Higher Order Derivatives in Costa's Entropy Power Inequality

Let $X$ be an arbitrary continuous random variable and $Z$ be an independent Gaussian random variable with zero mean and unit variance. For $t~>~0$, Costa proved that $e^{2h(X+\sqrt{t}Z)}$ is concave in $t$, where the proof hinged on the first and second order derivatives of $h(X+\sqrt{t}Z)$. Specifically, these two derivatives are signed, i.e., $\frac{\partial}{\partial t}h(X+\sqrt{t}Z) \geq 0$ and $\frac{\partial^2}{\partial t^2}h(X+\sqrt{t}Z) \leq 0$. In this paper, we show that the third order derivative of $h(X+\sqrt{t}Z)$ is nonnegative, which implies that the Fisher information $J(X+\sqrt{t}Z)$ is convex in $t$. We further show that the fourth order derivative of $h(X+\sqrt{t}Z)$ is nonpositive. Following the first four derivatives, we make two conjectures on $h(X+\sqrt{t}Z)$: the first is that $\frac{\partial^n}{\partial t^n} h(X+\sqrt{t}Z)$ is nonnegative in $t$ if $n$ is odd, and nonpositive otherwise; the second is that $\log J(X+\sqrt{t}Z)$ is convex in $t$. The first conjecture can be rephrased in the context of completely monotone functions: $J(X+\sqrt{t}Z)$ is completely monotone in $t$. The history of the first conjecture may date back to a problem in mathematical physics studied by McKean in 1966. Apart from these results, we provide a geometrical interpretation to the covariance-preserving transformation and study the concavity of $h(\sqrt{t}X+\sqrt{1-t}Z)$, revealing its connection with Costa's EPI.Comment: Second version submitted. https://sites.google.com/site/chengfancuhk

The capacity region of classes of product broadcast channels

We establish a new outer bound for the capacity region of product broadcast channels. This outer bound matches Marton's inner bound for a variety of classes of product broadcast channels whose capacity regions were previously unknown. These classes include product of reversely semi-deterministic and product of reversely more-capable channels. A significant consequence of this new outer bound is that it establishes, via an example, that the previously best known outer-bound is strictly suboptimal for the general broadcast channel. Our example is comprised of a product broadcast channel with two semi-deterministic components in reverse orientation.Comment: full version of isit pape

Centralized Coded Caching with User Cooperation

In this paper, we consider the coded-caching broadcast network with user cooperation, where a server connects with multiple users and the users can cooperate with each other through a cooperation network. We propose a centralized coded caching scheme based on a new deterministic placement strategy and a parallel delivery strategy. It is shown that the new scheme optimally allocate the communication loads on the server and users, obtaining cooperation gain and parallel gain that greatly reduces the transmission delay. Furthermore, we show that the number of users who parallelly send information should decrease when the users' caching size increases. In other words, letting more users parallelly send information could be harmful. Finally, we derive a constant multiplicative gap between the lower bound and upper bound on the transmission delay, which proves that our scheme is order optimal.Comment: 9 pages, submitted to ITW201

Information-Theoretic Limits of Bistatic Integrated Sensing and Communication

The bistatic integrated sensing and communication (ISAC) system model avoids the strong self-interference in a monostatic ISAC system by employing a pair of physically separated sensing transceiver and maintaining the merit of co-designing radar sensing and communications on shared spectrum and hardware. Inspired by the appealing benefits of bistatic radar, we study bistatic ISAC, where a transmitter sends a message to a communication receiver and a sensing receiver at another location carries out a decoding-and-estimation(DnE) operation to obtain the state of the communication receiver. In this paper, both communication and sensing channels are modelled as state-dependent memoryless channels with independent and identically distributed time-varying state sequences. We consider a rate of reliable communication for the message at the communication receiver as communication metric. The objective of this model is to characterize the capacity-distortion region, i.e., the set of all the achievable rate while simultaneously allowing the sensing receiver to sense the state sequence with a given distortion threshold. In terms of the decoding degree on this message at the sensing receiver, we propose three achievable DnE strategies, the blind estimation, the partial-decoding-based estimation, and the full-decoding-based estimation, respectively. Based on the three strategies, we derive the three achievable rate-distortion regions. In addition, under the constraint of the degraded broadcast channel, i.e., the communication receiver is statistically stronger than the sensing receiver, and the partial-decoding-based estimation, we characterize the capacity region. Examples in both non-degraded and degraded cases are provided to compare the achievable rate-distortion regions under three DnE strategies and demonstrate the advantages of ISAC over independent communication and sensing.Comment: 40 pages, 7 figure