An isotropic universal decentralized estimation scheme for a bandwidth constrained ad hoc sensor network

Consider a decentralized estimation problem whereby an ad hoc network of K distributed sensors wish to cooperate to estimate an unknown parameter over a bounded interval [-U,U]. Each sensor collects one noise-corrupted sample, performs a local data quantization according to a fixed (but possibly probabilistic) rule, and transmits the resulting discrete message to its neighbors. These discrete messages are then percolated in the network and used by each sensor to form its own minimum mean squared error (MMSE) estimate of the unknown parameter according to a fixed fusion rule. In this paper, we propose a simple probabilistic local quantization rule: each sensor quantizes its observation to the first most significant bit (MSB) with probability 1/2, the second MSB with probability 1/4, and so on. Assuming the noises are uncorrelated and identically distributed across sensors and are bounded to [-U,U], we show that this local quantization strategy together with a fusion rule can guarantee a MSE of 4U/sup 2//K, and that the average length of local messages is bounded (no more than 2.5 bits). Compared with the worst case Cramer-Rao lower bound of U/sup 2//K (even for the centralized counterpart), this is within a factor of at most 4 to the minimum achievable MSE. Moreover, the proposed scheme is isotropic and universal in the sense that the local quantization rules and the final fusion rules are independent of sensor index, noise distribution, network size, or topology. In fact, the proposed scheme allows sensors in the network to operate identically and autonomously even when the network undergoes changes in size or topology.     

Power scheduling of universal decentralized estimation in sensor networks

We consider the optimal power scheduling problem for the decentralized estimation of a noise-corrupted deterministic signal in an inhomogeneous sensor network. Sensor observations are first quantized into discrete messages, then transmitted to a fusion center where a final estimate is generated. Supposing that the sensors use a universal decentralized quantization/estimation scheme and an uncoded quadrature amplitude modulated (QAM) transmission strategy, we determine the optimal quantization and transmit power levels at local sensors so as to minimize the total transmit power, while ensuring a given mean squared error (mse) performance. The proposed power scheduling scheme suggests that the sensors with bad channels or poor observation qualities should decrease their quantization resolutions or simply become inactive in order to save power. For the remaining active sensors, their optimal quantization and transmit power levels are determined jointly by individual channel path losses, local observation noise variance, and the targeted mse performance. Numerical examples show that in inhomogeneous sensing environment, significant energy savings is possible when compared to the uniform quantization strategy.     

An indicator kriging method for distributed estimation in wireless sensor networks

The problem of distributed estimation in a wireless sensor network with unknown observation noise distribution is investigated, where each sensor only sends quantized data to a fusion center. The sensing field is modeled as a spatially random field. The objective was to accurately estimate a hidden parameter at the location where no sensor exists, while minimizing the total energy consumption. Driven by the lack of a prior knowledge of the sensing field and the existence of some outliers, an indicator kriging estimator is developed for distributed estimation under imperfect communication channels between the sensors and the fusion center. The tradeoff between estimation performance and energy consumption is formulated as an optimization problem, and a global search algorithm is proposed to approximate the solution. The results show that the proposed indicator kriging estimator achieves better performance than the inverse distance estimator and the simple averaging estimator. Moreover, the proposed search algorithm can schedule the sensors to reach the tradeoff. Copyright © 2012 John Wiley & Sons, Ltd.     

Position‐based adaptive quantization for target location estimation in wireless sensor networks using one‐bit data

The problem of target location estimation in a wireless sensor network is considered, where due to the bandwidth and power constraints, each sensor only transmits one‐bit information to its fusion center. To improve the performance of estimation, a position‐based adaptive quantization scheme for target location estimation in wireless sensor networks is proposed to make a good choice of quantizer' thresholds. By the proposed scheme, each sensor node dynamically adjusts its quantization threshold according to a kind of position‐based information sequences and then sends its one‐bit quantized version of the original observation to a fusion center. The signal intensity received at local sensors is modeled as an isotropic signal intensity attenuation model. The position‐based maximum likelihood estimator as well as its corresponding position‐based Cramér–Rao lower bound are derived. Numerical results show that the position‐based maximum likelihood estimator is more accurate than the classical fixed‐quantization maximum likelihood estimator and the position‐based Cramér–Rao lower bound is less than its fixed‐quantization Cramér‐Rao lower bound. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear coherent distributed estimation over unknown channels

We study linear distributed estimation with coherent multiple access channel model and MMSE fusion rule. The flat fading channels are assumed unknown at the fusion center and need to be estimated. We adopt a two-phase approach, which first estimates channels and then estimates the source signal, to minimize the MSE of the estimated signal. We study optimal power allocation under a total network power constraint. We consider the optimal power allocation scheme in which training power and data power for each sensor are optimized, and the equal power allocation scheme in which training power is optimized while data power for each sensor is set equal. In both schemes, the problem is formulated as a constrained optimization problem and analytical closed-form solution is obtained. Analytic results reveal that (i) with estimated channels, the MSE approaches to a finite nonzero value as the number of sensors increases; (ii) the optimal training powers are the same in both schemes; (iii) the MSE performance compared with the case when channels are known shows the penalty caused by channel estimation becomes worse as the number of sensors increases. Simulation results verify our findings.     

On the Joint Synchronization of Clock Offset and Skew in RBS-Protocol

Motivated by the necessity of having a good clock synchronization amongst the nodes of wireless ad-hoc sensor networks, the joint maximum likelihood (JML) estimator for clock phase offset and skew under exponential noise model for reference broadcast synchronization (RBS) protocol is formulated and found via a direct algorithm. The Gibbs sampler is also proposed for joint clock phase offset and skew estimation and shown to provide superior performance relative to JML- estimator. Lower and upper bounds for the mean-square errors (MSE) of JML-estimator and Gibbs Sampler are introduced in terms of the MSE of the uniform minimum variance unbiased (UMVU) estimator and the conventional best linear unbiased estimator (BLUE), respectively.     

BLUE‐Based Channel Estimation Technique for Amplify and Forward Wireless Relay Networks

The best linear unbiased estimator (BLUE) is most suitable for practical application and can be determined with knowledge of only the first and second moments of the probability density function. Although the BLUE is an existing algorithm, it is still largely unexplored and has not yet been applied to channel estimation in amplify and forward (AF)‐based wireless relay networks (WRNs). In this paper, a BLUE‐based algorithm is proposed to estimate the overall channel impulse response between the source and destination of AF strategy‐based WRNs. Theoretical mean square error (MSE) performance for the BLUE is derived to show the accuracy of the proposed channel estimation algorithm. In addition, the Cramér‐Rao lower bound (CRLB) is derived to validate the MSE performance. The proposed BLUE channel estimation algorithm approaches the CRLB as the length of the training sequence and number of relays increases. Further, the BLUE performs better than the linear minimum MSE estimator due to the minimum variance characteristic exhibited by the BLUE, which happens to be a function of signal‐to‐noise ratio.     

A minimum misadjustment adaptive FIR filter

The performance of an adaptive filter is limited by the misadjustment resulting from the variance of adapting parameters. This paper develops a method to reduce the misadjustment when the additive noise in the desired signal is correlated. Given a static linear model, the linear estimator that can achieve the minimum parameter variance estimate is known as the best linear unbiased estimator (BLUE). Starting from classical estimation theory and a Gaussian autoregressive (AR) noise model, a maximum likelihood (ML) estimator that jointly estimates the filter parameters and the noise statistics is established. The estimator is shown to approach the best linear unbiased estimator asymptotically. The proposed adaptive filtering method follows by modifying the commonly used mean-square error (MSE) criterion in accordance with the ML cost function. The new configuration consists of two adaptive components: a modeling filter and a noise whitening filter. Convergence study reveals that there is only one minimum in the error surface, and global convergence is guaranteed. Analysis of the adaptive system when optimized by LMS or RLS is made, together with the tracking capability investigation. The proposed adaptive method performs significantly better than a usual adaptive filter with correlated additive noise and tracks a time-varying system more effectively     

Energy efficiency of dense wireless sensor networks: to cooperate or not to cooperate

Decentralized detection in a network of wireless sensor nodes involves the fusion of information about a phenomenon of interest (PoI) from geographically dispersed nodes. In this paper, we investigate the problem of binary decentralized detection in a dense and randomly deployed wireless sensor network (WSN), whereby the communication channels between the nodes and the fusion center are bandwidth-constrained. We consider a scenario in which sensor observations, conditioned on the alternate hypothesis, are independent but not identically distributed across the sensor nodes. We compare two different fusion architectures, namely, the parallel fusion architecture (PFA) and the cooperative fusion architecture (CFA), for such bandwidth-constrained WSNs, where each sensor node is restricted to send a I-bit information to the fusion center. For each architecture, we derive expression for the probability of decision error at the fusion center. We propose a consensus flooding protocol for CFA and analyze its average energy consumption. We analyze the effects of PoI intensity, realistic link models, consensus flooding protocol, and network connectivity on the system reliability and average energy consumption for both fusion architectures. We demonstrate that a trade-off exists among spatial diversity gain, average energy consumption, delivery ratio of the consensus flooding protocol, network connectivity, node density, and Poll intensity in CFA. We then provide insight into the design of cooperative WSNs     

On rate-constrained distributed estimation in unreliable sensor networks

We study the problem of estimating a physical process at a central processing unit (CPU) based on noisy measurements collected from a distributed, bandwidth-constrained, unreliable, network of sensors, modeled as an erasure network of unreliable "bit-pipes" between each sensor and the CPU. The CPU is guaranteed to receive data from a minimum fraction of the sensors and is tasked with optimally estimating the physical process under a specified distortion criterion. We study the noncollaborative (i.e., fully distributed) sensor network regime, and derive an information-theoretic achievable rate-distortion region for this network based on distributed source-coding insights. Specializing these results to the Gaussian setting and the mean-squared-error (MSE) distortion criterion reveals interesting robust-optimality properties of the solution. We also study the regime of clusters of collaborative sensors, where we address the important question: given a communication rate constraint between the sensor clusters and the CPU, should these clusters transmit their "raw data" or some low-dimensional "local estimates"? For a broad set of distortion criteria and sensor correlation statistics, we derive conditions under which rate-distortion-optimal compression of correlated cluster-observations separates into the tasks of dimension-reducing local estimation followed by optimal distributed compression of the local estimates.     

Asymptotic Optimality Theory for Decentralized Sequential Hypothesis Testing in Sensor Networks

The decentralized sequential hypothesis testing problem is studied in sensor networks, where a set of sensors receive independent observations and send summary messages to the fusion center, which makes a final decision. In the scenario where the sensors have full access to their past observations, the first asymptotically Bayes sequential test is developed having the same asymptotic performance as the optimal centralized test that has access to all sensor observations. Next, in the scenario where the sensors do not have full access to their past observations, a simple but asymptotically Bayes sequential tests is developed, in which sensor message functions are what we call tandem quantizer, where each sensor only uses two different sensor quantizers with at most one switch between these two possibilities. Moreover, a new minimax formulation of optimal stationary sensor quantizers is proposed and is studied in detail in the case of additive Gaussian sensor noise. Finally, our results show that in the simplest models, feedback from the fusion center does not improve asymptotic performance in the scenario with full local memory, however, even a one-shot, one-bit feedback can significantly improve performance in the case of limited local memory.     

PAC vs. MAC for Decentralized Detection Using Noncoherent Modulation

In decentralized detection, local sensor observations have to be communicated to a fusion center through the wireless medium, inherently a multiple-access channel (MAC). As communication is bandwidth- and energy-constrained, it has been suggested to use the properties of the MAC to combine the sensor observations directly on the channel. Although this leads to an array-processing gain if the sensors' transmissions combine coherently on the channel, it has been shown that this is not the case when they combine noncoherently. We review known results for the coherent case and then analyze the noncoherent case based on a simple on/off scheme combined with optimal sensor “censoring.” Since the optimal forwarding function is not available, we also bound the performance using an equivalent communication problem and a centralized estimator to verify trends. We find that for noncoherent modulation, there is no processing gain using the MAC for decentralized detection, but compared to parallel-access channels (PACs) the MAC avoids the noncoherent combining loss. Still the performance of the MAC approach is only of diversity one, as the output of the MAC is approximately a zero-mean complex Gaussian random variable for a large number of sensor. The MAC performance can be increased by using multiple independent channels, each used as a MAC by all sensors, which we term diversity-MAC. This approach always outperforms the PAC scheme on Rayleigh fading channels, where the output is exactly Gaussian, but has inferior performance across random phase channels when few sensors are used, as the PAC does not create “artificial” fading.      

集中式BLUE滤波器在组网雷达中的应用

为解决组网雷达对目标跟踪中的量测非线性问题,提出基于最佳线性无偏估计器(BLUE)准则的融合滤波方法。建立以融合中心为原点的组网雷达对目标定位的量测方程,推导出极坐标系与球坐标系下跟踪目标的BLUE滤波模型。理论分析表明,集中式BLUE滤波架构在估计单个雷达量测转换误差统计特性的同时,还估计出雷达间量测转换误差的统计特性。因此,跟踪精度和置信度较分布式BLUE滤波方法有显著提高,计算量较其他算法也有明显优势。不同场景下的仿真分析证明:该方法在不同状态噪声水平下的表现优异,是一种很有竞争力的跟踪算法。     

Information Regularized Sensor Fusion: Application to Localization With Distributed Motion Sensors

We propose the information regularization principle for fusing information from sets of identical sensors observing a target phenomenon. The principle basically proposes an importance-weighting scheme for each sensor measurement based on the mutual information based pairwise statistical similarity matrix between sensors. The principle is applied to maximum likelihood estimation and particle filter based state estimation. A demonstration of the proposed regularization scheme in centralized data fusion of dense motion detector networks for target tracking is provided. Simulations confirm that the introduction of information regularization significantly improves localization accuracy of both maximum likelihood and particle filter approaches compared to their baseline implementations. Outlier detection and sensor failure detection capabilities, as well as possible extensions of the principle to decentralized sensor fusion with communication constraints are briefly discussed.     

Decision fusion in mobile wireless sensor networks using cooperative multiple symbol differential space time coding

In this article, we propose a cooperative scheme for differential space-time codes (DSTCs) to be applied for mobile wireless sensor networks (WSNs) in order to mitigate multipath fading effect. We assume that sensors make independent local decisions about the existing hypothesis and report their decisions to a fusion center, where the final decision is made. Sensors are divided into groups with two sensors each, where sensors in each pair cooperate to send their decisions as a DSTC. Differential modulation scheme, which does not require knowledge of the instantaneous fading gains, is considered to avoid the channel estimation overhead at the cooperating sensors and the fusion center. Channels between sensors and the fusion center are assumed independent identically distributed (i.i.d) Rayleigh fading channels. Moreover, Jakes-Clarkes’ channel model is considered to model the mobility of sensors and/or the fusion center. Since the complexity of the optimal fusion rule grows up exponentially with the observation interval, suboptimal fusion rules are derived and discussed. Finally, simulation results of the proposed cooperative scheme are provided and the detection capabilities of the derived decision fusion rules are compared.     

Covariance shaping least-squares estimation

A new linear estimator is proposed, which we refer to as the covariance shaping least-squares (CSLS) estimator, for estimating a set of unknown deterministic parameters, x, observed through a known linear transformation H and corrupted by additive noise. The CSLS estimator is a biased estimator directed at improving the performance of the traditional least-squares (LS) estimator by choosing the estimate of x to minimize the (weighted) total error variance in the observations subject to a constraint on the covariance of the estimation error so that we control the dynamic range and spectral shape of the covariance of the estimation error. The presented CSLS estimator is shown to achieve the Cramer-Rao lower bound for biased estimators. Furthermore, analysis of the mean-squared error (MSE) of both the CSLS estimator and the LS estimator demonstrates that the covariance of the estimation error can be chosen such that there is a threshold SNR below which the CSLS estimator yields a lower MSE than the LS estimator for all values of x. As we show, some of the well-known modifications of the LS estimator can be formulated as CSLS estimators. This allows us to interpret these estimators as the estimators that minimize the total error variance in the observations, among all linear estimators with the same covariance.     

Decentralized Detection With Censoring Sensors

In the censoring approach to decentralized detection, sensors transmit real-valued functions of their observations when "informative" and save energy by not transmitting otherwise. We address several practical issues in the design of censoring sensor networks including the joint dependence of sensor decision rules, randomization of decision strategies, and partially known distributions. In canonical decentralized detection problems involving quantization of sensor observations, joint optimization of the sensor quantizers is necessary. We show that under a send/no-send constraint on each sensor and when the fusion center has its own observations, the sensor decision rules can be determined independently. In terms of design, and particularly for adaptive systems, the independence of sensor decision rules implies that minimal communication is required. We address the uncertainty in the distribution of the observations typically encountered in practice by determining the optimal sensor decision rules and fusion rule for three formulations: a robust formulation, generalized likelihood ratio tests, and a locally optimum formulation. Examples are provided to illustrate the independence of sensor decision rules, and to evaluate the partially known formulations.     

Decentralized detection in sensor networks

In this paper, we investigate a binary decentralized detection problem in which a network of wireless sensors provides relevant information about the state of nature to a fusion center. Each sensor transmits its data over a multiple access channel. Upon reception of the information, the fusion center attempts to accurately reconstruct the state of nature. We consider the scenario where the sensor network is constrained by the capacity of the wireless channel over which the sensors are transmitting, and we study the structure of an optimal sensor configuration. For the problem of detecting deterministic signals in additive Gaussian noise, we show that having a set of identical binary sensors is asymptotically optimal, as the number of observations per sensor goes to infinity. Thus, the gain offered by having more sensors exceeds the benefits of getting detailed information from each sensor. A thorough analysis of the Gaussian case is presented along with some extensions to other observation distributions.     

Distributed Detection in Wireless Sensor Networks Using A Multiple Access Channel

Distributed detection in a one-dimensional (1-D) sensor network with correlated sensor observations, as exemplified by two problems-detection of a deterministic signal in correlated Gaussian noise and detection of a first-order autoregressive [AR(1)] signal in independent Gaussian noise, is studied in this paper. In contrast with the traditional approach where a bank of dedicated parallel access channels (PAC) is used for transmitting the sensor observations to the fusion center, we explore the possibility of employing a shared multiple access channel (MAC), which significantly reduces the bandwidth requirement or detection delay. We assume that local observations are mapped according to a certain function subject to a power constraint. Using the large deviation approach, we demonstrate that for the deterministic signal in correlated noise problem, with a specially chosen mapping rule, MAC fusion achieves the same asymptotic performance as centralized detection under the average power constraint (APC), while there is always a loss in error exponents associated with PAC fusion. Under the total power constraint (TPC), MAC fusion still results in exponential decay in error exponents with the number of sensors, while PAC fusion does not. For the AR signal problem, we propose a suboptimal MAC mapping rule which performs closely to centralized detection for weakly correlated signals at almost all signal-to-noise ratio (SNR) values, and for heavily correlated signals when SNR is either high or low. Finally, we show that although the lack of MAC synchronization always causes a degradation in error exponents, such degradation is negligible when the phase mismatch among sensors is sufficiently small     

Large system decentralized detection performance under communication constraints

The problem of decentralized detection in a sensor network subjected to a total average power constraint and all nodes sharing a common bandwidth is investigated. The bandwidth constraint is taken into account by assuming non-orthogonal communication between sensors and the data fusion center via direct-sequence code-division multiple-access (DS-CDMA). In the case of large sensor systems and random spreading, the asymptotic decentralized detection performance is derived assuming independent and identically distributed (iid) sensor observations via random matrix theory. The results show that, even under both power and bandwidth constraints, it is better to combine many not-so-good local decisions rather than relying on one (or a few) very-good local decisions.