Maximum Likelihood Detection Research Articles

Quantum Approximate Optimization Algorithm Based Maximum Likelihood Detection

Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices, where the quantum approximation optimization algorithm (QAOA) constitutes a promising candidate for demonstrating tangible quantum advantages based on NISQ devices. In this paper, we consider the maximum likelihood (ML) detection problem of binary symbols transmitted over a multiple-input and multiple-output (MIMO) channel, where finding the optimal solution is exponentially hard using classical computers. Here, we apply the QAOA for the ML detection by encoding the problem of interest into a level- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> QAOA circuit having <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2p$ </tex-math></inline-formula> variational parameters, which can be optimized by classical optimizers. This level- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> QAOA circuit is constructed by applying the prepared Hamiltonian to our problem and the initial Hamiltonian alternately in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> consecutive rounds. More explicitly, we first encode the optimal solution of the ML detection problem into the ground state of a problem Hamiltonian. Using the quantum adiabatic evolution technique, we provide both analytical and numerical results for characterizing the evolution of the eigenvalues of the quantum system used for ML detection. Then, for level-1 QAOA circuits, we derive the analytical expressions of the expectation values of the QAOA and discuss the complexity of the QAOA based ML detector. Explicitly, we evaluate the computational complexity of the classical optimizer used and the storage requirement of simulating the QAOA. Finally, we evaluate the bit error rate (BER) of the QAOA based ML detector and compare it both to the classical ML detector and to the classical minimum mean squared error (MMSE) detector, demonstrating that the QAOA based ML detector is capable of approaching the performance of the classical ML detector.

Performance Bound for MIMO Systems Using One-Bit ADCs Over Rayleigh Fading Channels

This study explores the performance of multiple-input multiple-output (MIMO) systems with one-bit analog-to-digital converters (ADCs) over Rayleigh fading channels. We particularly aim to characterize the pairwise error probability (PEP) achieved by maximum likelihood detection (MLD), which provides the optimal detection performance of the considered systems. First, the upper bound of the PEP is derived by leveraging Jensen’s inequality and the Chernoff bound. The asymptotic bounds of the PEP are then characterized to clarify the impact of the one-bit ADCs on the diversity order of the MIMO systems. Notably, the diversity order of the MIMO systems with one-bit ADCs is the same as the number of receive antennas only when transmitting two symbol vectors with the opposite sign. Except for this special case, the asymptotic bound of the PEP is saturated at a high signal-to-noise ratio due to the use of the one-bit ADCs. Using simulations, we verify that the analytical results coincide with the asymptotic behavior of the PEP of the MLD for MIMO systems with one-bit ADCs over Rayleigh fading channels.

2D Generalized Optical Spatial Modulation for MIMO-OWC Systems

In this paper, a novel two-dimensional (2D) generalized optical spatial modulation (GOSM) scheme is proposed for multiple-input multiple-output optical wireless communication (MIMO-OWC) systems. By grouping multiple successive time slots as one time block, 2D GOSM mapping can be performed not only in the space domain but also in the time domain. Specifically, two types of 2D GOSM mapping schemes are designed, including 2D-1 and 2D-2 GOSM mappings. Moreover, to address the high complexity issue of optimal joint maximum-likelihood (ML) detection and the noise amplification and error propagation issues of zero-forcing-based ML (ZF-ML) detection, a deep neural network (DNN)-aided detection scheme is further designed for 2D GOSM systems. Simulation results demonstrate the superiority of the proposed 2D GOSM scheme with deep learning-aided detection for high-speed and low-complexity MIMO-OWC systems. More specifically, a remarkable 3.4-dB signal-to-noise ratio (SNR) gain can be achieved by 2D GOSM in comparison to the conventional one-dimensional (1D) GOSM, when applying the DNN-aided detection.

Rectangular Differential OFDM With Index Modulation Using Coordinate Interleaving

Differential subcarrier index modulation (DSIM) is an emerging OFDM with index modulation (OFDM-IM) technique, which does not require channel state information (CSI) at the receiver. However, the DSIM scheme has two drawbacks of lower spectral efficiency and lacking transmit diversity. In this letter, by combining rectangular differential coding and coordinate interleaving orthogonal design, a novel rectangular differential OFDM-IM using coordinate interleaving (CI-OFDM-RDIM) scheme is proposed. In this scheme, the dispersion matrix is projected to a dispersion vector that is transmitted in a single symbol interval, so that CI-OFDM-RDIM improves spectral efficiency compared with DSIM. Since the symbols in two adjacent differential groups are interleaved, the proposed scheme obtains diversity gain. Furthermore, owing to the orthogonality of its processing matrix, a low-complexity decoupled maximum likelihood (D-ML) detector is proposed. Theoretical performance analysis and simulation results show that CI-OFDM-RDIM outperforms DSIM in terms of the bit error rate (BER) performance.

Error Performance Analysis of Multiuser Detection in Uplink-NOMA With Adaptive M-QAM

This letter provides a generalized performance analysis for the multi-user uplink-NOMA system with adaptive square quadrature amplitude modulation (QAM) over Rayleigh fading channels. Motivated by the massive IoT connections and unavailability of orthogonal resources for each node, we consider a multi-access scheme where multi-users with single-antenna transmit data to a multiple-antenna base station through the same resource block. By taking advantage of combining diversity paths with the proposed joint maximum-likelihood detector (JMLD), a closed form expression for the upper bound of bit error rate (BER) is obtained. Despite the number of users or the order of modulation, the analytical results endorsed via computer simulations reveal the ability of the MRC-JMLD detector to discard the error floor completely. Moreover, the simulation results show that the MRC-JMLD surpasses its counterparts significantly and ensures a full diversity order.

Achievable rate analysis of a generalized uplink asynchronous non-orthogonal multiple access with MSK-type signals

Non-orthogonal multiple access (NOMA) is a key technology for future wireless access. Recently, asynchronous NOMA (ANOMA) has attracted attention due to its large achievable rate. However, almost all work to date on NOMA employs linear modulation formats dedicated to linear amplifiers with low power efficiency. In this paper, we propose an uplink ANOMA scheme based on continuous phase modulation (CPM). In particular, a generalized system model with any number of users, response lengths, and frequency pulses is established, and the traditional ANOMA with linear waveforms (considering the nonlinearity of the power amplifier) is provided as a comparison. Furthermore, the achievable rate of each user and its asymptotic value under successive interference cancellation (SIC) and joint maximum likelihood (JML) detectors are derived respectively, and the achievable sum rates are also given. Numerical results show that the achievable rate converges to its asymptotic value. The nonlinear system can provide a larger achievable rate compared with the linear system. The relationship between the achievable sum rate of nonlinear systems and response length, transmit power, channel gain, frequency pulse, time offset and received signal-to-noise ratio (SNR) between users is also investigated via numerical simulations.

Weighted Alamouti Media-Based Modulation

A weighted Alamouti (WA) media-based modulation (WA-MBM) scheme is presented. In this scheme, the real and imaginary parts of the diagonal and anti-diagonal WA codes are interleaved and transmitted according to a well-designed MBM channel states activation approach, so that the spatial resource units in both diagonal and anti-diagonal directions of WA-MBM signal are fully exploited to achieve higher spectral efficiency. The maximum-likelihood (ML) detection and theoretical average bit error probability (ABEP) for WA-MBM are derived. Simulation results demonstrate obvious performance improvement of WA-MBM over some existing MBM schemes.

All-Digital LoS MIMO With Low-Precision Analog-to-Digital Conversion

Line-of-sight (LoS) multi-input multi-output (MIMO) systems exhibit attractive scaling properties with increase in carrier frequency: for a fixed form factor and range, the spatial degrees of freedom increase quadratically for 2D arrays, in addition to the typically linear increase in available bandwidth. In this paper, we investigate whether modern all-digital baseband signal processing architectures can be devised for such regimes, given the difficulty of analog-to-digital conversion for large bandwidths. We propose low-precision quantizer designs and accompanying spatial demultiplexing algorithms, considering <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> LoS MIMO with QPSK for analytical insight, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4 \times 4$ </tex-math></inline-formula> MIMO with QPSK and 16QAM for performance evaluation. Unlike prior work, channel state information is utilized only at the receiver (i.e., transmit precoding is not employed). We investigate quantizers with regular structure whose high-SNR mutual information approaches that of an unquantized system. We prove that amplitude-phase quantization is necessary to attain this benchmark; phase-only quantization falls short. We show that quantizers based on maximizing per-antenna output entropy perform better than standard Minimum Mean Squared Quantization Error (MMSQE) quantization. For spatial demultiplexing with severely quantized observations, we introduce the novel concept of virtual quantization which, combined with linear detection, provides reliable demodulation at significantly reduced complexity compared to maximum likelihood detection.

Successive Interference Cancellation on Frequency-Selective Channels With Mode-Dependent Gain

In ultra-long-haul optical communication systems, polarization-dependent gain from single-mode semiconductor optical amplifiers or mode-dependent gain from multi-mode erbium-doped fiber amplifiers causes a channel’s achievable information rate (AIR) using linear minimum-mean-square error (MMSE) detection to become substantially lower than the capacity of optimal maximum likelihood detection owing to loss of orthogonality between modes. However, such capacity loss can be mitigated using techniques that retain reasonable complexity, such as successive interference cancellation (SIC). In wireless systems, multicarrier transmission with many subcarriers transforms the frequency-selective channel into a set of uncoupled memoryless subchannels, and it is possible to obtain good performance using SIC on each subchannel independently. Multicarrier transmission in optical fibers, however, is susceptible to performance loss caused by Kerr nonlinearity. Therefore, we study methods for applying SIC on frequency-selective optical fiber channels using single-carrier transmission. We compare the theoretical performance of MMSE and SIC equalizers on frequency-selective optical fiber channels operating at 64 Gbaud with transmission distances up to 15,000 km and evaluate the penalties of SIC implementation. We consider single-mode fiber links with random artificial polarization-mode dispersion inserted along the link to enhance frequency diversity, as well as multi-core fiber links supporting 14 spatial and polarization modes. We show that SIC can increase outage capacity by 19%, but the AIR increase is reduced by almost half if error correction coding is not performed across modes. We confirm that SIC can adapt to fast channel perturbations with only a 3% decrease in AIR.

Orthogonal Frequency Division Multiplexing With Index Modulation Based on Discrete Hartley Transform in Visible Light Communications

This paper investigates optical orthogonal frequency division multiplexing (O-OFDM) systems with index modulation (IM) based on discrete Hartley transform (DHT) for visible light communications (VLC). An interesting trade-off between spectral efficiency (SE), energy efficiency, peak-to-average power ratio (PAPR) and bit error rate (BER) performance can be made by using the IM technique. And the DHT-based O-OFDM-IM systems can achieve higher SE compared with the DFT (discrete Fourier transform)-based counterparts. Employing DHT instead of DFT enables the removal of the Hermitian symmetry requirement, which contributes to transmitting more index bits and results in SE improvement. To acquire the same SE, the proposed systems can significantly lower the constellation order leading to better BER performance by contrast with the DFT-based counterparts. We employ DC (direct current) biasing and AC (Asymmetrically clipped) techniques to investigate the O-OFDM-IM systems in this paper. Simulation results indicate that SNR gains can be obtained by the DHT-based DCO/ACO-IM in comparison to the DFT-based counterparts. We also propose a novel reduced-complexity ML (maximum likelihood) detector specifically applicable to VLC. The new detector exhibits the same BER performance as the ML detector and performs better than the LLR (log-likelihood ratio) detector with reduced computational complexity.

Low Complexity Deep Learning-Assisted Golden Code Sphere-Decoding with Sorted Detection Subsets

Golden code is a space-time block coding (STBC) scheme that has spatial multiplexing gain over the Alamouti STBC which is widely used in modern wireless communication standards. Golden code has not been widely adopted in modern wireless standards because of its inherent high detection complexity. However, detection algorithms like the sphere-decoding with sorted detection subsets (SD-SDS) have been developed to lower this detection complexity. Literature indicates that the SD-SDS algorithm has lower detection complexity relative to the traditional sphere-decoding (SD) algorithm, for all signal-to-noise ratio (SNR) values. The SD-SDS algorithm exhibits low detection complexity at high SNR; however, at low SNR the detection complexity is higher. We propose a deep neural network (DNN) aided SD-SDS algorithm (SD-SDS-DNN) that will lower the Golden code's SD-SDS low SNR detection complexity, whilst maintaining the bit-error-rate (BER) performance. The proposed SD-SDS-DNN is shown to achieve a 75% reduction in detection complexity relative to SD-SDS at low SNR values for 16-QAM, whilst maintaining the BER performance. For 64-QAM, the SD-SDS-DNN achieves 99% reduction in detection complexity relative to the SD-SDS at low SNR, whilst maintaining the BER performance. The SD-SDS-DNN has also shown to achieve low detection complexity comparable to that of the Alamouti linear maximum likelihood (ML) detector for a spectral efficiency of 8 bits/s/Hz. For a spectral efficiency of 12 bits/s/Hz, the SD-SDS-DNN achieves a detection complexity that is 90% lower than the Alamouti linear ML detector.

An IRS-Aided GSSK Scheme for Wireless Communication System

In this letter, we propose a novel communication scheme that combines intelligent reflecting surface (IRS) with generalized space shift keying (GSSK), where the IRS adjusts the phase shift according to the incoming bits at any specific time slot and multiple antennas are selected at the receiver to convey the index signals. Besides, a practical pre-greedy aided maximum likelihood (PGML) detector is proposed to supplement this IRS-aided received GSSK (IRS-RGSSK) system with lower complexity compared with the maximum likelihood (ML) detector and better error performance compared with the greedy detector (GD). By comparing with the IRS-aided received spatial modulation (IRS-RSM), the theoretical and simulation results show that the proposed system has better bit error rate (BER) performance.

Differential Modulation in Massive MIMO With Low-Resolution ADCs

In this paper, we present a differential modulation and detection scheme for use in the uplink of a system with a large number of antennas at the base station, each equipped with low-resolution analog-to-digital converters (ADCs). We derive an expression for the maximum likelihood (ML) detector of a differentially encoded phase information symbol received by a base station operating in the low-resolution ADC regime. We also present an equal performing reduced complexity receiver for detecting the phase information. To increase the supported data rate, we also present a maximum likelihood expression to detect differential amplitude phase shift keying symbols with low-resolution ADCs. We note that the derived detectors are unable to detect the amplitude information. To overcome this limitation, we use the Bussgang Theorem and the Central Limit Theorem (CLT) to develop two detectors capable of detecting the amplitude information. We numerically show that while the first amplitude detector requires multiple quantization bits for acceptable performance, similar performance can be achieved using one-bit ADCs by grouping the receive antennas and employing variable quantization levels (VQL) across distinct antenna groups. We validate the performance of the proposed detectors through simulations and show a comparison with corresponding coherent detectors. Finally, we present a complexity analysis of the proposed low-resolution differential detectors

ML and MAP Device Activity Detections for Grant-Free Massive Access in Multi-Cell Networks

Device activity detection is one main challenge in grant-free massive access, which is recently proposed to support massive machine-type communications (mMTC). Existing solutions for device activity detection fail to consider inter-cell interference generated by massive IoT devices or important prior information on device activities and inter-cell interference. In this paper, given different numbers of observations and network parameters, we consider both non-cooperative device activity detection and cooperative device activity detection in a multi-cell network, consisting of many access points (APs) and IoT devices. Under each activity detection mechanism, we consider the joint maximum likelihood (ML) estimation and joint maximum a posterior probability (MAP) estimation of both device activities and interference powers, utilizing tools from probability, stochastic geometry, and optimization. Each estimation problem is a challenging non-convex problem, and a coordinate descent algorithm is proposed to obtain a stationary point. Each proposed joint ML estimation extends the existing one for a single-cell network by considering the estimation of interference powers, together with the estimation of device activities. Each proposed joint MAP estimation further enhances the corresponding joint ML estimation by exploiting prior distributions of device activities and interference powers. The proposed joint ML estimation and joint MAP estimation under cooperative detection outperform the respective ones under non-cooperative detection at the costs of increasing backhaul burden, knowledge of network parameters, and computational complexities. Numerical results show the substantial gains of the proposed designs over well-known existing designs and reveal the importance of explicit consideration of inter-cell interference, the value of prior information, and the advantage of AP cooperation in device activity detection.

A Novel Average Autoencoder-Based Amplify-and-Forward Relay Networks With Hardware Impairments

In this paper, we propose a novel Average autoencoder (AE)-based amplify-and-forward (AF) relay networks impacted by the I/Q imbalance (IQI) and additional hardware impairments (AHI), where the source and destination nodes are equipped with neural network (NN)-based encoder and decoder, while a conventional AF relay node assists the transmission. The average AE employs multiple small NN-based decoders at the destination node, each decoding a soft probabilistic output that is averaged to obtain the final soft probabilistic output at the destination node. By considering multiple small NN decoders, we reduce the implementation complexity significantly while improving the performance compared to the AE with a single large but NN-based decoder. Within this Average AE framework, we propose a coded modulation design (CMD) with zero-forcing-based IQI compensation that considers the availability of the channel state information (CSI) and IQI knowledge. However, the IQI and CSI need to be estimated separately. Thus, we also propose a CMD with no IQI compensation that requires only the CSI knowledge. Finally, we propose a differential CMD that removes the necessity of both the CSI and IQI knowledge. Under low signal-to-interference-and-noise-ratio regimes, we show that the proposed Average AE outperforms the optimal maximum likelihood detector by considerable margin.

Low-Complexity GSM Detection Based on Maximum Ratio Combining

Generalized spatial modulation (GSM) technology is an extension of spatial modulation (SM) technology, and one of its main advantages is to further improve band efficiency. However, the multiple active antennas for transmission also brings the demodulation difficulties at the receiver. To solve the problem of high computational complexity of the optimal maximum likelihood (ML) detection, two sub-optimal detection algorithms are proposed through reducing the number of transmit antenna combinations (TACs) detected at the receiver. One is the maximum ratio combining detection algorithm based on repetitive sorting strategy, termed as (MRC-RS), which uses MRC repetitive sorting strategy to select the most likely TACs in detection. The other is the maximum ratio combining detection algorithm, which is based on the iterative idea of the orthogonal matching pursuit, termed the MRC-MP algorithm. The MRC-MP algorithm reduces the number of TACs through finite iterations to reduce the computational complexity. For M-QAM constellation, a hard-limited maximum likelihood (HLML) detection algorithm is introduced to calculate the modulation symbol. For the M-PSK constellation, a low-complexity maximum likelihood (LCML) algorithm is introduced to calculate the modulation symbol. The computational complexity of these two algorithms for calculating the modulation symbol are independent of modulation order. The simulation results show that for GSM systems with a large number of TACs, the proposed two algorithms not only achieve almost the same bit error rate (BER) performance as the ML algorithm, but also can greatly reduce the computational complexity.

Space–time line coded spatial modulation

In this letter, a novel multiple-input multiple-output transceiver technique, named space–time line coded spatial modulation (STLC-SM) is proposed, where two modulation symbols are transmitted through one of the multiple transmit antennas determined by the incoming information bits after constellation-rotated STLC encoding. The receiver decodes the transmitted modulation symbols and the activated antenna index through an optimal joint maximum-likelihood detector. As a main result, the closed-form upper-bound of bit-error-rate of the proposed STLC-SM system in general antenna configurations is mathematically analysed. Finally, it is validated that the STLC-SM significantly outperforms the conventional STLC in terms of the bit-error-rate.

Efficient jamming attack against MIMO transceiver

This work addresses the issue of point-to-point intelligent jamming against MIMO architecture-based systems. The receiver uses an optimal Maximum Likelihood (ML) detector to extract the data stream from the received signal. Analyzing the receiver performance in the presence of a jamming signal, optimal attack strategies based on the pairwise error probability parameter as an objective function are designed, enabling the achievement of maximum degradation of the service quality. To perform the worst-case jamming of a legitimate transceiver, we assume that the jammer can eavesdrop on the Channel State Information (CSI). Depending on the jamming power budget constraint and the potential of jamming systems to estimate the CSI, different scenarios are envisaged, including imperfect CSI estimation cases. By minimizing the trace of Positive Semi-Definite (PSD) matrix products, the optimal jamming approaches that ensure maximum degradation of the legitimate link are designed. The disruption efficiency of the proposed approaches is comparable to the employment of the brute-force attack, but with a higher power level that is proportional to the number of jamming antennas in the case of partial knowledge of CSI, and inversely to the statistical expectation of the square root of the Wishart matrix's eigenvalues in the case of complete knowledge of CSI. To justify the relevance of the suggested policy, extensive simulations of the effect of these interventions are presented.

Performance Analysis of Regularized Convex Relaxation for Complex-Valued Data Detection

In this work, we study complex-valued data detection performance in massive multiple-input multiple-output (MIMO) systems. We focus on the problem of recovering an n-dimensional signal whose entries are drawn from an arbitrary constellation K⊂C from m noisy linear measurements, with an independent and identically distributed (i.i.d.) complex Gaussian channel. Since the optimal maximum likelihood (ML) detector is computationally prohibitive for large dimensions, many convex relaxation heuristic methods have been proposed to solve the detection problem. In this paper, we consider a regularized version of this convex relaxation that we call the regularized convex relaxation (RCR) detector and sharply derive asymptotic expressions for its mean square error and symbol error probability. Monte-Carlo simulations are provided to validate the derived analytical results.

Energy efficient data detection with low complexity for an uplink multi-user massive MIMO system

The current development of the Internet of Things (IoT) network in sixth-generation (6G) communication opens up various opportunities. When IoT devices with edge platforms connect to the telecommunication network, mobility, interferences, and intelligent device capacity issues might occur. The challenge with uplink MU-MIMO systems is data detection owing to noise, frequency allocations, and dense deployment of intelligent devices. The exponential complexity of extensive wireless networks makes optimal maximum likelihood detection impossible. Using MATLAB simulation-based analysis, we suggested an iterative data detection strategy based on a coordinate descent method (CDM) to reduce computing complexity while preserving an acceptable bit error rate (BER). Linear systems demand low co-channel interference (CCI) MU-MIMO uplink communications (CCI). A superior mean square error or BER performance is achieved using Maximum ratio combining with CDM. The proposed system outperforms the Richardson Method (RM), approximation message passing (AMP), and linear minimum mean square error (LMMSE) algorithms in terms of BER and complexity.