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IEEE 1588, built on the classical two-way message exchange scheme, is a popular clock synchronization protocol for packet-switched networks. Due to the presence of random queuing delays in a packet-switched network, the joint recovery of the clock skew and offset from the timestamps of the exchanged synchronization packets can be treated as a statistical estimation problem. In this paper, we address the problem of clock skew and offset estimation for IEEE 1588 in the presence of possible unknown asymmetries between the deterministic path delays of the forward master-to-slave path and reverse slave-to-master path, which can result from incorrect modeling or cyber-attacks. First, we develop lower bounds on the mean square estimation error for a clock skew and offset estimation scheme for IEEE 1588 assuming the availability of multiple master-slave communication paths and complete knowledge of the probability density functions (pdf) describing the random queuing delays. Approximating the pdf of the random queuing delays by a mixture of Gaussian random variables, we then present a robust iterative clock skew and offset estimation scheme that employs the space alternating generalized expectation-maximization (SAGE) algorithm for learning all the unknown parameters. Numerical results indicate that the developed robust scheme exhibits a mean square estimation error close to the lower bounds. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of clock skew and offset estimation (CSOE) for the IEEE 1588 precision time protocol. Built on the classical two-way message exchange scheme, IEEE 1588 is a prominent synchronization protocol for packet switched networks. Due to the presence of random queuing delays in a packet switched network, the joint recovery of clock skew and offset from the received packet timestamps can be viewed as a statistical estimation problem. Recently, assuming perfect clock skew information, minimax optimum clock offset estimators were developed for IEEE 1588. Building on this work, we first develop joint optimum invariant clock skew and offset estimators for IEEE 1588 for known queuing delay statistics and unlimited computational complexity. We then show that the developed estimators are minimax optimum, i.e., these estimators minimize the maximum skew normalized mean square estimation error over all possible values of the unknown parameters. Minimax optimum estimators that utilize information from past timestamps to improve accuracy are also introduced. The developed optimum estimators provide useful fundamental limits for evaluating the performance of CSOE schemes. These performance limits can aid system designers to develop algorithms with the desired computational complexity that achieve performance close to the performance of the optimum estimators. If a designer finds an approach with a complexity they find acceptable and which provides performance close to the optimum performance, they can use it and know they have near optimum performance. This is precisely the approach used in communications when comparing to capacity. [ABSTRACT FROM AUTHOR]

It is revealed that there is a link between the quantization approach employed and the dimension of the vector parameter which can be accurately estimated by a quantized estimation system. A critical quantity called inestimable dimension for quantized data (IDQD) is introduced, which does not depend on the quantization regions and the statistical models of the observations but instead depends only on the number of sensors and on the precision of the vector quantizers employed by the system. It is shown that the IDQD describes a quantization-induced fundamental limitation on the estimation capabilities of the system. To be specific, if the dimension of the desired vector parameter is larger than the IDQD of the quantized estimation system, then the Fisher information matrix for estimating the desired vector parameter is singular, and, moreover, there exist infinitely many nonidentifiable vector parameter points in the vector parameter space. Furthermore, it is shown that under some common assumptions on the statistical models of the observations and the quantization system, a smaller IDQD can be obtained, which can specify an even more limiting quantization induced fundamental limitation on the estimation capabilities of the system. [ABSTRACT FROM AUTHOR]

Signal amplitude estimation and detection from unlabeled quantized binary samples are studied, assuming that the order of the time indexes is completely unknown. First, maximum likelihood (ML) estimators are utilized to estimate both the permutation matrix and unknown signal amplitude under arbitrary but known signal shape and quantizer thresholds. Sufficient conditions are provided, under which an ML estimator can be found in polynomial time, and an alternating maximization algorithm is proposed to solve the general problem via good initialization. In addition, the statistical identifiability of the model is studied. Furthermore, an approximation of the generalized likelihood ratio test detector is adopted to detect the presence of the signal. In addition, an accurate approximation of the probability of successful permutation matrix recovery is derived, and explicit expressions are provided to reveal the relationship between the signal length and the number of quantizers. Finally, numerical simulations are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of robust clock phase offset estimation for the IEEE 1588 precision time protocol in the presence of unknown asymmetric network path delays. The presence of an unknown asymmetry between the path delays can lead to significant degradation in the performance of a phase offset estimation scheme. Assuming multiple master-slave communication paths are available, we first provide lower bounds on the best achievable performance for a phase offset estimation scheme in the presence of asymmetric master-slave communication paths. We then present a novel phase offset estimation scheme that employs the expectation-maximization algorithm to identify the asymmetric master–slave communication paths. After discarding information from these paths, we employ the minimax optimum vector location parameter estimator for estimating the phase offset. Simulation results are presented to show that the proposed phase offset estimation scheme exhibits performance close to the lower bounds in a wide variety of network scenarios. [ABSTRACT FROM PUBLISHER]

We consider the problem of downlink channel estimation for millimeter wave (mmWave) MIMO-OFDM systems, where both the base station (BS) and the mobile station (MS) employ large antenna arrays for directional precoding/beamforming. Hybrid analog and digital beamforming structures are employed in order to offer a compromise between hardware complexity and system performance. Different from most existing studies that are concerned with narrowband channels, we consider estimation of wideband mmWave channels with frequency selectivity, which is more appropriate for mmWave MIMO-OFDM systems. By exploiting the sparse scattering nature of mmWave channels, we propose a CANDECOMP/PARAFAC (CP) decomposition-based method for channel parameter estimation (including angles of arrival/departure, time delays, and fading coefficients). In our proposed method, the received signal at the MS is expressed as a third-order tensor. We show that the tensor has the form of a low-rank CP, and the channel parameters can be estimated from the associated factor matrices. Our analysis reveals that the uniqueness of the CP decomposition can be guaranteed even when the size of the tensor is small. Hence the proposed method has the potential to achieve substantial training overhead reduction. We also develop Cramér-Rao bound (CRB) results for channel parameters and compare our proposed method with a compressed sensing-based method. Simulation results show that the proposed method attains mean square errors that are very close to their associated CRBs and present a clear advantage over the compressed sensing-based method. [ABSTRACT FROM AUTHOR]

Advances in the area of active controls for wave energy converter (WEC) devices have uncovered a need for accurate estimates of the actual waves that will arrive at the WEC devices in the near future, typically several seconds ahead. This paper considers the parametric estimation of these impending waves by employing a network of spatially distributed ocean sensors, possibly of different types, whose measurements are noisy. We provide general expressions for the Fisher information matrix and the Cramer–Rao Bound (CRB) under white Gaussian noise. The CRB is a lower bound on the smallest mean-square estimation error, which is nearly achievable for a sufficient number of observations. The expressions are general in the sense that the sensor network could be made up of a single or multiple types of sensors. We present a low-complexity estimation method that employs linear optimizations coupled with periodograms and phase unwrapping. Numerical results demonstrating the accuracy of the presented estimation method relative to the CRB are provided. [ABSTRACT FROM PUBLISHER]

Conventional compressed sensing theory assumes signals have sparse representations in a known dictionary. Nevertheless, in many practical applications such as line spectral estimation, the sparsifying dictionary is usually characterized by a set of unknown parameters in a continuous domain. To apply the conventional compressed sensing technique to such applications, the continuous parameter space has to be discretized to a finite set of grid points, based on which a “nominal dictionary” is constructed for sparse signal recovery. Discretization, however, inevitably incurs errors since the true parameters do not necessarily lie on the discretized grid. This error, also referred to as grid mismatch, leads to deteriorated recovery performance. In this paper, we consider the line spectral estimation problem and propose an iterative reweighted method which jointly estimates the sparse signals and the unknown parameters associated with the true dictionary. The proposed algorithm is developed by iteratively decreasing a surrogate function majorizing a given log-sum objective function, leading to a gradual and interweaved iterative process to refine the unknown parameters and the sparse signal. A simple yet effective scheme is developed for adaptively updating the regularization parameter that controls the tradeoff between the sparsity of the solution and the data fitting error. Theoretical analysis is conducted to justify the proposed method. Simulation results show that the proposed algorithm achieves super resolution and outperforms other state-of-the-art methods in many cases of practical interest. [ABSTRACT FROM PUBLISHER]

In this letter, we study the problem of estimating a complex parameter in a wireless sensor network under individual power constraints and bounded channel uncertainty, where the channel state information is assumed to be imperfect at both the sensors and the fusion center. Transmit beamformers and a linear estimator that minimize the maximum mean square error are found. Although this problem is nonconvex, it can be relaxed to become a semidefinite programming problem by re-parameterization and semidefinite relaxation. Surprisingly, we find that relaxation is tight and a global optimal solution can be found. Finally, numerical simulations are performed to evaluate the performance of the robust estimator. [ABSTRACT FROM PUBLISHER]

The IEEE 1588 protocol has received recent interest as a means of delivering sub-microsecond level clock phase synchronization over packet-switched mobile backhaul networks. Due to the randomness of the end-to-end delays in packet networks, the recovery of clock phase from packet timestamps in IEEE 1588 must be treated as a statistical estimation problem. A number of estimators for this problem have been suggested in the literature, but little is known about the best achievable performance. In this paper, we describe new minimax estimators for this problem, that are optimum in terms of minimizing the maximum mean squared error over all possible values of the unknown parameters. Minimax estimators that utilize information from past timestamps to improve accuracy are also introduced. Simulation results indicate that significant performance gains over conventional estimators can be obtained via such optimum processing techniques. These minimax estimators also provide fundamental limits on the performance of phase offset estimation schemes. [ABSTRACT FROM AUTHOR]

Distributed estimation of a deterministic mean-shift parameter in additive zero-mean noise is studied when using quantized data in the presence of Byzantine attacks. Several subsets of sensors are assumed to be tampered with by adversaries using different attacks such that the compromised sensors transmit fictitious data. First, we consider the task of identifying and categorizing the attacked sensors into different groups according to distinct types of attacks. It is shown that increasing the number K of time samples at each sensor and enlarging the size N of the sensor network can both ameliorate the identification and categorization, but to different extents. As K\rightarrow\infty, the attacked sensors can be perfectly identified and categorized, while with finite but sufficiently large K, as N\rightarrow\infty, it can be shown that the fusion center can also ascertain the number of attacks and obtain an approximate categorization with a sufficiently small percentage of sensors that are misclassified. Next, in order to improve the estimation performance by utilizing the attacked observations, we consider joint estimation of the statistical description of the attacks and the parameter to be estimated after the sensors have been well categorized. When using the same quantization approach successfully employed without attacks, it can be shown that the corresponding Fisher Information Matrix (FIM) is singular. To overcome this, a time-variant quantization approach is proposed, which will provide a nonsingular FIM, provided that K\geq 2. Furthermore, the FIM is employed to provide necessary and sufficient conditions under which utilizing the compromised sensors in the proposed fashion will lead to better estimation performance when compared to approaches where the compromised sensors are ignored. [ABSTRACT FROM PUBLISHER]

In this paper, we describe new lower bounds on error variance of phase offset estimation schemes used in IEEE 1588 based synchronization. The motivation for this study is to determine the feasibility of providing microsecond-level time synchronization over mobile backhaul networks for the backend in 4G cellular systems. Many packet filtering/selection techniques for phase offset estimation have been proposed in synchronization literature, however, lower limits on the performance of such estimators have not yet been described. In this paper, we re-derive two Bayesian estimation bounds, namely the Ziv-Zakai and Weiss-Weinstien bounds, for use under a non-Bayesian formulation. This enables us to apply these bounds to the problem of phase offset estimation. Simulation results compare the performance of existing estimation schemes against these lower bounds under a variety of different network scenarios. [ABSTRACT FROM AUTHOR]

Abstract: MIMO radar with properly placed antennas that employs a coherent processing approach can provide superior MSE performance, as indicated by recent work. This paper demonstrates that the magnitude of these gains decreases with an increase in the product of the number of transmit and receive antennas if the antennas for the noncoherent system are also suitably placed, using a placement which is generally different from the one for the coherent processing approach. Initially, we study the systems without constraining the complexity and energy, where each added transmit antenna employs a fixed energy so that the total transmitted energy is allowed to increase as we increase the number of transmit antennas. Later we also look at constrained systems, where adding a transmit antenna splits the total system energy and the total number of antennas employed is restricted. A rigorous theorem is presented for the case of orthogonal signals in temporally and spatially white clutter-plus-noise, but numerical results for nonorthogonal signals and colored clutter-plus-noise follow a similar pattern. [Copyright &y& Elsevier]

A discretized version of a continuous optimization problem is considered for the case where data is obtained from a set of dispersed sensor nodes and the overall metric is a sum of individual metrics computed at each sensor. An example of such a problem is maximum-likelihood estimation based on statistically independent sensor observations. By ordering transmissions from the sensor nodes, a method for achieving a saving in the average number of sensor transmissions is described. While the average number of sensor transmissions is reduced, the approach always yields the same solution as the optimum approach where all sensor transmissions occur. The approach is described first for a general optimization problem. A maximum-likelihood target location and velocity estimation example for a multiple node noncoherent multiple-input multiple-output (MIMO) radar system is later described. In particular, for cases with N good quality sensors with ideal signals and sufficiently large signal-to-interference-plus-noise ratio (SINR), the average percentage of transmissions saved approaches 100% as the number of discrete grid points in the optimization problem Q becomes significantly large. In these same cases, the average percentage of transmissions saved approaches (Q-1)/ Q\times \100% as the number of sensors N in the network becomes significantly large. Similar savings are illustrated for general optimization (or estimation) problems with some sufficiently well-designed sensors. Savings can be even larger in some cases for systems with some poor quality sensors. [ABSTRACT FROM AUTHOR]

Abstract: Image fusion algorithms attempt to produce a single fused image that is more informative than any of the multiple source images used to produce the fused image. Analytical studies of image fusion performance have been lacking. Such studies can augment existing experimental studies by addressing some aspects that are difficult to study using experimental methods. Here, an estimation theory approach is employed using a mathematical model based on the observation that each different sensor can provide a different quality when viewing a given object in the scene. One sensor may be better for viewing one object and a different sensor may be better for viewing a different object. The model also acknowledges that distortion and noise will enter into the sensor observations. This model allows us to employ known estimation theory techniques to find the best possible fusion performance, measured in terms of the standard estimation theory measure of performance. This performance measure has not yet received attention in the image fusion community. Some interesting results include the demonstration that a particular weighted averaging approach is shown to yield optimum estimation performance for the model we focus on. It is also shown that it is important to employ a priori information that describes which sensor is able to provide a good view of the important objects in the scene. The essential aspects of some frequently employed fusion approaches are studied and the capabilities of these approaches are analyzed and compared to the best fusion algorithms. We hope this study will encourage further analytical studies of image fusion. [Copyright &y& Elsevier]

Image fusion is studied using a very basic and reasonable mathematical model for the observed images. The model attempts to characterize the statistical aspects of the problem, including the impact of random distortions, like noise. The image fusion problem is posed as an estimation problem where the best fusion algorithm minimizes the mean square error between the fused image and the true scene. The optimum image fusion approach is described for the case where all the parameters of the model are known. A robust image fusion approach is proposed for cases where various parameters of the model are unknown which may be the case in practice. It is shown that the robust image fusion approach will provide a mean square error which is always smaller than a given bound, thus limiting the loss from not knowing the exact model parameters. Further, our results imply that ignoring correlation between the noise from different sensors is a robust approach, a fact which has not been rigorously demonstrated elsewhere. Numerical results are presented which further verify the robustness of the proposed approach. [Copyright &y& Elsevier]