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

Signal detection is both a fundamental topic of data science and a great challenge for practical engineering. One of the canonical tasks widely investigated is detecting a sinusoidal signal of known frequency ω with time duration T:I(t)=Acos ω t+Γ(t), embedded within a stationary noisy data. The most direct, and also believed to be the most efficient, method is to compute the Fourier spectral power at ω:B=|2/T∫₀^T I(t)e^iωtdt|. Whether one can out-perform the linear Fourier approach by any other nonlinear processing has attracted great interests but so far without a consensus. Neither a rigorous analytic theory has been offered. We revisit the problem of weak signal, strong noise, and finite data length T=O(1), and propose a signal detection method based on resonant filtering. While we show that the linear approach of resonant filters yield a same signal detection efficiency in the limit of T→∞, for finite time length T=O(1), our method can improve the signal detection due to the highly nonlinear interactions between various characteristics of a resonant filter in finite time with respect to transient evolution. At the optimal match between the input I(t), the control parameters, and the initial preparation of the filter state, its performance exceeds the above threshold B considerably. Our results are based on a rigorous analysis of Gaussian processes and the conclusions are supported by numerical computations.