Duality between MIMO Source and Channel Coding

Sandeep Pradhan1
(Professor Kannan Ramchandran)
(DARPA) F30602-00-2-0538 and (NSF) CCR-0219722

We address duality in a variety of multiple-input-multiple-output (MIMO) source and channel coding problems of interest under different scenarios of "one-sided" inter-terminal collaboration at either the transmitter or at the receiver, including certain cases of duality between (1) broadcast channel coding and distributed source coding, and (2) multi-access channel coding and multiple-descriptions source coding. Our notion of duality in this project is in a functional sense, where the optimal encoder mapping for a MIMO source coding problem becomes identical to the optimal decoder mapping for the dual MIMO channel coding problem, and vice versa. For ease of illustration we give the formulation only for two-input-two-output systems, which can be easily extended to the MIMO case. We present the precise mathematical conditions under which these encoder-decoder mappings are swappable in the two dual MIMO problems, identifying the key roles played by the source distortion and channel cost measures respectively in the MIMO source and channel coding problems in capturing this duality.

Since the first observation by Shannon in 1959 [1] that source and channel coding problems can be studied as information-theoretic duals of each other, a number of researchers have made significant contribution to furthering this understanding, as noted in excellent textbooks such as [2,3]. In [4], this duality has been studied in the context of quantization and modulation. As multiuser information theory has gained significant interest recently, there has been a corresponding interest in extending this duality to these scenarios. Duality between source and channel coding with side information was first reported in [5], and later studied in [6].

A mathematical formulation of the functional duality between conventional point-to-point source and channel coding problems was given in [7], where the important roles of distortion and cost measures for these two problems were highlighted, inspired by seemingly unrelated work on the optimality of uncoded transmission in [8].

This concept was then extended to the case of source and channel coding with side information. The notion of duality addressed in [7] is in a functional sense, i.e., the optimal encoder-decoder mappings for one problem become the optimal decoder-encoder mappings for the dual problem. The precise conditions under which such swappable mappings are feasible involve dual relationships between distortion and cost measures in the source and channel coding problems respectively. Study of this functional duality serves two important purposes: (1) it provides new insights into these problems from the different perspectives of source and channel coding, and allows for cross-leveraging of advances in the individual fields; (2) more importantly, it provides a basis for sharing efficient constructions of the encoder and decoder functions in the two problems, e.g., through the use of structured algebraic codes, turbo-like codes, trellis-based codes, etc.

In this work we extend this notion of functional duality to more instances of MIMO source and channel coding problems. We study various MIMO structures admitting different scenarios of collaboration among multi-terminal inputs and/or outputs. (To keep the exposition simple, in this work we describe only two-input-two-output systems.) The collaboration scenarios we consider involve those where either the multi-terminal encoders or the multi-terminal decoders can collaborate, i.e., be joint, but not both. (The case of collaboration at both ends degenerates to point-to-point MIMO systems.) Under this one-sided collaboration abstraction, we address four problems of interest in source and channel coding: (1) distributed source coding; (2) broadcast channel coding; (3) multiple description source coding with no excess sum-rate; and (4) multiple access channel coding with independent message sets. In (1) and (4), the decoders collaborate, whereas in (2) and (3), the encoders collaborate. These four problems have been studied in the literature extensively. In this project we point out that for a given distributed source coding problem, under certain cases, we can obtain a specific dual broadcast channel coding problem and vice versa. Similarly, for a given multiple description coding problem, we can find a specific dual multiple access channel coding problem and vice versa.

C. E. Shannon, "Coding Theorems for a Discrete Source with a Fidelity Criterion," IRE Nat. Conv. Rec., Vol. 4, 1959.
I. Csiszar and J. Korner, Information Theory: Coding Theorems for Discrete Memoryless Sources, New York, Academic Press, 1981.
T. M. Cover and J. A. Thomas, Elements of Information Theory, New York, John Wiley and Sons, 1991.
M. V. Eyuboglu and G. D. Forney, "Lattice and Trellis Quantization with Lattice and Trellis-bounded Codebooks--High Rate Theory for Memoryless Sources," IEEE Trans. Information Theory, Vol. 39, January 1993.
J. Chou, S. S. Pradhan, and K. Ramchandran, "On the Duality between Distributed Source Coding and Data Hiding," Proc. Asilomar Conf. on Signals, Systems, and Computers, November 1999.
M. Chiang and T. M. Cover, "Unified Duality between Channel Capacity and Rate Distortion with Side Information," Proc. Int. Symp. Information Theory, Washington, DC, June 2001.
S. S. Pradhan, J. Chou, and K. Ramchandran, Duality between Source and Channel Coding with Side Information, UC Berkeley Electronics Research Laboratory, Memorandum No. UCB/ERL M01/34, December 2001.
M. Gastpar, B. Rimoldi, and M. Vetterli, "To Code or Not to Code," Proc. IEEE Int. Symp. Information Theory, Sorrento, Italy, June 2000.
1Professor, University of Michigan

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