CRLB Based Optimal Noise Enhanced Parameter Estimation Using Quantized Observations

Cataloged from PDF version of article.In this letter, optimal additive noise is characterized for parameter estimation based on quantized observations. First, optimal probability distribution of noise that should be added to observations is formulated in terms of a Cramer–Rao lower bound (CRLB) minimization problem. Then, it is proven that optimal additive “noise” can be represented by a constant signal level, which means that randomization of additive signal levels is not needed for CRLB minimization. In addition, the results are extended to the cases in which there exists prior information about the unknown parameter and the aim is to minimize the Bayesian CRLB (BCRLB). Finally, a numerical example is presented to explain the theoretical results

On Distributed Linear Estimation With Observation Model Uncertainties

We consider distributed estimation of a Gaussian source in a heterogenous bandwidth constrained sensor network, where the source is corrupted by independent multiplicative and additive observation noises, with incomplete statistical knowledge of the multiplicative noise. For multi-bit quantizers, we derive the closed-form mean-square-error (MSE) expression for the linear minimum MSE (LMMSE) estimator at the FC. For both error-free and erroneous communication channels, we propose several rate allocation methods named as longest root to leaf path, greedy and integer relaxation to (i) minimize the MSE given a network bandwidth constraint, and (ii) minimize the required network bandwidth given a target MSE. We also derive the Bayesian Cramer-Rao lower bound (CRLB) and compare the MSE performance of our proposed methods against the CRLB. Our results corroborate that, for low power multiplicative observation noises and adequate network bandwidth, the gaps between the MSE of our proposed methods and the CRLB are negligible, while the performance of other methods like individual rate allocation and uniform is not satisfactory

Performance Analysis for Time-of-Arrival Estimation with Oversampled Low-Complexity 1-bit A/D Conversion

Analog-to-digtial (A/D) conversion plays a crucial role when it comes to the design of energy-efficient and fast signal processing systems. As its complexity grows exponentially with the number of output bits, significant savings are possible when resorting to a minimum resolution of a single bit. However, then the nonlinear effect which is introduced by the A/D converter results in a pronounced performance loss, in particular for the case when the receiver is operated outside the low signal-to-noise ratio (SNR) regime. By trading the A/D resolution for a moderately faster sampling rate, we show that for time-of-arrival (TOA) estimation under any SNR level it is possible to obtain a low-complexity $1$-bit receive system which features a smaller performance degradation then the classical low SNR hard-limiting loss of $2/\pi$ ($-1.96$ dB). Key to this result is the employment of a lower bound for the Fisher information matrix which enables us to approximate the estimation performance for coarsely quantized receivers with correlated noise models in a pessimistic way

Performance Analysis for Time-of-Arrival Estimation with Oversampled Low-Complexity 1-bit A/D Conversion

Analog-to-digtial (A/D) conversion plays a crucial role when it comes to the design of energy-efficient and fast signal processing systems. As its complexity grows exponentially with the number of output bits, significant savings are possible when resorting to a minimum resolution of a single bit. However, then the nonlinear effect which is introduced by the A/D converter results in a pronounced performance loss, in particular for the case when the receiver is operated outside the low signal-to-noise ratio (SNR) regime. By trading the A/D resolution for a moderately faster sampling rate, we show that for time-of-arrival (TOA) estimation under any SNR level it is possible to obtain a low-complexity $1$-bit receive system which features a smaller performance degradation then the classical low SNR hard-limiting loss of $2/\pi$ ($-1.96$ dB). Key to this result is the employment of a lower bound for the Fisher information matrix which enables us to approximate the estimation performance for coarsely quantized receivers with correlated noise models in a pessimistic way

Noise enhanced parameter estimation using quantized observations

Ankara : The Department of Electrical and Electronics Engineering and the Institute of Engineering and Sciences of Bilkent University 2010.Thesis (Master's) -- Bilkent University, 2010.Includes bibliographical references leaves 54-58.In this thesis, optimal additive noise is characterized for both single and multiple parameter estimation based on quantized observations. In both cases, first, optimal probability distribution of noise that should be added to observations is formulated in terms of a Cramer-Rao lower bound (CRLB) minimization problem. In the single parameter case, it is proven that optimal additive “noise” can be represented by a constant signal level, which means that randomization of additive signal levels (equivalently, quantization levels) are not needed for CRLB minimization. In addition, the results are extended to the cases in which there exists prior information about the unknown parameter and the aim is to minimize the Bayesian CRLB (BCRLB). Then, numerical examples are presented to explain the theoretical results. Moreover, performance obtained via optimal additive noise is compared to performance of the commonly used dither signals. Furthermore, mean-squared error (MSE) performances of maximum likelihood (ML) and maximum a-posteriori probability (MAP) estimates are investigated in the presence and absence of additive noise. In the multiple parameter case, the form of the optimal random additive noise is derived for CRLB minimization. Next, the theoretical result is supported with a numerical example, where the optimum noise is calculated by using the particle swarm optimization (PSO) algorithm. Finally, the optimal constant noise in the multiple parameter estimation problem in the presence of prior information is discussed.Balkan, Gökçe OsmanM.S

On Distributed Estimation for Resource Constrained Wireless Sensor Networks

We study Distributed Estimation (DES) problem, where several agents observe a noisy version of an underlying unknown physical phenomena (which is not directly observable), and transmit a compressed version of their observations to a Fusion Center (FC), where collective data is fused to reconstruct the unknown. One of the most important applications of Wireless Sensor Networks (WSNs) is performing DES in a field to estimate an unknown signal source. In a WSN battery powered geographically distributed tiny sensors are tasked with collecting data from the field. Each sensor locally processes its noisy observation (local processing can include compression, dimension reduction, quantization, etc) and transmits the processed observation over communication channels to the FC, where the received data is used to form a global estimate of the unknown source such that the Mean Square Error (MSE) of the DES is minimized. The accuracy of DES depends on many factors such as intensity of observation noises in sensors, quantization errors in sensors, available power and bandwidth of the network, quality of communication channels between sensors and the FC, and the choice of fusion rule in the FC. Taking into account all of these contributing factors and implementing a DES system which minimizes the MSE and satisfies all constraints is a challenging task. In order to probe into different aspects of this challenging task we identify and formulate the following three problems and address them accordingly: 1- Consider an inhomogeneous WSN where the sensors\u27 observations is modeled linear with additive Gaussian noise. The communication channels between sensors and FC are orthogonal power and bandwidth-constrained erroneous wireless fading channels. The unknown to be estimated is a Gaussian vector. Sensors employ uniform multi-bit quantizers and BPSK modulation. Given this setup, we ask: what is the best fusion rule in the FC? what is the best transmit power and quantization rate (measured in bits per sensor) allocation schemes that minimize the MSE? In order to answer these questions, we derive some upper bounds on global MSE and through minimizing those bounds, we propose various resource allocation schemes for the problem, through which we investigate the effect of contributing factors on the MSE. 2- Consider an inhomogeneous WSN with an FC which is tasked with estimating a scalar Gaussian unknown. The sensors are equipped with uniform multi-bit quantizers and the communication channels are modeled as Binary Symmetric Channels (BSC). In contrast to former problem the sensors experience independent multiplicative noises (in addition to additive noise). The natural question in this scenario is: how does multiplicative noise affect the DES system performance? how does it affect the resource allocation for sensors, with respect to the case where there is no multiplicative noise? We propose a linear fusion rule in the FC and derive the associated MSE in closed-form. We propose several rate allocation schemes with different levels of complexity which minimize the MSE. Implementing the proposed schemes lets us study the effect of multiplicative noise on DES system performance and its dynamics. We also derive Bayesian Cramer-Rao Lower Bound (BCRLB) and compare the MSE performance of our porposed methods against the bound. As a dual problem we also answer the question: what is the minimum required bandwidth of the network to satisfy a predetermined target MSE? 3- Assuming the framework of Bayesian DES of a Gaussian unknown with additive and multiplicative Gaussian noises involved, we answer the following question: Can multiplicative noise improve the DES performance in any case/scenario? the answer is yes, and we call the phenomena as \u27enhancement mode\u27 of multiplicative noise. Through deriving different lower bounds, such as BCRLB,Weiss-Weinstein Bound (WWB), Hybrid CRLB (HCRLB), Nayak Bound (NB), Yatarcos Bound (YB) on MSE, we identify and characterize the scenarios that the enhancement happens. We investigate two situations where variance of multiplicative noise is known and unknown. We also compare the performance of well-known estimators with the derived bounds, to ensure practicability of the mentioned enhancement modes