Maximum Likelihood Channel Research Articles

Channel Estimation and Pilot Design for Uplink Sparse Code Multiple Access System based on Complex-Valued Sparse Autoencoder

Channel estimation is one of the most important aspects of wireless communication. Especially in Sparse Code Multiple Access (SCMA) system, the accuracy of channel estimation has a significant impact on decoding performance. Various methods, so far, have been developed for channel estimation. Most of these methods regard channel estimation as a parameter estimation problem of linear models. However, these methods require lots of time-frequency resources to ensure high estimation accuracy. In massive connection scenarios, high pilot overhead makes the spectrum resource more scarce. Therefore, the drawback of conventional channel estimation methods limits the further improvement of system capacity in the Internet of Things(IoT) when time-frequency resources is restricted. To address this problem, in this paper, we propose an efficient channel estimation scheme and sparse pilot structure design method in SCMA system based on complex-valued sparse autoencoder which is effective to learn features of wireless channel. Complex-valued sparse autoencoder is a kind of neural network with complex-valued weights. It contains two parts: encoder and decoder. In our work, the encoder part is used to realize pilot design. Channel estimation is implemented by the decoder. Complex-valued weights obtained from training are used as baseband pilots. Compared with maximum likelihood channel estimation (MLE) of linear model, the proposed method can achieve higher channel estimation accuracy with more sparse pilot structure. The bit-error rates performance of the SCMA receiver in our work is very close to that of the perfect channel state information (CSI).

Deep Learning for Estimation and Pilot Signal Design in Few-Bit Massive MIMO Systems

Estimation in few-bit MIMO systems is challenging, since the received signals are nonlinearly distorted by the low-resolution ADCs. In this paper, we propose a deep learning framework for channel estimation, data detection, and pilot signal design to address the nonlinearity in such systems. The proposed channel estimation and data detection networks are model-driven and have special structures that take advantage of domain knowledge in the few-bit quantization process. While the first data detection network, B-DetNet, is based on a linearized model obtained from the Bussgang decomposition, the channel estimation network and the second data detection network, FBM-CENet and FBM-DetNet respectively, rely on the original quantized system model. To develop FBM-CENet and FBM-DetNet, the maximum-likelihood channel estimation and data detection problems are reformulated to overcome the indeterminant gradient issue. An important feature of the proposed FBM-CENet structure is that the pilot matrix is integrated into the weight matrices of its channel estimator. Thus, training the proposed FBM-CENet enables a joint optimization of both the channel estimator at the base station and the pilot signal transmitted from the users. Simulation results show significant performance gains in estimation accuracy by the proposed deep learning framework.

Blind Equalization and Channel Estimation in Coherent Optical Communications Using Variational Autoencoders

We investigate the potential of adaptive blind equalizers based on variational inference for carrier recovery in optical communications. These equalizers are based on a low-complexity approximation of maximum likelihood channel estimation. We generalize the concept of variational autoencoder (VAE) equalizers to higher order modulation formats encompassing probabilistic constellation shaping (PCS), ubiquitous in optical communications, oversampling at the receiver, and dual-polarization transmission. Besides black-box equalizers based on convolutional neural networks, we propose a model-based equalizer based on a linear butterfly filter and train the filter coefficients using the variational inference paradigm. As a byproduct, the VAE also provides a reliable channel estimation. We analyze the VAE in terms of performance and flexibility over a classical additive white Gaussian noise (AWGN) channel with inter-symbol interference (ISI) and over a dispersive linear optical dual-polarization channel. We show that it can extend the application range of blind adaptive equalizers by outperforming the state-of-the-art constant-modulus algorithm (CMA) for PCS for both fixed but also time-varying channels. The evaluation is accompanied with a hyperparameter analysis.

A Study of Channel Time-Domain Response on Equalization for Reproducing a Double-Layer Magnetic Recording Medium

To increase the recording density of hard disk drives (HDD), 3-D magnetic recording using microwave-assisted magnetic recording (MAMR) with multiple recording layers is attracting much attention as a next-generation recording method. Developing a signal processing method for 3-D magnetic recording using the multiple conventional recording layers and an magnetoresistive (MR) head reading system is desired. In this article, we evaluate 2-D partial response maximum likelihood (PRML) channels by the bit error rate (BER) performance, where the transfer function of the PR channel is determined by the response of the in-phase or differential-phase bits between layers in a double-layer magnetic recording medium. The results show that a BER of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-4}$ </tex-math></inline-formula> can be achieved by the PRML channel determined by the differential-phase response with the signal-to-noise ratio (SNR) greater than 26.0 dB for system noise.

Computationally Efficient Maximum Likelihood Channel Estimation for Coarsely Quantized Massive MIMO Systems

We consider computationally efficient maximum likelihood (ML) channel estimation for massive multiple-input multiple-output (MIMO) systems using coarsely quantized measurements obtained from low-resolution analog-to-digital converters (ADCs) at the receivers. We first devise a computationally efficient ML estimator (referred to as CQML) by using the cyclic optimization and majorization-minimization (MM) techniques. Then, we show the connections between CQML and the conventional unquantized ML estimator. Numerical examples are provided to demonstrate the effectiveness and computational efficiency of the proposed channel estimation algorithm.

Efficient Majorization-Minimization-Based Channel Estimation for One-Bit Massive MIMO Systems

We consider efficient channel estimation for massive multiple-input multiple-output (MIMO) systems using one-bit analog-to-digital converters (ADCs) with antenna-varying thresholds at the receivers. We introduce a computationally efficient majorization-minimization (MM) based maximum likelihood (ML) channel matrix estimator (referred to as 1bMM-ML), which maximizes the one-bit likelihood function iteratively by solving simple linear least squares problems. Moreover, to take into account the low-rank property of the millimeter-wave (mmWave) massive MIMO channels, we add a nuclear-norm based penalty term to the negative log-likelihood function and solve the resulting problem efficiently using the MM approach (referred to as 1bMM-LR). To further enhance the channel estimation performance, we consider an angular-domain channel model and introduce a hybrid approach (referred to as 1bLR-RELAX), which combines 1bMM-LR with an existing parametric algorithm called 1bRELAX, to recover the angular-domain channel parameters. Numerical examples are provided to demonstrate the effectiveness and efficiency of the proposed channel estimation algorithms.

Evaluation of Quantum Annealer Performance via the Massive MIMO Problem

Quantum annealing offers an appealing route to handle large-scale optimization problems. Existing Quantum Annealing processing units are readily available via cloud platform access for solving Quadratic Unconstrained Binary Optimization (QUBO) problems. In particular, the novel D-Wave Advantage device has been recently released. Its performance is expected to improve upon the previous state-of-the-art D-Wave 2000Q annealer, due to higher number of qubits and the Pegasus topology. Here, we present a comparative study via an ensemble of Maximum Likelihood (ML) Channel Decoder problems for MIMO scenarios in Centralized Radio Access Network (C-RAN) architectures. The main challenge for exact optimization of ML decoders with ever-increasing demand for higher data rates is the exponential increase of the solution space with problem sizes. Since current 5G solutions mainly use approximate methodologies, Kim et al. leveraged Quantum Annealing for large MIMO problems with Phase Shift Keying and Quadrature Amplitude Modulation scenarios. Here, we extend their work and analyze experiments for more complex modulations and larger MIMO antenna array sizes. By implementing the extended QUBO formulae on the novel annealer architecture, we uncover the limits of state-of-the-art quantum optimization for the massive MIMO ML decoder. We report on the improvements and discuss the uncovered limiting factors learned from the 64-QAM extension. We include the enhanced evaluation of raw annealer sampling via implementation of post-processing methods in the comparative analysis between D-Wave 2000Q and the D-Wave Advantage system.

Exploring Embeddings for MIMO Channel Decoding on Quantum Annealers

Quantum Annealing provides a heuristic method leveraging quantum mechanics for solving Quadratic Unconstrained Binary Optimization problems. Existing Quantum Annealing processing units are readily available via cloud platform access for a wide range of use cases. In particular, a novel device, the D-Wave Advantage has been recently released. In this paper, we study the applicability of Maximum Likelihood (ML) Channel Decoder problems for MIMO scenarios in centralized RAN. The main challenge for exact optimization of ML decoders with ever-increasing demand for higher data rates is the exponential increase of the solution space with problem sizes. Since current 5G solutions can only use approximate methodologies, Kim et al. [1] leveraged Quantum Annealing for large MIMO problems with phase shift keying and quadrature amplitude modulation scenarios. Here, we extend upon their work and present embedding limits for both more complex modulation and higher receiver / transmitter numbers using the Pegasus P16 topology of the D-Wave Advantage system.

On the relaxed maximum-likelihood blind MIMO channel estimation for orthogonal space-time block codes

This paper concerns the maximum-likelihood channel estimation for MIMO systems with orthogonal space-time block codes when the finite alphabet constraint of the signal constellation is relaxed. We study the channel coefficients estimation subspace generated by this method. We provide an algebraic characterisation of this subspace which turns the optimization problem into a purely algebraic one and more importantly, leads to several interesting analytical proofs. We prove that with probability one, the dimension of the estimation subspace for the channel coefficients is deterministic and it decreases by increasing the number of receive antennas up to a certain critical number of receive antennas, after which the dimension remains constant. In fact, we show that beyond this critical number of receive antennas, the estimation subspace for the channel coefficients is isometric to a fixed deterministic invariant space which can be easily computed for every specific OSTB code.

DFT-based low-complexity optimal cell ID estimation in NB-IoT

This paper deals with cell identifier (ID) estimation for narrowband-Internet of things (NB-IoT) system. It is suggested to transform the usual maximum likelihood (ML) estimator expression in order to highlight a discrete Fourier transform (DFT), which can be computed with fast algorithms. Therefore, the proposed method is a DFT-based low-complexity cell ID estimator that can be qualified as optimal in the ML sense. The principle is extended to the practical case where the channel is unknown and must be estimated. In this scenario, the concentrated likelihood function needs to be maximized, in which the ML channel estimate is a function of the unknown cell ID parameter. This operation only involves a few additional multiplications. Simulation results reveal that the performance of the proposed method actually matches the optimal one of the ML cell ID estimator. Furthermore, the technique is robust to residual frequency offset up to several hundreds of Hertz. We also show that the mean square error of channel estimation reaches its Cramér-Rao bound (CRB).

Spatial Modulation for Terahertz Communication Systems With Hardware Impairments

Terahertz (THz) communication has emerged as a promising technology that efficiently alleviates the scarcity of the frequency resources, and is capable of providing Terabit-per-second (Tbps) data transmission. However, the hardware impairments, which are usually omitted in the low-rate systems, have a detrimental impact on the THz communication link. These hardware imperfections, including in-phase/quadrature (I/Q) imbalance, phase noise and nonlinearities of the power amplifier, can be approximately modelled as complex-Gaussian distortions at both the transmitter and the receiver. In this paper, the channel estimator and signal detector (CE/SD) design of the THz-band spatial modulation (THz-SM) system is investigated, where the counterparts for classical low-rate SM are no longer applicable. More specifically, a dual-stage maximum-likelihood (ML) channel estimator is proposed containing two phases of candidate acquirement and exhaustive search. Then a novel low-complexity mean-least-squares (MLS) channel estimator is developed by considering the hardware impairments in an averaging manner. Besides, the optimal detector for THz-SM is designed by taking hardware impairments into account. Simulation results validate that the proposed CE/SD design is capable of attaining reliable data transmission in THz-SM with hardware imperfections.

On Sparse Channel Estimation in Aeronautical Telemetry

This paper examines the application of sparse estimation techniques for the estimation of a discrete-time equivalent multipath channel in the aeronautical telemetry context. The sensing matrix comprises samples of shaped offset QPSK-TG (a continuous phase modulation) based on the pilot bit sequence currently defined in the aeronautical telemetry standard. Representative algorithms from the three broad classes of sparse estimators were examined side by side using computer simulations to estimate the postequalizer bit error rate (BER). Ideal and nonideal frequency offset synchronization were assumed in the simulations. The results show that the performance of the matching pursuit (MP) algorithms seemed to be better suited to this application in the sense that no additional steps were required and the postequalizer BER of the best MP algorithm was slightly better than that of the other sparse estimation techniques. In the case of both ideal and nonideal frequency offset synchronization, the postequalizer BER achieved by the generalized orthogonal MP algorithm was approximately 1.5 dB better than that obtained using the nonsparse-constrained maximum likelihood channel estimate.

Iterative Approaches for Massive MIMO Uplink Processing Under Imperfect Channel Conditions

Obtaining channel state information in massive multiple-input multiple-output (MIMO) systems is challenging because of significant multiple-access interference and very limited resources available for channel estimation. Most existing equalization schemes for massive MIMO systems assume perfect channel knowledge at the receivers. When channel estimation error is present, however, significant performance degradation will be experienced, which has been demonstrated in the literature. In this paper, we model the maximum-likelihood channel estimation as the sum of real channel and random channel estimation error, and develop a minimum mean-square error (MMSE) based turbo equalization method conditioned on the channel estimate. We demonstrate that our newly developed iterative processing approach is a general expression, whereas existing methods can be viewed as special cases. We further derive an MMSE-sorted QR decomposition (MMSE-SQRD) based turbo equalization to further improve the performance for massive MIMO systems under large user numbers and severe channel estimation error conditions. Numerical results and performance comparison show that the proposed MMSE-based turbo equalization scheme greatly outperforms the existing schemes. With a large number of users being simultaneously served, our proposed MMSE-SQRD-based scheme can further enhance system performance over the MMSE-based turbo equalization scheme.

Short Packets Over Block-Memoryless Fading Channels: Pilot-Assisted or Noncoherent Transmission?

We present nonasymptotic upper and lower bounds on the maximum coding rate achievable when transmitting short packets over a Rician memoryless block-fading channel for a given requirement on the packet error probability. We focus on the practically relevant scenario in which there is no a priori channel state information available at the transmitter or at the receiver. An upper bound built upon the min-max converse is compared with two lower bounds: the first one relies on a noncoherent transmission strategy in which the fading channel is not estimated explicitly at the receiver and the second one employs pilot-assisted transmission (PAT) followed by maximum-likelihood channel estimation and scaled mismatched nearest-neighbor decoding at the receiver. Our bounds are tight enough to unveil the optimum number of diversity branches that a packet should span so that the energy per bit required to achieve a target packet error probability is minimized, for a given constraint on the code rate and the packet size. Furthermore, the bounds reveal that noncoherent transmission is more energy efficient than PAT, even when the number of pilot symbols and their power is optimized. For example, in Rayleigh fading, for the case when a coded packet of 168 symbols is transmitted using a channel code of rate 0.48-bits/channel use, over a block-fading channel with block size equal to eight symbols, PAT requires an additional 1.2 dB of energy per information bit to achieve a packet error probability of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> compared with a suitably designed noncoherent transmission scheme. Finally, we devise a PAT scheme based on punctured tail-biting quasi-cyclic codes and ordered-statistics decoding, whose performance is close (1-dB gap at 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> packet error probability) to the ones predicted by our PAT lower bound. This shows that the PAT lower bound provides useful guidelines on the design of actual PAT schemes.

Estimation and Compensation for RF Chain Mismatch in UCA OAM Radio Systems

This letter presents a calibration technique for uniform circular array orbital angular momentum systems that can estimate and compensate for gain/phase mismatch among radio-frequency (RF) chains in the transmitter and the receiver simultaneously. The basic idea is that the RF chain mismatch parameters can be estimated from a mathematical model of channel estimates when the ideal channel matrix is known a priori . It is shown that the maximum-likelihood (ML) channel estimation error becomes a zero-mean complex Gaussian random variable, and then, a joint ML solution for the RF mismatch parameters is derived.

On channel estimation for power line communication systems in the presence of impulsive noise

In this paper, two maximum-likelihood (ML) channel estimators that are robust and perform consistent channel estimation in the presence of impulsive noise (IN) for power line communication (PLC) systems are proposed and analyzed. The two estimators differ on the way they exploit the estimated IN to determine the coefficients of the channel impulse response. In the first approach, the channel estimator treats the estimated IN as a deterministic quantity; in the second as a random quantity. The performances of both estimators are analyzed and numerically evaluated. As it will be shown, between the two proposed estimators, the one that is based on the random approach outperforms the deterministic one in all typical PLC scenarios. However, the deterministic approach-based estimator can perform consistent channel estimation regardless of the IN behavior with less computational effort and becomes an efficient channel estimation strategy in situations where high computational complexity cannot be afforded.

Receiver for IEEE 802.11ah in Interference Limited Environments

Recently, IEEE 802.11ah standard has been proposed to extend the range of wireless local area network operating in the sub-1-GHz frequency band. This standard along with other protocols can provide communication services to the Internet of Things applications. However, in future, this band is also expected to be crowded like 2.45 GHz ISM band and cause interference to other devices operating in the same band. For a communication channel affected by additive white Gaussian noise, the least square (LS)-based estimator and Euclidean distance-based Viterbi decoder give optimal performance. However, the receiver’s performance with LS estimator followed by the Viterbi decoder degrades for high interference affected communication channels. In this paper, a new orthogonal frequency division multiplexing-based receiver structure operating in high interference environment is proposed. The proposed receiver is based on nonparametric maximum likelihood channel estimation followed by Viterbi decoder. The Viterbi decoder’s branch metric is updated based on the distribution of residual error. The proposed receiver structure is tested on IEEE 802.11ah-based receiver in two different type of additive interference: 1) IEEE 802.15.4 device and 2) impulsive noise. Both simulations and real-world experimental results on standard compliant platform show that the proposed algorithm performs better in terms of bit error rate than other receivers in all the considered interference models. Additionally, we also derive analytical expression for the probability of symbol error.

Analysis of the maximum likelihood channel estimator for OFDM systems in the presence of unknown interference

This paper is a theoretical analysis of the maximum likelihood (ML) channel estimator for orthogonal frequency-division multiplexing (OFDM) systems in the presence of unknown interference. The following theoretical results are presented. Firstly, the uniqueness of the ML solution for practical applications, i.e., when thermal noise is present, is analytically demonstrated when the number of transmitted OFDM symbols is strictly greater than one. The ML solution is then derived from the iterative conditional ML (CML) algorithm. Secondly, it is shown that the channel estimate can be described as an algebraic function whose inputs are the initial value and the means and variances of the received samples. Thirdly, it is theoretically demonstrated that the channel estimator is not biased. The second and the third results are obtained by employing oblique projection theory. Furthermore, these results are confirmed by numerical results.

FSO channel estimation for OOK modulation with APD receiver over atmospheric turbulence and pointing errors

In the free-space optical (FSO) links, atmospheric turbulence and pointing errors lead to scintillation in the received signal. Due to its ease of implementation, intensity modulation with direct detection (IM/DD) based on ON–OFF-keying(OOK) is a popular signaling scheme in these systems. For long-haul FSO links, avalanche photo diodes (APDs) are commonly used, which provide an internal gain in photo-detection, allowing larger transmission ranges, as compared with PIN photo-detector (PD) counterparts. Since optimal OOK detection at the receiver requires the knowledge of the instantaneous channel fading coefficient, channel estimation is an important task that can considerably impact the link performance. In this paper, we investigate the channel estimation issue when using an APD at the receiver. Here, optimal signal detection is quite more delicate than in the case of using a PIN PD. In fact, given that APD-based receivers are usually shot-noise limited, the receiver noise will have a different distribution depending on whether the transmitted bit is ‘0’ or ‘1’, and moreover, its statistics are further affected by the scintillation. To deal with this, we first consider minimum mean-square-error (MMSE), maximum a posteriori probability (MAP) and maximum likelihood (ML) channel estimation over an observation window encompassing several consecutive received OOK symbols. Due to the high computational complexity of these methods, in a second step, we propose an ML channel estimator based on the expectation–maximization (EM) algorithm which has a low implementation complexity, making it suitable for high data-rate FSO communications. Numerical results show that for a sufficiently large observation window, by using the proposed EM channel estimator, we can achieve bit error rate performance very close to that with perfect channel state information. We also derive the Cramer–Rao lower bound (CRLB) of MSE of estimation errors and show that for a large enough observation window, this CRLB can be adequately tight.

A Maximum-Likelihood Channel Estimator for Self-Interference Cancelation in Full-Duplex Systems

Operation of full-duplex systems requires efficient mitigation of the self-interference signal caused by the simultaneous transmission/reception. In this paper, we propose a maximum-likelihood (ML) approach to jointly estimate the self-interference and intended channels by exploiting its own known transmitted symbols and both the known pilot and unknown data symbols from the other intended transceiver. The ML solution is obtained by maximizing the ML function under the assumption of Gaussian received symbols. A closed-form solution is first derived, and subsequently, an iterative procedure is developed to further improve the estimation performance at moderate-to-high signal-to-noise ratios (SNRs). We establish the initial condition to guarantee the convergence of the iterative algorithm to the ML solution. In the presence of considerable phase noise from the oscillators, a phase noise estimation method is proposed and combined with the ML channel estimator to mitigate the effects of the phase noise. Illustrative results show that the proposed methods offer good cancelation performance close to the Cramer–Rao bound (CRB).