Maximum Likelihood Detection Problem 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.

Linear and Deep Neural Network-Based Receivers for Massive MIMO Systems With One-Bit ADCs

The use of one-bit analog-to-digital converters (ADCs) is a practical solution for reducing cost and power consumption in massive Multiple-Input-Multiple-Output (MIMO) systems. However, the distortion caused by one-bit ADCs makes the data detection task much more challenging. In this paper, we propose a two-stage detection method for massive MIMO systems with one-bit ADCs. In the first stage, we present several linear receivers based on the Bussgang decomposition that show significant performance gains over conventional linear receivers. Next, we reformulate the maximum-likelihood (ML) detection problem to address its non-robustness. Based on the reformulated ML detection problem, we propose a model-driven deep neural network-based detector, namely OBMNet, whose performance is comparable with an existing support vector machine-based receiver, albeit with a much lower computational complexity. A nearest-neighbor search method is then proposed for the second stage to refine the first stage solution. Unlike existing search methods that typically perform the search over a large candidate set, the proposed search method generates a limited number of most likely candidates and thus limits the search complexity. Numerical results confirm the low complexity, efficiency, and robustness of the proposed two-stage detection method.

A Low-Complexity Maximum Likelihood Detector for the Spatially Modulated Signals: Algorithm and Hardware Implementation

Aiming to increase the transmission data rate, the spatial modulation (SM) scheme maps each block of information bits to a digitally modulated symbol and additionally a transmit antenna index in a multiple transmit antenna system. In this brief, we focus on the maximum likelihood detection (MLD) scheme that detects SM signals with low computational complexity. By reformulating the MLD problem, we obtain a new mathematical expression as the MLD solution. To avoid the numerically unstable division operation, we apply the Givens rotation to realize this new expression and obtain a low-complexity MLD algorithm. We also propose a coordinate rotation digital computer-based hardware architecture to detect SM signals with 64-quaternary amplitude modulated (QAM) symbols over the correlated channel with eight transmit antennas and four receive antennas. The VLSI implementation results under the TSMC 90-nm CMOS technology reveal that our division-free architecture requires 70.5K gates and provides detection throughput 432.68 Mb/s, while operating at frequency 384.6 MHz. Our design is the first hardware architecture to achieve the exact MLD performance and is easily modified to detect SM signals with arbitrary ${M}$ -ary QAM symbols and arbitrary numbers of transmit and receive antennas.

A Low-Complexity Detection Algorithm for Uplink Massive MIMO Systems Based on Alternating Minimization

In this letter, we propose an algorithm based on the alternating minimization technique to solve the uplink massive multiple-input multiple-output (MIMO) detection problem. The proposed algorithm is specifically designed to avoid any matrix inversion and any computations of the Gram matrix at the receiver. The algorithm provides a lower complexity compared to the conventional minimum mean square error detection technique, especially when the total number of user equipment antennas (across all users) is close to the number of base station antennas. The idea is that the algorithm re-formulates the maximum-likelihood detection problem as a sum of convex functions based on decomposing the received vector into multiple vectors. Each vector represents the contribution of one of the transmitted symbols in the received vector. Alternating minimization is used to solve the new formulated problem in an iterative manner with a closed-form solution update in every iteration. Simulation results demonstrate the efficacy of the proposed algorithm in the uplink massive MIMO setting for both coded and uncoded cases.

A Variational Inference-Based Detection Method for Repetition Coded Generalized Spatial Modulation

In this paper, we consider a simple coding scheme for spatial modulation, where the same set of active transmit antennas is repeatedly used over consecutive multiple transmissions. Based on a Gaussian approximation, an approximate maximum likelihood (ML) detection problem is formulated to detect the indices of active transmit antennas. We show that the solution to the approximate ML detection problem can achieve a full coding gain. Furthermore, we develop a low-complexity iterative algorithm to solve the problem with low complexity based on a well-known machine learning approach, i.e., variational inference. Simulation results show that the proposed algorithm can have a near ML performance. A salient feature of the proposed algorithm is that its complexity is independent of the number of active transmit antennas, whereas an exhaustive search for the ML problem requires a complexity that grows exponentially with the number of active transmit antennas.

Efficient Recovery Algorithm for Discrete Valued Sparse Signals Using an ADMM Approach

Motivated by applications in wireless communications, in this paper we propose a reconstruction algorithm for sparse signals whose values are taken from a discrete set, using a limited number of noisy observations. Unlike conventional compressed sensing algorithms, the proposed approach incorporates knowledge of the discrete valued nature of the signal in the detection process. This is accomplished through the alternating direction method of the multipliers which is applied as a heuristic to decompose the associated maximum likelihood detection problem in order to find candidate solutions with a low computational complexity order. Numerical results in different scenarios show that the proposed algorithm is capable of achieving very competitive recovery error rates when compared with other existing suboptimal approaches.

MIMO Detection and Equalization for Single-Carrier Systems Using the Alternating Direction Method of Multipliers

In this letter, a multiple-input multiple-output detection algorithm based on the alternating direction method of the multipliers (ADMM) is proposed for single-carrier transmissions in time dispersive channels. The ADMM is applied as a heuristic to solve the maximum likelihood detection problem for arbitrary signal constellations formulated in the frequency domain. This approach allows most of the decoding steps to be implemented in the frequency domain with an overall reduced complexity cost. Simulation results show that the performances close to the matched filter bound can be attained in large problem sizes, even in overloaded scenarios.

A DETECTION TECHNIQUE OF SIGNAL IN MIMO SYSTEM

MIMO techniques are based on multiple antennae in receiving and transmitting signals and also used in multipath propagation for the transformation of entire channel into many independent virtual channels. In MIMO system multiple antennae can increase the spectral efficiency/ reliability of radio channel without increasing bandwidth or transmit power. Commercially, it is not feasible in case of MIMO systems. So, simple and efficient receiver that can harness MIMO architecture benefits without draining mobile receiver battery power or long time to decode transmitted symbols was required. In this paper problem of receiver design for MIMO system in spatial multiplexing scheme that is Maximum likelihood detection problem also known as NP hard combinatorial optimization problem, which need an exponential search over the space of all possible transmitted symbols in order to find closest point in Euclidean sense to received symbols, has been considered. A metaheuristic algorithm for detection of MIMO wireless system based on the Ant colony optimization (ACO) technique using MATLAB give the best solution to the problem and find the optimal path for the receivers.

Maximum-Likelihood Noncoherent PAM Detection

Sequence detection offers improved error-rate performance over conventional symbol-by-symbol detection when channel knowledge is not available at the receiver end. However, maximum-likelihood (ML) noncoherent sequence detection is proven to be notoriously intractable in many communication settings. In this work, we develop a new ML sequence detector for pulse-amplitude modulation (PAM) or quadrature-amplitude modulation (QAM) transmissions in unknown Rayleigh fading. Our detector identifies the ML sequence with overall polynomial complexity. This is possible via an auxiliary-angle approach that unlocks a low-rank property of the ML detection problem, reduces the exponential-size set of solution sequences to a polynomial-size set of candidates, and guarantees that the ML sequence is always contained in this substantially smaller set.

MIMO Detection by Lagrangian Dual Maximum-Likelihood Relaxation: Reinterpreting Regularized Lattice Decoding

This paper considers lattice decoding for multi-input multi-output (MIMO) detection under PAM constellations. A key aspect of lattice decoding is that it relaxes the symbol bound constraints in the optimal maximum-likelihood (ML) detector for faster implementations. It is known that such a symbol bound relaxation may lead to a damaging effect on the system performance. For this reason, regularization was proposed to mitigate the out-of-bound symbol effects in lattice decoding. However, minimum mean square error (MMSE) regularization is the only method of choice for regularization in the present literature. We propose a systematic regularization optimization approach considering a Lagrangian dual relaxation (LDR) of the ML detection problem. As it turns out, the proposed LDR formulation is to find the best diagonally regularized lattice decoder to approximate the ML detector, and all diagonal regularizations, including the MMSE regularization, can be subsumed under the LDR formalism. We show that for the 2-PAM case, strong duality holds between the LDR and ML problems. Also, for general PAM, we prove that the LDR problem yields a duality gap no worse than that of the well-known semidefinite relaxation method. To physically realize the proposed LDR, the projected subgradient method is employed to handle the LDR problem so that the best regularization can be found. The resultant method can physically be viewed as an adaptive symbol bound control wherein regularized lattice decoding is recursively performed to correct the decision. Simulation results show that the proposed LDR approach can outperform the conventional MMSE-based lattice decoding approach.

Optimal detection of a repeated tuple of reference fragments in a quasi-periodic sequence

An a posteriori (off-line) approach to the problem of maximum likelihood detection of a repeated tuple of reference fragments in a numerical quasi-per iodic sequence is studied. The problem is considered under the following assumptions The total amount of fragments in a sequence is unknown. The number of sequence members corresponding to the beginning of a fragment is a determined (not random) value. The sequence available for observation is distorted by an additive Gaussian uncorrelated noise. The problem can be reduced to a test of a single mean hypothesis about a random Gaussian vector. A notable aspect of the problem is the fact that the cardinality of the set of tested hypotheses grows exponentially as the vector dimension (the length of the observed sequence) is increased. The search for the maximum likelihood hypothesis is shown to be equivalent to the search for arguments that provide the maximum of a special objective function. The search for the maximum of the function is demonstrated to be accomplished by solving a basic extremal problem. The basic extremal problem is shown to be solvable in polynomial time. The exact solving algorithm is presented and proved to provide optimal detection of a repeated tuple. Results of numerical simulation that show the detection algorithm to be noise-proof are given.

A Linear Fractional Semidefinite Relaxation Approach to Maximum-Likelihood Detection of Higher-Order QAM OSTBC in Unknown Channels

This paper considers the blind maximum-likelihood (ML) detection problem for orthogonal space-time block codes (OSTBCs) in multiple-input multiple-output flat-fading channels. While the blind ML detection problem for general space-time codes is difficult to solve, it has been shown that for OSTBCs with constant modulus constellations, the blind ML detection problem can be formulated as a discrete quadratic program, and then handled by a powerful convex approximation technique known as semidefinite relaxation (SDR). In this paper, we turn our attention to the case of higher order QAM OSTBCs. Due to the nonconstant modulus nature of higher order QAM signals, the blind ML detection problem turns out to be a discrete Rayleigh quotient maximization problem, and as a result the current SDR technique is no longer directly applicable. We propose a linear fractional SDR (LFSDR) approach to this problem. This approach first relaxes the higher order QAM blind ML detection problem into a quasi-convex problem, followed by a simple solution approximation procedure. In general, quasi-convex problems are computationally more complex to solve than convex problems, but we show that an optimum solution of our quasi-convex problem can be efficiently obtained by solving a convex semidefinite program. The approximation accuracy of the proposed approach relative to other possible relaxation approaches is also studied. Simulation results are presented to demonstrate that the proposed LFSDR-based blind ML detector outperforms some existing suboptimal detectors and can yield promising performance even with a small to moderate number of code blocks.

Partial regularisation approach for detection problems in underdetermined linear systems

The maximum likelihood detection problem in many underdetermined linear communications systems can be described as an underdetermined integer least squares (ILS) problem. To solve it efficiently, a partial regularisation approach is proposed. The original underdetermined ILS problem is first transformed to an equivalent overdetermined ILS problem by using part of the transmit vector to do the regularisation. Then the overdetermined ILS problem is solved by conventional sphere decoding algorithms. Simulation results indicate that this approach can be much more efficient than other approaches for any square constellation higher than 4QAM.

An EM based Joint Data Detection and Channel Estimation Incorporating with Initial Channel Estimate

In this letter, we derive an expectation-maximization (EM) algorithm to solve the maximum-likelihood (ML) detection problem over multiple input multiple output (MIMO) channels. The difference from the existing EM algorithms for the ML detection problem is that the proposed EM algorithm does not require the probability density function (pdf) of channel parameters. Thus, the proposed EM algorithm is applicable for any type of MIMO channels (e.g., Rician or Rayleigh) without any modification.

Optimal detection of a recurring tuple of reference fragments in a quasiperiodic sequence

An a posteriori (off-line) approach to the problem of maximum likelihood detection of a repeated tuple of reference fragments in a numerical quasi-per iodic sequence is studied. The problem is considered under the following assumptions

An Efficient Maximum-Likelihood Detector for Four-Transmit-Antenna Quasi-Orthogonal Space-Time Block Code

This letter deals with computationally efficient maximumlikelihood (ML) detection for the quasi-orthogonal space-time block code (QOSTBC) with four transmit antennas. The proposed ML detector uses a permutation based real-valued equivalent channel matrix representation. As a result, the complexity of ML detection problem is moderated from O(2|A|2) to O(4|A|), where |A| is modulation order. Numerical results show that the proposed ML detector provides ML performance and achieves greatly high computational savings.

Efficient maximum likelihood detector for MIMO systems with small number of streams

The problem of maximum likelihood detection (MLD) of overlayed data symbols, which occurs in MIMO communications and interference mitigation applications, is addressed. A simple algorithm is presented that reduces the complexity of MLD by a factor of the constellation size compared to the complexity assumed in other publications. Although the complexity remains exponential in the number of transmit antennas, this reduction in complexity makes implementation of full-set MLD for MIMO systems with two-three streams feasible. The main difference compared to suboptimal schemes such as QRD-M and sphere decoding is that the result is exactly equivalent to full-set MLD. An efficient implementation is also presented.

Integer QP relaxation-based algorithms for intercarrier-interference reduction in OFDM systems

Orthogonal frequency-division multiplexing (OFDM) modulation can be utilized to deal with severe channel conditions without complex equalization. However, in a fast-fading channel, Doppler spread caused by user mobility destroys the orthogonality among subcarriers, prompting intercarrier interference (ICI). In this paper, the OFDM ICI reduction problem is formulated as a combinatorial optimization problem. Two relaxation methods are proposed to relax the maximum-likelihood detection problem into convex quadratic programming (QP) problems. To further reduce computational complexity, the QP problems are solved by limiting the search to the two-dimensional subspace. A low-bit descent search can also be employed to improve the system performance. The extension to higher-order quadrature amplitude modulation (QAM) OFDM systems is also addressed. Performance results are given which demonstrate that the integer QP relaxation-based algorithms provide excellent performance with reasonable computational complexity.

Sphere-constrained ML detection for frequency-selective channels

The maximum-likelihood (ML) sequence detection problem for channels with memory is investigated. The Viterbi algorithm (VA) provides an exact solution. Its computational complexity is linear in the length of the transmitted sequence, but exponential in the channel memory length. On the other hand, the sphere decoding (SD) algorithm also solves the ML detection problem exactly, and has expected complexity which is a low-degree polynomial (often cubic) in the length of the transmitted sequence over a wide range of signal-to-noise ratios. We combine the sphere-constrained search strategy of SD with the dynamic programming principles of the VA. The resulting algorithm has the worst-case complexity determined by the VA, but often significantly lower expected complexity

Polynomial-constrained detection using a penalty function and a differential-equation algorithm for MIMO systems

In this letter, we develop a family of approximate maximum-likelihood (ML) detectors for multiple-input multiple-output systems by relaxing the ML detection problem using constellation-specific polynomial constraints. The resulting constrained optimization problem is solved using a penalty function approach. Moreover, to escape from the local minima, which improves the detection performance, a differential equation algorithm using classical mechanics is proposed. Simulation results show that the polynomial constrained detector performs better than least-squares (LS) detector.