Far-field Localization Research Articles

Quantum limits of position and polarizability estimation in the optical near field

Optical near fields are at the heart of various applications in sensing and imaging. We investigate dipole scattering as a parameter estimation problem and show that optical near fields carry more information about the location and the polarizability of the scatterer than the respective far fields. This increase in information originates from, and occurs simultaneously with, the scattering process itself. Our calculations also yield the far-field localization limit for dipoles in free space. Published by the American Physical Society 2024

Enhanced Far-field Localization Scheme Using Multi-RIS and Efficient Beam Sweeping

Enhanced Far-field Localization Scheme Using Multi-RIS and Efficient Beam Sweeping

Far-Field Localization for RIS Empowered Wireless Systems Leveraging Beamforming

Reconfigurable intelligent Surfaces (RISs) have lately received a lot of interest due to their capability of providing a smarter controlled radio environment. This paper investigates the use of RIS beamforming to attain user equipment (UE) localization in far-field propagation with two localization schemes in millimeter-wave (mmWave) with orthogonal frequency division multiplexing (OFDM) signaling. The first scheme adopts a single RIS that leverages the time of arrival (ToA) and angle of departure (AoD) measurements. For this model, the impact of various parameters on the localization accuracy are examined. In addition, the Cramer-Rao Lower Bound (CRLB) for the positioning, i.e., the position error bounds (PEB), is derived to be used as benchmark. Also, an efficient multi-RIS aided localization scheme is proposed, in which the use of only AoDs to estimate the UE position is shown to be possible, whereas this was challenging without using multi-RIS scenario. The main contribution of this paper is relieving the complexity of localization algorithm by eliminating the necessity of ToA estimation where consensus-based fusion of AoDs estimations is used. The estimation of AoD from the RIS to the UE is obtained by using the RIS as a beamformer to scan the working area with successive beams. The highest received signal at UE from a particular beam will determine the AoD. The performance of the two localization models - discussed in this paper - are evaluated via extensive numerical simulations. For the first model, the simulation results and PEB demonstrated that increasing the number of beams that swept the region of interest has a significant impact on the accuracy, while using RIS with a large number of elements demonstrated marginal influence on the localization accuracy for a large number of RIS elements. Furthermore, the results reveal that the localization accuracy of the proposed multi-RIS localization scheme outperforms the performance of the first model by increasing the number of RISs and the number of beams. In addition, the proposed scheme benefits from being computationally efficient due to the release of ToA estimation in the first model that requires the computation of inverse fast Fourier transform (IFFT) of the oversampled version of the received signal for a sufficient number of transmissions and fine-tuning using search optimization to attain accurate estimation

An improved functional beamforming algorithm for far-field multi-sound source localization based on Hilbert curve

In view of the mismatch problem between the sound sources and searching grid points (non-ideal case), an improved functional beamforming (FBF) algorithm based on the Hilbert curve, so-called HFBF, for far-field localization of multi-sound sources is presented in this paper. In contrast with the traditional FBF, the HFBF may generate the searching grids on a sound source plane and optimize them by reducing the searching plane size, adjusting the function order and the focusing vector direction, so as to improve the efficiency of grid searching and the estimation accuracy of sound source parameters. A computational procedure of the HFBF is proposed and applied to identify two independent sound sources. In terms of the locating error and the estimated error of sound pressure level (SPL), the performances of four algorithms are compared by simulations and experiments. The results suggest that the accuracies of the location coordinates and the estimated SPL values are significantly improved by the Hilbert-based grid searching. The conclusions can be drawn that, for multi-source localization, the HFBF can somewhat break through the Rayleigh limit. If the exponent value is selected properly, the estimated SPL error of the HFBF can be less than 0.1 dB. The HFBF has a larger range of available exponent values, which is negatively correlated with the frequency of a sound source. Generally speaking, in the non-ideal cases, the proposed HFBF is more accurate and effective than the FBF. It can be applied to a large-scale space, such as aircraft, train, traffic environment, etc., for far-field localization of multi-sound sources in engineering.

Mixed Near-Field and Far-Field Localization and Array Calibration With Partly Calibrated Arrays

The problem of passive localization of mixed near-field (NF) and far-field (FF) source signals in the presence of array gain-phase uncertainties is addressed. A new algorithm is aimed to use partly calibrated nonuniform linear arrays (NLAs), in which only three sensors have been fully-calibrated. Most of the existing algorithms deal with this problem by exploiting uniform linear arrays (ULAs). Moreover, they assume a simplified source-array model, in which the propagation magnitude scaling is completely neglected and the spatial phase difference is approximated by Taylor’s polynomial. As an opposite, the proposed algorithm is employed to accommodate a more general situation: the exact spatial geometries and nonuniform linear arrays. In the proposed algorithm, three cumulant matrices are firstly defined to construct two matrix pencils. Unambiguous range and angle parameter estimates of the NF sources are then obtained from the generalized eigenvalues of the two defined matrix pencils. After that, these estimates are utilized to calibrate array gain-phase errors. Finally, a spectrum-MUSIC like approach is applied to accomplish the angle estimation for the FF sources. The new algorithm is shown to be readily simple and effective and will be verified both mathematically and numerically.

Semidefinite relaxation method for unified near-Field and far-Field localization by AOA

This paper investigates the unified near-field and far-field localization using angle-of-arrival measurements, where the knowledge that the source is in the near-field or far-field is not required. We propose a semidefinite relaxation method to solve this problem by using the modified polar representation of the source. By approximating the measurement model, we formulate the localization problem as a weighted least squares problem, which is a non-convex constrained optimization problem. This problem is then relaxed to a convex semidefinite program (SDP) by applying semidefinite relaxation. The relaxed SDP problem is tightened by adding a set of second-order cone constraints to reach a tighter mixed semidefinite and second-order cone program. The proposed method provides an accurate position estimate for the near-field source, and yields a precise direction of arrival estimate for the far-field source. To reduce the estimation bias introduced by the approximation of the measurement model, we further derive the analytical expression of the bias and subtract it from the solution to obtain an approximately unbiased estimate. Both simulations and theoretical analysis show that the proposed method reaches the Cramer-Rao lower bound when the noise is not large under Gaussian noise, and has a comparable bias with the maximum likelihood estimator.

Anapole mediated giant photothermal nonlinearity in nanostructured silicon

Featured with a plethora of electric and magnetic Mie resonances, high index dielectric nanostructures offer a versatile platform to concentrate light-matter interactions at the nanoscale. By integrating unique features of far-field scattering control and near-field concentration from radiationless anapole states, here, we demonstrate a giant photothermal nonlinearity in single subwavelength-sized silicon nanodisks. The nanoscale energy concentration and consequent near-field enhancements mediated by the anapole mode yield a reversible nonlinear scattering with a large modulation depth and a broad dynamic range, unveiling a record-high nonlinear index change up to 0.5 at mild incident light intensities on the order of MW/cm2. The observed photothermal nonlinearity showcases three orders of magnitude enhancement compared with that of unstructured bulk silicon, as well as nearly one order of magnitude higher than that through the radiative electric dipolar mode. Such nonlinear scattering can empower distinctive point spread functions in confocal reflectance imaging, offering the potential for far-field localization of nanostructured Si with an accuracy approaching 40 nm. Our findings shed new light on active silicon photonics based on optical anapoles.

Nanometer Accuracy in Cryogenic Far-Field Localization Microscopy of Individual Molecules.

We demonstrate the nanometer accuracy of far-field fluorescence localization microscopy at a temperature of 1.8 K using near-infrared and red fluorophores bonded to double-stranded DNA molecules (10.2 nm length). Although each fluorophore was localized with a 1 nm lateral precision by acquiring an image at one axial position within the focal depth of ±0.7 μm, the distance between the two fluorophores on the lateral plane (Dxy) was distributed from 0 to 50 nm. This systematic error was mainly due to detecting with the large focal depth the dipole emission from orientationally fixed fluorophores. Each fluorophore was localized with precisions of ±1 nm (lateral) and simultaneously ±11 nm (axial) by acquiring images every 100 nm in the axial direction from -900 to 900 nm. By correcting the dipole orientation effects, the distribution of Dxy was centered around the DNA length. The average and standard deviation of Dxy were 10 and 5 nm.

Unified Near-Field and Far-Field Localization for AOA and Hybrid AOA-TDOA Positionings

Point positioning of a signal source is feasible if it is not far from the sensors and direction of arrival (DOA) localization is only applicable if it is distant. Point positioning and DOA localization employ different estimation models and prior knowledge about the source range is often not available to decide which model is appropriate. This paper introduces the modified polar representation to unify the localization of a source using angle of arrival (AOA) regardless if it is near or far. From the Gaussian AOA measurements, we utilize the hybrid Bhattacharyya-Barankin (HBB) bound to illustrate it is not possible to obtain the Cartesian coordinates of a distant source when applying the near-field model, and derive the DOA bias of a not so distant source when using the far-field model. An iterative maximum likelihood estimator (MLE) is next derived under the modified polar representation with a single model, where the HBB bound confirms the stable behavior of the estimator regardless it is near or far. The algorithm yields a position if the source is close and a DOA if it is distant. A preliminary solution to initialize the MLE using semidefinite relaxation is also proposed. The HBB bound, the analysis and the algorithm are extended for hybrid AOA-TDOA localization.

A geometrical closed form solution for RSS based far-field localization: Direction of Exponent Uncertainty

In this study, a new powerful geometrical closed-form solution called Direction of Exponent Uncertainty (DEU) is proposed for received signal strength (RSS) based far-field localization when path loss exponent (PLE) and transmit power are both unknown. The uncertainty in the PLE due to environmental factors is a significant challenge for RSS based localization. DEU is built after careful investigation of geometrical behaviors of differential received signal strength circles, i.e. the locus of possible location of the emitter when transmit power is unknown. It is shown that the uncertainty in the PLE corresponds to a linear uncertainty for the location of the emitter in two dimensional space. This critical observation creates a basis for the sensor to move towards the emitter without estimating the emitter location after only three measurements. Furthermore, with only four different measurements, it is possible to effectively estimate the location of the emitter as well as the PLE by means of intersection of DEUs. Intersection of DEUs attains Cramer Rao Lower Bound with a dramatically reduced execution time compared to nonlinear least squares estimator. DEU is also proposed as an efficient route planning tool for moving sensors such as unmanned aerial vehicles.

Fire Localization Based On Range-Range-Range Model for Limited Interior Space

Fire localization problem is studied based on temperature data taken by wireless sensor arrays and a novel range-range-range (RRR) model is proposed to overcome shortcomings, which exists in the current range-point-range (RPR) model in this paper. For a single sensor array composed of four sensors deployed with a square, three angle estimates on fire bearing can be obtained using far-field localization technology. These angle estimates are used to get their statistical mean and variance at a single time. Based on the statistical features, we propose two fire localization methods under the RRR frame, which are angle bisector and nonlinear filtering methods. For the angle bisector method, a recursive formula of the mean and variance is presented in time series so that global angle estimates can be used. Furthermore, a fire coordinate estimate, which is actually the center of estimated-range circle, can be taken by use of intersecting two angle bisectors from two sensor arrays. Moreover, the estimation of a radius for the estimated fire region is also realized. In order to improve localization accuracy and robustness of fire estimation to non-Gaussian noise component, the fire localization is taken as a nonlinear bearing-only tracking issue for the case where the covariance of measurement noise is unknown and a specific variational Bayesian adaptive square-cubature Kalman filter is proposed to estimate the coordinate of the center. These proposed algorithms not only provide some new points of view on the fire localization for limited interior space, but are helpful for practical fire fighting applications.

Using conventional fluorescent markers for far‐field fluorescence localization nanoscopy allows resolution in the 10‐nm range

We present a novel technique of far-field localization nanoscopy combining spectral precision distance microscopy with widely used fluorochromes like the Green Fluorescent Protein (GFP) derivatives eGFP, EmGFP, Yellow Fluorescent Protein (YFP) and eYFP, synthetic dyes like Alexa 488 and Alexa 568, as well as fluoresceine derivates. Spectral precision distance microscopy allows the surpassing of conventional resolution limits in fluorescence far-field microscopy by precise object localization after the optical isolation of single signals in time. Based on the principles of this technique, our novel nanoscopic method was realized for laser optical precision localization and image reconstruction with highly enhanced optical resolution in intact cells. This allows for spatial assignment of individual fluorescent molecules with nanometre precision. The technique is based on excitation intensity dependent reversible photobleaching of the molecules used combined with fast time sequential imaging under appropriate focusing conditions. A meaningful advantage of the technique is the simple applicability as a universal tool for imaging and investigations to the major part of already available preparations according to standard protocols. Using the above mentioned fluorophores, the positions of single molecules within cellular structures were determined by visible light with an estimated localization precision down to 3 nm; hence distances in the range of 10-30 nm were resolved between individual fluorescent molecules allowing to apply different quantitative structure analysis tools.

Auditory localization of nearby sources. III. Stimulus effects.

A series of experiments has examined the auditory localization of a nearby (< 1 m) sound source under four conditions: (1) a fixed-amplitude condition where loudness-based distance cues were available; (2) a monaural condition where the contralateral ear was occluded by an ear-plug and muff; (3) a high-pass condition where the stimulus bandwidth was 3 Hz to 15 kHz; and (4) a low-pass condition where the stimulus bandwidth was 200 Hz to 3 kHz. The results of these experiments were compared to those of a previous experiment that measured localization performance for a nearby broadband, random-amplitude source [Brungart et al., J. Acoust. Soc. Am. 106, 1956-1968 (1999)]. Directional localization performance in each condition was consistent with the results of previous far-field localization experiments. Distance localization accuracy improved slightly in the fixed-amplitude condition relative to the earlier broadband random-amplitude experiment, especially near the median plane, but was severely degraded in the monaural condition. Distance accuracy was also found to be highly dependent on the low-frequency energy of the stimulus: in the low-pass condition, distance accuracy was similar to that in the broadband condition, while in the high-pass condition, distance accuracy was significantly reduced. The results suggest that low-frequency interaural level differences are the dominant auditory distance cue in the proximal region.