Detection of weak signals under arbitrary noise distributions

TL;DR

This work proposes a hybrid framework that combines a lightweight neural network with the Rao detection framework, and offers a robust and interpretable solution for signal detection across diverse applications.

Abstract

Detecting weak signals buried in complex, non-Gaussian noise is a fundamental challenge in science and engineering, with applications ranging from radar systems and communications to industrial monitoring and gravitational wave detection. The Rao detector, a key concept in this domain, achieves asymptotically optimal performance as the number of measurements increases, but requires precise knowledge of the data's statistical properties, often relying on simplified noise models. We propose a hybrid framework that combines a lightweight neural network with the Rao detection framework to address this limitation. The neural network, trained on noise-only data, learns the optimal multivariate nonlinearity, transforming noisy data to enhance signal detectability. The newly introduced LRao detector then fully extracts the signal information, achieving asymptotically optimal performance even under challenging noise conditions. Validated on both simulated and real-world magnetic sensor data, our method significantly outperforms conventional approaches. By bridging data-driven techniques with model-based signal processing, it offers a robust and interpretable solution for signal detection across diverse applications.

Figures (8)

• Figure 1: Sketch of the appraoch and its performance.a, A lightweight neural network is trained to optimally process the complicated, non-Gaussian data. By minimizing the cost function we introduce in this work, the neural network learns to represent the data in a way that is most suitable for further processing, no longer containing any outliers. When combined with an additive signal model, the neural network is trained exclusively on noise-only data, which is often available in abundance. b, As the cost function is minimized, the performance of the proposed LRao detector applied to the transformed data gradually improves until it reaches that of the asymptotically optimal Rao detector, which, in contrast to our method, requires precise knowledge of the data PDF. The cost function does not heuristically penalize outliers, instead, it arises naturally as the (negative) linear Fisher information. This quantity formally measures the information content in the transformed data that is easily extractable, without further complicated processing.
• Figure 2: Noise samples drawn from a mixture of Gaussians.a, In our task to estimate and detect a shift in a vector of IID noise samples, the original distribution (mix auf Gaussiens, red) is transformed to a new distribution (blue) using the score function (black), thereby maximizing the LFI that the LBLUE can extract. b, Means and the 95 percentiles of the LBLUE estimates for both the original and the transformed data. Notably, the LBLUE applied to the transformed data attains the CRLB locally, but yields biased estimates when considered globally. The LBLUE applied to the original data is unbiased globally, however, its variance lies above the CRLB. c, LRao detection statistics and threshold. Here, the LRao detector applied to the score-transformed data, which is equivalent to the Rao detector on the original data, demonstrates a higher local probability of detection compared to the LRao detector on the original data. This can be seen by comparing the color-filled areas between the blue and the red curves, respectively, above and below the horizontal dashed line representing the threshold.
• Figure 3: Simulated Cauchy noise.a, The periodic signal to detect and the signal corrupted by additive heavy-tailed Cauchy noise. For improved visibility, the scale of the ordinate is linear within the interval $[-1,1]$ and logarithmic outside of the interval. b, The trace of the LFI, estimated at the output of the neural network, normalized by the trace of the FI. c, The simulated and the asymptotically optimal ROC curves for different SNRs for a sequence length of $N=128$. The LRao detector does not yet attain its asymptotic performance. d, The simulated and analytical distributions of the LRao detector. Under $\mathcal{H}_1$, the LRao detection statistic is slightly shifted to the left compared to the analytical asymptotic distribution. e, The ROC curves for $N=1024$. The LRao detector approaches its asymptotic performance, as can also be seen in f.
• Figure 4: Experimental data from magnetic sensors.a, The spikey and dependent noise data captured by a magnetic sensor, available in the supplementary material of kay2013fundamentals. The data sequence of length 10000.0 is split into smaller sequences of length 128.0, forming a data set of 78.0 sequences. A nested cross-validation procedure is used to evaluate the performance of our detector, which aims to detect a weak periodic signal that is added in post-processing. b, ROC curves and areas under the curves with 95 percentiles (given in the brackets) for an input SNR of -25.5dB. The proposed LRao detector in combination with the lightweight convolutional neural network (CNN+LRao) clearly outperforms the reference methods. The CNN+LRao detector attains its asymptotic performance in good approximation, and is therefore optimal to the degree that the LFI at the network output matches the original data's FI. We emphasize the effectiveness of our approach despite the limited amount of training data.
• Figure 5: Normalized histogram and Gaussian fit of the prewhitened magnetic sensor data. Parts of the data can be well approximated by a Gaussian distribution.