Phase reconstruction is in general a non-trivial problem when it comes to devices where the reference is not accessible. A non-convex iterative optimization algorithm is proposed in this paper in order to reconstruct the phase in reference-less spherical multiprobe measurement systems based on a rotating arch of probes. The algorithm is based on the reconstruction of the phases of self-transmitting devices in multiprobe systems by taking advantage of the on-axis top probe of the arch. One of the limitations of the top probe solution is that when rotating the measurement system arch, the relative phase between probes is lost. This paper proposes a solution to this problem by developing an optimization iterative algorithm that uses partial knowledge of relative phase between probes. The iterative algorithm is based on linear combinations of signals when the relative phase is known. Phase substitution and modal filtering are implemented in order to avoid local minima and make the algorithm converge. Several noise-free examples are presented and the results of the iterative algorithm analyzed. The number of linear combinations used is far below the square of the degrees of freedom of the non-linear problem, which is compensated by a proper initial guess. With respect to noisy measurements, the top probe method will introduce uncertainties for different azimuth and elevation positions of the arch. This is modelled by considering the real noise model of a low-cost receiver and the results demonstrate the good accuracy of the method. Numerical results on antenna measurements are also presented. Due to the numerical complexity of the algorithm, it is limited to electrically small- or medium-size problems.
Reconstruction of Relative Phase of Self-Transmitting Devices by Using Multiprobe Solutions and Non-Convex Optimization