Signal-to-Noise Ratio Calculation and Analysis for Weak Signal Detection in the Phase Noise

The test signal-to-noise ratio （TSNR） for weak signal detection is analyzed when the useful signal is modeled in the phase of the complex data. Firstly， considering the zero-mean complex Gaussian noise in complex domain， the corresponding phase noise's distribution functions and numerical characteristics are discussed. Closed-form algorithms are developed for noise characteristics approximation under the assumption of weak signals. Secondly， considering the flat fading channel， i.e.， fixed signal amplitude， the TSNR of weak signal detection based on multiple samples is derived and the approximation calculation is developed. Then， the TSNR and bit error ratio of the Rayleigh fading channel are studied， where the binary phase modulation is employed as an example. Finally， the proposed TSNR formulas and their closed-form approximations are demonstrated by numeral simulations. This paper offers valuable advices to the research on the signal detection in the phase domain.