Suboptimal Maximum Likelihood Research Articles

A Signal Matching Method of In-Orbit Calibration of Altimeter in Tracking Mode Based on Transponder

In this paper, a matching method for altimeter and transponder signals in Sub-optimal Maximum Likelihood Estimate (SMLE) tracking mode is proposed. In the in-orbit calibration of the altimeter in SMLE tracking mode using the reconstructive transponder, it is necessary to separate the forwarding signal from the ground echo signal. At the same time, the fluctuations in the received signal of the altimeter, which are caused by the forwarding signal of the transponder, can be eliminated. The transponder generates a bias when measuring the arrival time of the transmitting signal from the altimeter and embeds this bias in both the transponder-recorded data and the altimeter-recorded data. Therefore, the two sets of data have one-to-one correspondence, and they are superimposed using the sliding sum method. Moreover, the distance between the altimeter and the transponder is a parabolic geometric relationship, and the outliers are eliminated by the fitting error minimization decision, and the transponder signal is separated from the ground echo. The final altimeter transmitting–receiving signal path is obtained. Furthermore, the principles underlying this method can be used for any transponder that can adjust the response signal delay during calibration.

Low-Complexity Channel and Carrier Frequency Offset Estimation for Burst-Mode CPM Using Optimized Training Waveforms

We present algorithms of the channel and carrier frequency offset (CFO) estimation for burst-mode continuous phase modulation (CPM) transmission with optimized training waveforms, which are obtained through minimizing the Cramer-Rao bound (CRB). Wirtinger calculus, which provides a framework for complex-valued signal processing, is used to simplify the derivations of CRBs and maximum likelihood estimation. Unlike linear modulation, the training waveform is optimized not only with respect to the modulated training sequence but also the sampling delay and sampling rate due to the continuous phase of CPM waveforms. In order to reduce the computation complexity of the optimization, we derive a closed-form expression for efficient computation of the objective function. The numerical results demonstrate the significant performance improvement of the optimized waveforms compared to the training sequences of previous works. Based on that, we propose two schemes for CFO estimation. The first scheme is a suboptimal maximum likelihood estimation of the CFO and it achieves a better trade-off between the estimation accuracy and complexity compared to optimal maximum likelihood estimation. The second scheme is based on the channel estimation, which leads to much lower implementation complexity than the other estimators. Finally, the measured bit error rate reveals the considerable gain of the estimation algorithm with the optimized training waveform.

A Study of the Technology Used to Distinguish Sea Ice and Seawater on the Haiyang-2A/B (HY-2A/B) Altimeter Data

When the Haiyang-2B (HY-2B) was launched into space to form a star network with the Haiyang-2A (HY-2A), it provided new data sources for the sea ice research of the Earth’s polar regions. The ability of altimeter echoes to distinguish sea ice and sea water is usable in operational ice charting. In this research study, the level 1B (L1B) data of HY-2A/B altimeter from November 2018 was used to analyze the altimeter waveforms from the polar regions. The Suboptimal Maximum Likelihood Estimation (SMLE) and Offset Center of Gravity (OCOG) tracking packages could maintain the waveform characteristics of diffused and quasi-specular surfaces by comparison. Also, they could be utilized to distinguish sea ice from seawater in the polar regions. It was determined that the types of echoes obtained from the seawater were diffuse. Also, some “ocean-like” waveform data had existed for the old ice formations in the Arctic regions during the study period. The types of echoes obtained from Arctic sea ice were found to be mainly quasi-specular. In the present study, three methods (Threshold segmentation, K-nearest-neighbor (KNN), and Lib-Support Vector machine (LIBSVM)) with four waveform parameters (Automatic Gain Control (AGC) and Pulse Peaking (PP) values of the Ku and C Bands) were adopted to distinguish between the sea ice and seawater areas. The accuracy rate of the separation results for the LIBSVM except band Ku from HY-2B ALT was found to be less than 40% in Antarctic. Meanwhile, the other two methods were observed to have maintained the waveforms correctly at accuracy rates of approximately 80% in Antarctic and the Arctic. In addition, the observed distinguishing errors were located in the regions of the old ice of the Arctic region. In addition, due to the summer melting processes, the large number of ice floes and the snow cover had made it difficult to distinguish the seawater and sea ice in the Antarctic regions.

Bayesian Helmholtz Stereopsis with Integrability Prior

Helmholtz Stereopsis is a 3D reconstruction method uniquely independent of surface reflectance. Yet, its sub-optimal maximum likelihood formulation with drift-prone normal integration limits performance. Via three contributions this paper presents a complete novel pipeline for Helmholtz Stereopsis. First, we propose a Bayesian formulation replacing the maximum likelihood problem by a maximum a posteriori one. Second, a tailored prior enforcing consistency between depth and normal estimates via a novel metric related to optimal surface integrability is proposed. Third, explicit surface integration is eliminated by taking advantage of the accuracy of prior and high resolution of the coarse-to-fine approach. The pipeline is validated quantitatively and qualitatively against alternative formulations, reaching sub-millimetre accuracy and coping with complex geometry and reflectance.

Suboptimal maximum likelihood detection of on-off keying for a wireless optical communication system.

This paper investigates a detection scheme without channel state information for wireless optical communication systems. Employing conventional on-off keying signals, we supposed that conditional probability density function P(r|0) is much bigger than P(r|1) when r<0. Under this assumption, the suboptimal maximum likelihood scheme is obtained by utilizing the probability density function without channel information. Theoretical analysis shows the performance of the proposed scheme is close to the maximum likelihood symbol-by-symbol detection. Compared with the maximum likelihood symbol by symbol detection, Monte Carlo simulations show that the performance of the proposed scheme is about 0.62 dB loss for a gamma-gamma channel with a Rytov variance of 1 at the signal-to-noise ratio of 2 dB, but the efficient algorithm makes the real-time implementation of detection based on maximum likelihood feasible. Besides, the experiment is set up under 2 Gbps, and the experimental results match well with that of the theory and simulation.

Iterative noncoherent block detection of coded MPSK for cooperative relay systems

Maximum likelihood (ML) noncoherent block detection techniques are investigated for block-coded MPSK modulation in cooperative decode-and-forward relay systems over slow fading channels. A decision-directed iterative Viterbi algorithm (IVA) is derived for a suboptimal ML noncoherent detection. Simulation results show that the IVA can approach the error performances of the exhaustive detection method but at a lower complexity.

Optimum Detection of Two-Dimensional Carrier Modulations With Linear Phase Noise Using Received Amplitude and Phase Information and Performance Analysis

We consider the design of the optimum detector for two-dimensional amplitude/phase modulated signals received in additive white Gaussian noise (AWGN) and a Gaussian distributed phase reference error (PRE) due to imperfect carrier phase estimation. We propose a novel approach of using the amplitude and phase information of the received signal, based on viewing the AWGN as an equivalent additive observation phase noise (AOPN) whose statistics is derived in our earlier work. This allows the AOPN to be combined with PRE, and the maximum $a posterior$ probability (MAP)/maximum-likelihood (ML) detection scheme to be readily derived in amplitude-phase form. This amplitude-phase approach is simpler and more convenient than the conventional method of using the in-phase and quadrature components of signals in phase noise. For $M$ -ary phase-shift keying, which only has one ring of signal points, the ML detector here turns out to be the same as the conventional minimum Euclidean distance (MED) detector which is derived without taking the PRE into account, and leads to angular bisector decision boundaries (DB). However, for constellations which have multiple rings, e.g., $M$ -ary quadrature amplitude modulation ( $M$ -QAM) and amplitude phase-shift keying ( $M$ -APSK), the ML detector here is very computationally inefficient. Thus, simpler and closed-form approximations to the ML detector for equiprobable signals are given, which can be easily implemented online. All these simplified ML detectors are shown via simulations to perform almost the same as the exact one and perform much better than the MED detector. The approximate ML DB for both 8-star QAM and rotated 8-star QAM are illustrated as examples and shown to be not necessarily straight lines. As the variance of PRE or the signal-to-noise ratio (SNR) or both increases, the DB between two signal rings asymptotically becomes circular. This leads to an annular sector as the decision region for each signal point. For annular-sector decision regions, simple, accurate, and closed-form approximations to the symbol error probability (SEP) are obtained for both 8-star QAM and the rotated case, 16QAM and even general $M$ -APSK. These expressions provide explicit insight into how the PRE variance affects the performance. Within a wide range of PRE variances, our SEP approximations agree very well with the Monte Carlo simulations for all SNR values of interest. For the special case of an $M$ -APSK constellation which has the same number of points and the same phase values on each ring, such as 8-star QAM, one of the suboptimal ML detectors further simplifies to a structure that performs ring detection and phase detection separately, and the decision regions are always annular sectors.

Signal Detection in MIMO-OFDM Systems Based on SSDE Algorithm

Device-to-device communication enables to improve the application performance of multi-input multi-output (MIMO) technology. In a multi-input multi-output orthogonal frequency division multiplexing (MIMO-OFDM) system, its performance is largely reflected by the signal detection algorithm used in receiver. As a sub-optimal maximum likelihood (ML) detection method, the Selective Spanning with Fast Enumeration algorithm can be successfully applied in MIMO-OFDM systems with high-order modulation. However, its Fast Enumeration scheme calculates constellation points based on fixed formula, which tends to yield pseudo constellation points outside of constellation maps, and consequently cannot work well in low-order modulation. To address this, a selective spanning with direct enumeration (SSDE) algorithm is proposed in this paper. Simulation results proved that the SSDE can achieve similar detection performance at a much lower computational cost in comparison with the ML method. The performance in terms of bit error rate (BER) obtained by SSDE method is also superior to those from the Minimum mean square error and Zero forcing detection algorithms with a huge savings in computational load. By adjusting the parameters used in the SSDE, the tradeoff between BER and computation complexity can be flexibly changed to satisfy specific design requirements in different applications.

Parameter Estimation of Linear-FM with Modified sMLE for Radar Signal Active Cancelation Application

This study examined a radar signal active cancelation technique, which is a theoretical way of achieving stealth by employing a baseband process that involves sampling the incoming hostile radar signal, analyzing its characteristics, and generating countermeasure signals to cancel out the linear-FM signal of the hostile radar signal reflected from the airborne target. To successfully perform an active cancelation, the effects of errors in the countermeasure signal were first analyzed. To generate the countermeasure signal that requires very fast and accurate processing, the down-sampling technique with the suboptimal maximum likelihood estimation (sMLE) scheme was proposed to improve the speed of the estimation process while preserving the estimation accuracy. The simulation results showed that the proposed down-sampling technique using a 2048 FFT size yields substantial power reduction despite its small FFT size and exhibits similar performance to the sMLE scheme using the 32768 FFT size.

Pilot-assisted data recovery for OFDM systems with blind selected mapping scheme and space-frequency block coding

Pilot-assisted blind selected mapping (PB-SLM) scheme for multiple input single output (MISO) orthogonal frequency division multiplexing (OFDM) systems is proposed. At the receiver, individual values of autocorrelation of two consecutive pilot sub-channel responses are evaluated for all the phase sequences available at the transmitter. The phase sequence index corresponding to the highest value of autocorrelation is considered as the side information (SI) and used for data recovery. Due to the robust phase sequence selection based on pilot symbols, data decoding performance of the PB-SLM scheme is substantially better than that of the recent blind SLM (B-SLM) scheme for MISO OFDM systems with space-frequency block coding, which is based on sub-optimal maximum likelihood of data symbols, and almost identical with that by the simplified SLM scheme with perfect SI. Moreover, computational complexity is significantly lower than that of the B-SLM scheme.

Performance analysis of a suboptimum fast frequency-hopped/M-ary frequency-shift-keying maximum-likelihood receiver over Rician-fading channels with composite effects of partial-band noise jamming and multitone jamming

Bit-error rate (BER) expressions of a suboptimum maximum-likelihood (ML) receiver with the composite effect of partial-band noise jamming and multitone jamming are derived for fast frequency-hopped systems employing M-ary frequency-shift-keying modulation over Rician-fading channels. These BER expressions can be used to study the system performance under Rayleigh fading as a special case. Owing to the fact that the computation of the logarithm of the modified Bessel function is non-linear in nature, one of the existing approximations is required to derive the BER expressions for the analysed system. A suboptimum ML receiver structure is therefore proposed and analysed. The analytical results are compared with simulation results. Under various jamming and fading conditions, the proposed suboptimum ML receiver is found to be able to suppress interference effectively.

Noncoherent Maximum Likelihood Classification of Quadrature Amplitude Modulation Constellations: Simplification, Analysis, and Extension

This paper is concerned with classification of quadrature amplitude modulation (QAM) signals by using noncoherent maximum likelihood (ML) schemes, which are robust to frequency mismatch and phase shift. To reduce the high complexity involved in optimal noncoherent ML classification, we propose two suboptimal alternatives that perform well under various conditions. In particular, one suboptimal scheme, derived from high signal-to-noise ratio (SNR) approximation, is shown to provide close-to-optimal performance. We also present a general performance analysis for noncoherent ML classification of K types of QAM constellations, ending up with simplified integration for K ; 5. In addition, we investigate the asymptotic behavior of noncoherent ML classification assuming the number of samples used increases to infinity. The asymptotic performance of optimal as well as suboptimal noncoherent ML classification is established.

Adaptive recursive M-robust system parameter identification using the QQ-plot approach

A new adaptive algorithm for the robust estimation of parameters of linear dynamic discrete-time systems in the presence of non-Gaussian impulsive noise within a measurement sequence is proposed in this study. Starting from the theory of robust estimation, a simple, adaptive, practically applicable robust approximate maximum likelihood algorithm is derived that, in the cases of contaminated normal distribution of measurement noise, demonstrates a high level of efficiency. The QQ-plot technique, combined with data cleaning based on the robustified winsorisation technique, is used as a framework for the classification of sorted data into the class of regular normally distributed data and the class of irregular data belonging to the contaminating distribution with a variance that is much greater than nominal. The link between the QQ-plot technique and a specific linear regression is established, so that the estimation of statistical parameters of the contaminated measurement distribution is performed using the least-squares technique. Then, the suboptimal maximum likelihood criterion is defined, and the system parameter estimation problem is solved robustly, using the proposed recursive robust parameter estimation scheme. Simulation results illustrate the discussion and show the efficiency of the proposed adaptive recursive parameter estimation algorithm in the presence of glint spikes or outliers.

Iterative Suboptimal Maximum Likelihood Receiver for Nonlinearly Distorted SC-FDMA Symbols

Iterative Suboptimal Maximum Likelihood Receiver for Nonlinearly Distorted SC-FDMA Symbols

Error probability analysis of FFH/MFSK receivers over frequency-selective Rician-fading channels with partial-band-noise jamming

An optimum maximum-likelihood (ML) receiver structure for a fast frequency-hopped M-ary frequency-shift keying system over frequency-selective Rician-fading channels against partial-band-noise jamming and additive white Gaussian noise is proposed. In addition, we propose two suboptimum ML receivers and derive their corresponding analytical bit-error rate expressions.

Impact of imperfect channel estimation on the performance of amplify-and-forward relaying

In this paper, we investigate the error rate performance of amplify-and-forward (AF) relaying with imperfect channel estimation. We consider a single-relay scenario with orthogonal and non-orthogonal AF (OAF and NAF) cooperative protocols. Two pilot-symbol-assisted receiver architectures are studied: in the mismatched-coherent receiver, the complex fading channel coefficients (i.e., both phase and amplitude) are estimated based on a linear minimum-mean-squared-error estimation approach and fed to a coherent sub-optimal maximum likelihood decoder as if the channels were perfectly known. In the partially-coherent receiver, channel amplitude is ignored and phase is estimated by a phase locked loop. For both receiver types, we analyze the achievable diversity orders for cooperative protocols under consideration and quantify the impact of channel estimation through the derivation of pairwise error probability. Our performance analysis reveals that a second order diversity order is obtained for the considered single-relay scenario indicating that full diversity is extracted. Our simulation results demonstrate that the performance degradation due to channel estimation with respect to the genie bound (i.e., perfect channel state information) is as small as 1.1 dB based on the employed detector. Performance results further show that partially-coherent receiver presents a similar performance to mismatched-receiver for sufficiently large loop SNRs although channel amplitude is completely ignored.

Maximum likelihood DOA estimation in unknown colored noise fields

Direction-of-arrival (DOA) estimation in unknown noise environments is an important but challenging problem. Several methods based on maximum likelihood (ML) criteria and parameterization of signals or noise covariances have been established. Generally, to obtain the exact ML (EML) solutions, the DOAs must be jointly estimated along with other noise or signal parameters by optimizing a complicated nonlinear function over a high-dimensional problem space. Although the computation complexity can be reduced via derivation of suboptimal approximate ML (AML) functions using large sample assumption or least square criteria, nevertheless the AML estimators still require multi-dimensional search and the accuracy is lost to some extent. A particle swarm optimization (PSO) based solution is proposed here to compute the EML functions and explore the potential superior performances. A key characteristic of PSO is that the algorithm itself is highly robust yet remarkably simple to implement, while processing similar capabilities as other evolutionary algorithms such as the genetic algorithm (GA). Simulation results confirm the advantage of paring PSO with EML, and the PSO-EML estimator is shown to significantly outperform AML-based techniques in various scenarios at less computational costs.

A Suboptimal Maximum-Likelihood Receiver for FFH/BFSK Systems With Multitone Jamming Over Frequency-Selective Rayleigh-Fading Channels

Maximum-likelihood (ML) receivers have been investigated for the fast frequency-hopped binary frequency-shift-keying (FFH/BFSK) spread-spectrum communications system with the worst case of the band multitone jamming and the additive white Gaussian noise over frequency-selective Rayleigh-fading channels. A suboptimal ML diversity-combining receiver has been theoretically analyzed based upon an approximation of the natural logarithm of the modified Bessel function under a correlated two-path frequency-selective channel model. The performance of the receiver compared with that of other diversity-combining receivers, in terms of the bit error rate, has been scrutinized. Computer simulations confirm that our numerical analysis is valid, and the proposed suboptimal ML receiver outperforms other receivers.

Detection Schemes for Weak Signals in First-Order Moving Average of Impulsive Noise

In this paper, the detection of weak signals in additive noise described by the first-order moving average (FOMA) of an impulsive process is considered. Specifically, decision regions of the maximum likelihood (ML) and suboptimum ML (S-ML) detectors are derived in the FOMA noise model, and specific examples of the ML and S-ML decision regions are obtained. The ML and S-ML detectors are employed in the antipodal signaling system and compared in terms of bit error rate in an impulsive noise environment. Numerical results show that the S-ML detector, despite its reduced complexity and simpler structure, exhibits practically the same performance as the optimum ML detector. It is also observed that the performance gap between detectors for FOMA and independent and identically distributed noise becomes larger as the degree of noise impulsiveness increases

Interpolation Based Maximum-Likelihood (ML) Detection of Asynchronous Servo Repeatable Run Out (RRO) Data

This paper demonstrates the use of asynchronous maximum-likelihood (ML) detection to improve the detection performance of coded servo repeatable run out data. A suboptimal ML algorithm based on an absolute value metric is presented. This paper compares the performance of the ML algorithms with an asynchronous bit by bit (BBB) detection algorithm. Simulation results quantify the performance improvement over the BBB algorithm. A gain correction algorithm is also proposed to allow the asynchronous ML detection performance to be less sensitive to gain errors. The efficacy of the gain correction algorithm is quantified via simulations