A Bayesian Framework for Collaborative Multi-Source Signal Detection

This paper introduces a Bayesian framework to detect multiple signals embedded in noisy observations from a sensor array. For various states of knowledge on the communication channel and the noise at the receiving sensors, a marginalization procedure based on recent tools of finite random matrix theory, in conjunction with the maximum entropy principle, is used to compute the hypothesis selection criterion. Quite remarkably, explicit expressions for the Bayesian detector are derived which enable to decide on the presence of signal sources in a noisy wireless environment. The proposed Bayesian detector is shown to outperform the classical power detector when the noise power is known and provides very good performance for limited knowledge on the noise power. Simulations corroborate the theoretical results and quantify the gain achieved using the proposed Bayesian framework.

💡 Research Summary

The paper presents a Bayesian detection framework for identifying multiple signal sources embedded in noisy observations collected by a sensor array. The authors start by highlighting the limitations of classical energy‑based detectors, which assume perfect knowledge of the noise power and degrade sharply when this assumption is violated. To address this, they formulate the detection problem as a binary hypothesis test—(H_0) (no signal) versus (H_1) (signal present)—and place prior probability distributions on the unknown channel matrix and the noise variance.

For the channel, an i.i.d. complex Gaussian prior is adopted, reflecting a lack of specific spatial correlation information. For the noise power, two scenarios are considered: (i) exact knowledge, leading to a delta‑function prior, and (ii) limited or no knowledge, for which a log‑uniform (or, when partial moments are known, a Gamma) prior is used. These choices follow the maximum‑entropy principle, ensuring that the priors are the most non‑committal given the available constraints.

The central technical contribution lies in the marginalization of the likelihood functions under each hypothesis. By leveraging recent results from finite random matrix theory—specifically the exact eigenvalue distributions of complex Wishart matrices—the authors avoid asymptotic large‑(N) approximations and obtain closed‑form expressions for the marginal likelihoods. Under (H_0) the observation matrix follows a pure complex Wishart distribution, while under (H_1) it follows a “signal‑plus‑noise” Wishart mixture. Integrating over the priors yields an explicit Bayes factor

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