Calculation of Weighted Geometric Dilution of Precision

Abstract

To achieve high accuracy in wireless positioning systems, both accurate measurements and good geometric relationship between the mobile device and the measurement units are required. Geometric dilution of precision (GDOP) is widely used as a criterion for selecting measurement units, since it 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 necessarily guarantee the selection of the optimal four measurement units with minimum GDOP. The conventional matrix inversion method for GDOP calculation demands a large amount of operation and causes high power consumption. To select the subset of the most appropriate location measurement units which give the minimum positioning error, we need to consider not only the GDOP effect but also the error statistics property. In this paper, we employ the weighted GDOP (WGDOP), instead of GDOP, to select measurement units so as to improve the accuracy of location. The handheld global positioning system (GPS) devices and mobile phones with GPS chips can merely provide limited calculation ability and power capacity. Therefore, it is very imperative to obtain WGDOP accurately and efficiently. This paper proposed two formations of WGDOP with less computation when four measurements are available for location purposes. The proposed formulae can reduce the computational complexity required for computing the matrix inversion. The simpler WGDOP formulae for both the 2D and the 3D location estimation, without inverting a matrix, can be applied not only to GPS but also to wireless sensor networks (WSN) and cellular communication systems. Furthermore, the proposed formulae are able to provide precise solution of WGDOP calculation without incurring any approximation error.

Weighted Geometric Dilution of Precision Calculations with Matrix Multiplication

“…In order to verify the superior properties of the proposed formula, we compare the results of WGDOP calculation accuracy for all methods and matrix inversion method. Figure 2 shows cumulative distribution functions (CDFs) of the average WGDOP residual for the proposed formula, the previously proposed method [ 30 ] and Rprop-based algorithm for Type 6 [ 27 ] when four measurements are available. From simulation results, the proposed method and matrix inversion method provide nearly identical WGDOP estimation.…”

“…To improve the WGDOP accuracy effectively and efficiently, the author of this paper proposed the closed-form solutions for two WGDOP formations for the case of processing four measurements with different variances and one of the measurements with better accuracy than the others [ 30 ]. When exactly four measurements are used, the formula provides the best computational efficiency.…”

Geometric Dilution of Precision of the GNSS for Mars (Gnss Fatima)

“…The value c is a constant which defines the speed of light and t b defines the deviation of the time bases of the space and user segments. The value v ri defines the noise of the measured pseudorange (Chen et al 2013). In the computation it is necessary to define the formal precision by the determination of values P xx , P yy , P zz , P tt .…”

“…As mentioned, the last expression of value H represents the geometric matrix (Chen et al 2013). According to the lowest exponentiation of the algorithm of direct visibility, the solution of equation 12can be expressed as ( )…”

Base Station Selection for Hybrid TDOA/RTT/DOA Positioning in Mixed LOS/NLOS Environment

“…Therefore, according to the definitions, GDOP is defined as the ratio of positioning error to positioning signal measurement error [ 22 ]. For the independent non-identical measurement errors, the GDOP calculated using WLS is called weighted GDOP [ 30 , 31 ] and the GDOP of the hybrid TDOA, RTT and DOA positioning is given by the trace of the inverse of the matrix …”