Corresponding Cramer-Rao Bound Research Articles

A novel calibration algorithm and its effects on heading measurement accuracy of a low-cost magnetometer

The Earth's magnetic heading plays a critical role in numerous applications, including navigation and orientation. The digital compass, which is made with a three-axis magnetometer, has proven to be a very cost-effective tool for providing measurements of the Earth’s magnetic heading. To obtain accurate heading measurements for end users, magnetometer calibration is a vital step to mitigate magnetic measurement distortions caused by hard-iron and soft-iron effects. However, current calibration approaches are often complex and lack performance evaluation at different noise levels. In this paper, we present a novel and efficient calibration algorithm that is conceptually simpler than most existing calibration methods and can be easily implemented. We hypothesized that a low-cost magnetometer as used in a smartphone would achieve 1-degree heading measurement accuracy after calibration using our new algorithm. Before applying this new algorithm to real-world data, we verified its effectiveness through numerical simulations. Our study demonstrated that, as the signal-to-noise ratio (SNR) increased, the root-mean-squared errors (RMSEs) of the estimated calibration parameters approached the corresponding Cramer-Rao bounds (CRBs). Experimental results showed that a magnetic heading accuracy of 1.37 degrees was attained for a smartphone located in Texas, U.S.A., which was more accurate than the heading accuracy obtained using the smartphone’s internal calibrated data. This study suggests that our calibration approach can allow low-cost magnetometers to provide highly accurate heading measurements, offering a cost-effective solution for various applications, including precise navigation.

Structured Nyquist Correlation Reconstruction for DOA Estimation With Sparse Arrays

Sparse arrays are known to achieve an increased number of degrees-of-freedom (DOFs) for direction-of-arrival (DOA) estimation, where an augmented virtual uniform array calculated from the correlations of sub-Nyquist spatial samples is processed to retrieve the angles unambiguously. Nevertheless, the geometry of the derived virtual array is dominated by the specific physical array configurations, as well as the deviation caused by the practical unforeseen circumstances such as detection malfunction and missing data, resulting in a quite sensitive model for virtual array signal processing. In this paper, we propose a novel sparse array DOA estimation algorithm via structured correlation reconstruction, where the Nyquist spatial filling is implemented on the physical array with a compressed transformation related to its equivalent filled array to guarantee the general applicability. While the unknown correlations located in the whole rows and columns of the augmented covariance matrix lead to the fact that strong incoherence property is no longer satisfied for matrix completion, the structural information is introduced as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a prior</i> to formulate the structured correlation reconstruction problem for matrix reconstruction. As such, the reconstructed covariance matrix can be effectively processed with full utilization of the achievable DOFs from the virtual array, but with a more flexible constraint on the array configuration. The described estimation problem is theoretically analyzed by deriving the corresponding Cram´er-Rao bound (CRB). Moreover, we compare the derived CRB with the performance of the virtual array interpolation-based algorithm. Extensive simulations are conducted to demonstrate the effectiveness of the proposed DOA estimation algorithm in terms of achievable DOFs, resolution, and estimation accuracy.

Handover Count Based MAP Estimation of Velocity With Prior Distribution Approximated via NGSIM Data-Set

In this paper, we propose a maximum-a-posteriori probability (MAP) based velocity estimation technique in which the prior distribution is defined by current location of the user. Motivation of this work is to improve accuracy of the existing velocity estimation techniques which are either solely based on cellular network measurements or location specific information. Our objective is to exploit both cellular measurements and location information in Bayesian sense; thus, to jointly address the critical applications of mobility management in Heterogeneous-Networks (HetNets), and intelligent transportation system. Here we assume that the Next Generation Simulation (NGSIM) data set for velocity is available at the current location and can be utilized to approximate the prior distribution. Additional information in form of prior distribution function is then exploited to improve the minimum variance unbiased (MVU) estimate of velocity which is based on handover count measurements. Since MVU estimate is a random variable, we first formulate its density function parameterized over the actual velocity. Next, we follow Bayesian approach to accommodate both prior distribution and parametric density function in deriving posterior density function of velocity. Finally, we derive expression of the MAP estimator considering various standard distribution functions which best fit to the density function obtained from NGSIM data set. In order to quantify the quality of estimate, we derive its variance and the corresponding Cramer-Rao-bound (CRB) on the minimum error variance. Numerical results demonstrate that the proposed estimator which incorporates NGSIM data set is asymptotically efficient and outperforms other classical handover count based estimation techniques.

Exploiting Information About the Structure of Signals of Opportunity for Passive Radar Performance Increase

In this paper, we study a passive radar system employing digital communication signals of opportunity. We develop the frequency domain model for the received observations of a passive radar and investigate the problem of joint target position and velocity estimation. We first consider the case where we know exactly the transmitted signals. This performance is compared with that for the case of known transmitted signal form, but where the signals have unknown parameters which include the unknown information bits. Next, we consider the case where we treat the samples of the received signal as deterministic unknowns. Finally, we consider the case where the samples of the received signals in a discrete-time formulation are modeled as being random with known distribution. In each case we derive the corresponding Cramer-Rao bound (CRB). Our focus is on the impact of employing extra information on the estimation performance. We show that for orthogonal frequency division multiplexing (OFDM) signals, knowing the signal form provides numerical performance results close to those for the completely known signal case (active radar) when the number of samples per bit is large enough. It is also shown that incorporating the direct path signals in the received observations can have a positive influence on estimating target parameters. The estimation performance improves with an increasing direct path signal strength for all cases.

Channel impulse response‐based source localization in a diffusion‐based molecular communication system

Molecular source localization finds its applications in future healthcare systems, including proactive diagnostics. This work localizes a molecular source in a diffusion based molecular communication (DbMC) system via a minimal set of passive anchor nodes and a fusion center. Two methods are presented which both utilize (the peak of) the channel impulse response measurements to uniquely localize the source, under the assumption that the molecular source of interest lies within the open convex‐hull of the sensor/anchor nodes. The first method is a one‐shot, triangulation‐based approach which estimates the unknown location of the molecular source using least‐squares method. The second method is an iterative approach, which utilizes the gradient‐descent control law to minimize a non‐convex cost function. The corresponding Cramer‐Rao bound (CRB) is also derived. Simulation results reveal that: (a) the gradient‐descent method outperforms the triangulation method (in terms of mean squared error performance) for a wide range of values of signal‐to‐noise ratio; (b) the gradient‐descent method converges to the true source location uniformly (in less than 100 iterations).

Comprehensive Effects of Imperfect Synchronization and Channel Estimation on Known Interference Cancellation

Known interference (KI) is commonly used in many scenarios, where the original interference data is known to the receiver in advance. Unfortunately, due to the impacts of hardware imperfections and propagation environment, the received KI may suffer frequency offset, phase noise, and multi-path propagation with different time delays and channel gains. However, perfect time-frequency synchronization and channel estimation are hard to implement in practice, making it challenging to cancel the KI completely. In this paper, the comprehensive effects of synchronization and channel impairments on KI cancellation are investigated. First, the normalized time synchronization error is modeled as the inter-symbol interference (ISI), and the frequency synchronization error and phase noise are modeled as the inter-frequency interference (IFI). Subsequently, the closed-form expression of KI cancellation ratio (KICR) is derived for the multi-path block fading channel, by jointly considering imperfect time-frequency synchronization and channel estimation, as well as phase noise. Furthermore, assuming the channel and synchronization parameters estimation reach their corresponding Cramer-Rao bounds (CRBs), the upper bound of KICR and the lower bound of channel capacity loss are analyzed for the single-path channel. Simulation results confirm the theoretical analysis that the KICR is more sensitive to phase noise compared to time-frequency synchronization and channel estimation errors. In addition, it is also proved that lower KI power is less sensitive to the implementation imperfections.

Scoring-Based ML Estimation and CRBs for Reverberation, Speech, and Noise PSDs in a Spatially Homogeneous Noise Field

Hands-free speech systems are subject to performance degradation due to reverberation and noise. Common methods for enhancing reverberant and noisy speech require the knowledge of the speech, reverberation and noise power spectral densities (PSDs). Most literature on this topic assumes that the noise power spectral density (PSD) matrix is known. However, in many practical acoustic scenarios, the noise PSD is unknown and should be estimated along with the speech and the reverberation PSDs. In this article, the noise is modeled as a spatially homogeneous sound field, with an unknown time-varying PSD multiplied by a known time-invariant spatial coherence matrix. We derive two maximum likelihood estimators (MLEs) for the various PSDs, including the noise: The first is a non-blocking-based estimator, that jointly estimates the PSDs of the speech, reverberation and noise components. The second MLE is a blocking-based estimator, that blocks the speech signal and estimates the reverberation and noise PSDs. Since a closed-form solution does not exist, both estimators iteratively maximize the likelihood using the Fisher scoring method. In order to compare both methods, the corresponding Cramer-Rao Bounds (CRBs) are derived. For both the reverberation and the noise PSDs, it is shown that the non-blocking-based CRB is lower than the blocking-based CRB. Performance evaluation using both simulated and real reverberant and noisy signals, shows that the proposed estimators outperform competing estimators, and greatly reduce the effect of reverberation and noise.

Noise Suppression for Direction of Arrival Estimation in Co-located MIMO Sonar.

Noise suppression capacity in multiple-input multiple-output (MIMO) sonar signal processing is derived under the assumption of white Gaussian noise. However, underwater noise mainly includes white Gaussian noise and colored noise. There exists a certain correlation between the noise signals received by each MIMO sonar array element. The performance of traditional direction-of-arrival (DOA) estimation methods decreases obviously in complex marine noise. In this paper, we propose a marine environment noise suppression method for MIMO applied to multiple targets’ DOA estimation. The noise field can be decomposed into a symmetric noise component and an asymmetric noise component. We use the covariance matrix imaginary component to pre-estimate the signal sources, then use the dimension reduction transformation to reconstruct the real component of the covariance matrix. The Toeplitz technique is utilized to reduce the correlation of the reconstructed covariance matrix. Thus, the subspace decomposition-based techniques such as multiple signal classification (MUSIC) can be used for multiple targets’ DOA estimation. To reduce the computational complexity of the methods, search-free direction-finding techniques such as the estimation of signal parameters via rotational invariance techniques (ESPRIT) can be utilized. As a result, the proposed methods can achieve better direction-finding performance in the condition of limited snapshots with lower computational cost. The corresponding Cramer-Rao bound (CRB) is deduced and the signal-to-noise ratio (SNR) gain obtained by dimension reduction processing is discussed. Simulation results also show the superiority of the proposed method over the existing methods.

Impersonation Detection in Line-of-Sight Underwater Acoustic Sensor Networks

This work considers a line-of-sight underwater acoustic sensor network (UWASN) consisting of $M$ underwater sensor nodes randomly deployed according to uniform distribution within a vertical half-disc (the so-called trusted zone). The sensor nodes report their sensed data to a sink node on water surface on a shared underwater acoustic (UWA) reporting channel in a time-division multiple-access (TDMA) fashion, while an active-yet-invisible adversary (so-called Eve) is present in the close vicinity who aims to inject malicious data into the system by impersonating some Alice node. To this end, this work first considers an additive white Gaussian noise (AWGN) UWA channel, and proposes a novel, multiple-features based, two-step method at the sink node to thwart the potential impersonation attack by Eve. Specifically, the sink node exploits the noisy estimates of the distance, the angle of arrival, and the location of the transmit node as device fingerprints to carry out a number of binary hypothesis tests (for impersonation detection) as well as a number of maximum likelihood hypothesis tests (for transmitter identification when no impersonation is detected). We provide closed-form expressions for the error probabilities (i.e., the performance) of most of the hypothesis tests. We then consider the case of a UWA with colored noise and frequency-dependent pathloss, and derive a maximum-likelihood (ML) distance estimator as well as the corresponding Cramer-Rao bound (CRB). We then invoke the proposed two-step, impersonation detection framework by utilizing distance as the sole feature. Finally, we provide detailed simulation results for both AWGN UWA channel and the UWA channel with colored noise. Simulation results verify that the proposed scheme is indeed effective for a UWA channel with colored noise and frequency-dependent pathloss.

Transmit Array Interpolation for DOA Estimation via Tensor Decomposition in 2-D MIMO Radar

In this paper, we propose a two-dimensional (2D) joint transmit array interpolation and beamspace design for planar array mono-static multiple-input-multiple-output (MIMO) radar for direction-of-arrival (DOA) estimation via tensor modeling. Our underlying idea is to map the transmit array to a desired array and suppress the transmit power outside the spatial sector of interest. In doing so, the signal-tonoise ratio is improved at the receive array. Then, we fold the received data along each dimension into a tensorial structure and apply tensor-based methods to obtain DOA estimates. In addition, we derive a close-form expression for DOA estimation bias caused by interpolation errors and argue for using a specially designed look-up table to compensate the bias. The corresponding Cramer-Rao Bound (CRB) is also derived. Simulation results are provided to show the performance of the proposed method and compare its performance to CRB.

Near‐Field Source Localization Using Spherical Microphone Arrays

A new method is proposed for joint range and bearing (azimuth and elevation) estimation of multiple near-field acoustic sources using observations collected by a spherical microphone array. First, Spherical Fourier transform (SFT) is used to construct the array signal model in the spherical harmonics domain to decouple range and bearing information. Then the relation among the spherical harmonics of three adjacent degrees is exploited to build the recursive relationship of the signal subspace. Using Eigenvalue decomposition (EVD), bearings are estimated based on the eigenvalues and simultaneously the steering matrix can be represented by the signal subspace. Finally, range is estimated using the energy ratios of the elements of the steering matrix in the spherical harmonics domain. The algorithm can avoid parameter pairing and multi-dimensional searching. It has lower computational complexity than that of the Multiple signal classification (MUSIC) method. The performance is evaluated by Monte-Carlo simulations and the estimation root meansquare errors are compared to the corresponding Cramer-Rao bounds (CRBs) and those of MUSIC range estimates, which demonstrate the validity of the proposed algorithm.

基于无源定位观测方程的一类伪线性加权最小二乘定位闭式解及其理论性能分析

Utilizing a type of special measurement equation from the passive location problem, a novel theoretical analysis framework, which transforms nonlinear localization equations into pseudo-linear equalities and locates the source with a closed-form solution, is presented. An algebraic model is constructed for pseudo-linearizing the nonlinear localization equations, and the corresponding pseudo-linear weighted least-squares solution, namely Pwls-a, is provided under the assumption that system errors (i.e., observer position and velocity perturbations) are not present. Moreover, the estimation covariance matrix of solution Pwls-a is derived through a first-order perturbation analysis method, and its theoretical performance is proved to attain the corresponding Cramer-Rao bound (CRB). Furthermore, the statistical covariance matrix is also mathematically analyzed in a case in which system errors exist; the result shows that its asymptotical estimation variance cannot reach the relevant CRB in the presence of system errors. As a consequence, a pseudo-linear weighted least-squares solution, namely Pwls-b, is proposed to mitigate the effects of system errors, and its statistical performance is proved to be asymptotically equal to the CRB under the existence of system errors. In addition, the solution Pwls-b is extended into a scenario in which multiple targets are present. Finally, a location scenario with AOA/TDOA/GROA measurements is used as an example to describe the application of the pseudo-linearization method. Simulation experiments are conducted to show the effectiveness of the theoretical analysis and the advantages of the location solutions in this paper.

Time Domain Correlation Based Joint Channel Estimation for Multi-Cell Cooperative OFDM Networks

In this paper, we focus on joint channel estimation (JCE) of multi-cell networks, which is the basis for application of multi-cell cooperative. For overcoming the weakness that the existing multi-cell JCEs need power delay profile (PDP) knowledge of all multiple cells remained the same and known by receiver or pilot sequence sets of all cells being identical and other limitations, the paper presents a time domain correlation based JCE algorithm for multi-cell cooperative network when PDPs of multiple cells are not same and unknown and derives the corresponding Cramer-Rao bound (CRB). Then, by using Generalized Akaike Information Criterion (GAIC) to estimate PDPs of all cells for reducing the signal space of channel estimation, the paper further optimizes this JCE algorithm. Simulation results illustrate the proposed algorithms have good mean square error (MSE) performance, and corresponding space frequency block coded (SFBC) cooperative multi-cell transmission system have the good bit error rate (BER) performance too. 

Cooperative Spectrum Sensing With Location Information

In this paper, we address the problem of cooperatively detecting a primary user (PU) among multiple cognitive users (CUs) when their location information is available at a CU base station. For fast detection, each CU reports a power estimate, based on one-snapshot observation of the radio environment, to the CU base station. A generalized likelihood ratio test (GLRT) is developed at the CU base station to first estimate the transmit power of the PU and then form a test variable for detection. The maximum likelihood estimator (MLE) of the unknown transmit power is discussed and analyzed to offer insight into the proposed cooperative spectrum-sensing scheme. In addition, a weighted average estimator (WAE) is proposed, which is computationally more efficient than the MLE. Asymptotic analysis for the proposed GLRT is presented. Performance of the MLE and WAE is examined along with the corresponding Cramer-Rao bound (CRB). Extensive comparisons between the proposed GLRTs and the hard- and soft-decision based spectrum sensing methods are provided, which show the effectiveness of the proposed detector.

Digital Suppression of Spurious PLL Tones in A/D Converters

In a nonideal PLL circuit, leakage of the reference signal into the control line produces spurious tones. When the distorted PLL signal is used in an analog-to-digital converter (ADC), it injects the spurious tones into the sampled data. These distortions are particularly harmful for wideband applications, such as spectrum sensing, since they affect the detection of vacant frequency bands. This paper analyzes this distortion effect in some detail and proposes estimation and filtering techniques in the digital domain to clean the data and remove the PLL leakage effects. Rather than remove the PLL sidebands by perfecting the circuitry, the proposed approach focuses on compensating their effect on the sampled data by using signal processing compensation algorithms. We study the performance of the proposed estimation algorithms and compare it with the corresponding Cramer-Rao bound (CRB). Such digitally based approaches are cost effective since the cost of perfecting the analog circuits can be prohibitive due to regular variations and imperfections in circuit fabrication processes.

Adaptive Distributed Estimation of Signal Power from One-Bit Quantized Data

We examine distributed estimation of the average power of a random signal in wireless sensor networks (WSNs). Due to stringent bandwidth/power constraints, each sensor quantizes its observation into one bit of information and sends the quantized data to a fusion center, where the signal power is estimated. We firstly introduce two fixed quantization (FQ) schemes, with the first using a single threshold and the second employing a pair of symmetric thresholds. The maximum likelihood (ML) estimators associated with the two FQ schemes are developed, and their corresponding Cramer-Rao bounds (CRBs) are analyzed. We show that the FQ approach, especially the second one, can achieve an estimation performance close to that of a clairvoyant estimator using unquantized data if the optimum quantization threshold is available; however, the optimum threshold is dependent on the unknown signal power, and as the threshold deviates from its optimum value, the performance degrades rapidly. To cope with this difficulty, we propose a distributed adaptive quantization (AQ) approach by which the threshold is dynamically adjusted from one sensor to another in a way such that the threshold converges to the optimum threshold. Our analysis shows that the proposed AQ approach is asymptotically optimum, yielding an asymptotic CRB equivalent to that of the FQ approach with optimum threshold.

Noncoherent Synchronization of UWB-IR Multiple-Antenna Multipath Channels

This paper deals with the maximum-likelihood (ML) noncoherent data-aided (e.g., no blind) synchronization of multiple-antenna ultrawideband impulse-radio (UWB-IR) terminals that operate over broadband channels and are affected by multipath fading with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> priori unknown number of paths and path-gain statistics. The synchronizer that we developed achieves the ML data-aided joint estimate of the number of paths and their arrival times (e.g., time delays), without requiring any a priori knowledge and/or a posteriori estimate of the amplitude (e.g., module and sign) of the channel gains. The ultimate performance of the proposed synchronizer is evaluated (in closed form) by developing the corresponding Cramer-Rao bound (CRB), and the analytical conditions for achieving this bound are provided. The performance gain for the synchronization accuracy of multipath-affected UWB-IR signals arising from the exploitation of the multiple-antenna paradigm is (analytically) evaluated. Furthermore, a low-cost sequential implementation of the proposed synchronizer is detailed. It requires an all-analog front-end circuitry composed of a bank of sliding-window correlators, whose number is fully independent from the number of paths comprising the underlying multiple-antenna channel. Finally, the actual performance of the proposed synchronizer is numerically tested under both the signal acquisition and tracking operating conditions.

Fast Iterative Maximum-Likelihood Algorithm (FIMLA) for Multipath Mitigation in the Next Generation of GNSS Receivers

In this paper, we efficiently solve the maximum likelihood (ML) time-delay estimation problem for GNSS signals in a multipath environment. Exploiting the GNSS signal model structure and the spreading code periodicity, we develop an efficient implementation of the Newton iterative likelihood maximization method by finding simple analytical expressions for the first and second derivatives of the likelihood function. The proposed fast iterative ML algorithm (FIMLA) for timedelay estimation, which uses the correlation function of the received signal with its local replica, is shown to be an attractive technique for mitigation of closely-spaced multipath arrivals. For the future modernized GPS and the European Galileo signals based on binary offset carrier (BOC) waveforms, the correlation function has multiple positive and negative peaks leading to potential tracking ambiguities. Instead of the standard crosscorrelation, we propose an implementation characterized by a different choice of the local replica so as to cancel the sub-carrier phase, thus eliminating ambiguities. The asymptotic performance of FIMLA is analyzed by deriving the corresponding Cramer-Rao bound (CRB). Representative simulation examples are included to illustrate the FIMLA is performance for delay estimations in the presence of multipath for both C/A code and BOC signals.

Distributed Adaptive Quantization for Wireless Sensor Networks: From Delta Modulation to Maximum Likelihood

We consider distributed parameter estimation using quantized observations in wireless sensor networks (WSNs) where, due to bandwidth constraint, each sensor quantizes its local observation into one bit of information. A conventional fixed quantization (FQ) approach, which employs a fixed threshold for all sensors, incurs an estimation error growing exponentially with the difference between the threshold and the unknown parameter to be estimated. To address this difficulty, we propose a distributed adaptive quantization (AQ) approach, which, with sensors <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sequentially</i> broadcasting their quantized data, allows each sensor to adaptively adjust its quantization threshold. Three AQ schemes are presented: (1) AQ-FS that involves distributed delta modulation (DM) with a fixed stepsize, (2) AQ-VS that employs DM with a variable stepsize, and (3) AQ-ML that adjusts the threshold through a maximum likelihood (ML) estimation process. The ML estimators associated with the three AQ schemes are developed and their corresponding Cramer-Rao bounds (CRBs) are analyzed. We show that our 1-bit AQ approach is asymptotically optimum, yielding an asymptotic CRB that is only pi/2 times that of the clairvoyant sample-mean estimator using <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">unquantized</i> observations.

Performance of blind channel estimation algorithms for space-frequency block coded multi-carrier code division multiple access systems

The problem of blind channel estimation for downlink space-frequency block coded multi-carrier code division multiple access (SFBC MC-CDMA) schemes is considered. For these schemes, the authors first develop a system model for complex modulated signals, which reduces the multichannel estimation problem to a single-input single-output problem. Then, they present an intuitive subspace-based channel estimation method along with the corresponding necessary and sufficient conditions under which the channel estimate is unique (within a complex scalar). Their studies highlight two interesting properties of SFBC MC-CDMA systems: (i) there is no antenna order ambiguity (also known as permutation ambiguity) even though only one spreading code is assigned to each user; (ii) channel identifiability is guaranteed, regardless of the channel zeros location. They also establish the unbiasedness of the channel estimates and derive closed-form expressions for the mean-square-error of the estimates as well as the corresponding Cramer-Rao bound (CRB). In the derivation of the CRB, they suggest a novel approach which assumes the knowledge of only the spreading code of desired user. This approach results in a tighter bound than the CRB derived based on the knowledge of all users' signatures.