Calibrating the error from sensor position uncertainty in TDOA-AOA localization

Efficient closed-form solution for target localization using TDOA measurements in the presence of sensor position errors

It is well known that the presence of sensor position errors may lead to failure of farfield/nearfield wideband beamformers for microphone arrays. Therefore, many efforts have been made to design wideband beamformers for microphone arrays robust against sensor position errors in recent years. This paper presents the findings obtained on the study and analysis of the effects of sensor position errors to the performance of farfield/nearfield wideband beamformers for microphone arrays. Several useful properties have been derived through the conduct of a comprehensive statistical analysis, including the bias and variance analysis of array response, to provide in-depth insights into the robustness characteristics of farfield and nearfield wideband beamformers in the presence of sensor position errors. The analysis does not assume any specific array geometry and is thus applicable to arrays with arbitrary geometries. Numerical examples obtained via Monte Carlo simulations are presented to verify the proposed theoretical analysis. This paper is helpful to better understand the effects of sensor position errors on wideband beamformers and useful in microphone array design.

An auxiliary source-based near field source localization method with sensor position error

The sensor position uncertainties and synchronization offsets can cause substantial performance degradation if the sensors are not properly calibrated. This paper investigates the localization of a constant velocity moving target and the self-calibration of sensors using a sequence of range and azimuth measurements observed at successive instants. A theoretical study by the Cramer-Rao Lower Bound (CRLB) reveals that the sensor positions can only be self-calibrated when there are at least two sensors and synchronization offsets can be handled by joint estimation. A low complexity sequential closed-form solution is proposed to estimate the target position and velocity first, and the coordinates of each sensor and synchronization offset afterward. While less intuitive, the analysis shows that the closed-form solutions for both the target and sensor parameters can reach the CRLB accuracy under small Gaussian noise. We also develop a semidefinite programming (SDP) solution by semidefinite relaxation (SDR) for joint localization and calibration from the Maximum Likelihood formulation, which exhibits higher noise tolerance than the closed-form solution. Simulations validate the analysis and the performance of the proposed methods.

The demand for accurate localization in complex environments continues to increase despite the difficulty in extracting positional information from measurements are not yet complete. Like any other process, localization also has security requirements. The use of ultra-wideband (UWB) indoor localization systems have recently grown quickly in industries, with a reliable, fast, and have high accuracy performances. In particular, time difference-of-arrival (TDOA) is one of the widely used localization models. However, as TDOA measurement errors increase, the accuracy of the localization decreases. The accuracy of the TDOA measurement is influenced the accuracy of the localization and affected by multiple factors such as time synchronization, errors in sensor positions, missing data is caused by network attack. To reduce the influence of a sensor measurement error in localization, this paper proposes an improved localization algorithm for source localization using deep learning to address a TDOA measurements error or missing data in an asynchronous localization called DeepTAL. Unlike the conventional algorithm, DeepTAL can obtain highly accurate localization data in the presence of TDOA measurement errors or missing data. The algorithm starts with TDOA measurements without time synchronization. A network based on Long Short-Term Memory (LSTM) is then applied to achieve a stronger learning and better representation of the determined target state and TDOA prediction. The network can express features more abstractly at higher levels and increase recognition accuracy. After that, the target node is accurately located by TDOAs. We implement the DeepTAL algorithm on an asynchronous localization system with UWB signals. The experiments show that the proposed DeepTAL algorithm is efficient in improving the localization precision as measurement errors or missing data.

Source localization is an important field of research and the received signal strength (RSS)-based method is of particular interest. Further, by exploiting the sparsity of localization problem, the compressive sensing (CS) can be applied to considerably decrease the number of RSS measurements, especially in multi-source scenario. However, most existing CS-based localization methods usually neglect some practical issues. In particular, the sensor positions are assumed be known exactly, while in practice they may not be accurate. Additionally, a uniform Gaussian noise assumption is made that the noise variances of sensors are identical, but in practice the noise should be nonuniform. When these assumptions are violated, the localization performance will deteriorate dramatically. To address such issues, in this paper, we formulate the source localization based on superimposed RSS measurements as a sparse signal recovery problem. Moreover, by regarding the sensor positions as adjustable parameters, the inaccurate sensor positions can be refined through the adjustment of parameters. Following this idea, we develop a novel iterative algorithm for joint signal recovery and parameter optimization based on the variational expectation-maximization algorithm. Consequently, the sensor position uncertainty can be alleviated and thus the signal recovery performance will be improved greatly. Meanwhile, it is also capable of learning the variance of nonuniform noise. Extensive simulation results demonstrate the superiority of the proposed method.

When sensor position errors exist, the performance of recently proposed interference-plus-noise covariance matrix (INCM)-based adaptive beamformers may be severely degraded. In this paper, we propose a weighted subspace fitting-based INCM reconstruction algorithm to overcome sensor displacement for linear arrays. By estimating the rough signal directions, we construct a novel possible mismatched steering vector (SV) set. We analyze the proximity of the signal subspace from the sample covariance matrix (SCM) and the space spanned by the possible mismatched SV set. After solving an iterative optimization problem, we reconstruct the INCM using the estimated sensor position errors. Then we estimate the SV of the desired signal by solving an optimization problem with the reconstructed INCM. The main advantage of the proposed algorithm is its robustness against SV mismatches dominated by unknown sensor position errors. Numerical examples show that even if the position errors are up to half of the assumed sensor spacing, the output signal-to-interference-plus-noise ratio is only reduced by 4 dB. Beam patterns plotted using experiment data show that the interference suppression capability of the proposed beamformer outperforms other tested beamformers.

This paper proposes an improved two-step weighted least squares (TSWLS) algorithm for target location using time difference of arrival (TDOA) with sensor position errors. The performance of the TSWLS method is improved in a simple way through re-selecting reference sensor and rotating co-ordinate system, and the improved TSWLS(I-TSWLS) algorithm remains to be closed-form and approximately efficient as the TSWLS method. Simulation results show that the I-TSWLS algorithm can achieve the CRLB easily before higher noise threshold, and it exhibits lower estimation bias and better robustness with increasing sensor position errors.

This article proposes two algorithms for the localization or motion analysis of a moving target, by using a number of stationed sensors observing the frequency of the emitted target signal. This is a challenging problem due to the complicated non-linear relation between the target location parameters and the observations. The first algorithm is closed-form through algebraic evaluation that is computationally efficient. The second is derived from semidefinite programming by relaxationthat is robust against noise. The situations of having target frequency known but erroneous and completely unknown are investigated, as well as the presence of sensor position errors. Sequential algorithms for motion analysis are proposed in addition to the two for reducing latency and complexity. The algorithms are analyzed and their performance is able to achieve the CRLB accuracy under Gaussian noise.

Source localization in a wireless sensor network (WSN) is sensitive to the sensors' positions. In practice, due to mobility, the receivers' positions may be known inaccurately, leading to non-negligible degradation in source localization estimation performance. The goal of this paper is to develop a semidefinite programming (SDP) method using time-difference-of arrival (TDOA) and frequency-difference-of-arrival (FDOA) by taking the sensor position uncertainties into account. Specifically, we transform the commonly used maximum likelihood estimator (MLE) problem into a convex optimization problem to obtain an initial estimation. To reduce the coupling between position and velocity estimator, we also propose an iterative method to obtain the velocity and position, by using weighted least squares (WLS) method and SDP method, respectively. Simulations show that the method can approach the Cramér-Rao lower bound (CRLB) under both mild and high noise levels.

The combined effects of mutual coupling, gain and phase errors, and sensor position errors have great negative impact on the direction-finding performance of the MUSIC algorithm. This paper investigates the calibration of the array errors induced by the three errors. Based on the array errors calibration algorithm (algorithm 1) presented by See, two improved array error-calibrating algorithms are presented. One (algorithm 2) uses the sparseness of mutual coupling matrix, and the other (algorithm 3) utilizes more special structures of mutual coupling matrix. Although the three algorithms have the same computation mode and theoretical framework, the latter two algorithms have much higher precision than the former one, as more properties of mutual coupling matrix are utilized in the two algorithms, and if the array manifold is uniform linear array or uniform circular array, algorithm 3 could have more obvious advantages. On the other hand, the three algorithms can be generalized to the situation when the calibration source azimuths have deviations. The three extended algorithms can not only calibrate the array errors, but also offset the deviations of calibration source azimuths simultaneously. Finally, the performances of the three algorithms are compared both in the absence and presence of the deviations of calibration source azimuths. Many simulation experiments demonstrate that the estimation precision of array errors can be obviously improved if more properties of mutual coupling matrix are effectively used.

Uniform linear array is a kind of very important underwater acoustic detection equipment, it mainly divided into two kinds: rigid array and flexible array. Rigid array is the best means to obtain ideal detection result, but the layout and recovery is difficult. Flexible array is used more popular for its easy to operation, but the array shape is difficult to maintain. Shape distortion of the flexible linear array will impact the DOA estimation performance tremendously. Based on theoretical analysis, simulation studies and experimental data processing, this paper proposed an elements position correction method. The correction method of array distortion based on the equation of distance between two points in Euclidean 3D space. As shown in the figure, in order to estimate the actual spatial location of each element of the tow line array, two calibration sources were mounted on the ship rigidly, the absolute geographic coordinates is set to know based GPS. Then we can write 2N distance equations which between the two calibration sources and the array hydrophones: (xi − x01)2 + (yi − y01)2 + (zi − z01)2 = R2i1 (i = 1,2,···, N)(xi − x02)2 + (yi − y02)2 + (zi − z02)2 = R2i2 (i = 1,2,···, N) where (x01, y01, z01)and(x02, y02, z02) are the coordinates of calibration sources, (xi, yi, zi) are the coordinates of hydrophones, Ri1, Ri2 are the estimated distance between the calibration sources and the hydrophones. There are total 3N variable for N hydrophones' space coordinates, where it is assumed that each hydrophone depths can be read from the depth sensor, then the variable remain are 2N. We can get the 2N variable through solve the above 2N range equations, then the array shape can be calibrated. The signal propagation delay variation caused by sensor position turbulence are analyzed, the delay variation will direct impact on the steering vector which used in beamforming processor. The acoustic calibration method is given, which using two calibration sound sources mounted under the carrier transmit default acoustic signal, the hydrophones of the flexible linear array then receive these acoustic signals and gathered by the process center, the process center make an estimate of the actual coordinates of the sensors use the data, and fix the steering vector, complete the sensor position calibration. The simulation frequency is 750Hz, the sound speed of water is 1500m/s, the true target direction is 37 degrees, eight elements receiving array, array element spacing is one meter (half wavelength). For the sake of comparison, we execute the simulation with the same receive data use three array shapes: supposed shape, real shape and estimated shape respectively. The simulation based conventional beamforming(CBF) and adaptive beamforming (ABF) algorithm. The simulation results and field data processing proved that, after a certain accuracy of shape distortion correction, when processing with CBF and ABF processor, both the DOA estimation accuracy and 3dB mainlobe width can be significantly improved. Through the compare research on CBF and ABF beamforming algorithm with array shape distortion or not, we can see the CBF is more robust for array shape distortion. Though above analysis is limited that elements should in the same plane coordinate, but the result is still general, can be extended to three-dimensional space.

A mathematical method is developed for the reconstruction of the magnetic field of a source with sensors arranged in a rectangular array on a plane. A prolate spheroidal shell encloses the source. The magnetic field of the source is expressed in prolate spheroidal coordinates and the prolate spheroidal expansion coefficients are determined by the application of the derived selective functions to the x component of the measured magnetic field. In a simulation with 192 sensors, the magnetic field is accurately reconstructed for sample sources in the presence of sensor noise, sensor position, and orientation error. The reconstruction error is the least for sources with significant dipolar component.

High-resolution array processing algorithms for source localization are known to be sensitive to errors in the model for the sensor-array spatial response. In particular, unknown gain, phase, and mutual coupling as well as errors in the sensor positions can seriously degrade the performances of array-processing algorithms. This paper describes a calibration algorithm that estimates the calibration matrix consisting of the unknown gain, phase, and mutual-coupling coefficients as well as the sensor positions using a set of calibration sources in known locations. The estimation of the various parameters is based on a maximum likelihood approach. Cramer-Rao lower-bound (CRB) expressions for the sensor positions and the calibration matrix parameters are also derived. Numerical results are shown to illustrate the potential usefulness of the proposed calibration algorithm toward better accuracy and resolution in parametric array-processing algorithms.

Phenomenological modeling coupled with statistical signal processing has been shown to significantly improve capabilities for discriminating unexploded ordnance (UXO) from benign clutter using electromagnetic induction (EMI) sensor data. The general premise underlying the majority of these coupled approaches is that a phenomenological model is fit to the measured data, and the parameters estimated from this model inversion, which characterize the interrogated target, are utilized in subsequent statistical signal processing algorithms to classify the target as either UXO or clutter. A potential limitation of this coupled approach is that the inversion has been shown to be sensitive to uncertainty associated with the sensor positions. When the measurement positions are uncertain, the inversion results are more variable, and consequently, discrimination performance degrades. In this letter, a Bayesian methodology is applied to estimate the desired features from the measured data. This method explicitly acknowledges that uncertainty in the sensor positions exists and incorporates this knowledge to find the maximum-likelihood feature estimates by integrating over the uncertain measurement positions. Due to the high dimensionality of the integration, Monte Carlo integration, a statistical technique to estimate the value of an integral, is employed. Simulation results show that this Bayesian approach in mitigating sensor position uncertainty produces features with lower variability and, therefore, provides improved discrimination performance.