On Power Allocation for Parallel Gaussian Broadcast Channels with Common Information
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Research Article On Power Allocation for Parallel Gaussian Broadcast Channels with Common Information Ramy H. Gohary1, 2 and Timothy N. Davidson1 1 Department
of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada Research Centre, Industry Canada, Ottawa, ON, Canada
2 Communications
Correspondence should be addressed to Ramy H. Gohary, [email protected] Received 28 October 2008; Accepted 13 March 2009 Recommended by Sergiy Vorobyov This paper considers a broadcast system in which a single transmitter sends a common message and (independent) particular messages to K receivers over N unmatched parallel scalar Gaussian subchannels. For this system the set of all rate tuples that can be achieved via superposition coding and Gaussian signalling (SPCGS) can be parameterized by a set of power loads and partitions, and the boundary of this set can be expressed as the solution of an optimization problem. Although that problem is not convex in the general case, it will be shown that it can be used to obtain tight and efficiently computable inner and outer bounds on the SPCGS rate region. The development of these bounds relies on approximating the original optimization problem by a (convex) Geometric Program (GP), and in addition to generating the bounds, the GP also generates the corresponding power loads and partitions. There are special cases of the general problem that can be precisely formulated in a convex form. In this paper, explicit convex formulations are given for three such cases, namely, the case of 2 users, the case in which only particular messages are transmitted (in both of which the SPCGS rate region is the capacity region), and the case in which only the SPCGS sum rate is to be maximized. Copyright © 2009 R. H. Gohary and T. N. Davidson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction Consider a broadcast communication scenario in which a single transmitter wishes to send a combination of (independent) particular messages that are intended for individual users and a common message that is intended for all users [1]. Such broadcast systems can be classified according to the probabilistic model that describes the communication channels between the transmitter and the receivers. A special class of broadcast channels is the class of degraded channels, in which the probabilistic model is such that the signals received by the users form a Markov chain. Using this Markovian property, a coding scheme that can attain every point in the capacity region for this class of channels was developed in [2]. If, however, the received signals do not form a Markov chain, the broadcast channel is said to be nondegraded, and the coding scheme developed in [2] does not apply directly to this case. Although degraded channels are useful in modelling single-input single-output broadcast systems,
many practical systems give ris
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