Theoretical Analyses of Detectors for Additive Noise Channels with Mean-Variance Uncertainty under Nonlinear Expectation Theory

In classical information theory, both the form and performance of the optimal detector for additive noise channels can be precisely derived, based on the assumption that the channel noise follows a specific probability distribution or a mixture of known distributions, or that the exact distribution exists but is unknown.

In this paper, we extend the analyses of detectors for additive noise channel to the situation where the probability model for analyzing channels is uncertain, utilizing nonlinear expectation theory.

We consider two types of distribution uncertainties: one with no mean uncertainty but with variance uncertainty, and another with both mean and variance uncertainties.

We derive the optimal threshold detectors for binary input additive noise channel under the nonlinear expectation optimal criterion for both scenarios and provide their explicit forms.

Our findings reveal that mean uncertainty significantly influences the form of the optimal detector, whereas variance uncertainty does not.

Additionally, we propose an estimation method for the uncertain parameters of the channel noise.

Finally, we present theoretical analyses and simulated performance results of the newly derived optimal threshold detectors, and compare these results with the performance of optimal detector under classical information theory, which assumes a deterministic probability model.

The results of experiments show that our new detection methods outperform conventional methods in most scenarios with uncertain probability models, showing the practical relevance of our theoretical contributions.