Integer estimation methods for GPS ambiguity resolution: an applications oriented review and improvement

Abstract

The integer least squares (ILS) problem, also known as the weighted closest point problem, is highly interdisciplinary but no algorithm can find its global optimal integer solution in polynomial time. We first outline two suboptimal integer solutions, which can be important either in real-time communication systems or to solve high dimensional GPS integer ambiguity unknowns. We clarify that the popular sorted QR suboptimal estimator, usually known to be invented by Wübben et al. [42], was first discussed by Xu et al. [51]. We then focus on the most efficient algorithms to search for the exact integer solution. We show that the combined algorithm proposed by Fincke and Pohst [8] and Schnorr and Euchner [29], which is well known to be the most powerful algorithm for solving the ILS problem, is much faster than LAMBDA in the sense that the ratio of integer candidates to be checked by the combined algorithm to those by LAMBDA can be theoretically expressed by r m , where r ≤ 1 and m is the number of integer unknowns. Finally, we further improve the searching efficiency of the most powerful combined algorithm by implementing two sorting strategies, which can either be used for finding the exact integer solution or for constructing a suboptimal integer solution. Test examples clearly demonstrate that the improved methods can perform significantly better than the most powerful combined algorithm to simultaneously find the optimal and second optimal integer solutions, if the ILS problem cannot be well reduced.

Mixed Integer Linear Models

“…In case that some prior knowledge on the lower and upper bounds of z is available, one can also formulate the corresponding 0 1 (mixed) integer models. For more details on these specific models, the reader is referred to Xu et al (1995Xu et al ( , 2000 and Xu (1998Xu ( , 2002b.…”

“…is the error vector of the observations y. The first mathematically rigorous treatment of solving the integer unknowns in geodesy was made by Teunissen (1993), followed by Xu et al (1995) with a different approach. For more details on GNSS systems, the reader may refer to, for example, Hofmann-Wellenhof et al (1992), Seeber (1993), and Parkinson and Spilker (1996).…”

“…Although (2) has been well known in geodetic literature as the standard GPS observation model, the author feels that the terminology of GPS observation model may hardly be understood by the people who do not work on GPS. Since the interest in (2) is obviously interdisciplinary (e.g., integer programming, geometry of numbers, communication theory, cryptography, and crystallography), Xu et al (1995Xu et al ( , 2000 and Xu (1998Xu ( , 2002bXu ( , 2003bXu ( , 2006 …”

Impact of Decorrelation on Success Rate Bounds of Ambiguity Estimation

“…The most popular way to calculate the IB success rate is sorting the diagonal entries of Qââ by ascending order. Xu et al (2012) also examined another more complicated sorting algorithm called Vertical Bell Labs Layered SpaceTime (V-BLAST) and reported having a better performance and heavier computation burden.…”

“…Recently, the importance of permutation has been systematically studied. Xu et al (2012) compared the impact of different permutation strategies on decorrelation performance. The permutation procedure has been applied to the LLL method, e.g.…”