Sphere Decoding Research Articles

A Low‐Complexity and High‐Decoding Performance Scheme for the MIMO‐SCMA System

Sparse code multiple access (SCMA) has been proposed to obtain high capacity and support massive connections. When combined with the multiple‐input multiple‐output (MIMO) techniques, the spectrum efficiency of the SCMA system can be further improved. However, the detectors of the MIMO‐SCMA system have high computational complexity. For the maximum likelihood (ML) detection, though it is optimal decoding algorithm for the MIMO‐SCMA system, the detection complexity would grow exponentially with the number of both the antennas and users increase. In this paper, we consider a space‐time block code (STBC) based MIMO‐SCMA system where SCMA is used for multiuser access. Besides, we propose a low‐complexity utilizing joint message passing algorithm (JMPA) detection, which narrowing the range of superimposed constellation points, called joint message passing algorithm based on sphere decoding (S‐JMPA). But for the S‐JMPA detector, the augment of the amount of access users and antennas leads to the degradation of decoding performance, the STBC is constructed to compensate the performance loss of the S‐JMPA detector and ensure good bit error rate (BER) performance. The simulation results show that the proposed method achieves a close error rate performance to ML, JMPA, and a fast convergence rate. Moreover, compared to the ML detector, it also significantly reduces the detection complexity of the algorithms.

Near-Optimal Performance With Low-Complexity ML-Based Detector for MIMO Spatial Multiplexing

In Spatial Multiplexing MIMO systems, many powerful non-linear detection techniques as sphere decoding have emerged to overcome the performance limitations of linear detection techniques. However, these non-linear techniques suffer from high complexity that increases dramatically with the number of antennas and the modulation order. Hence, they cannot be implemented on highly parallel hardware architecture and are thus not suitable for real-time high data rate transmission. In this letter, a new detection technique is proposed to approach the optimal performance obtained by Maximum Likelihood (ML) detector without increasing the complexity significantly. This detector is denoted by OSIC-ML since it combines two techniques: the Ordered Successive Interference Cancellation (OSIC) and the ML. The proposed OSIC-ML detector shows a near-optimal performance at very low complexity even with large scale MIMO and imperfect channel estimation, where this complexity can be efficiently controlled to achieve the desired complexity-performance tradeoff.

A New Method of Integer Parameter Estimation in Linear Models With Applications to GNSS High Precision Positioning

The high precision positioning of global navigation satellite systems (GNSS), which essentially corresponds to solving the integer least-squares (ILS) problem in the integer phase ambiguity estimation, has emerged as a key problem for the development of industrial internet-of-things (IIoT). In this paper, a novel paradigm for solving the ILS problem is introduced to the integer phase ambiguity estimation. Different from the traditional paradigm of ILS which only involves one parameter to characterize the trade-off between performance and complexity, the proposed ILS paradigm entails two parameters named as the initial searching size K ≥ 1 and the standard deviation σ > 0, thus introducing extra degrees of freedom to facilitate the system trade-off. Based on it, explicit analysis can be carried out for a mathematically tractable trade-off, where great potentials could be further exploited for the high precision positioning of GNSS. The equivalent searching algorithm (ESA) is proposed, which achieves the same performance as the classic Fincke-Pohst strategy in sphere decoding (SD) but with tractable complexity measured by the number of visited nodes in the searching stage. Moreover, the candidate protection mechanism is given to further upgrade the equivalent searching algorithm, which makes it not only an optimal but also a sub-optimal ILS estimator given the flexible setup of K.

Distributed Learning Assisted Fronthaul Compression for Multi-Antenna C-RAN

This paper investigates the uplink reception in the cloud radio access network (C-RAN), sometimes called centralized-RAN. C-RAN is an architecture for cellular networks which consists of many remote radio units (RRUs) connected to a central processor (CP). Due to the prohibitive complexity of computations, the most efficient uplink C-RAN schemes are challenging to be implemented in practical systems. Using deep neural networks (DNNs), we propose a new and low complex method for uplink C-RAN subject to some quantization rules. This is the first work that uses DNNs to mimic the C-RAN system to the best of our knowledge. Our architecture’s objective, called QDNet, is to jointly optimize the processing done at the RRUs, which considers the quantization constraints and the processing done at the CP side. Inspired by the projected gradient descent algorithm, QDNet is designed as a distributed DNN with sparse connections. The performance of QDNet is compared to current solutions such as the zero-forcing (ZF) equalizer and the sphere decoder (SD). In some scenarios, experiment results show that our scheme achieves 2 dB SNR gain over the linear ZF with the same or lower computational complexity. It also achieves near-optimal performance compared to the SD algorithm, especially for low-to-moderate fronthaul link capacity and many RRUs in the C-RAN system.

Cellular V2X With D2D Communications for Emergency Message Dissemination and QoS Assured Routing in 5G Environment

Cellular Vehicle to Everything (C-V2X) in 5G communications has attracted researchers’ attention due to advantages such as higher throughput, data rate and lower delay, etc. However, achieving Quality of Service (QoS) in disseminating emergency messages is a challenging task due to frequent topology changes and concurrent link losses between vehicles. To address this issue, we propose the QoS guaranteed Routing and Emergency message Dissemination in Device-to-Device(D2D) communication (Q-REDD) assisted C-V2X. Primarily, we perform Stable Matching based Routing that assures low end-to-end delay via selecting the best forwarder. The Q-REDD method adopts the Enhanced Sphere Decoder Like (ESDL) algorithm to select the best device for establishing effective D2D communication in the C-V2X environment. If the requested content is not present in the nearby device, the discoverer device requests a 5G base station through a nearby pedestrian. To reduce broadcast storms, Q-REDD performs Chaotic Crow Search Algorithm (CCSA). To validate the performance of Q-REDD, we perform simulations on Omnet++4.6 and SUMO 0.21.0 simulator in terms of Packet Delivery Ratio (PDR), End to End delay, Throughput, and Emergency Information Coverage. The obtained results demonstrate that our method enhances the Throughput and PDR up to 30%, reduces End to End delay by 30%, and enhances Emergency Information Coverage by 23%, compared to the existing methods; GTLQR, GPSR, MIR, ${ETD}^{2}$ , ALQ and TBED.

Reconfigurable Intelligent Surface-Aided Single-Input Single-Output K-Complex Symbol Golden Codeword-Based Modulation

Small reconfigurable intelligent surfaces (RISs) have greater potential for flexibility and ubiquity compared to large RISs. In the literature, the theoretical average bit error probability (ABEP) of a phase-adjustable element access-point-based RIS (AP-RIS) is derived by exploiting the central-limit theorem for large surface size. Hence, the analysis does not hold for small surface sizes. In this paper, we first formulate a simple closed-form expression for the ABEP of a phase-adjustable element AP-RIS with $M$ -ary quadrature amplitude modulation ( $M$ QAM). Compared to the results available in literature, this expression is valid even for small RISs. Second, we investigate AP-RIS-aided single-input single-output $K$ -complex symbol Golden codeword-based modulation, which encodes/transmits $K~M$ QAM symbols in each of $K$ consecutive independently faded time slots. The encoding ensures that each symbol experiences $K$ different fadings, hence resulting in a $K$ -fold increase in diversity order, while the AP-RIS maximizes the received signal-to-noise ratio (SNR) in each slot and yields significant SNR gains with increasing size. An increase in $K$ yields significant SNR gains for small RISs. A theoretical bound on the ABEP is formulated and validated by simulation. Finally, a detector based on sorted symbol set sphere decoding is devised.

Hardware Implementation and Performance Analysis of Improved Sphere Decoder in Spatially Correlated Massive MIMO Channels

Hardware Implementation and Performance Analysis of Improved Sphere Decoder in Spatially Correlated Massive MIMO Channels

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.

A Capacity Achieving MIMO Detector Based on Stochastic Sampling

Spatial-multiplexing multiple-input multiple-output (MIMO) systems have been developed and enhanced over the past two decades. In particular, a great amount of effort has gone towards development of capacity achieving detectors with affordable computational complexity. The developed detectors may be broadly divided into two classes: (i) deterministic sampling, such as list sphere decoding detector; and (ii) stocastic sampling, such as those based on Markov chain Monte Carlo (MCMC) search schemes. This paper proposes a novel detection scheme that is based on stochastic sampling, but is fundamentally different from the MCMC detectors. While MCMC follows a set of sequential sampling steps, hence, the sample sets obtained are highly correlated, the method proposed in this paper takes stochastic samples that are completely independent. This new approach of stochastic sampling leads to a detector with significantly reduced complexity. It also allows reduction in the detector latency.

Fixed-Complexity Tree Search Schemes for Detecting Generalized Spatially Modulated Signals: Algorithms and Hardware Architectures

In the generalized spatial modulation (GenSM) multiple input multiple output (MIMO) system, each block of data bits is mapped to a set of spatially multiplexed (SMX) symbols and an index of transmit antenna combination (TAC) of active antennas. The difficulty for the GenSM MIMO receiver is to detect the SMX symbols and TAC index simultaneously. Recently, we applied the conventional sphere decoding algorithm (SDA) successively to achieve the exact maximum likelihood detection (MLD) of GenSM MIMO signals. The SDA scheme suffers from variable computational complexity and leads to hardware detectors with variable throughput rate. Instead, fixed-complexity tree search algorithms, e.g., reduced fixed sphere decoding (rFSD), with nearly MLD performance are employed to facilitate hardware implementation. Here, we propose to apply the rFSD successively and design a hardware architecture for detecting GenSM MIMO signals under the scenario of 5 transmit antennas, 2 transmit radio frequency chains, 4 receive antennas, and 64-QAM SMX symbols. The VLSI implementation results under the TSMC 90nm CMOS technology reveal that our architecture requires 276.7K gates and provides detection throughput 1.613 Gbps, while operating at 322.6 MHz. Compared with other related architectures, our architecture provides higher detection throughput rate and is of better hardware efficiency.

Likelihood ascent search augmented sphere decoding receiver for MIMO systems using M‐QAM constellations

MIMO systems employing sphere decoding (SD) algorithm are known to achieve near maximum likelihood (ML) performance at a reduced complexity by restricting the candidate search space to a sphere of a certain radius. The performance of SD depends on the precise estimation of its soft output. In this paper, a low complexity modified Likelihood Ascent Search (LAS) algorithm is proposed to be used within a SD receiver in order to precisely estimate the counter-hypothesis for its winner candidates. The LAS algorithm is modified to search for the best counter-hypothesis in only one-half of the signal lattice thereby improving the performance of MIMO receiver. Our results challenge the popular perception that for a SD receiver a large number of candidates within the search sphere is essential for good performance. Instead, it is shown that accurate estimation of the counter-hypothesis is equally important and in fact, the performance of the proposed augmented SD receiver with only single candidate approaches that of a classical SD with multiple candidates. Bit error rate performance of the proposed method when compared with the existing research works on soft output generation for the same number of candidates shows that our proposed method outperforms them by upto 3 dB.

Learnable MIMO Detection Networks Based on Inexact ADMM

In this article, we present a new iterative MIMO detection algorithm based on inexact alternating direction method of multipliers. Each iteration is considered as a neural network layer with learnable parameters, which are optimized by the stochastic gradient descent algorithm with a training data set of the received vectors and the ground truth transmitted signals. Numerical results show that the proposed algorithm outperforms the existing learnable detection network and it achieves near-optimal performance close to the sphere decoder in the case of a large number of receive antennas.

Performance evaluation for low complexity cascaded Sphere Decoders using K best detection algorithm

In this letter a new detection scheme combining the conventional Sphere decoders and K best detection algorithm is proposed. This algorithm leverages the Sphere decoder (SD) results to use smaller values of K in K best algorithm to achieve better performance. Post K best detection algorithm Darlington pairs of SD-K SD, SD-K1 SD, and K-K1 SD are used to get the final detection results. The proposed K best detection algorithm finds out smallest K paths by counting and sorting the bits of every candidate, which is much simpler as used in conventional sphere decoders. The computational complexity has been calculated in terms of time taken to visit the average number of K paths taken by Conventional SDs K-SDs and Darlington pair of SDs. Extensive Monte Carlo simulations are used to demonstrate that the proposed approach exhibits significant performance gain over conventional SD schemes in terms of bit error rate and computational time. Besides, a complete analytical approach is provided to validate the simulation results.

On the Complexity Reduction of Uplink Sparse Code Multiple Access for Spatial Modulation

Multi-user spatial modulation (SM) assisted by sparse code multiple access (SCMA) has been recently proposed to provide uplink high spectral efficiency transmission. The message passing algorithm (MPA) is employed to detect the transmitted signals, which suffers from high complexity. This paper proposes three low-complexity algorithms for the first time to the SM-SCMA. The first algorithm is referred to as successive user detection (SUD), while the second algorithm is the modified version of SUD, namely modified SUD (MSUD). Then, for the first time, the tree-search of the SM-SCMA is constructed. Based on that tree-search, another variant of the sphere decoder (SD) is proposed for the SM-SCMA, referred to as fixed-complexity SD (FCSD). SUD provides a benchmark for decoding complexity at the expense of bit-error-rate (BER) performance. Further, MSUD slightly increases the complexity of SUD with a significant improvement in BER performance. Finally, FCSD provides a near-optimum BER with a considerable reduction of the complexity compared to the MPA decoder and also supports parallel hardware implementation. The proposed algorithms provide flexible design choices for practical implementation based on system design demands. The complexity analysis and Monte-Carlo simulations of the BER are provided for the proposed algorithms.

Fast Multibit Decision Polar Decoder for Successive-Cancellation List Decoding

Successive-cancellation list (SCL) decoding for polar codes has the disadvantage of high latency owing to serial operations. To improve the latency, several algorithms with additional circuits have been proposed, but the area becomes larger. This paper proposes a fast multibit decision method having-high area efficiency based on the SCL decoding algorithm. First, multiple bits can be determined to reduce clock cycles using new nodes represented by the information bits and frozen bits. We propose the new nodes called the combined nodes and the other node in this paper. The combined nodes that combine redundant operations of the fast-simplified SC (fast-SSC) algorithm can increase area efficiency. The other node with bit patterns other than the node types of the fast-SSC algorithm performs an 8-bit multibit decision to reduce the number of decoding cycles. Latency is further reduced by applying a sphere decoding method to the other node. In addition, a sorter is proposed to reduce the critical path delay. As a large number of path metrics causes sorter delays, the proposed sorter can achieve high throughput with the small area. The proposed (1024, 512) SCL decoder showed negligible performance degradation in the simulation using Matlab and was synthesized using 65 nm CMOS technology. The proposed decoder achieves about 1.3Gbps with the small area. As a result, the area-throughput efficiency is at least 1.4 times higher than the state-of-the-art works over 1 Gbps.

Model Predictive Control of a Three-Phase Two-Level Four-Leg Grid-Connected Converter Based on Sphere Decoding Method

To achieve optimal control of four-leg grid-connected converters in terms of switching frequency and current tracking, the finite-control-set model-predictive-control method (FCS-MPC) can be applied. In this method, the cost function (CF) considers tracking control of grid current, filter capacitance voltage, and converter-side current as well as switching frequency before allocating different weights in its calculations. Thus, multiobjective optimization is achieved by trying to find the optimal switching sequence that minimizes the CF. However, as the horizon length is increased, the solution search enlarges exponentially, soon requiring an exhaustive search through each of many candidates. To alleviate this computational burden, this article presents an FCS-MPC method that is based on a new node-comparison sphere decoding method (NC-SDM), thereby reformulating the CF minimization problem into an integral-least-squares problem. The proposed NC-SDM reduces the computational burden associated with longer horizons by excluding as many suboptimal solutions from the candidates as possible. It does this by continuously comparing the length of two paths corresponding to each node of the search tree, and always taking the branch with shorter length. The final length of the total path is set as the initial radius after superposing all the path lengths. As a result, the initial radius estimation is much smaller than that in the Babai method and the computational cost is further reduced. Finally, simulation and experimental results validate the feasibility and suitability of the proposed method.

Advantages of Spectrally Efficient Frequency Division Multiplexing Over Orthogonal Frequency Division Multiplexing

An analysis on Spectrally Efficient Frequency Division Multiplexing (SEFDM) is contrast with Orthogonal Frequency Division Multiplexing (OFDM) considering the impact on Peak to Average Power Ratio (PAPR) and nonlinearities within fibre. With respect to OFDM the sub-carriers in SEFDM signals are compressed adjacent to each other at a rate of frequency lesser than the symbol rate. At the receiver end we have utilized the Sphere Decoder which is used to recover the data to remunerate the Interference created by the compressed signals (ICI) faced in the system. This research shows the advantages by using SEFDM and evaluates its achievement. PAPR. when compared with OFDM, while effects of non-linear fibres are considered. The use of various formats of modulation going from 4-QAM to 32-QAM, shows that the SEFDM signals have a noteworthy increment in the transmission length with respect to ordinary signals.

A Near-Optimal and Low-Complex Joint Multiuser Detection for QCSS-MC-CDMA System

Joint maximum likelihood (ML) multiuser detection for quasi-synchronous multicarrier code division multiple access (MC-CDMA) system has an excessive computational complexity, and it increases exponentially as the number of users and bits in each of the modulation symbols increases. Recently, sphere decoding (SD) has been adopted for multiuser detection whose complexity is polynomial in nature with the number of users, and it does not rely on the number of bits in the modulation. In this article, first we provide a construction of large size quasi-complementary sequence sets (QCSSs), and then a low-complexity optimum SD algorithm is proposed to detect the multiuser communication in QCSS-based MC-CDMA (QCSS-MC-CDMA) system. The complexity analysis has shown that the proposed SD-based QCSS-MC-CDMA receiver for multiuser communication provides lower complexity than that of ML-based QCSS-MC-CDMA receiver. We also show that peak to average power ratio of the proposed system is lesser than that of complete complementary code (CCC) based MC-CDMA (CCC-MC-CDMA) system. The proposed system overcomes the fractional delay problem which is noticed for CCC-MC-CDMA receiver. The bit error rate performance of the proposed SD-based QCSS-MC-CDMA system is compared with maximum ratio combining based MC-CDMA, ML-based MC-CDMA, frequency domain equalization based MC-CDMA, and CCC-MC-CDMA receivers over Rayleigh frequency selective fading channels. The work is validated through Monte Carlo simulation results.

Multi-User Detection Based on Expectation Propagation for the Non-Coherent SIMO Multiple Access Channel

We consider the non-coherent single-input multiple-output (SIMO) multiple access channel with general signaling under spatially correlated Rayleigh block fading. We propose a novel soft-output multi-user detector that computes an approximate marginal posterior of each transmitted signal using only the knowledge about the channel distribution. Our detector is based on expectation propagation (EP) approximate inference and has polynomial complexity in the number of users, number of receive antennas and channel coherence time. We also propose two simplifications of this detector with reduced complexity. With Grassmannian signaling, the proposed detectors outperform a state-of-the-art non-coherent detector with projection-based interference mitigation. With pilot-assisted signaling, the EP detector outperforms, in terms of symbol error rate, some conventional coherent pilot-based detectors, including a sphere decoder and a joint channel estimation–data detection scheme. Our EP-based detectors produce accurate approximates of the true posterior leading to high achievable sum-rates. The gains of these detectors are further observed in terms of the bit error rate when using their soft outputs for a turbo channel decoder.

Enhanced Diversity-Achieving Quadrature Spatial Modulation With Fast Decodability

In this paper, based on the architecture of diversity-achieving quadrature spatial modulation (DA-QSM) transmission scheme, we propose an Enhanced DA-QSM (EDA-QSM) scheme. In the proposed EDA-QSM scheme, by using Kronecker product extension to the eight dispersion matrices of Sezginer-Sari-Biglieri (SSB) code, the size of the dispersion matrices set is doubled compared with DA-QSM. Consequently, EDA-QSM is shown to have higher spectral efficiency as well as nonvanishing determinant (NVD) property without requiring any parameter optimization. Furthermore, EDA-QSM with an appropriate ordering of the dispersion matrices is a block-orthogonal space-time code (BOSTC), which then makes the four real variables corresponding to the same block are independent and orthogonal. By utilizing this feature, for the detection of EDA-QSM we propose a state codewords matched block-by-block sphere decoding (SD) algorithm, which significantly decreases the decoding complexity of ML decoder. Simulation results of bit error rate (BER) performance show that the new EDA-QSM scheme outperforms some newly presented diversity-achieving SM schemes.