Alexander James Grant

Rate-splitting multiple access for discrete memoryless channels

It is shown that the encoding/decoding problem for any asynchronous M-user discrete memoryless multiple-access channel can be reduced to corresponding problems for at most 2M-1 single-user discrete memoryless channels. This result, which extends a similar result for Gaussian channels, reduces the seemingly hard task of finding good multiple-access codes to the much better understood task of finding good codes for single-user channels. As a by-product, some interesting properties of the capacity region of M-user asynchronous discrete memoryless channels are derived.

Source coding for a simple network with receiver side information

We consider the problem of source coding with receiver side information for the simple network proposed by R. Gray and A. Wyner in 1974. In this network, a transmitter must reliably transport the output of two correlated information sources to two receivers using three noiseless channels: a public channel which connects the transmitter to both receivers, and two private channels which connect the transmitter directly to each receiver. We extend Gray and Wyners original problem by permitting side information to be present at each receiver. We derive inner and outer bounds for the achievable rate region and, for three special cases, we show that the outer bound is tight.

Rate distortion with Side-Information at many receivers

We present a new inner bound for the admissible rate region of the t-stage successive-refinement problem with side-information. We also present a new upper bound for the rate-distortion function for lossy-source coding with multiple receivers and side-information. A single-letter characterisation of this rate-distortion function is a long-standing open problem, and it is widely believed that the tightest upper bound is provided by Theorem 2 of Heegard and Bergers paper “Rate Distortion when Side Information may be Absent,” IEEE Trans. Inform. Theory, 1985. We give a counterexample to Heegard and Bergers result.

Characterising correlation via entropy functions

Characterising the capacity region for a network can be extremely difficult. Even with independent sources, determining the capacity region can be as hard as the open problem of characterising all information inequalities. The majority of computable outer bounds in the literature are relaxations of the Linear Programming bound which involves entropy functions of random variables related to the sources and link messages. When sources are not independent, the problem is even more complicated. Extension of Linear Programming bounds to networks with correlated sources is largely open. Source dependence is usually specified via a joint probability distribution, and one of the main challenges in extending linear program bounds is the difficulty (or impossibility) of characterising arbitrary dependencies via entropy functions. This paper tackles the problem by answering the question of how well entropy functions can characterise correlation among sources. We show that by using carefully chosen auxiliary random variables, the characterisation can be fairly “accurate”.