In the classical compress-and-forward relay scheme developed by Cover and El Gamal, the decoding process operates in a successive way: the destination first decodes the compression of the relay's observation and then decodes the original message of the source. Recently, several modified compress-and-forward relay schemes were proposed, where the destination jointly decodes the compression and the message, instead of successively. Such a modification on the decoding process was motivated by realizing that it is generally easier to decode the compression jointly with the original message, and more importantly, the original message can be decoded even without completely decoding the compression. Thus, joint decoding provides more freedom in choosing the compression at the relay. However, the question remains in these modified compress-and-forward relay schemes-whether this freedom of selecting the compression necessarily improves the achievable rate of the original message. It has been shown by El Gamal and Kim in 2010 that the answer is negative in the single-relay case. In this paper, it is further demonstrated that in the case of multiple relays, there is no improvement on the achievable rate by joint decoding either. More interestingly, it is discovered that any compressions not supporting successive decoding will actually lead to strictly lower achievable rates for the original message. Therefore, to maximize the achievable rate for the original message, the compressions should always be chosen to support successive decoding. Furthermore, it is shown that any compressions not completely decodable even with joint decoding will not provide any contribution to the decoding of the original message. The above phenomenon is also shown to exist under the repetitive encoding framework recently proposed by Lim et al., which improved the achievable rate in the case of multiple relays. Here, another interesting discovery is that the improvement is not a result of repetitive encoding, but the benefit of delayed decoding after all the blocks have been finished. The same rate is shown to be achievable with the simpler classical encoding process of Cover and El Gamal with a block-by-block backward decoding process.

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Consider a memoryless relay channel, where the relay is connected to the destination with an isolated bit pipe of capacity C0. Let C(C0) denote the capacity of this channel as a function of C0. What is the critical value of C0 such that C(C0) first equals C(∞)? This is a long-standing open problem posed by Cover and named "The Capacity of the Relay Channel," in Open Problems in Communication and Computation, Springer-Verlag, 1987. In this paper, we answer this question in the Gaussian case and show that C(C0) can not equal to C(∞) unless C0 = ∞, regardless of the SNR of the Gaussian channels. This result follows as a corollary to a new upper bound we develop on the capacity of this channel. Instead of "single-letterizing" expressions involving information measures in a high-dimensional space as is typically done in converse results in information theory, our proof directly quantifies the tension between the pertinent n-letter forms. This is done by translating the information tension problem to a problem in high-dimensional geometry. As an intermediate result, we develop an extension of the classical isoperimetric inequality on a highdimensional sphere, which can be of interest in its own right.

repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span," J. Math. Phys., vol. 39, pp. 6459-6489, 1998. [22] E. Bollt, "Stability of order: An example of chaos "near" a linear map,"Int. J. Bifurcat. Chaos, vol. 9, no. 10, pp. 2081-2090, 1999. Clock-Gating and Its Application to Low Power Design of Sequential CircuitsQing Wu, Massoud Pedram, and Xunwei Wu

We consider the discrete memoryless symmetric primitive relay channel, where, a source X wants to send information to a destination Y with the help of a relay Z and the relay can communicate to the destination via an error-free digital link of rate R 0 , while Y and Z are conditionally independent and identically distributed given X. We develop two new upper bounds on the capacity of this channel that are tighter than existing bounds, including the celebrated cut-set bound. Our approach significantly deviates from the standard information-theoretic approach for proving upper bounds on the capacity of multi-user channels. We build on the blowing-up lemma to analyze the probabilistic geometric relations between the typical sets of the n-letter random variables associated with a reliable code for communicating over this channel. These relations translate to new entropy inequalities between the n-letter random variables involved.As an application of our bounds, we study an open question posed by (Cover, 1987), namely, what is the minimum needed Z-Y link rate R * 0 in order for the capacity of the relay channel to be equal to that of the broadcast cut. We consider the special case when the X-Y and X-Z links are both binary symmetric channels. Our tighter bounds on the capacity of the relay channel immediately translate to tighter lower bounds for R * 0 . More interestingly, we show that when p → 1/2, R * 0 ≥ 0.1803; even though the broadcast channel becomes completely noisy as p → 1/2 and its capacity, and therefore the capacity of the relay channel, goes to zero, a strictly positive rate R 0 is required for the relay channel capacity to be equal to the broadcast bound. Existing upper bounds on the capacity of the relay channel, and the cut-set bound in particular, would rather imply R * 0 → 0, while achievability schemes require R * 0 → 1. We conjecture that R * 0 → 1 as p → 1/2.