Towards a Unified Theory of GPS Ambiguity Resolution
Towards a Unified Theory of GPS Ambiguity Resolution
复制标题
DOI：
10.5081/jgps.2.1.1
复制
发表时间：
2003-06
期刊：
影响因子：
--
通讯作者：
P. Teunissen
中科院分区：
文献类型：
--
作者：
P. Teunissen
在这篇特邀文章中，将简要回顾作者在过去十年中发展起来的整数估计理论，并从1993年引入LAMBDA方法开始。本文讨论了三种不同但密切相关的模糊估计器。它们是整数估计量、整数孔径估计量和整数等变估计量。整数估计量是整数孔径估计量，整数孔径估计量是整数等变估计量。然而，反过来不一定是正确的。因此，在三种类型的估计量中，整数估计量是最具限制性的。它们的拉入区域是平移不变的、不相交的，并且完全覆盖了模糊空间。众所周知的例子有整数舍入、整数自举和整数最小二乘。一类限制较少的估计量是整数孔径估计量。它们的牵引区域只服从三个条件中的两个。它们仍然是平移不变的和分析的，但它们不需要完全覆盖歧义空间。因此，整数孔径估计量具有混合性质，具有整数或非整数结果。整数孔径估计量的例子有比值估计量和差分估计量。整数等变估计量是这三类中限制较小的一类。这些估计量只满足三个条件中的一个，即平移不变量的条件。因此，整数等变估计的结果总是被重新赋值。对于每一类估计量，我们也给出了最优估计量。虽然通常假设为高斯情况，但结果是针对浮点解的任意概率密度函数提出的。在高斯情况下，最优整数估计量是整数最小二乘估计量。所使用的最优性准则是使正确整数估计的概率最大化，即所谓的成功率。高斯情况下的最优整数孔径估计量是当整数最小二乘残差位于最优孔径拉入区域时，只返回整数最小二乘解的估计量。该区域由浮点解的概率密度函数和整数最小二乘残差的概率密度函数控制。拉入区域的孔径由用户定义的孔径参数控制。所使用的最优性准则是在给定一个固定的、用户定义的不正确整数估计概率的情况下，使正确整数估计的概率最大化。当成功率和失败率之和等于1时，最优整数孔径估计量与最优整数孔径估计量相等。最佳整数等变估计量是所有整数的无限加权和。权重由浮点解的概率密度函数与其整数移位副本序列之比确定。所使用的最优性准则是使均方误差最小。因此，最佳整数等变估计器在精度方面总是优于浮点解决方案。
In this invited contribution a brief review will be presented of the integer estimation theory as developed by the author over the last decade and which started with the introduction of the LAMBDA method in 1993. The review discusses three different, but closely related classes of ambiguity estimators. They are the integer estimators, the integer aperture estimators and the integer equivariant estimators. Integer estimators are integer aperture estimators and integer aperture estimators are integer equivariant estimators. The reverse is not necessarily true however. Thus of the three types of estimators the integer estimators are the most restrictive. Their pull-in regions are translational invariant, disjunct and they cover the ambiguity space completely. Well-known examples are integer rounding, integer bootstrapping and integer least-squares. A less restrictive class of estimators is the class of integer aperture estimators. Their pull-in regions only obey two of the three conditions. They are still translational invariant and disjunct, but they do not need to cover the ambiguity space completely. As a consequence the integer aperture estimators are of a hybrid nature having either integer or non-integer outcomes. Examples of integer aperture estimators are the ratio-testimator and the differencetestimator. The class of integer equivariant estimators is the less restrictive of the three classes. These estimators only obey one of the three conditions, namely the condition of being translational invariant. As a consequence the outcomes of integer equivariant estimators are always realvalued. For each of the three classes of estimators we also present the optimal estimator. Although the Gaussian case is usually assumed, the results are presented for an arbitrary probability density function of the float solution. The optimal integer estimator in the Gaussian case is the integer least-squares estimator. The optimality criterion used is that of maximizing the probability of correct integer estimation, the so-called success rate. The optimal integer aperture estimator in the Gaussian case is the one which only returns the integer least-squares solution when the integer least-squares residual resides in the optimal aperture pull-in region. This region is governed by the probability density function of the float solution and by the probability density function of the integer least-squares residual. The aperture of the pull-in region is governed by a userdefined aperture parameter. The optimality criterion used is that of maximizing the probability of correct integer estimation given a fixed, user-defined, probability of incorrect integer estimation. The optimal integer aperture estimator becomes identical to the optimal integer estimator in case the success rate and the fail rate sum up to one. The best integer equivariant estimator is an infinite weighted sum of all integers. The weights are determined as ratios of the probability density function of the float solution with its train of integer shifted copies. The optimality criterion used is that of minimizing the mean squared error. The best integer equivariant estimator therefore always outperforms the float solution in terms of precision.