Multiuser Detection Scheme Research Articles

Novel Hybrid SOR- and AOR-Based Multi-User Detection for Uplink M-MIMO B5G Systems

The Internet of Things (IoT) is one of the most important wireless sensor network (WSN) applications in 5G systems and requires a large amount of wireless data transmission. Therefore, massive multiple-input multiple-output (M-MIMO) has become a crucial type of technology and trend in the future of beyond fifth-generation (B5G) wireless network communication systems. However, as the number of antennas increases, this also causes a significant increase in complexity at the receiving end. This is a challenge that must be overcome. To reduce the BER, confine the computational complexity, and produce a form of detection suitable for 4G and B5G environments simultaneously, we propose a novel multi-user detection (MUD) scheme for the uplink of M-MIMO orthogonal frequency division multiplexing (OFDM) and universal filtered multi-carrier (UFMC) systems that combines the merits of successive over-relaxation (SOR) and accelerated over-relaxation (AOR) named mixed over-relaxation (MOR). Herein, we divide MOR into the initial and collaboration stages. The former will produce the appropriate initial parameters to improve feasibility and divergence risk. Then, the latter achieves rapid convergence and refinement performance through alternating iterations. The conducted simulations show that our proposed form of detection, compared with the BER performance of traditional SOR and AOR, can achieve 99.999% and 99.998% improvement, respectively, and keep the complexity at O(N2). It balances BER performance and complexity with fewer iterations.

Multi-Service Oriented Joint Channel Estimation and Multi-User Detection Scheme for Grant-Free Massive MTC Networks

Multi-Service Oriented Joint Channel Estimation and Multi-User Detection Scheme for Grant-Free Massive MTC Networks

Prefiltering-based reduced rank multiuser detection in uplink generalized spatial modulation massive MIMO system

Exploiting generalized spatial modulation in massive MIMO system, we develop an efficient uplink multiuser detection scheme. Different from the maximum likelihood detection algorithm, the spatial and temporal data in the proposed detection scheme are decoded in a sequential manner: At the front end of base station receiver, a bank of spatial filters is first employed to mitigate the multiuser interference using the benefits of large-scale antenna array. There then, a reduced rank generalized Eigen-value decomposition-based algorithm is proposed to identify the antennas activated at each user terminal. In the second stage, a least-squares-based multiuser detector is employed to decode the multiplexed temporal symbols. Both theoretical analysis and computer simulations are conducted to evaluate system performance in terms of correct identification probability and overall symbol error rate. Simulation results demonstrate that the proposed algorithm outperforms most of the existing algorithms yet the complexity is extensively reduced.

Temporal Correlation Enhanced Multiuser Detection for Uplink Grant-Free NOMA

Compressed sensing has been identified as a good candidate for user detection in grant-free non-orthogonal multiple access (NOMA) by exploiting the inherent sparsity of user activity. However, most of the existing works do not fully utilize the temporal correlation in NOMA and rely heavily on the unrealistic assumption that the number of active users is known in advance. To address these issues, we propose a temporal correlation enhanced multiuser detection scheme to achieve efficient and practical multiuser detection. Firstly, using 1 bit memory to piggyback the information whether the active users still have data to transmit, the base station can realize whether the active users in current time slot will turn to be silent or remain active. Then, to make explicit use of the temporal correlation of active user sets, a cross validation based adaptive subspace pursuit (CVASP) algorithm, is developed by utilizing the reported information on prior active users. The proposed CVASP is a highly practical algorithm that does not require any prior knowledge of the user sparsity level, as the cross validation technique could properly determine the stopping condition. Extensive simulation results demonstrate that the proposed mechanism could achieve a superier performance without requiring any prior knowledge.

Bayesian Learning-Based Multiuser Detection for Grant-Free NOMA Systems

Grant-Free Non-Orthogonal Multiple Access (GF-NOMA) is considered as a promising technology to support the massive connectivity of Machine-Type Communications (MTC). The design of efficient and high-performance multi-user detection (MUD) scheme is a challenging issue of GF-NOMA, especially when the number of active users is unknown and relatively high. This paper adopts Sparse Bayesian Learning (SBL) approaches to solve the MUD problem of GF-NOMA in MTC. The MUD problem within a certain access slot is formulated as a Single Measurement Vector (SMV) model and efficiently solved via SBL-based methods. To further improve the MUD performance, we set up a Multiple Measurement Vector (MMV) model and develop block SBL-based MUD methods, by exploiting the temporal correlation of user activity over successive access slots. Then to extend the usage of the aforementioned algorithms to the scenarios with relatively high, or quasi-sparse, user activity, we propose novel SBL-based MUD algorithms via post sparse error recovery methodology, for both the SMV and MMV problem models. Simulation results show that the proposed SBL-based MUD algorithms achieve substantial performance gain over traditional ones, especially when the number of active users is unknown and relatively high.

M-ARY CDMA SCHEME WITH CCSK MODULATION AND PARALLEL INTERFERENCE CANCELLATION TECHNIQUE

In this paper, we tackle the problem of implementation a parallel interference cancellation scheme for multi-user detection (MUD) in M-ary CCSK-modulated CDMA system. Complexity remains the main obstacle to the practical implementation of MUD and a factor limiting, if not the very possibility, then the feasibility of using high order M-ary modulation formats for applications such as low-probability-of-intercept systems with a large spreading factor or, myriads of small self-powered IoT devices. The design of MUD algorithms for M-ary systems leads to a new round of growth in the receiver complexity comparing with the binary modulation. Demodulator has a major impact on the complexity of CDMA system. Cyclic code-shift keying (CCSK) is a modulation technique which is designed to reduce the complexity of M-ary signaling. In this, each symbol is a circularly shifted version of a single code sequence. Assuming synchronization, the receiver cyclically correlates the input signal plus noise with the base sequence and estimates the position of the correlation peak. The preliminary stage of the proposed scheme is a conventional multi-channel CDMA receiver. The preliminary estimates of the data is multiplied by user codes and amplitude estimates in the spreader, and then fed to the adder to generate the MAI estimates. After the latter are subtracted from the group signal, K-dimensional vector of cleared user signals is passed to the matched filter bank to form the refined estimates of the data. At each subsequent stage, the estimates of the output data of the previous stage are used as input data. The complexity gain is the ratio of the number of computations for finding cyclic convolution and, the amount of computations for finding M linear convolutions, i.e., M/log2M. Since the performance of the algorithm is sensitive to the reliability of the preliminary decisions, there are strict criteria for the selection of spreading codes: the length which must be an integer power of 2, the large family size, good periodic autocorrelation and good cross-correlation properties. To examine the proposed algorithm using computer simulation, we selected minimax periodic and odd-periodic complementary codes, the properties of which are close to the properties of the codes used in the CCSK scheme implemented in Link-16 protocol tactical data networks JTIDS and MIDS. Our study shows that the gain over the conventional receiver increases as the SNR increases, achieving 15 dB for BER 10-5. The system with complementary codes outperform system based on minimax PN-codes, achieving a target bit error rate at a lower SNR.

Bitwise Early Termination of Multiuser Detection for IDMA Systems

This letter presents an early termination scheme for iterative multiuser detection in interleave division multiple access (IDMA) systems. Instead of handling an entire frame as a whole, each bit in the frame is separately considered. If further iterations are found to be dispensable in determining a bit, the estimate of the bit is immediately finalized, and all the calculations associated with the bit are bypassed afterward, diminishing the computational complexity. To confirm the dispensability, all log-likelihood ratios (LLRs) associated with a bit are examined. Once their magnitudes exceed a threshold, their statistical measures are unlikely to change even after a few more iterations, implying that further iterations are unnecessary. As a countermeasure for infrequent behaviors deviating from such an expectation, the signs of the LLRs in two consecutive iterations are compared. If they all match, the bitwise procedure is terminated. For the remaining bits not fulfilling the criterion, the procedure continues as usual. As a result, in the realm of practical signal-to-noise ratios, a significant number of iterations and operations can be discarded while sustaining the near-optimal error rate.

An Efficient Code Domain NOMA Scheme with Enhanced Spectral and Energy Efficiency for Networks Beyond 5G

Non-Orthogonal Multiple Access (NOMA) is an efficient multiple access scheme which augments the capacity and overall throughput in LTE and 5G networks. Tackling the high system throughput and device connectivity of multiple users with different channel state information is a key challenge at this juncture. The joint detection and decoding using parity check polar coding (PCPC) and the utilization of Sparse Code Multiple Access (SCMA) in Code domain NOMA improves throughput supporting an overloaded number of users. Also, Multiple-Input Multiple-Output (MIMO)-SCMA scheme has potential in providing spectrum and energy efficiency. MIMO-SCMA superimposes multiple users in the code domain and utilizes the channel gain difference between multiplexed users. Hence, this paper aims to propose a proficient code domain NOMA scheme, the PCPC SCMA multiplexed over Orthogonal Frequency Code Division Multiple Access (OFCDMA), implementing a joint detection and decoding with user scheduling. Signals of numerous users are superimposed and transmitted over a channel. At the receiver, Multi-User Detection (MUD) exploiting the Message Passing Algorithm (MPA) is used to identify the desired user. To ensure proper resource utilization of multiple data blocks, the users are scheduled by implementing the Invasive Weed Optimization (IWO) technique along with the MPA. The proposed scheme, which engages an IWO-MPA based MUD scheme, improves the convergence rate, in terms of the number of iterations thereby leading to lower complexity, as well as reduces the Bit Error Rate (BER). Computer simulations reveal that the proposed scheme achieves high spectrum and energy efficiency, higher throughput and overloaded user fairness compared with recent researches.

Joint User Activity Identification and Channel Estimation for Grant-Free NOMA: A Spatial–Temporal Structure-Enhanced Approach

Exploiting the sparse nature of user activity, compressed sensing (CS) has been a powerful technique to realize efficient user detection in grant-free nonorthogonal multiple access (NOMA). However, most of the existing CS-based multiuser detection schemes merely independently incorporate the temporal correlation in frame-based transmission or spatial correlation induced by multiantenna reception, leading to unsatisfactory user detection performance. Driven by the observation in the CS theory that the signal recovery performance could be enhanced by an increased number of sparse vectors with a common support set, in this article, we propose a novel joint user activity identification and channel estimation (JUICE) framework by integrating the temporal correlation of active user sets with multiantenna reception, which could achieve superior user detection performance. Specifically, we first formulate the JUICE as a Kronecker CS (KCS) problem to model the CS measurement process, by fully extracting the spatial-temporal structure of user activity. Then, based on the mined spatial-temporal structure of user activity, an adaptive subspace pursuit algorithm is developed, i.e., spatial-temporal structure enhanced adaptive subspace pursuit (STS-ASP), which could realize efficient multiuser detection. A distinct advantage of the proposed algorithm is that it does not require any prior knowledge (e.g., the number of active users and the noise level), by adaptively acquiring the number of active users and employing the cross-validation technique to appropriately terminate the iterative procedures. Extensive experimental evaluation is conducted, and the results corroborate the superiority of the proposed framework compared with the existing CS-based multiuser detection methods.

An Error Rate Comparison of Power Domain Non-Orthogonal Multiple Access and Sparse Code Multiple Access

Non-orthogonal Multiple Access (NOMA) has been envisioned as one of the key enabling techniques to fulfill the requirements of future wireless networks. The primary benefit of NOMA is higher spectrum efficiency compared to Orthogonal Multiple Access (OMA). This paper presents an error rate comparison of two distinct NOMA schemes, i.e., power domain NOMA (PD-NOMA) and Sparse Code Multiple Access (SCMA). In a typical PD-NOMA system, successive interference cancellation (SIC) is utilized at the receiver, which however may lead to error propagation. In comparison, message passing decoding is employed in SCMA. To attain the best error rate performance of PD-NOMA, we optimize the power allocation with the aid of pairwise error probability and then carry out the decoding using generalized sphere decoder (GSD). Our extensive simulation results show that SCMA system with “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5\times 10$ </tex-math></inline-formula> ” setting (i.e., ten users communicate over five subcarriers, each active over two subcarriers) achieves better uncoded BER and coded BER performance than both typical “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\times 2$ </tex-math></inline-formula> ” and “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\times 4$ </tex-math></inline-formula> ” PD-NOMA systems in uplink Rayleigh fading channel. Finally, the impacts of channel estimation error on SCMA, SIC and GSD based PD-NOMA and the complexity of multiuser detection schemes are also discussed.

Sparse code multiple access on the generalized frequency division multiplexing

Recent advances in the communication systems culminated in a new class of multiple access schemes, named non-orthogonal multiple access (NOMA), where the main goal is to increase the spectrum efficiency by overlapping data from different users in a single time-frequency resource used by the physical layer. NOMA receivers can resolve the interference among data symbols from different users, increasing the overall system spectrum efficiency without introducing symbol error rate (SER) performance loss, which makes this class of multiple access techniques interesting for future mobile communication systems. This paper analyzes one promising NOMA technique, called sparse code multiple access (SCMA), where C users can share U<C time-frequency resources from the physical layer. Initially, the SCMA and orthogonal frequency division multiplexing (OFDM) integration is considered, defining a benchmark for the overall SER performance for the multiple access technique. Furthermore, this paper proposes the SCMA and generalized frequency division multiplexing (GFDM) integration. Since GFDM is a highly flexible non-orthogonal waveform that can mimic several other waveforms as corner cases, it is an interesting candidate for future wireless communication systems. This paper proposes two approaches for combining SCMA and GFDM. The first one combines a soft equalizer, called block expectation propagation (BEP), and a multi-user detection (MUD) scheme based on the sum-product algorithm (SPA). This approach achieves the best SER performance, but with the significant increment of the complexity at the receiver. In the second approach, BEP is integrated with a simplified MUD, which is an original contribution of this paper, aiming for reducing the receiver’s complexity at the cost of SER performance loss. The solutions proposed in this paper show that SCMA-GFDM can be an interesting solution for future mobile networks.

Likelihood Ascent Search Detection for Coded Massive MU-MIMO Systems to Mitigate IAI and MUI

The main aim of massive multiuser multiple-input multiple-output (MU-MIMO) system is to improve the throughput and spectral efficiency in 5G wireless networks. The performance of MU-MIMO system is severely influenced by inter-antenna interference (IAI) and multiuser interference (MUI). The IAI occurs due to space limitations at each user terminal (UT) and the MUI is added when one UT is in the vicinity of another UT in the same cellular network. IAI can be mitigated through a precoding scheme such as singular value decomposition (SVD), and MUI is suppressed by an efficient multiuser detection (MUD) schemes. The maximum likelihood (ML) detector has optimal performance; however, it has a highly complex structure and involves the need of a large number of computations especially in massive structures. Thus, the neighborhood search-based algorithm such as likelihood ascent search (LAS) has been found to be a better alternative for mitigation of MUI as it results in near optimal performance with low complexity. Most of the recent papers are aimed at eliminating either MUI or IAI, whereas the proposed work presents joint SVD precoding and LAS MUD to mitigate both IAI and MUI. The proposed scheme can achieve a near-optimal performance with smaller number of matrix computations.

Approximate Message Passing-Based Detection for Asynchronous NOMA

A novel low-complexity multiuser detection scheme is proposed for the asynchronous non-orthogonal multiple access (NOMA). Directly applying the conventional symbol-synchronous detection techniques for the asynchronous NOMA would result in high complexity. For low-complexity, we divide one frame with a large number of symbols into several small blocks. Furthermore, for real-time and low-delay processing under the scenario with a large number of users, each block is independently processed in a parallel way by an approximate message passing-based algorithm. Furthermore, a jointly detection technique is proposed to address the large detection error in the boundary of the blocks, which is induced by the non-interaction between neighbor blocks. Simulation results demonstrate that the proposed detection scheme can achieve near optimal performance with a much lower complexity.

Design of Massive Multiuser MIMO System to Mitigate Inter Antenna Interference and Multiuser Interference in 5G Wireless Networks

Massive Multi-user Multiple Input Multiple Output (MU‒MIMO) system is aimed to improve throughput and spectral efficiency in 5G communication networks. Inter-antenna Interference (IAI) and Multi-user Interference (MUI) are two major factors that influence the performance of MU–MIMO system. IAI arises due to closely spaced multiple antennas at each User Terminal (UT), whereas MUI is generated when one UT comes in the vicinity of another UT of the same cellular network. IAI can be mitigated by the use of a pre-coding scheme such as Singular Value Decomposition (SVD) and MUI can be cancelled through efficient Multi-user Detection (MUD) schemes. The highly complex and optimal Maximum Likelihood (ML) detector involves a large number of computations, especially when in massive structures. Therefore, the local search-based algorithm such as Likelihood Ascent Search (LAS) has been found to be a better alternative for mitigation of MUI, as it results in near optimal performance using lesser number of matrix computations. Most of the literature have been aimed at mitigating either IAI or MUI, whereas the proposed work presents SVD pre-coding and LAS MUD to mitigate both IAI and MUI. Simulation results indicate that the proposed scheme can attain near-optimal bit error rate (BER) performance with fewer computations.

Multiuser Detection for Nonlinear MIMO Uplink

For the multiple-input multiple-output (MIMO) uplink employing high-order quadrature amplitude modulation (QAM) signaling and with nonlinear high power amplifiers (HPAs) at mobile users’ transmitters, the existing multiuser detection methods can no longer be applied. We propose a novel nonlinear multiuser detection scheme for the nonlinear MIMO uplink. Specifically, we adopt an effective B-spline parameterization of the nonlinear transmit HPAs and derive an efficient and accurate algorithm to identify the nonlinear MIMO uplink channel, including the nonlinear B-spline model of the nonlinear transmit HPAs and the estimate of the linear MIMO channel matrix. Moreover, as the direct result of this nonlinear MIMO channel identification, the B-spline inverse model of nonlinear transmit HPAs can readily be identified. The nonlinear multiuser detection can be effectively implemented by the zero-forcing linear detection based on the estimated linear MIMO channel and followed by compensating the nonlinear distortion of the nonlinear transmit HPAs based on the estimated B-spline inverse model. An extensive simulation investigation is performed to demonstrate the effectiveness of our proposed nonlinear multiuser detection scheme for nonlinear MIMO uplink with high-order QAM signaling.

SLLR‐MDIC: Spatial log‐likelihood multiuser detection and interference cancelation scheme for OFDM‐IDMA

SummaryWireless communication systems have gained huge attraction from research community, industrial, and academic field due to their significant impact on improving the communication efficiency, ease of deployment, and cost‐effective solution for real‐time communication. In this field of wireless communication, cellular communications have grown rapidly due to their daily usages and advantages. This increased demand of cellular communication systems has led to the evolution of 3G, 4G, and 5G communication systems, which in turn demands for higher efficiency and better bandwidth utilization. Due to heavy usage of network communication, multiple users may cause interference which subsequently may lead to the performance degradation which could be addressed using multiuser detection scheme. However, several schemes have been introduced for improving the system performance, but multiple access (MA) still remains a challenging task. Hence, in this work, we present a novel approach called Spatial Log‐Likelihood Multiuser Detection and Interference Cancelation (SLLR‐MDIC) that uses interleaving division multiple access (IDMA) to improve the communication and developed a multiuser detection approach using spatial log‐likelihood ratios. Further, we have developed orthogonal frequency‐division multiple access (OFDM)–IDMA‐based interference cancelation scheme in multiple access to improve the performance using rake receiver based approach. The performance of SLLR‐MDIC scheme is compared with existing techniques of multiuser detection in terms of bit‐error rate (BER) and symbol error rate (SER). The experimental analysis shows that proposed approach achieves improved performance when compared with existing techniques.

Sparsity Learning-Based Multiuser Detection in Grant-Free Massive-Device Multiple Access

In this paper, we study the multiuser detection (MUD) problem for a grant-free massive-device multiple access (MaDMA) system, where a large number of single-antenna user devices transmit sporadic data to a multi-antenna base station (BS). Specifically, we put forth two MUD schemes, termed random sparsity learning multiuser detection (RSL-MUD) and structured sparsity learning multiuser detection (SSL-MUD) for the time-slotted and non-time-slotted grant-free MaDMA systems, respectively. In RSL-MUD, active users generate and transmit data packets with random sparsity. In SSL-MUD, we introduce a sliding-window-based detection framework, and the user signals in each observation window naturally exhibit structured sparsity. We show that by exploiting the sparsity embedded in the user signals, we can recover the user activity state, the channel, and the user data in a single phase, without using pilot signals for channel estimation and/or active user identification. To this end, we develop a message-passing-based statistical inference framework for the BS to blindly detect the user data without any prior knowledge of the identities and the channel state information (CSI) of active users. The simulation results show that our RSL-MUD and SSL-MUD schemes significantly outperform their counterpart schemes in both reducing the transmission overhead and improving the error behavior of the system.

Multi-user detection for the downlink of NOMA systems with multi-antenna schemes and power-efficient amplifiers

Non-orthogonal multiple access (NOMA) schemes have been recognized as a promising multiple access method for the fifth generation (5G) communication systems. In NOMA, multiple users can share the same physical channel, which allows for a substantial increase in capacity gains when comparing to conventional orthogonal multiple access (OMA) techniques. This paper considers the downlink transmission of NOMA schemes combined with single-carrier (SC) signals. We present a multi-antenna transmission technique that is compatible with highly efficient power amplifiers without requiring any pre-processing. Furthermore, a multi-user detection scheme based on iterative frequency-domain equalization (FDE) is presented. We show that besides the high power-efficiency of the transmission scheme, the system’s performance can be close to the matched filter bound (MFB).

An Adaptive Bayesian Computation Model for CDMA Multiuser Detection using Evolutionary Computation Scheme

Demand of wireless communication is increasing drastically where CDMA (Code Division Multiple Access) is considered as most promising technique for real-time communication. However, due to extreme utilization of these technologies several challenges occur such as interference and packet drop resulting in poor communication. To address these issues, multi-user detection schemes have been adopted widely which are based on the filtering techniques and MMSE (Minimum Mean Square Error) based multiuser detection schemes. In this work we address these issues and proposed a novel approach of multi-user detection for asynchronous CDMA using combined optimization and evolutionary computation. Bayesian computation model is applied which helps to compute the Log Likelihood Ratio (LLR) using Monte Carlo simulation. Later genetic algorithm is incorporated to obtain the optimal solution for LLR probability resulting in reliable communication. An extensive simulation study is presented which shows significant improvement in the performance when compared with state-of-art multiuser detection schemes.

Gaussian Message Passing for Overloaded Massive MIMO-NOMA

This paper considers a low-complexity Gaussian message passing (GMP) Multi-User Detection (MUD) scheme for a coded massive multiple-input multiple-output (MIMO) system with non-orthogonal multiple access (massive MIMO-NOMA), in which a base station with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> antennas serves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}$ </tex-math></inline-formula> sources simultaneously in the same frequency. Both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> are large numbers, and we consider the overloaded cases with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}&gt;N_{s}$ </tex-math></inline-formula> . The GMP for MIMO-NOMA is a message passing algorithm operating on a fully-connected loopy factor graph, which is well understood to fail to converge due to the correlation problem. The GMP is attractive as its complexity order is only linearly dependent on the number of users, compared to the cubic complexity order of linear minimum mean square error (LMMSE) MUD. In this paper, we utilize the large-scale property of the system to simplify the convergence analysis of the GMP under the overloaded condition. We prove that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">variances</i> of the GMP definitely converge to the mean square error (MSE) of the LMMSE multi-user detection. Second, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">means</i> of the traditional GMP will fail to converge when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}/N_{s}&lt; (\sqrt {2}-1)^{-2}\approx 5.83$ </tex-math></inline-formula> . Therefore, we propose and derive a new convergent GMP called scale-and-add GMP (SA-GMP), which always converges to the LMMSE multi-user detection performance for any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{u}/N_{s}&gt;1$ </tex-math></inline-formula> , and show that it has a faster convergence speed than the traditional GMP with the same complexity. Finally, the numerical results are provided to verify the validity and accuracy of the theoretical results presented.