Approximate iterative Least Squares algorithms for GPS positioning

The efficient implementation of positioning algorithms is investigated for Global Positioning System (GPS) and Differential GPS (DGPS). This is particularly important for smart phones with battery limitations. With the help of the information from base stations, Assisted GPS (AGPS) and DGPS can do the positioning more efficiently and more precisely than GPS. In order to do the positioning, the pseudoranges between the receiver and the satellites are required. The most commonly used algorithm for position computation from pseudoranges is non-linear Least Squares (LS) method. Linearization is done to convert the non-linear system of equations into an iterative procedure, which requires the solution of a linear system of equations in each iteration, i.e. linear LS method is applied iteratively. CORDIC-based approximate rotations are used while computing the QR decomposition for solving the LS problem in each iteration. By choosing accuracy of the approximation, e.g. with a chosen number of optimal CORDIC angles per rotation, the LS computation can be simplified. The accuracy of the positioning results is compared for various numbers of required iterations and various approximation accuracies using real GPS data. The results show that very coarse approximations are sufficient for a reasonable positioning accuracy. Therefore, the presented method reduces the computational complexity significantly and is highly suitable for hardware implementation.