Wen-Xuan Lang

Wen-Xuan Lang, Shaoshi Yang, Jianhua Zhang et al.

A mathematical framework for information-theoretic analysis is established, with a new viewpoint of describing transmitted messages and communication channels by the nonlinear expectation theory, beyond the framework of classical probability theory. The major motivation of this research is to emphasize the probabilistic distribution uncertainty within the ever increasingly complex communication networks, where random phenomena are often nonstationary, heterogeneous, and cannot be characterized by a single probability distribution. Based on the nonlinear expectation theory, in this paper we first explicitly define several fundamental concepts, such as nonlinear information entropy, nonlinear joint entropy, nonlinear conditional entropy and nonlinear mutual information, and establish their basic properties. Secondly, by using the strong law of large numbers under sublinear expectations, we propose a nonlinear source coding theorem, which shows that the nonlinear information entropy is the upper bound of the achievable coding rate of sources whose distributions are uncertain under the maximum error probability criterion, and determines a cluster point of the coding rate of such sources under the minimum error probability criterion. Thirdly, we propose a nonlinear channel coding theorem, which gives the explicit expression of the upper bound under the maximum error probability criterion and a cluster point under the minimum error probability criterion, respectively, for the achievable coding rate of communication channels whose distributions are uncertain. Additionally, we propose a nonlinear rate-distortion source coding theorem, proving that the rate distortion function based on the nonlinear mutual information is a cluster point of the lossy compression performance of uncertain-distribution sources under the minimum expected distortion criterion.

Wen-Xuan Lang, Guiying Yan, Zhi-Ming Ma

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 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 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.

Jialiang Fu, Wen-Xuan Lang

In this paper, we establish a new law of large numbers with the rate of convergence for special partial sums in a probability space. The proof relies on nonlinear expectation theory, as the uncertainty of random variables in the special partial sums induces the sublinearity of the expectation. As an application, we apply the new theorem to analyze the feedback channel-based detection problem of non-i.i.d. input signals in communication systems. Specifically, we investigate the convergence rates of the upper probabilities of the detection errors within the sublinear expectation space.

Jialiang Fu, Wen-Xuan Lang

Classical energy detection (ED) methods for cognitive radio (CR) have addressed noise uncertainty as deviations in noise power and signal uncertainty as variability in signal characteristics, which use probabilistic methods and assume fixed probability distributions for both. In practical scenarios, due to the uncertainty in probability models and the significant variation of primary signals encountered by receivers across different radio technologies, wireless environments exhibit not only distributional uncertainty but also substantial signal variety. In this paper, we develop a generalized formulation of energy detection based on nonlinear expectation theory, where both the signal and noise distributions are uncertain. We utilize the $G$-normal distribution to characterize channel noise. Moreover, to capture practical signal variety, the absolute values of transmitted signal random variables are assumed to lie within a bounded range $[\underlineσ_X,\overlineσ_X]$. The worst-case detection performance is then characterized by a double supremum, meaning over all admissible distributions and all possible signal realizations. We derive estimations for the minimum and the maximum detection error probabilities, and demonstrate the validity of the results through numerical simulations. The proposed model generalizes the classical theoretical analysis of energy detection and offers a potential theoretical foundation for robust detection and information-theoretic analysis under distributional uncertainty.