Relay with Private Messages

Introduction

The traditional relay channel was introduced by van der Meulen [1] and analyzed by Cover and El Gamal [2]. In this three-terminal model, two nodes communicate with the help of a third node. This model and its derivatives and variations have spawned a large amount of research in the past few years.

The motivation in this work is to shed light on the operational cost of relaying in terms of loss of rate for other communications. Assume that a node has the ability to both communicate with the nodes in its vicinity, as well as act as a relay for them. Intuitively, the limited channel resources will lead to a tradeoff, thus helping other nodes will incur a cost in terms of the rate available to the node's own communications. The goal of this work is to quantify the nature of this cost, using a model that we denote Relay Channel with Private Messages (RCPM).

The three nodes, indexed 1,2,3, are denoted source, relay, and destination. There are three simultaneous communications: between source and destination (assisted by the relay), as well as individual (private) communications between the source and relay, and between the relay and destination.

In this work [3] we find achievable rate regions as well as outer bounds on the capacity region, for the discrete memoryless relay channel with private messages. We then study the Gaussian versions of this channel and achievable rate regions are characterized. Numerical results are provided that give insights into the trade-offs between private messaging and relayed messaging in this hybrid three-node network.

Overview of Analysis

The system model for the discrete memoryless channel is depicted in Fig. 2 below.

In our analysis we use new combinations of coding strategies inspired by the MAC channel with generalized feedback, and Marton's approach to the broadcast channel. We derive achievable rates for the discrete memoryless and Gaussian RCPM, and outer bounds for the DMC case. The discrete memoryless and Gaussian RCPM generalize their counterparts in the original relay channel and relay-broadcast channels.

In this work we use regular encoding and backward decoding [4]. Backward decoding has been used previously for degraded relay channel in [5] and more recently for the general relay channel with partial decode-and-forward in [6]. As a by-product of this work, we demonstrate that backward decoding does not improve the achievable rate of a non-degraded relay channel employing compress-and-forward scheme. This work is also related to, and is in a way a generalization of, [7,8].

For the detailed equations of achievable rate regions, the interested reader is referred to [3]. Both DMC and Gaussian cases have been investigated. In Figure 3 below, we show the performance of a degraded Gaussian RCPM with 10 dB SNR at the relay input and 5 dB SNR at the destination input. Contour plots of the achievable rate region in the R13-R23 plane are shown to demonstrate the trade-off between the relayed rate and one of the private rates.

References:

1. E. C. van der Meulen, ``Three-terminal communication channels,'' Adv. Appl. Probab., vol. 3, pp. 120--154, 1971.

2. T. Cover and A. E. Gamal, ``Capacity theorems for the relay channel,'' IEEE Trans. Inform. Theory, vol. 25, no. 5, pp. 572--584, 1979.

3. R. Tannious and A. Nosratinia, ``Relay Channel with Private messages,'' IEEE Trans. Information Theory, vol. 53, no. 10, Oct. 2007, pp. 3777 - 3785.

4. G. Kramer, M. Gastpar and P. Gupta, ``Cooperative strategies and capacity theorems for relay networks'' IEEE Trans. on Information Theory, vol. 51, no. 9, pp. 3037--3063, Sep. 2005.

5. C.-M. Zing, F. Kuhlmann and A. Buzo, ``Achivability proof of some multiuser channel coding theorems using backward decoding'' IEEE Trans. on Information Theory, vol. 35, no. 6, pp. 1160--1165, Nov. 1989.

6. H.-F. Chong, M. Motani and K. H. Carg, ``Generalized backward decoding strategies for the relay channel'' IEEE Trans. on Information Theory, vol. 53, no. 1, pp. 394--401, Jan. 2007.

7. Y. Liang and V. V. Veeravalli, ``Cooperative relay broadcast channels'' IEEE Trans. on Information Theory, vol. 53, no. 3, pp. 900--928, Mar. 2007.

8. Y. Liang and G. Kramer, ``Rate regions for relay broadcast channels'' IEEE Trans. on Information Theory, accepted.

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