[Paper Review] Capacity of a Class of Broadcast Relay Channels

Arash Behboodi, Pablo Piantanida|arXiv (Cornell University)|May 4, 2010

Cooperative Communication and Network Coding4 references4 citations

TL;DR

This paper derives new inner bounds on the capacity region of a broadcast relay channel (BRC) with two relays, unifying and improving upon prior work using Marton coding, backward decoding, and message recombination. It establishes the capacity for semi-degraded and degraded Gaussian BRCs with common relays, demonstrating tightness of the proposed region in these cases.

ABSTRACT

Consider the broadcast relay channel (BRC) which consists of a source sending information over a two user broadcast channel in presence of two relay nodes that help the transmission to the destinations. Clearly, this network with five nodes involves all the problems encountered in relay and broadcast channels. New inner bounds on the capacity region of this class of channels are derived. These results can be seen as a generalization and hence unification of previous work in this topic. Our bounds are based on the idea of recombination of message bits and various effective coding strategies for relay and broadcast channels. Capacity result is obtained for the semi-degraded BRC-CR, where one relay channel is degraded while the other one is reversely degraded. An inner and upper bound is also presented for the degraded BRC with common relay (BRC-CR), where both the relay and broadcast channel are degraded which is the capacity for the Gaussian case. Application of these results arise in the context of opportunistic cooperation of cellular networks.

Motivation & Objective

• To derive tighter inner bounds on the capacity region of the broadcast relay channel (BRC) with two relays and a common relay.
• To unify and improve upon existing results from Kramer et al. and previous BRC studies.
• To establish the capacity for semi-degraded and degraded Gaussian BRCs with common relays.
• To develop a general achievable rate region combining Marton coding, backward decoding, and message recombination techniques.

Proposed method

• Derives a novel inner bound using a combination of Marton coding, backward decoding, and message recombination strategies.
• Introduces auxiliary random variables (U, V, X1, X2, etc.) to model dependencies across source, relays, and destinations.
• Applies backward decoding at receivers to handle interference and improve rate bounds.
• Uses time-sharing via a random variable Q to optimize the rate region.
• Employs mutual information and entropy inequalities to bound achievable rates, particularly involving Y1, Y2, Z1, Z2, and X1, X2.
• Applies the power entropy inequality and Gaussian channel properties to derive tight bounds for the Gaussian case.

Experimental results

Research questions

• RQ1 What is the capacity region of a broadcast relay channel with two relays and a common relay?
• RQ2 How can coding strategies from broadcast and relay channels be combined to improve achievable rates?
• RQ3 Can the proposed inner bound achieve capacity for specific classes of BRCs?
• RQ4 What is the role of message recombination and backward decoding in enhancing performance in BRCs?
• RQ5 How do the bounds behave in the semi-degraded and degraded Gaussian cases?

Key findings

• The proposed inner bound strictly improves upon previous results by Kramer et al. and earlier BRC studies.
• The capacity is achieved for the semi-degraded BRC with common relay, where one relay channel is degraded and the other is reversely degraded.
• For the degraded BRC with common relay, the inner and outer bounds coincide in the Gaussian case, establishing capacity.
• The inner bound includes the Marton region for broadcast channels as a special case.
• The derived region is tight for the Gaussian case due to matching inner and outer bounds.
• The use of backward decoding and message recombination significantly enhances the achievable rate region.