Extracting hidden weak sinusoidal signal with short duration from noisy data: Analytical theory and computational realization

Zhang Ying¹, Zhang Zhaoyang^{1, 2}, Qian Hong³, Hu Gang^{1, †}

(color online) Definition and computation of separation of probability distributions SPD with signal from that without signal. (a) Schematic figure on signal prediction and definition of SPD. At we have , and for , respectively. We thus most probably predict signal when while no signal for . And the overall probability of correct predictions (including correct predictions of signal and no signal) reads . (b) SPD_I for the input I(t) of Eqs. (3) and (6) (red colored plots) and SPD_x for the output of Eqs. (10) and (5), (11), (14) (blue colored plots) plotted against T. SPD is identified by theoretical and rigorous derivations (continuous curves) and numerical computations (dots). In particular, ratio SPD is rather larger than 1 for small Tʼs.