A Multiscale Approach for Enhancing Weak Signal Detection

TL;DR

This work addresses the challenge of detecting time-varying, weak signals in noisy environments by extending stochastic resonance (SR) with a robust double-threshold detector and applying SR in the multiscale domain via wavelet transforms. It develops a probabilistic estimation framework for the hidden signal $ heta$ based on threshold-exceedance probabilities $p_a$ and $p_b$, including a variance-weighted fusion and non-parametric/wavelet-based extensions for non-constant signals. The authors provide analytic estimators, Fisher-information-based insights, and extensive simulations across 1D and 2D data, demonstrating that SR in the multiscale domain substantially lowers the required noise level and AMSE, outperforming traditional single-threshold SR methods. The approach offers a practical, scalable method for robust weak-signal detection with potential applications in diverse disciplines where non-stationary signals are embedded in noise.

Abstract

Stochastic resonance (SR), a phenomenon originally introduced in climate modeling, enhances signal detection by leveraging optimal noise levels within non-linear systems. Traditional SR techniques, mainly based on single-threshold detectors, are limited to signals whose behavior does not depend on time. Often large amounts of noise are needed to detect weak signals, which can distort complex signal characteristics. To address these limitations, this study explores multi-threshold systems and the application of SR in multiscale applications using wavelet transforms. In the multiscale domain signals can be analyzed at different levels of resolution to better understand the underlying dynamics. We propose a double-threshold detection system that integrates two single-threshold detectors to enhance weak signal detection. We evaluate it both in the original data domain and in the multiscale domain using simulated and real-world signals and compare its performance with existing methods. Experimental results demonstrate that, in the original data domain, the proposed double-threshold detector significantly improves weak signal detection compared to conventional single-threshold approaches. Its performance is further improved in the frequency domain, requiring lower noise levels while outperforming existing detection systems. This study advances SR-based detection methodologies by introducing a robust approach to weak signal identification, with potential applications in various disciplines.

Figures (9)

• Figure 1: Stochastic resonance with threshold detector: schematic representation of a double-threshold detector and signal input $\theta$. The detector produces indicators $-1$ and $1$ in response to the events $X < a$ and $X > b$, where $X = \theta + \epsilon_i$. These event indicators are used to compute an estimator $\hat{\theta}$ for the signal $\theta$
• Figure 2: Wavelet decomposition of a Doppler signal of length $1024$ ($J = \log_2(1024) = 10$), illustrating the amplification of the wavelet-transformed signal beyond the two thresholds $a = -1$ and $b = 1$. The decomposition progresses across levels $j = 1$ to $5$. At each level, the signal is separated into smoothing (black) and detail (blue) components. The red lines indicate the boundaries where the wavelet decomposition occurs, visually marking the separation between components at different levels of resolution. This figure demonstrates how the signal structure evolves with finer decompositions.
• Figure 3: Visualization of Fisher Information (FI): Sensitivity to thresholds, signal amplitude, and noise levels. The left panel illustrates the dependence of FI on lower and upper thresholds ($a$ and $b$) for a constant signal ($\theta = 2$) under a fixed noise level ($\sigma = 1$). The right panel depicts how FI varies with signal amplitude ($\theta$) and noise level ($\sigma$), given fixed thresholds ($a = -2$ and $b = 2$). The red curves highlight the maximum FI, revealing the influence of parameter tuning on signal detectability.
• Figure 4: An overview of the procedures used for the detection of (1D and 2D) weak signals in the data acquisition and multiscale(frequency/wavelet) domains. In the original data domain, thresholding is applied to the original signal, while in the multiscale domain, thresholding is applied to the wavelet-transformed data.
• Figure 5: One-dimensional signal recovery performance with original data domain (a) and frequency/multiscale domain (b). The graph on (a) shows the original signals as well as thresholds and a noise sample. The second graph shows the recovered signals from the sub-, the sup-, and the double-threshold detectors. In the frequency domain, (b), noise contamination is conducted in the wavelet domain (second graph) and recovery from the three detectors is shown on the third graph.