Optimal fractional non-coherent detector for high-sensitivity GNSS receivers robust against residual frequency offset and unknown bits

This paper addresses the problem of finding the optimal non-coherent detector or Post Detection Integration (PDI) technique to acquire weak signals in the context of High-Sensitivity Global Navigation Satellite System (HS-GNSS) receivers. This detector is derived using the Generalized Likelihood Ratio Test (GLRT) in the presence of data bits and variations in the carrier phase. The resulting detector is difficult to implement in practice because the amplitude of the signal must be known a priori. Two approximations of the resulting detector, which depend on the signal-to-noise ratio, are proposed, namely the Non-coherent PDI (NPDI) and non-quadratic NPDI (NPDInq) techniques. From this result, we prove that in general the NPDInq technique is the best option to detect weak signals in HS-GNSS receivers. In addition, a new statistical characterization of the NPDInq technique is proposed, which improves the approach used in the literature by applying the central limit theorem.