Abstract
The Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm and many of its variants have been widely used by cryptography, multiple-input-multiple-output (MIMO) communication systems and carrier phase positioning in global navigation satellite system (GNSS) to solve the integer least squares (ILS) problem. In this paper, we propose an n-dimensional LLL reduction algorithm (n-LLL), expanding the Lovász condition in LLL algorithm to n-dimensional space in order to obtain a further reduced basis. We also introduce pivoted Householder reflection into the algorithm to optimize the reduction time. For an m-order positive definite matrix, analysis shows that the n-LLL reduction algorithm will converge within finite steps and always produce better results than the original LLL reduction algorithm with n > 2. The simulations clearly prove that n-LLL is better than the original LLL in reducing the condition number of an ill-conditioned input matrix with 39% improvement on average for typical cases, which can significantly reduce the searching space for solving ILS problem. The simulation results also show that the pivoted reflection has significantly declined the number of swaps in the algorithm by 57%, making n-LLL a more practical reduction algorithm.
Highlights
With the rapid development of the Beidou System (BDS), the Galileo system, the GlobalPositioning System (GPS) and the GLONASS system, the Global Navigation Satellite System (GNSS)is serving more and more people with higher positioning accuracy [1,2]
Xu used Cholesky decomposition to calculate the Z-transformation matrix [10]; Chang modified the Gauss transformation in Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) with symmetric pivoting strategy [11]; Xu proposed the parallel Cholesky-based reduction method using minimum pivoting strategy [12] and Hassibi was the first to introduce LLL algorithm into integer ambiguity resolution [13]
We propose the n-dimensional LLL reduction algorithm (n-LLL) reduction algorithm, which inherits the basic outline of the LLL reduction algorithm and strengths the constraint of the order of basis vectors
Summary
With the rapid development of the Beidou System (BDS), the Galileo system, the Global. The main computational effort in carrier-phase precise positioning is to resolve the carrier phase integer ambiguity which is contaminated by all kind of noises during the signal propagation. The great breakthrough of LAMBDA algorithm is bringing the “decorrelation” process into integer ambiguity resolution, dividing the whole process into: estimation, decorrelation ( known as Z-transformation), search and back transformation. For real-time applications, the decorrelation process is curial to reduce search effort. Xu used Cholesky decomposition to calculate the Z-transformation matrix [10]; Chang modified the Gauss transformation in LAMBDA with symmetric pivoting strategy [11]; Xu proposed the parallel Cholesky-based reduction method using minimum pivoting strategy [12] and Hassibi was the first to introduce LLL algorithm into integer ambiguity resolution [13]. The original LLL reduction algorithm uses Gram-Schmidt orthogonalization to generate orthogonal basis, which involves O(nlog(B))-bit integer. The new algorithm causes no significant increase in computational efforts because of the pivoted reflection, which is able to reduce as much as 57% swaps in the algorithm
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