Improved HLLL Lattice Basis Reduction Algorithm to Solve GNSS Integer Ambiguity

Abstract

Recently, lattice theory has been widely used for integer ambiguity resolution in the Global Navigation Satellite System (GNSS). When using lattice theory to deal with integer ambiguity, we need to reduce the correlation between lattice bases to ensure the efficiency of the solution. Lattice reduction is divided into scale reduction and basis vector exchange. The scale reduction has no direct impact on the subsequent search efficiency, while the basis vector exchange directly impacts the search efficiency. Hence, Lenstra-Lenstra-Lovász (LLL) is applied in the ambiguity resolution to improve the efficiency. And based on Householder transformation, the HLLL improved method is also used. Moreover, to improve the calculation speed further, a Pivoting Householder LLL (PHLLL) method based on Householder orthogonal transformation and rotation sorting is proposed here. The idea of PHLLL method is as follows: First, a sort matrix is introduced into the lattice basis reduction process to sort the original matrix. Then, the sorted matrix is used for Householder transformation. After transformation, it needs to be sorted again, until the diagonal elements in the matrix meet the ascending order. In addition, when using the Householder image operator for orthogonalization, the old column norm is modified to obtain a new norm, reducing the number of column norm calculations. Compared with the LLL reduction algorithm and HLLL reduction algorithm, the experimental results show that the PHLLL algorithm has higher reduction efficiency and effectiveness. The theoretical superiority of the algorithm is proved.