Inequalities for the Dependent Gaussian Noise Channels Based on Fisher Information and Copulas

Abstract

Considering the Gaussian noise channel, Costa [4] investigated the concavity of the entropy power when the input signal and noise components are independent. His argument was connected to the first-order derivative of the Fisher information. In real situations, however, the noise can be highly dependent on the main signal. In this paper, we suppose that the input signal and noise variables are dependent. Then, some well-known copula functions are used to define their dependence structure. The first- and second-order derivatives of Fisher information of the model are obtained. Then, by using these derivatives, we will generalize two inequalities based on the Fisher information and a functional that is closely associated to Fisher information for the case when the input signal and noise variables are dependent. We will also show that the previous results for the independent case are recovered as special cases of our result. Several applications are provided to support the usefulness of our results. Finally, the channel capacity of the Gaussian noise channel model with dependent signal and noise is studied.

Costa’s concavity inequality for dependent variables based on the multivariate Gaussian copula

“…Now, under the same conditions as in Corollary 1, according to the relations (51) and (52), the first-order derivative of the Fisher information, simply follows by setting and in (34). This coincides with the result in [4], where a direct proof of (53) is provided.…”

“…simply follows by setting m = 1 and p j (y; t) = p (y; t) in (34). This coincides with the result in [4], where a direct proof of (53) is provided.…”

“…Copula theory was first introduced in [14] in order to achieve the connection between a joint PDF and its marginals. In [4], the authors extended two inequalities based on the Fisher information when the input signal and noise components are dependent and their dependence structure is modeled by several well-known copulas. There are several families of copulas with different dependence structures.…”