A General Multi-Dimensional GNSS Signal Processing Scheme Based on Multicomplex Numbers

Modern global navigation satellite system (GNSS) broadcast signals on several frequencies allowing advanced applications, such as the estimation of the ionospheric delays and fast converging precise point positioning (PPP) algorithms. Four components are currently broadcast by Galileo whereas five signals are transmitted by third generation Beidou satellite navigation system (BDS) satellites. The availability of such components calls for the development of advanced algorithms fully exploiting the benefits of multi-frequency GNSS signals. A possible approach for multi-frequency GNSS signal processing is based on the so-called meta-signal paradigm, where components from different frequencies are treated as a single entity. A general framework for jointly processing 2M signals is developed using multicomplex numbers that are multidimensional extensions of complex numbers. A set of GNSS signals from different frequencies is at first mapped into a single meta-signal represented using multicomplex numbers. Then, single frequency GNSS processing is generalized to the multi-dimensional case using the theory of multicomplex numbers. A multicomplex cross ambiguity function (CAF) is derived and used for the design of multi-frequency acquisition and tracking algorithms. Theoretical results are supported by experiments where four radio frequency (RF) frontends are used to capture Galileo signals from different frequencies. The four signals are then processed jointly using the algorithms developed.