In this paper, we discuss a number of high-resolution direction finding methods for determining the two-dimensional directions of arrival of a number of plane waves, impinging on a sensor array. The array consists of triplets of sensors that are identical, as an extension of the 1D ESPRIT scenario to two dimensions. New algorithms are devised that yield the correct parameter pairs while avoiding an extensive search over the two separate one-dimensional parameter sets. EDICS: 5.2.2., 5.2.4. PRELIMINARIES The data modelConsider m sensor triplets, each composed of three identical sensors with unknown gain and phase patterns, which may vary from triplet to triplet. For every triplet, the displacement vectors d xy and d xz between its components are required to be the same (and, for convenience, are assumed to be orthogonal). This way, assigning the three sensor of each triplet to each of the sensor-arrays X, Y, Z, respectively, three identical although displaced arrays are obtained. This is a direct extension of the 1D ESPRIT scenario to two dimensions (see also [9]). Impinging on every array are d narrowband non-coherent signals s k t, The subspace approachMatrix polynomials of the form E − αF;α ∈ | C ; are called matrix pencils. Forming the pencilsit is seen that, in the noise free case, numbers λ = λ i and µ = µ j , i; j = 1; 2; : : : ; d , that reduce the rank of the pencil by one are equal to φ −1 i and θ −1 j respectively. With square data matrices, these rank reducing numbers are the generalized eigenvalues of the matrix pairs X ; Y and X ; Z.With noise present, however, a large number of samples are taken to improve accuracy. As a result, X, Y and Z will not be square. Noise will also increase the rank of the pencils, and this will introduce new rank reducing numbers. By computing a Total Least Squares projection of the data matrices (see e.g., [10] on
Global Navigation Satellite System (GNSS) Receiver Autonomous Integrity Monitoring (RAIM) is essential for safety-of-life and liability critical applications. This paper discusses two fundamentally different ways to assess the integrity risk of an operation with RAIM, based on a different amount of information available : the expected (or average) performance that is computed using the GNSS models only and the real-time (or actual) performance, which also uses information on the internal status of a GNSS receiver. It is shown both theoretically and by simulation that the real-time integrity risk significantly exceeds the expected risk after the detection and exclusion of a failing satellite. Therefore, while most published RAIM algorithms base their performance assessment on the expected performance only, this is only correct when the requirements allow the risk evaluation to be averaged over multiple operations. However, when the GNSS integrity requirement is to be applied on a ' per operation' basis, real-time integrity measures are more appropriate.
Performance verification of GNSS-based safety-of-life systems requires demonstration that theprobability of an excessive positioning error is as small as 10 -7 or even smaller. Integrity concepts are based on the requisite that the system provides conservative performance estimates for alert generation. The GNSS Integrity Monitoring and Analysis Tool (GIMAT) developed by the authors focuses on data-based analyses to verify whether this is indeed the case. Since the tails of the error distributions must be investigated, it is impractical to simply use observed cases of performance over-estimation. Such events are -by design -extremely rare and would require many decades of data collection. Nevertheless, system verification requires assessment of a system's integrity performance within limited time. GIMAT therefore exploits Extreme Value Theory (EVT), which enables extrapolation of the observed distribution's tail into the misleading information domain. The paper presents a short introduction into the mathematical theory of EVT in the context of integrity verification, and contains real-life results that are obtained from the analysis of EGNOS data. Based on two months of data, it is shown that EGNOS is compliant with ICAO requirements for this period, and that EVT is successful in predicting integrityperformance of navigation systems at high confidence levels.
Global navigation satellite systems (GNSS) are widely used for safety-of-life positioning applications. Such applications require high integrity, availability, and continuity of the positioning service. Integrity is assessed by the definition of a protection level, which is an estimation of the maximum positioning error at extremely low probability levels. The emergence of multi-frequency civilian signals and the availability of satellite-based augmentation systems improve the modeling of ionospheric disturbances considerably. As a result, in many applications the tropospheric delay tends to become one of the limiting factors of positioning—especially at low elevation angles. The currently adopted integrity concepts employ a global constant to model the variance of the residual tropospheric delay error. We introduce a new approach to derive residual tropospheric delay error models using the extreme value analysis technique. Seventeen years of global numerical weather model fields are analyzed, and new residual error models are derived for some recently developed tropospheric delay models. Our approach provides models that consider both the geographical location and the seasonal variation of meteorological parameters. Our models are validated with a 17-year-long time series of zenith tropospheric delay estimates as provided by the International GNSS Service. The results show that the developed models are still conservative, while the maximal residual error of the tropospheric delay is still improved by 39–55%. This improvement yields higher service availability and continuity in safety-of-life applications of GNSS.
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