Cramer-Rao Lower Bound Analysis Research Articles

Heterogeneous Transfer Learning for Wi-Fi Indoor Positioning Based Hybrid Feature Selection.

This paper presents the application of heterogeneous transfer learning (HetTL) methods which consider hybrid feature selection to reduce the training calibration effort and the noise generated by fingerprint duplicates obtained from multiple Wi-Fi access points. The Cramer–Rao Lower Bound analysis (CRLB) was also applied to evaluate and estimate a lower limit for the variance of a parameter estimator used to analyze positioning performance. We developed two novel algorithms for feature selection in fingerprint-based indoor positioning problems (IPP) to enhance positioning performance in the target domain with the HetTL. The algorithms comprised two scenarios: (i) a principal component analysis-based approach (PCA-based) and (ii) a hybrid approach that takes both PCA and correlation effect analysis into account (hybrid scenario). Accordingly, a new feature vector was constructed by retaining only the most significant predictors, and the most efficient feature dimensions were also determined by using a hybrid-based approach. Experimental results showed that the hybrid-based proposed algorithm has the minimum mean absolute error. The CRLB analysis also showed that the number of Wi-Fi access points could affect the lower bound location estimation error; however, identifying the most significant predictors is an effective approach to improve positioning performance.

Cramer-Rao Lower Bound Analysis of Data Fusion for Fingerprinting Localization in Non-Line-of-Sight Environments

This paper proposes a novel framework for analyzing the localization accuracy of data fusion for fingerprinting approaches in non-line-of-sight (NLOS) environments. Using simulation data generated for two very different NLOS environments (a suburban area of $3.3km\times 3.3km$ in Santa Clara, California, and a mountainous area of $11.4km\times 11.4km$ in the Caspian region), we establish novel channel models for measurement differences of three data types (received signal strength indicator (RSS), time of arrival (TOA) and direction of arrival (DOA)) at $K$ neighboring nodes of an arbitrary node. The crucial point is that the modeling errors for each of the three data types are shown to be jointly Gaussian distributed. Based on these measurement difference models, Cramer-Rao Lower Bound (CRLB) is used as a benchmark to evaluate $K$ -nearest neighbor (KNN) and Weighted $K$ -Nearest Neighbor (WKNN). It is shown that the proposed CRLB analyses can be employed to evaluate fingerprinting systems with various designs (such as different data types and fusion options) and different configurations (such as densities of reference nodes and numbers of anchor nodes) in diverse NLOS environments(such as suburban and mountainous regions).

Novel Current Unbalance Estimation and Diagnosis Algorithms for Condition Monitoring With Wireless Sensor Network and Internet of Things Gateway

This article presents two novel algorithms for the estimation and diagnosis of the current unbalance factor (CUF) for three-phase power systems from single period of three-phase acquired data samples. The CUF is evaluated by an algorithm named circular phase shift (CPS), and the three-phase parameters are estimated by a circular cross-correlation (CCC) algorithm for unbalance diagnosis. The estimated CUF along with multisensory data is transmitted and monitored through a remote web server for diagnosis and detection of incipient three-phase power systems faults including three-phase machines. The CPS algorithm estimation has an accuracy that exceeds 95% for CUF values exceeding 5%. The CCC time-complexity and the Cramer–Rao Lower Bound analysis are presented for performance evaluation. The CCC algorithm outperforms the IEEE-Standard-1057 estimation method in both phase accuracy and processing memory requirements compatible with low-cost microcontrollers. Experimental results on condition monitoring of industrial induction machines (1.5 to 7.5 KW) are also presented with custom designed 2.4-GHz wireless sensor network and an IEEE 802.11 Internet of Things gateway with multisensory data which carries out the effectiveness of the system.

Performance evaluation and analysis for dipole source localization with lateral line sensor arrays

The unique mechanosensory lateral line system of fish inspires a novel and promising sensing scheme in dealing with the challenges that autonomous underwater vehicles (AUVs) face in underwater localization and navigation. In this paper, the quantitative and method-independent Cramer–Rao lower bound (CRLB) model is established to evaluate the practical performance of a lateral line sensor array (LLSA) for dipole source localization by considering the excitation pattern mismatches and the measurement noises. The effective localization area of the LLSA is studied under varying array lengths and sensor densities. Localizations for the LS (least squares) method and the MUSIC (multiple signal classification) method are compared in simulations. It is shown that the longer and denser the LLSA, the larger the effective localization area. Typically, the too separated sensor-to-sensor spacing deteriorates the near-body localization performance. The effective localization area obtained by employing the LS method matches the CRLB analysis, which also proves the validity of the analytical CRLB model. Besides, the MUSIC method presents smaller effective localization areas than the LS method. Physical experiments are also conducted and agree well with the CRLB analysis. Further study shows that the measurement noises have an equivalent effect on the excitation pattern mismatches, and can be effectively suppressed by choosing a large number of snapshots. In addition, the CRLB gives the perceptual distance that is consistent with the observation. At last, guidance on the design of a simplified LLSA with the least number of sensors and an optimal sensor-to-sensor spacing is provided for AUVs.

CRLB Analysis for a Robust TRN Based on a Combination of RNN and PF

Terrain-referenced navigation (TRN) is a technology that estimates the position of an aircraft by comparing the terrain elevation on a digital elevation model (DEM) and the altitude measured by an altimeter. The particle filter (PF)-based TRN has been widely used for unmanned aerial vehicles (UAVs) operating at a high altitude. Even though TRN performs well in rough and unique terrains, its performance degrades on flat and repetitive terrain. For this study, a robust PF-based TRN was designed, which uses a recurrent neural network (RNN)-based deep learning method to function on flat and repetitive terrains. Noise covariances and the measurement model of the PF were also trained. An RNN is an artificial neural network that recognizes patterns from a time-series. Due to the difficulty in theoretically verifying that the model generated by the deep learning method was designed optimally, we performed a Cramer–Rao lower bound (CRLB) analysis to evaluate how close the proposed method was to the optimal design. We also used a perturbation error model of an interferometric radar altimeter (IRA) as the measurement noise covariance needed for an accurate CRLB analysis.

Wireless Interrogation of Implantable SAW Sensors.

Implantable sensors provide long-term, accurate physiological measurements after a minimally invasive procedure, particularly when designed as transponders. Wireless interrogation of deeply implanted transponders with RF remains a challenge due to the high loss at the skin-air interface and large tissue RF absorption. This paper presents a system for wirelessly interrogating surface acoustic wave (SAW) sensors implanted in the main pulmonary artery (PA), where the pressure (PAP) is a very important parameter in the management of heart failure patients. The proposed PAP monitoring system consists of an implantable SAW pressure sensor integrated with an antenna and anchor in a housing, an external antenna and an electronic interrogator. The PAP is determined by measuring the frequency of the echo signal from SAW sensor accurately. An asymmetric antenna was designed and integrated with the sensor. The combination of simulation, theoretical calculation and phantom measurement indicates that the path loss to the implant location, about 6cm below the skin, is around 25dB. A portable interrogator was designed based on a dual conversion receiver and single echo high frequency sampling approach to assess achievable frequency estimation accuracy predicted by Cramer-Rao Lower Bound (CRLB) analysis. The system was characterized using a high quality (Q) factor SAW sensor, fabricated at wafer level, wire-connected to the interrogator via an attenuator to simulate path loss. The signal-to-noise ratio (SNR) of captured echo signals was calculated and used in CRLB analysis. The analysis indicates that without using signal post processing, the sensor sensitivity has to be at least 440Hz/mmHg in order to achieve a target 1mmHg accuracy. Although the current sensor sensitivity is only 200Hz/mmHg, the in vivo measurement showed that acceptable accuracy can be obtained by signal post processing. The results from an invasive catheter tip transducer measured simultaneously with the SAW sensor showed that the differences in pulse pressure and relative mean pressure are 0.8mmHg and 1.4mmHg, respectively. The accuracy could be further improved by increasing the sensor Q factor and sensitivity and reducing path loss.

Cramér–Rao Bounds for Complex-Valued Independent Component Extraction: Determined and Piecewise Determined Mixing Models

This paper presents Cram\'er-Rao Lower Bound (CRLB) for the complex-valued Blind Source Extraction (BSE) problem based on the assumption that the target signal is independent of the other signals. Two instantaneous mixing models are considered. First, we consider the standard determined mixing model used in Independent Component Analysis (ICA) where the mixing matrix is square and non-singular and the number of the latent sources is the same as that of the observed signals. The CRLB for Independent Component Extraction (ICE) where the mixing matrix is re-parameterized in order to extract only one independent target source is computed. The target source is assumed to be non-Gaussian or non-circular Gaussian while the other signals (background) are circular Gaussian or non-Gaussian. The results confirm some previous observations known for the real domain and bring new results for the complex domain. Also, the CRLB for ICE is shown to coincide with that for ICA when the non-Gaussianity of background is taken into account. %unless the assumed sources' distributions are misspecified. Second, we extend the CRLB analysis to piecewise determined mixing models. Here, the observed signals are assumed to obey the determined mixing model within short blocks where the mixing matrices can be varying from block to block. However, either the mixing vector or the separating vector corresponding to the target source is assumed to be constant across the blocks. The CRLBs for the parameters of these models bring new performance bounds for the BSE problem.

Performance Analysis of Parameter Estimation in Electromagnetic Induction Data

We derive the Cramer–Rao lower bound (CRB) for target tensor amplitudes and target location parameters that can be estimated from electromagnetic induction (EMI) measurements of a target. In deriving the bound, no restrictions are placed on the target type, target orientation, quantity, or location of the measurement positions, nor the geometry of the EMI sensor or sensor array. The analysis is applicable to both a scanned sensor and a stationary array, as well as both time- and frequency-domain sensors. We show how the bound varies as a function of target type, orientation, depth, and signal-to-noise ratio. In addition, we illustrate the ways to use the CRB and the Fisher information matrix to analyze the relationships between the tensor amplitudes and location parameters. We show results of the CRB analysis applied to an experimental frequency-domain Georgia Tech sensor for a few target types for a nominal 2-D scan geometry as well as the Time-domain Electromagnetic Multi-sensor Towed Array Detection System. We also provide an algorithm for estimating the target location and tensor that achieves the CRB.

Measuring Cerebral Hypoperfusion Induced by Hyperventilation Challenge With Intravoxel Incoherent Motion Magnetic Resonance Imaging in Healthy Volunteers

The aim of this study was to demonstrate the feasibility to assess cerebral hypoperfusion with a hyperventilation (HV) challenge protocol using intravoxel incoherent motion (IVIM) magnetic resonance imaging. Magnetic resonance imaging experiments were performed on 10 healthy volunteers at 1.5 T, with a diffusion IVIM magnetic resonance imaging protocol using a set of b-values optimized by Cramer-Rao Lower Bound analysis. Hypoperfusion was induced by an HV maneuver. Measurements were performed in normoventilation and HV conditions. Biexponential curve fitting was used to obtain the perfusion fraction (f), pseudodiffusion coefficient (D*), and the product fD* in gray matter (GM) regions of interest (ROIs). Regional cerebral blood flow in the same ROIs was also assessed with arterial spin labeling. The HV challenge led to a diminution of IVIM perfusion-related parameters, with a decrease of f and fD* in the cerebellum (P = 0.03 for f; P = 0.01 for fD*), thalamus GM (P = 0.09 for f; P = 0.01 for fD*), and lenticular nuclei (P = 0.03 for f; P = 0.02 for fD*). Mean GM cerebral blood flow (in mL/100 g tissue/min) measured with arterial spin labeling averaged over all ROIs also decreased (normoventilation: 42.7 ± 4.1 vs HV: 33.2 ± 2.2, P = 0.004) during the HV challenge. The optimized IVIM protocol proposed in the current study allows for measurements of cerebral hypoperfusion that might be of great interest for pathologies diagnosis such as ischemic stroke.

Node localization over small world WSNs using constrained average path length reduction

Small world characteristics have been observed and studied in social networks in the context of information dissemination in important applications like disease spread among communities. However, the significance of small world characteristics in wireless sensor network (WSN) applications has hitherto not been investigated. This work investigates the significance of introducing small world characteristics in a conventional WSN for improving the node localization accuracy. In this context, a novel constrained iterative average path length reduction algorithm is proposed to introduce small world characteristics into a conventional WSN. This method utilizes a novel frequency selective approach for the introduction of small world phenomena. The frequency selective approach utilizes two frequency bands for link addition and pruning to create an optimized small world WSN. Subsequently, a constrained least squares node localization method over the small world WSN is developed using both model-based and model-free signal propagation approaches. The proposed constrained iterative average path length reduction method provides reduced average path length and a high average clustering coefficient when compared to conventional WSN. Constrained least squares node localization over the small world WSN yields improved localization performance when compared to node localization over conventional WSN. Performance analysis in terms of localization error, power consumption, bandwidth and anchors required for node localization is also discussed. Cramer-Rao lower bound analysis for node location parameters is also used to provide insights into the error bounds obtained using the proposed method. The experimental results obtained are motivating enough for the introduction of small world phenomena in conventional WSN for applications like node localization and related applications.

The Error Propagation Analysis of the Received Signal Strength-Based Simultaneous Localization and Tracking in Wireless Sensor Networks

Simultaneous localization and tracking (SLAT) in wireless sensor networks (WSNs) involves tracking the mobile target while calibrating the nearby sensor node locations. In practice, localization error propagation (EP) phenomenon will arise, due to the existence of the latest tracking error, target mobility, measurement error, and reference node location errors. In this case, the SLAT performance limits are crucial for the SLAT algorithm design and WSN deployment, and the study of localization EP principle is desirable. In this paper, we focus on the EP issues for the received signal strength-based SLAT scheme, where the measurement accuracy is assumed to be spatial-temporal-domain doubly random due to the target mobility, environment dynamics, and different surroundings at different reference nodes. First, the Cramer–Rao lower bound (CRLB) is derived to unveil both the target tracking EP and the node location calibration EP. In both cases, the EP principles turn out to be in a consistent form of the Ohm’s Law in circuit theory. Second, the asymptotic CRLB analysis is then presented to reveal that both EP principles scale with the inverse of sensor node density. Meanwhile, it is shown that, the tracking and calibration accuracy only depends on the expectation of the measurement precision. Third, the convergence conditions, the convergence properties, and the balance state of the target tracking EP and the location calibration EP are examined to shed light on the EP characteristics of the SLAT scheme Finally, numerical simulations are presented to corroborate the EP analysis.

Derivation of CRLB for Wireless Capsule Endoscope Localization Using Received Signal Strength

This paper presents a theoretical analysis for location estimation of a wireless capsule endoscope (WCE) in the small-intestine region. Under three different shadowing scenarios by assuming standard deviation of the shadowing with the constant variance, parameter-dependent variance, or correlated and parameter-dependent variance, analytical formulas are derived for the Cramer–Rao lower bound (CRLB) for capsule positioning when distance and azimuth angle measurements are employed to estimate the location of the capsule. The CRLB analysis can be employed to quantify the best achievable location estimation performance. We also present some numerical results to show the CRLB for three scenarios. The results show that in the most realistic scenario, when the shadowing is spatially correlated and distance dependent and the standard deviation of noise is 7 dB, the maximum square root of CRLB is 3.8 mm. Moreover, the square root of CRLB experiences its minimum value of 2 mm, when the capsule is away from its first location about half of the capsule length. Overall, the result of this paper suggests that it is possible to use received signal strength such as measured by a radar system for location estimation of a WCE.

Novel Robust Direction-of-Arrival-Based Source Localization Algorithm for Wideband Signals

Source localization for wideband signals using acoustic sensor networks has drawn much research interest recently. The maximum-likelihood is the predominant objective for a wide variety of source localization approaches, and we have previously proposed an expectation-maximization (EM) algorithm to solve the source localization problem. In this paper, we tackle the source localization problem based on the realistic assumption that the sources are corrupted by spatially-non-white noise. We explore the respective limitations of our recently proposed algorithm, namely EM source localization algorithm, and design a new direction-of-arrival (DOA) estimation based (DEB) source localization algorithm. We also derive the Cramer-Rao lower bound (CRLB) analysis and the computational complexity study for the aforementioned source localization schemes. Through Monte Carlo simulations and our derived CRLB analysis, it is demonstrated that our proposed DEB algorithm significantly outperforms the previous EM method in terms of both source localization accuracy and computational complexity.

Lower bounds of accuracy for enhanced mode-S distributed sensor networks

The aim of this study is to analyse the accuracy of passive location systems by using the CramerRao lower bound (CRLB). The study is focused on sensor networks derived from the well-known Multilateration (MLAT) systems, used for aircraft and vehicle surveillance. The measurement used in the standard MLAT configuration is the Time Difference Of Arrival, but it can be improved by using other types of measurement like angle of arrival, round trip delay or time integration. Here, the CRLB is used to model the system accuracy performance using different measurements and related to the target kinematics, to evaluate a possible system evolution. In this frame, the CRLB analysis is useful to define the basic limitations and the advantages of different system architectures as well as to optimise existing architectures by adding new measurement capabilities to some receiving stations.

EM-Based Adaptive Frequency Domain Estimation of Doppler Shifts with CRLB Analysis for CDMA Systems

Combating time and frequency selectivity in wireless channels is one of the most challenging tasks in next generation wireless networks. In this paper, we propose an adaptive estimation algorithm to estimate Doppler shifts in a direct sequence code division multiple access (DS-CDMA) radio system with multiple Doppler subpaths. By modeling doubly selective channels using a basis expansion model (BEM), an expectation-maximization (EM) algorithm based adaptive estimation method is developed to extract accurate Doppler shift information. The Cramer-Rao lower bound (CRLB) analysis is conducted to study the performance bound of the proposed estimation algorithm. Based on the estimated Doppler shift results, a frequency domain equalizer (FDE) based receiver architecture is developed to exploit Doppler diversity in the frequency domain. Our analysis and simulation results demonstrate that this receiver architecture features a low complexity while still achieving a good performance compared with traditional CDMA receivers.

Cramer–Rao Lower Bound Analysis of Vegetation Height Estimation With Random Volume Over Ground Model and Polarimetric SAR Interferometry

The random volume over ground model, which is based on a simple description of the electromagnetic wave interaction with a vegetated media, allows one to define techniques to estimate the vegetation height from the polarimetric interferometric synthetic aperture radar observations. We discuss this issue with the Cramer-Rao lower bound (CRLB) which provides a minimal bound of the variance independently of the estimation if it is unbiased. The usefulness of this approach is illustrated on three different examples. The first example deals with the influence of a priori knowledge of some physical parameters. We show that reducing the number of unknown physical parameters does not necessarily improve the CRLB. The second example focus on the efficiency of the Cloude et al. height estimator. On the considered data, this estimator reaches the CRLB when the standard deviation equals approximately one meter. In the third example we optimize the radar baseline.

Optimization of source and detector configurations based on Cramer–Rao lower bound analysis

Optimization of source and detector (SD) arrangements in a diffuse optical tomography system is helpful for improving measurements' sensitivity to localized changes in imaging domain and enhancing the capacity of noise resistance. We introduced a rigorous and computationally efficient methodology and adapt it into the diffuse optics field to realize the optimizations of SD arrangements. Our method is based on Cramer-Rao lower bound analysis, which combines the diffusion-forward model and a noise model together. This method can be used to investigate the performance of the SD arrangements through quantitative estimations of lower bounds of the standard variances of the reconstructed perturbation depths and values. More importantly, it provides direct estimations of parameters without solving the inverse problem. Simulations are conducted in the reflection geometry to validate the effectiveness of the method on selections of the optimized SD sets, with a fixed number of sources and detectors, from an SD group on a planar probe surface. The impacts of different noise levels and target perturbation depths are considered in the simulations. It is demonstrated that the SD sets selected by this method afford better reconstructed images. This methodology can be adapted to other probe surfaces and other imaging geometries.

Time-Reversal Ground-Penetrating Radar: Range Estimation With Cramér–Rao Lower Bounds

In this paper, first, a new range-estimation technique using time reversal (TR) for ground-penetrating-radar (GPR) applications is presented. The estimator is referred to as the TR/GPR range estimator. The motivation for this paper comes from the need of accurately estimating the location of underground objects such as landmines or unexploded ordinance for safe clearance. Second, the Cramer-Rao lower bound (CRLB) for the performance of the TR/GPR range estimator is derived and compared with the CRLB for the conventional matched filter (MF). The CRLB analysis shows that the TR/GPR range estimator has the potential to achieve higher accuracy in estimating the location of the target than that of the conventional MF estimator. Third, the proposed TR/GPR estimator is tested using finite-difference time-domain simulations, where the surface-based reflection GPR is modeled using an electromagnetic transverse-magnetic (TM) mode formulation. In our simulations, the TR/GPR estimator outperforms the conventional MF approach by up to 5-dB reduction in mean square error at signal-to-noise ratios ranging from -20 to 20 dB for dry-soil environments.

Using Cramer–Rao theory as spectrometer design tool aimed at quantitative complex-spectrum analysis

In this paper the use of the Cramer-Rao lower bound (CRLB) to aid in the design of diagnostic systems dealing with complex spectra is discussed. Specifically it is discussed how a priori knowledge, used to improve the estimation, can be included in the CRLB analysis. This provides improved predictions for the spectrometer characteristics that will lead to an optimal system.

Effects of Parametric Constraints on the CRLB in Gain and Phase Estimation Problems

The problem of estimating the direction-independent gain and phase characteristics of sensor arrays requires a boundary condition to solve the phase ambiguity of the solution. It has become common practice to use the constraint that the phase of the first element of the array is zero. By Cramer-Rao lower bound (CRLB) analysis, we show analytically for calibration on a single point source of an array of identical elements that the variance of the phase estimates decreases by a factor 2p/(p-1) under the boundary condition that the sum of all phases equals zero, where p is the number of elements in the array. We also show that this constraint is the one that among all possible constraints minimizes the total variance on all estimated parameters. Our analysis suggests that this conclusion also holds for arbitrary source models and arrays of nonidentical elements. This statement is confirmed by repeating the CRLB analysis in simulation, showing that the CRLB under this constraint coincides within the numerical accuracy with the minimal CRLB