Geometric Dilution Of Precision Research Articles

Dilution of Position Calculation for Accuracy Improvement in Wireless Location Systems

Geometric dilution of precision (GDOP) represents the geometric effect on the relationship between measurement error and positioning determination error. In the calculation of GDOP value, the maximum volume method does not guarantee the optimal selection of the four measurement units. The conventional method for calculating GDOP is to use matrix inversion to all subsets. In this paper, we employ GDOP using the matrix inversion method to select appropriate base stations (BSs) in cellular communication systems. The proposed BS selection criterion performs better than the random subsets of four or five BSs chosen from all seven BSs. The performances of MS location strongly depend on the relative position of the MS and BSs. Therefore, it is very important to select a subset with the most appropriate BSs rapidly and reasonably before positioning.

A Study on GPS GDOP Approximation Using Support-Vector Machines

Global Positioning System (GPS) has extensively been used in various fields. Geometric Dilution of Precision (GDOP) is an indicator showing how well the constellation of GPS satellites is geometrically organized. GPS positioning with a low GDOP value usually gains better accuracy. However, the calculation of GDOP is a time- and power-consuming task that involves complicated transformation and inversion of measurement matrices. When selecting from many GPS constellations the one with the lowest GDOP for positioning, methods that can fast and accurately obtain GPS GDOP are imperative. Previous studies have shown that numerical regression on GPS GDOP can get satisfactory results and save many calculation steps. This paper deals with the approximation of GPS GDOP using statistics and machine learning methods. The technique of support vector machines (SVMs) is mainly focused. This study compares the performance of several methods, such as linear regression, pace regression, isotonic regression, SVM, artificial neural networks, and genetic programming (GP). The experimental results show that SVM and GP have better performance than others.

Performance Improvement of GPS GDOP Approximation Using Recurrent Wavelet Neural Network

One of the most important factors affecting the precision of the performance of a GPS receiver is the relative positioning of satellites to each other. Therefore, it is essential to choose appropriate accessible satellites utilized in the calculation of GPS positions. Optimal subsets of satellites are determined using the least value of their Geometric Dilution of Precision (GDOP). The most correct method of calculating GPS GDOP uses inverse matrix for all combinations and selecting the lowest ones. However, the inverse matrix method, especially when there are so many satellites, imposes a huge calculation load on the processor of the GPS navigator. In this paper, the rapid and precise calculation of GPS GDOP based on Recurrent Wavelet Neural Network (RWNN) has been introduced for selecting an optimal subset of satellites. The method of NNs provides a realistic calculation approach to determine GPS GDOP without any need to calculate inverse matrix.

Optimal allocation of multi-sensor passive localization

To improve multi-sensor passive localization precision, the optimal allocation form of multi-sensor bearing-only localization is analyzed from the perspective of the geometric dilution of precision (GDOP) of the least squares (LS) algorithm. This paper indicates that whether the target lies inside the closed region formed by the sensors or not, the optimal allocation is when each adjacent cut angle between a sensor pair is identical, and all sensors are located on the border of the circle that has the target (which is also the origin of the coordinate system) as its center. When there is no restriction on the adjacent cut angle in the above allocation, the optimal estimate of the LS algorithm in the sense of minimum variance can be acquired. When the LS algorithm achieves the optimal estimate and the sensors present the optimal allocation, the respective consistencies are analyzed. Simulation results verify the analysis of the optimal allocation form above, which can be used in multi-sensor passive localization algorithms based on sensor management.

Adaptive sensor selection in wireless sensor networks for target tracking

Sensor selection in wireless sensor networks (WSN) is considered for target tracking. A decentralised estimation scenario is considered based on the extended information filter. A cost function using the geometrical dilution of precision (GDOP) measure is derived for active sensor selection. Accordingly, a new adaptive sensor selection (ASS) algorithm is proposed in which the number of active sensors is adaptively determined based on the absolute local innovations vector. Simulation results show that the tracking accuracy of the ASS is comparable to that of the other algorithms, but with less energy consumption.

Efficient solution of additional base stations in time-of-arrival positioning systems

An efficient solution of additional base stations (BSs) in time-of-arrival positioning systems, which aims to minimise the geometric dilution of precision, is presented based on generalised eigenvalue decomposition. A new indicator of geometry, the geometric consistency factor, is also introduced. The presented solution can save substantial computations for the geometry optimisation of additional BSs. Numerical simulations demonstrate the effectiveness of the solution for different numbers of additional BSs.

KNOWLEDGE-BASED METHODS FOR OPTIMUM APPROXIMATION OF GEOMETRIC DILUTION OF PRECISION

Global Positioning System (GPS) satellites signal processing to obtain all in view satellite measurements and to use them to find a solution and to do integrity monitoring forms a major component of the load on the receiver's processing element. If processing capability is limited there is restriction on the number of measurements which can be obtained and processed. Alternatively, the number of measurements can be restricted and the resulting saving in load on the processor can be used to offer more spare processing time which can be used for other user specific requirements. Thus if m visible satellites can provide measurements only n measurements can be used (n < m). The arrangement and the number of GPS satellites influence measurement accuracy. Dilution of Precision (DOP) is an index evaluating the arrangement of satellites. Geometric DOP (GDOP) is, in effect, the amplification factor of pseudo-range measurement errors into user errors due to the effect of satellite geometry. The GDOP approximation is an essential feature in determining the performance of a positioning system. In this paper, knowledge-based methods such as neural networks and evolutionary adaptive filters are presented for optimum approximation of GDOP. Without matrix inversion required, the knowledge-based approaches are capable of evaluating all subsets of satellites and hence reduce the computational burden. This would enable the use of a high-integrity navigation solution without the delay required for many matrix inversions. Models validity is verified with test data. The results are highly effective techniques for GDOP approximation.

A new location precision model for multi-static over-the-horizon radar system

In this paper, a multi-static system working in an active way is made up of ionospheric oblique backscatter sounding system (IOBSS) and two separate receiving stations, which adopts discontinuous wave mechanism. We have advanced a new model that contains skywave condition to locate over-the-horizon targets. We use a single quasi-parabolic (QP) ionosphere model and an analytic ray-tracing program to obtain the coordinate registration (CR) index, which changes skywave group range to ground range. Also, IOBSS and other two receiving stations use this distance information to locate the target which is far away from the system. The analytic expression for the geometric dilution of precision (GDOP) under different station deployments are obtained, which shows GDOP is influenced by the system measurement precision, the stations’ coordinates, and CR index. By computer simulation, we find that GDOP of isosceles right triangle deployment is smaller than that of line deployment and location precision will be improved with increasing base line length. The results indicate that this model is practicable with an acceptable range of error (less than 500 m under certain conditions in this paper).

Effects of Pseudolite Positioning on DOP in LAAS

In this paper, effects on DOP (Dilution of Precision) due to augmentation of Global Positioning System (GPS) with pseudolites are investigated. For this purpose, a typical Local Area Augmentation System (LAAS) scenario is consi-dered by placing pseudolites in various positions. It is found that only properly located pseudolites can improve the DOP. DOP values with two pseudolites located on either side of the run way are found to be the best. Geometric DOP (max) was found to be nearly 4 due to only GPS and came down to approximately 2 due to augmentation with two pseudolites. Implementation aspects of Bayes and Kalman filters while estimating DOP values are also examined.

Wireless Location Tracking Algorithms for Environments with Insufficient Signal Sources

Location estimation and tracking for the mobile devices have attracted a significant amount of attention in recent years. The network-based location estimation schemes have been widely adopted based on the radio signals between the mobile device and the base stations. The location estimators associated with the Kalman filtering techniques are exploited to both acquire location estimation and trajectory tracking for the mobile devices. However, most of the existing schemes become inapplicable for location tracking due to the deficiency of signal sources. In this paper, two predictive location tracking algorithms are proposed to alleviate this problem. The predictive location tracking (PLT) scheme utilizes the predictive information obtained from the Kalman filter in order to provide the additional signal inputs for the location estimator. Furthermore, the geometric-assisted PLT (GPLT) scheme incorporates the geometric dilution of precision (GDOP) information into the algorithm design. Persistent accuracy for location tracking can be achieved by adopting the proposed GPLT scheme, especially with inadequate signal sources. Numerical results demonstrate that the GPLT algorithm can achieve better precision in comparison with other network-based location tracking schemes.

A Fast Satellite Selection Algorithm: Beyond Four Satellites

Satellite selection is an important executive logic to prevent unnecessary navigation signals in the first place for Global Positioning System (GPS) receivers with limited number of channels and real-time processing power, such as those used in mobile phones, cars, and space crafts. In this paper, we propose a fast satellite selection algorithm to select more than four satellites based on the optimal geometries, which can obtain the smallest geometric dilution of precision (GDOP) values. The main idea of this fast algorithm is to select a subset of all satellites in view whose geometry is the most similar to the optimal geometry. Computer simulation shows that the consumed time of this algorithm is very close to that of the quasi-optimal satellite selection algorithm and obviously lower than that of the traditional optimal satellite selection algorithm to minimize GDOP factor, but the increased GDOP values relative to the minimal GDOP values are much smaller than those of the quasi-optimal satellite selection algorithm.

GDOP Analysis for Positioning System Design

Geometric dilution of precision (GDOP) has been widely used as an accuracy metric for navigation and tracking systems. Since high accuracy in a positioning system requires both accurate measurement of the range and a good geometric relationship between the mobile device and the measuring points, the analysis of GDOP is an essential feature in determining the performance of a positioning system. In this paper, we perform GDOP analysis to obtain concise analytical expressions for a number of scenarios, which are generally applicable to geometries where the mobile device is surrounded by base stations. Comparison of the analytical results with simulations using the typical geometries of indoor positioning systems shows good agreement, except when the mobile position is close to a base station. This effect is a consequence of the ranging errors being a significant proportion of the range in short-range tracking systems and discontinuities in GDOP at the base station. The results provide useful information for the design and testing of tracking systems, as well as for the determination of the geometric deployment of base stations for good GDOP in the coverage area.

Target localisation techniques and tools for multiple-input multiple-output radar

This study presents a comparative study of coherent and non-coherent target localisation techniques for multiple-input multiple-output (MIMO) radar systems with widely distributed elements. Performance is evaluated based on closed-form solutions developed for the best linear unbiased estimator (BLUE) for each of the localisation methods. These estimators afford insights into the relation between radar locations, target location and localisation accuracy. In particular, the means squared error of the BLUE is factored into a term dependent on signal and processing characteristics and a term dependent on sensor locations. The latter is referred to as geometric dilution of precision (GDOP). The best achievable accuracy for the coherent case is obtained, and a comparative study with the non-coherent case is presented. MIMO radar systems with coherent processing are shown to benefit from a gain because of coherent processing among sensors. This gain is referred to as coherent localisation gain, and it is proportional to the ratio of the signal carrier frequency to the effective bandwidth (a large ratio for typical signals). The footprint of multiple transmit/receive sensors results in a gain, referred to as MIMO gain, for both processing techniques. The MIMO gain is proportional to the product of the number of transmitting and receiving sensors. Analysis of the MIMO gain through the use of GDOP contour maps demonstrate the achievable accuracy at various target locations for a given layout of sensors.

ETP/GDOP Behavior Study for N-Sensors Arrays ina Multilateration Radar System

In this paper, we evaluated the ETP (Expected Theoretical Precision) and GDOP (Geometric Dilution Of Precision) enhancement related to the number of sensors in a Multilateration radar system. An introduction about the principles of the Multilateration radar system basis operation is described, then, the formulation for evaluation the ETP/GDOP of the 3D positioning is shown. We observed that the ETP and GDOP enhance with the increase of the number of sensors. A substantial improvement was obtained until nine sensors but, for more sensors that improvement is reduced. Results for a 75km×75km area are shown, including LAM (Local Area Multilateration) and WAM (Wide Area Multilateration) settings and different values of the aircraft height (5m for LAM surface, 5000m and 8000m for WAM). Additional parameters are shown in order to evaluate the system quality. These parameters are the Expected Theoretical Precision Gain (ETP Gain), Homogenization Level (HL) and Percentage Over a Reference Value (PORV). Due to the proportionality between the ETP and GDOP, only ETP results are shown. In the simulations the same settings of the sensors was used; 3ns for instrumental error and 27m for antenna height. These are typically values for real Multilateration radar system used for the air traffic control.

Joint time difference of arrival/angle of arrival position finding in passive radar

Several target position finding methods are proposed in various papers mainly regarding sensor networks. However, the problem of position finding in passive radar systems is somewhat different from the general case of sensor networks. Generally, in a passive radar system, there are few receivers located at short distances when compared with the it distance from the target. In this case, a problem known as geometric dilution of precision (GDP) occurs, which considerably increases the error of many proposed methods. This phenomenon can even make the position finding equations non-solvable. Regarding this fact, we have developed a least mean square error (LMSE) based method, which uses both the angle of arrival (AOA) and time difference of arrival (TDOA) measurements to estimate almost precisely the position of very distant targets in a passive radar system. A simple equation is derived here to compare the TDOA and AOA accuracies. Then, it is shown that whenever the accuracies of these two measurements are comparable, the proposed method estimates the target position more precisely than the conventional AOA-only and TDOA-only methods. It also avoids the non-convergence behaviour encountered in TDOA-only methods.

SSSD-Based Mobile Positioning: On the Accuracy Improvement Issues in Distance and Location Estimations

The stationary signal-strength-difference (SSSD)-based positioning technique has been proposed as a cost-effective, low-complexity solution. The positioning accuracy with this technique, however, is limited, as a result of the large error in the distance. In this paper, we contribute a simple and efficient method for the accuracy improvement issues in the distance. Starting from the concept of the global positioning system (GPS) pseudorange (PD) equation, a complete set of distance difference (DD) is formulated. In addition, we then correct the error in the distance by going in the set of DD, based on the least-squares (LS) sense. The theoretical model is established and verified via experimental data gathered in an urban environment. Results clearly show that the proposed method performs well when encountering a large error in the distance. On top of this, we investigate the accuracy improvement issues in the location estimation and propose an iterative LS algorithm by using the corrected distances. The positioning accuracy is studied based on the analysis of different location errors, and the effect of geometric dilution of precision (GDOP) is also investigated. The improvement of the positioning accuracy is clearly demonstrated through the experimental data collected at 55 different test points with 25 133 samples, which is outstanding.

Wireless Location Estimation With the Assistance of Virtual Base Stations

In recent years, wireless location estimation has attracted a significant amount of attention in different areas. Various types of radio signals are applied for the development of location-estimation algorithms. In this paper, the range measurements acquired from the received time-based information are adopted, and the modified least square (LS) method is utilized to process the raw data and to finally locate the target object. Practical issues, such as the nonline-of-sight (NLOS) errors and the geometric dilution of precision (GDOP) effect, are of concern. The NLOS error will cause a large nonnegative bias while measuring the propagation delay, which will lead to an unreliable result for location estimation. On the other hand, a large GDOP value corresponds to a poor geometric topology, which will result in inferior performance by adopting most of the existing location algorithms. The proposed location-estimation algorithms with virtual base stations (VBSs) will both mitigate the influence from the NLOS errors by imposing the geometric constraints and reduce the GDOP effect by incorporating the assisted VBSs. Two iterative schemes are proposed, including the center-of-gravity-based VBS (VBS-CG) and the minimal GDOP-based VBS (VBS-MG) algorithms, to determine the required number and the locations of the assisted VBSs. The proposed VBS algorithms are compared with other existing location-estimation schemes via simulations. The performance of the VBS-MG algorithm is observed to outperform the other schemes, particularly under the environments with larger NLOS errors and poor geometric layouts.

A closed-form formula for GPS GDOP computation

Geometric dilution of precision (GDOP) is often used for selecting good satellites to meet the desired positioning precision. An efficient closed-form formula for GDOP has been developed when exactly four satellites are used. It has been proved that increasing the number of satellites for positioning will always reduce the GDOP. Since most GPS receivers today can receive signals from more than four satellites, it is desirable to compute GDOP efficiently for the general case. Previous studies have partially solved this problem with artificial neural network (ANN). Though ANN is a powerful function approximation technique, it needs costly training and the trained model may not be applicable to data deviating too much from the training data. Using Newton’s identities from the theory of symmetric polynomials, this paper presents a simple closed-form formula for computing GDOP with the inputs used in previous studies. These inputs include traces of the measurement matrix and its second and third powers, and the determinant of the matrix.

Geometric dilution of precision for GPS single-point positioning based on four satellites *

To improve the positioning accuracy in GPS point positioning, the geometric dilution of precision (GDOP) including HDOP, VDOP, TDOP, PDOP is commonly considered. The properties of the DOP for the GPS satellite navigation system are studied and the coordinate system is improved in order to decrease the amount of variables. In the end, by simulation and discussing the results, the corresponding conclusions are presented.

Target Localization Accuracy Gain in MIMO Radar-Based Systems

This paper presents an analysis of target localization accuracy, attainable by the use of MIMO (Multiple-Input Multiple-Output) radar systems, configured with multiple transmit and receive sensors, widely distributed over a given area. The Cramer-Rao lower bound (CRLB) for target localization accuracy is developed for both coherent and non-coherent processing. Coherent processing requires a common phase reference for all transmit and receive sensors. The CRLB is shown to be inversely proportional to the signal effective bandwidth in the non-coherent case, but is approximately inversely proportional to the carrier frequency in the coherent case. We further prove that optimization over the sensors' positions lowers the CRLB by a factor equal to the product of the number of transmitting and receiving sensors. The best linear unbiased estimator (BLUE) is derived for the MIMO target localization problem. The BLUE's utility is in providing a closed form localization estimate that facilitates the analysis of the relations between sensors locations, target location, and localization accuracy. Geometric dilution of precision (GDOP) contours are used to map the relative performance accuracy for a given layout of radars over a given geographic area.