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Fast integer least-squares estimation for GNSS high-dimensional ambiguity resolution using lattice theory

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An Erratum to this article was published on 13 October 2011

Abstract

GNSS ambiguity resolution is the key issue in the high-precision relative geodetic positioning and navigation applications. It is a problem of integer programming plus integer quality evaluation. Different integer search estimation methods have been proposed for the integer solution of ambiguity resolution. Slow rate of convergence is the main obstacle to the existing methods where tens of ambiguities are involved. Herein, integer search estimation for the GNSS ambiguity resolution based on the lattice theory is proposed. It is mathematically shown that the closest lattice point problem is the same as the integer least-squares (ILS) estimation problem and that the lattice reduction speeds up searching process. We have implemented three integer search strategies: Agrell, Eriksson, Vardy, Zeger (AEVZ), modification of Schnorr–Euchner enumeration (M-SE) and modification of Viterbo-Boutros enumeration (M-VB). The methods have been numerically implemented in several simulated examples under different scenarios and over 100 independent runs. The decorrelation process (or unimodular transformations) has been first used to transform the original ILS problem to a new one in all simulations. We have then applied different search algorithms to the transformed ILS problem. The numerical simulations have shown that AEVZ, M-SE, and M-VB are about 320, 120 and 50 times faster than LAMBDA, respectively, for a search space of dimension 40. This number could change to about 350, 160 and 60 for dimension 45. The AEVZ is shown to be faster than MLAMBDA by a factor of 5. Similar conclusions could be made using the application of the proposed algorithms to the real GPS data.

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References

  • Agrell E, Eriksson T, Vardy A, Zeger K (2002) Closest point search in lattices. IEEE Trans Inf Theory 48: 2201–2214

    Article  Google Scholar 

  • Babai L (1986) On Lováasz’ lattice reduction and the nearest lattice point problem. Combinatorica 6: 1–13

    Article  Google Scholar 

  • Bartkewitz T (2009) Improved lattice basis reduction algorithms and their efficient implementation on parallel systems. Diploma thesis, Ruhr University, Germany

  • Buist PJ (2007) The baseline constrained LAMBDA method for single epoch, single frequency attitude determination applications. In: Proceedings of ION GPS, p 12

  • Chang X, Yang X, Zhou T (2005) MLAMBDA: a modified LAMBDA algorithm for integer least-squares estimation. J Geod 79: 552–565

    Article  Google Scholar 

  • Chen D (1994) Development of a fast ambiguity search filtering (FASF) method for GPS carrier phase ambiguity resolution. UCGE Reports 20071, PhD dissertation

  • Chen D, Lachapelle G (1995) A comparison of the FASF and least-squares search algorithms for on-the-fly ambiguity resolution, navigation. J Inst Navig 42(2): 371–390

    Google Scholar 

  • Cohen CE (1996) Attitude determination. In: Parkinson BW, Spilker JJ (eds) Global positioning system: theory and applications II. AIAA, Washington, pp 519–538

    Google Scholar 

  • Cohen H (1995) A course in computational algebraic number theory. Springer, Berlin

    Google Scholar 

  • Coveyou RR, Macpherson RD (1967) Fourier analysis of uniform random number generations. J ACM 14(1): 100–119

    Article  Google Scholar 

  • Damen MO, El Gamal H, Caire G (2003) On maximum-likelihood detection and the search for the closest lattice point. IEEE Trans Inf Theory 49(10): 2389–2402

    Article  Google Scholar 

  • Fincke U, Pohst M (1985) Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math Comput 44(170): 463–471

    Article  Google Scholar 

  • Frei E, Beutler G (1990) Rapid static positioning based on the fast ambiguity resolution approach “FARA”: theory and first results. Manuscr Geod 15: 325–356

    Google Scholar 

  • Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore

    Google Scholar 

  • Grafarend EW (2000) Mixed integer-real valued adjustment (IRA) problems. GPS Solut 4: 31–45

    Article  Google Scholar 

  • Grotschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization. Springer, Berlin

    Book  Google Scholar 

  • Guruswami V, Micciancio D, Regev O (2005) The complexity of the covering radius problem. Comput Complex 14: 90–121

    Article  Google Scholar 

  • Hanrot G, Stehle D (2007) Improved analysis of Kannan’s shortest lattice vector algorithm. In: Advances in cryptology—crypto 2007, proceedings. A. Menezes, vol 4622, pp 170–186

  • Hassibi A, Boyed S (1998) Integer parameter estimation in linear models with applications to GPS. IEEE Trans Signal Proc 46: 2938–2952

    Article  Google Scholar 

  • Hassibi B, Vikalo H (2005) On the sphere-decoding algorithm I. expected complexity. IEEE Trans Signal Proc 53: 2806–2818

    Article  Google Scholar 

  • Hermite C (1850) Extraits de lettres de M. Hermite à M. Jacobi sur diff′erents objets de la th′eorie des nombres. J Reine Angew Math 40: 279–290

    Article  Google Scholar 

  • Kannan R (1983) Improved algorithms for integer programming and related lattice problems. Paper presented at the conference proceedings of the annual ACM symposium on theory of computing, pp 193–206

  • Kannan R (1987) Algorithmic geometry of numbers. Annu Rev Comput Sci 16: 231–267

    Article  Google Scholar 

  • Kim D, Langley RB (2000) A search space optimization technique for improving ambiguity resolution and computational efficiency. Earth Planets Space 52(10): 807–812

    Google Scholar 

  • Korkine A, Zolotareff G (1873) Sur les formes quadratiques. Math Annalen 6(3): 366–389 (in French)

    Article  Google Scholar 

  • Lagrange JL (1773) Recherches d’arithmetique. Nouveaux Memoires de l’Academie de Berlin

  • Langley RB, Beutler G, Delikaraoglou D, Nickerson B, Santerre R, Vanicek P, Well DE (1984) Studies in the application of the GPS to differential positioning. Technical Report No. 108, University of New Brunswick, Canada

  • Lenstra AK, Lenstra HW, Lovász L (1982) Factoring polynomials with rational coefficients. Math Ann 261: 513–534

    Article  Google Scholar 

  • Liu LT, Hsu HT, Zhu YZ, Ou JK (1999) A new approach to GPS ambiguity decorrelation. J Geod 73: 478–490

    Article  Google Scholar 

  • Micciancio D, Goldwasser S (2002) Complexity of lattice problems: a cryptographic perspective. Kluwer international series in engineering and computer science, vol 671. Kluwer Academic Publishers, Boston

    Google Scholar 

  • Minkowski H (1905) Diskontinuitätsbereich für arithmetische Äquivalenz. J Rreine Angew Math 129: 220–274 (in German)

    Article  Google Scholar 

  • Mow WH (2003) Universal lattice decoding: principle and recent advances. Wirel Commun Mobile Comput 3(5): 553–569

    Article  Google Scholar 

  • Nguyen PQ, Stehlé D (2004) Low-dimensional lattice basis reduction revisited (extended abstract). In: Proceedings of the 6th international algorithmic number theory symposium (ANTS-VI), lecture notes in computer science, vol 3076. Springer, Berlin, pp 338–357

  • Nguyen PQ, Stehlé D (2009) An LLL algorithm with quadratic complexity. SIAM J Comput 39(3): 874–903

    Article  Google Scholar 

  • Nguyen PQ, Stern J (2001) The two faces of lattices in cryptology. In: Proceedings of CALC ’01, lecture notes in computer science, vol 2146. Springer, Berlin, pp 146–180

  • Pohst M (1981) On the computation of lattice vector of minimal length, successive minima and reduced bases with applications. ACM SIGSAM Bull 15: 37–44

    Article  Google Scholar 

  • Pujol X, Stehle D (2008) Rigorous and efficient short lattice vectors enumeration. In: Advances in cryptology—Asiacrypt 2008 J Pieprzyk, vol 5350, pp 390–405

  • Schnorr CP, Euchner M (1994) Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math Program 66: 181–199

    Article  Google Scholar 

  • Steinfeld R, Pieprzyk J, Wang H (2007) Lattice-based threshold changeability for standard Shamir secret-sharing schemes. IEEE Trans Inf Theory 53: 2542–2559

    Article  Google Scholar 

  • Teunissen PJG (1993) Least-squares estimation of the integer GPS ambiguities. Invited lecture, section IV theory and methodology, IAG general meeting, Beijing, China. Also in Delft Geodetic Computing Centre LGR series, No. 6, p 16

  • Teunissen PJG (1994) A new method for fast carrier phase ambiguity estimation. In: Proceedings of the IEEE PLANS’94, Las Vegas, NV, 11–15 April 1994, pp 562–573

  • Teunissen PJG (1995) The least-squares ambiguity decorrelation adjustment: a method for fast GPS ambiguity estimation. J Geod 70: 65–82

    Article  Google Scholar 

  • Teunissen PJG (1996) An analytical study of ambiguity decorrelation using dual frequency code and carrier phase. J Geod 70: 515–528

    Google Scholar 

  • Teunissen PJG (1997) A canonical theory for short GPS baselines, parts I–IV. J Geod 71:320–336, 389–401, 486–501, 513–525

    Google Scholar 

  • Teunissen PJG (1998a) GPS carrier phase ambiguity fixing concepts. In: Teunissen P, Kleusberg A (eds) GPS for geodesy, 2nd edn. Springer, Berlin, pp 317–388

    Chapter  Google Scholar 

  • Teunissen PJG (1998b) Success probability of integer GPS ambiguity rounding and bootstrapping. J Geod 72: 606–612

    Article  Google Scholar 

  • Teunissen PJG, De Jonge PJ, Tiberius CC (1997) The least-squares ambiguity decorrelation adjustment: its performance on short GPS baselines and short observation spans. J Geod 71: 589–602

    Article  Google Scholar 

  • Viterbo E, Boutros J (1999) A universal lattice code decoder for fading channels. IEEE Trans Inf Theory 45(5): 1639–1642

    Article  Google Scholar 

  • Wei Z (1986) Positioning with NAVSTAR, the global positioning system. Report No. 370, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, OH

  • Xu PL (1998) Mixed integer observation models and integer programming in geodesy. J Geod Soc Jpn 44: 169–187

    Google Scholar 

  • Xu PL (1999) Spectral theory of constrained second-rank symmetric random tensors. Geophys J Int 138(1): 1–24

    Article  Google Scholar 

  • Xu PL (2001) Random simulation and GPS decorrelation. J Geod 75: 408–423

    Article  Google Scholar 

  • Xu PL (2002) Isotropic probabilistic models for directions, planes and referential systems. Proc R Soc Lond Ser A 458(2024): 2017–2038

    Article  Google Scholar 

  • Xu PL (2006) Voronoi cells, probabilistic bounds and hypothesis testing in mixed integer linear models. IEEE Trans Inf Theory 52: 3122–3138

    Article  Google Scholar 

  • Xu PL, Grafarend E (1996) Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors. Geophys J Int 127(3): 744–756

    Article  Google Scholar 

  • Xu PL, Cannon E, Lachapelle G (1995) Mixed integer programming for the resolution of GPS carrier phase ambiguities. Paper presented at IUGG95 assembly, Boulder, 2–14 July

  • Zhao W, Giannakis GB (2005) Sphere decoding algorithms with improved radius search. IEEE Trans Commun 53(7): 1104–1109

    Article  Google Scholar 

  • Zhou Y (2010) A new practical approach to GNSS high-dimensional ambiguity decorrelation. GPS Solut. doi:10.1007/s10291-010-0192-6

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Authors and Affiliations

  1. Department of Surveying and Geomatics Engineering, College of Engineering, University of Tehran, Tehran, Iran

    S. Jazaeri & M. A. Sharifi

  2. Department of Surveying Engineering, Faculty of Engineering, University of Isfahan, 81746-73441, Isfahan, Iran

    A. R. Amiri-Simkooei

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  1. S. Jazaeri
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  2. A. R. Amiri-Simkooei
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  3. M. A. Sharifi
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Correspondence to S. Jazaeri.

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An erratum to this article can be found at http://dx.doi.org/10.1007/s00190-011-0518-3.

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Jazaeri, S., Amiri-Simkooei, A.R. & Sharifi, M.A. Fast integer least-squares estimation for GNSS high-dimensional ambiguity resolution using lattice theory. J Geod 86, 123–136 (2012). https://doi.org/10.1007/s00190-011-0501-z

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