entry from N t dimensional vector.
To solve the problem in (4), it requires more computations for large values of N t . However, computationally efficient algorithms based on sphere decoding methods [16] are available for small scale MIMO systems. Since these approaches are ineffective for large MIMO systems, the literature [29,30] contains a number of sub-optimal algorithms.
The problem in (4) can be solved using complex Bayesian approximate message passing approach. In this process, first we need to compute the marginalized distribution and then, this is converted to element wise data detection problem. In order to improve the performance in the proposed enhanced LAMA algorithm, a few modifications are incorporated in the traditional LAMA. The modifications are firstly, to reduce the complexity per iteration, the N r × N t channel matrix is transformed to N t × N t dimensional matrix using Gram matrix = H . Secondly, LAMA performance is reduced in the finite dimensional systems due to large scale user fading. To overcome this, message damping techniques are introduced in this research work and finally, the detection and decoding are performed at each iteration.

Enhanced LAMA
The sum-product message passing algorithm is used in [31] in order to arrive at an effective inference process. This algorithm is simplified by assuming a Gaussian distribution for the marginal densities of the messages p(ŝ l |s l , ) ∼ CN(s l , ) so that with the prior distribution in (1), the posterior distribution for the transmit symbol s l can be expressed as is the normalization constant and it is chosen Let S be the random variable, Ξ(ŝ l , ) be the conditional mean and the message mean can be defined from (5) as follows and Υ(ŝ l , ) be the conditional variance and it can be defined according to (5)

as message variance is
In the massive MU-MIMO systems, enhanced LAMA is one of the most effective data detection algorithms. Especially, for practical channels the proposed method provides better results due to addition of Gram matrix and the message damping techniques to the traditional LAMA. Based on the damped second moment estimate and damped signal variance estimate the signal to be updated at each iteration. The summary of algorithm is presented as follows LLRs are often used to convey reliability information about coded bits. To boost the error rate efficiency, it is passed back and forth iteratively between data detector and channel decoder. The intrinsic LLRs for soft-input soft-output detector according to [32].
for all bits l = 1, 2, … , N t , q = 1, 2, … , Q in the label and the prior LLRs for channel decoder is From (13) and (14), the Extrinsic LLRs can be computed as The extrinsic LLRs denote the reliability information for each x l,q encoded bit and then deliver it to the channel decoder that computes the new prior LLRs of the detector for the next iteration. The channel decoder provides final decisions for the information bits based on the LLRs at the channel decoder output after a certain number of iterations.

Complexity Analysis
The complexity of the proposed method depends on the preprocessing and the loop. In preprocessing, the complexity mainly depends on gram matrix O(N 2 t N r ) and matched filter O (N t N r ) . The total number of multiplications needed for preprocessing is number. So, to implement the proposed enhanced LAMA method, total number of multiplications required as O((N 2 t N r + N 2 t + N t N r + 2N t ) + (N 2 t + 7N t )i). In Gauss-Seidel method, the complexity depends on the preprocessing (equivalent Gram matrix and matched filter) and LLR computations. The computational complexity of preprocessing requires O(N 3 t + N 2 t N r + N t N r ) , and LLR computation requires O((i + 1)N 2 t + 4N t ) . Total number of multiplications required to implement the Gauss-Seidel method is

Simulation Results
The symbol error rate (SER) performance for various system ratios are simulated against signal-to-noise ratio (SNR) are provided to compare the proposed Enhanced LAMA with traditional LAMA, Matrix Inversion based methods such as Neumann [11], Gauss-Seidel (GS) [12] and Box version methods such as alternating direction method of multipliers based on the infinity norm (ADMIN) [18], optimal co-ordinate descent (OCD) [17]. The classical MMSE algorithm is also included as a benchmark for comparison. The performance of different techniques are evaluated for various number of users and BS using different modulation techniques are shown in Figs. 1, 2, 3 and 4. Throughout the paper, 10 iterations are considered. The simulation results of the proposed Enhanced LAMA with other existing methods for 16 users and 64 BS antennas using QPSK are depicted in Fig. 1. For a fair comparison we compare all the existing methods with the proposed enhanced LAMA method at a fixed SNR of 7dB. Proposed enhanced LAMA has a SER of 0.94 × 10 −5 , which is more than twice as good as LAMA's SER of 2.19 × 10 −5 , three times as good as OCD's SER of 2.812 × 10 −5 , more than seven times better than ADMIN's SER of 7.19 × 10 −5 , ten and fourteen times better than GS and MMSE SERs of 10.31 × 10 −5 and 14.06 × 10 −5 , and many orders better compared to neumann's SER of 11.49 × 10 −3 . It is worth noting that we are comparing results at 7 dB, if we go higher the difference will be even more. Also, at a SER of 10 −4 , Enhanced LAMA has a 0.23 dB gain compared to existing LAMA and 1 dB gain compared to GS. Figure 2 illustrates the performance analysis of the various algorithms with 32 users and 256 BS using 16QAM. In this case, all the existing schemes provides nearly same results but Enhanced LAMA gives slightly higher performance. At a particular SNR say 10dB, Enhanced LAMA has SER of 14.22 × 10 −5 and LAMA has 22.03 × 10 −5 . So, there is 1.5 times better error rate performance compared to traditional LAMA. It is clearly seen that a quarter dB gain is achieved by the proposed algorithm than the LAMA at 10 −3 . In Fig. 3, proposed enhanced LAMA performance is evaluated by considering 16 users and 16 BS elements with other existing schemes using BPSK. MMSE, GS, and ADMIN are having similar SER performance, and also they are slightly better than OCD and much better than Neumann. LAMA and Enhanced LAMA outperforms others conveniently but Enhanced LAMA has little edge over LAMA which is huge in Large scale MIMO systems.
Further, proposed enhanced LAMA performance is explored for 64 users and 128 BS antennas using 64QAM is shown in Fig. 4. At SNR of 19dB, the proposed algorithm has SER of 1.6 × 10 −2 , LAMA has SER of 4.8 × 10 −2 and MMSE has SER of 8.2 × 10 −2 , thereby having three and five times better error rate performance. At SER of 10 −1 , Enhanced LAMA contains approximately 1 dB and 1.6 dB higher gain than LAMA and MMSE. It is key to note that from all the Figs. 1, 2, 3 and 4, our proposed method is providing consistent results.
The variation of vector error rate (VER) versus SNR for 16 users and 32 BS antennas using BPSK is shown in Fig. 5. In that, the proposed method provides superior performance than the existing methods. Further in Fig. 6, the variation of VER versus SNR is depicted for 32 users and 128 BS antennas using 16 QAM modulation scheme. From both these Figs. 5 and 6, we concluded that irrespective of number of users, BS antennas and modulation scheme, the proposed enhanced LAMA provides better performance than the existing methods Proposed enhanced LAMA performance is evaluated over different number of BS antennas with corresponding modulation schemes are shown in Figs. 7, 8 and 9. The simulation results of 32 users using QPSK modulation scheme is exploited for our proposed method are shown in Fig. 7. The SER of 10 −3 is achieved at 11.12 dB when N r = 32 , whereas it obtained at 7.60 dB when N r = 64 , 4.29 dB for N r = 128 , 1.18 dB and − 1.76 dB for N r = 256 and N r = 512 respectively. We can notice a significant gain of 3.52 dB by doubling N r from 32 to 64, also a gain of 3.31 dB by augmenting N r to 128, a 3.11 dB of gain is achieved by increasing N r to 256 and finally, a gain of 2.94 dB is obtained by using 512 receiver antennas. We have also seen that a whooping gain of 12.88 dB by varying receiver antennas from 32 to 512. A consistent gain is observed by increasing number of receiver antennas which is feasible at BS. From Fig. 8, when number of receivers are 16 and 32, the SER is large due to low system ratio and it reduces exponentially for different BS antennas of 64, 128 and 256 respectively due to higher system ratio. The performance analysis of proposed enhanced LAMA with 64 users using QPSK is shown in Fig. 9. For a fair comparison, we consider a particular SNR say 4dB, SER of 2.35 × 10 −1 is obtained for N r = 64 , by doubling to 128 SER falls 10 times i.e., 3.76 × 10 −2 , again increasing to 256 SER reduces 10 times i.e., 1.71 × 10 −3 and now just by doubling to 512, SER has decreased drastically to 1.56 × 10 −6 . One can notice that until and unless system ratio is greater than two, SER is large for all the algorithms. SER falls significantly for higher system ratios, hence it is advised to maintain optimal system ratio to reap the benefits of the proposed enhanced LAMA algorithm.

Conclusion
Large Scale MIMO technology plays a crucial role in the upcoming wireless systems. In this research work, the Enhanced LAMA algorithm is proposed to get the better spectral efficiency in large scale MIMO systems. In the proposed enhanced LAMA, the Gram matrix is used instead of the channel matrix to reduce the per iteration complexity and also deployed the message damping techniques to eliminate the large scale user fading for practical channels. The simulation results show that the proposed enhanced LAMA outperforms the existing schemes such as Gauss-Seidel (GS), Neumann, optimal coordinate descent (OCD), alternating direction method of multipliers based on the infinity norm (ADMIN), MMSE and traditional LAMA.
Funding Not available.
Data availability Enquiries about data availability should be directed to the authors.