Research On Method Of Integer Ambiguity Estimation With Lattice Theory

With the construction and development of satellite navigation system, the application area ofsatellite positioning is enlarging and consumers’ requirements for the precision and reliability ofpositioning results are growing. In the process of high-precision positioning, the correctness ofinteger ambiguity resolution is the key factor in guaranteeing the positioning precision. Thisdissertation studied the integer ambiguity estimation based on the lattice theory in math. Themain achievement and creative points are as follows:1. In order to obtain the final positioning results, the mixed integer model which doesn’t onlycontain unknown parameters of real numbers, but also contains unknown parameters ofinteger numbers should be transformed to integer model first. Then two kinds oftransforming process are discussed based on least squares criterion and Bayesian criterion.And it is proved that the integer models obtained with the two criterions are consistent.2. The LLL decorrelation algorithm uses integer Gram Schmidt transformation to compute.Due to the impact of round-off error, it usually fails to decorrelate in circumstances of highdimensions. To solve this problem, an integer block orthogonal transformation method isproposed and an LLL decorrelation algorithm is designed based on this transformation.With computation and analysis, it proves that the new method can greatly improve thedecorrelation.3. The definition, character and two famous problems of lattice are studied thoroughly. Afteranalyzing the integer least squares model of ambiguity resolution, lattice referring toambiguity resolution can be made to conduct triangular decomposition of covariance matrix.The search and fix of ambiguity is equivalent to the nearest vector problem in lattice. Onthis basis, the ambiguity resolution algorithm based on lattice is proposed.4. The LLL reduction base and two algorithms are studied. Since there is a diversity of latticebase, the quality of lattice base will affect the efficiency and success rate of CVP resolutionin lattice. For the ambiguity resolution based on lattice, a most suitable group should bechose according to lattice reduction. After analyzing the LLL reduction base, the principle,process and ultimate goal are mastered. At the same time, two methods based on LLLreduction base of Gram Schmidt and Householder transformation are studied.5. An extended LLL reduction base is proposed. The restriction of its length reduction isstricter than LLL reduction which can guarantee the realization of length reduction in rangesof all base vectors. For fulfillment of E-LLL reduction, the Householder transformationbased on systematic rotation is designed. It guarantees that the basic vector can be arranged from short to long according to the length of orthogonal vectors which improves thereduction efficiency. Then the HE-LLL reduction algorithm is proposed based on thistransformation. This algorithm conducts size and length reduction of basic vector with theassist of systematic rotation transformation which can guarantee the acquisition of E-LLLreduction base finally. Using observed data for computation, it proves that HE-LLLalgorithm not only improves reduction effect, but also improves the reduction efficiency.6. An improved BKZ reduction algorithm is proposed. BKZ reduction disparts the lattice baseon the basis of LLL reduction and guarantees that basic vector in each block satisfies KZreduction. But the reduction effect relies on the size of blocks. The bigger the blocks, thebetter the effect while the worse the efficiency. In order to promote the efficiency of BKZreduction, the HE-LLL is used to optimize. With analysis of observed and simulated data,the computational efficiency of promoted BKZ algorithm is obviously better than BKZreduction, and the base reduction effect is better the HE-LLL reduction algorithm in theory.7. The VB-SD and SE-SD algorithms based on depth-first searching mode are studied. Andthe impact of lattice reduction process on searching space is analyzed. It proves that theprocess of lattice size reduction cannot improve the searching space, whereas only lengthreduction can affect the searching efficiency and success rate. Analysis with Observed datain different circumstances proves the conclusion above.8. The K-best searching algorithm based on HE-LLL reduction is proposed. Since thedepth-first searching algorithm needs complex iteration process, the efficiency is low inhigh-dimensional circumstance. The sphere searching algorithm based on breadth-firstmode which is represented by K-best algorithm, searches only k candidates, therefore it hasrelatively stable complexity, while leading this to be a suboptimal algorithm. In demand ofambiguity estimation, an improvement has been made on aspects of lattice base quality andsearching radius restriction. Observed data is used for computation analysis. Results haveshown that the method proposed in this paper only requires relatively small value of k toobtain the optimal ambiguity resolution and the efficiency is stable which will not changemuch with the increase of dimensions. It suits for high-dimensional ambiguity resolution.9. Voronoi cell, as a convex geometry structure which has dual concept, can be used to studyand analyze related questions in lattice. After analyzing the corresponding Voronoi cell inlattice vectors, the problem of ambiguity CVP resolution is transformed to resolving thecorresponding vector in the Voronoi cell whose target vector is at the origin. Using theVoronoi-related vector to construct the Voronoi cell correspond to the origin, the ambiguityCVP resolution method based on the Voronoi cell is designed on this basis. Using a set ofsimulated two-dimensional data, the ambiguity resolution process of the algorithm is analyzed and the stability of the results based on the Voronoi cell is estimated. Several setsof observed data is chosen to further prove the correctness of the algorithm.