1 Lossy Multicasting over Binary Symmetric Broadcast Channels O. Y. Bursalioglu 1 , M. Fresia 2 , G. Caire 1 , and H. V. Poor 3 Abstract—Lossy multicasting of a set of independent, discrete-time, continuous-amplitude source components un- der the Mean Square Error distortion criterion over binary symmetric broadcast channels is investigated. We consid- ered the practically appealing concatenation of successive refinement source coding with broadcast coding and, specif- ically, time-sharing of linear binary codes. Three different system optimization criteria are formulated for the lossy multicasting problem. The resulting system optimization is fairly general and applies to a variety of combinations of successive refinement source codes and channel codes. In particular, we investigate in depth the system optimization for a class of channel optimized quantization with suc- cessive refinement, obtained by using standard embedded scalar quantizers and linear mapping of the (redundant) quantizer bit-planes onto channel codewords by using a systematic Raptor encoder. This scheme is referred to as Quantization with Linear Index Coding (QLIC). Unlike existing literature on progressive transmission with unequal error protection or channel optimized quantization, we focus on the regime of moderate-to-large code block length and leverage the power of modern sparse-graph codes with iterative Belief Propagation decoding. In this regime, the system optimization takes on the form of simple convex programming that reduces to linear programming for QLIC. The performance of QLIC compares favorably with respect to the state of the art channel optimized quantization in the conventional setting of a single Gaussian source over a binary symmetric channel. For the multicast scenario, we quantified the performance gap incurred by the practical QLIC design with respect to ideal source and channel codes. Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs- permissions@ieee.org. 1 O. Y. Bursalioglu and G. Caire are with the Ming Hsieh Dept. of Electrical Engineering, University of Southern California, Los Angeles, CA, 90089, USA (E-mail: bursalio,caire@usc.edu, Phone: 1(213)740-4683 and Fax: 1(213)740-8729) 2 M. Fresia is with Intel Mobile Communications, Munich, Germany. (E-mail: maria.fresia@intel.com, Phone: +49 89 998853-24663) 3 H. V. Poor is with Dept. of Electrical Engineering, Princeton University, Princeton, NJ, 08544, USA. (E-mail: poor@princeton.edu, Phone : 1(609)258-6868). This work was supported in part by the National Science Foundation under Grants CNS-09-05398 and CCF-10-16671 and NeTS-NOSS 0722073. I. I NTRODUCTION A lossy multicast scenario where L ≥ 1 users wish to receive the same source, with possibly different distor- tion levels is considered. Analog broadcasting systems are able to send simultaneously the same source to a potentially unlimited number of receivers, with distortion levels that depend on the user channel conditions. Analog transmission finds its theoretical justification in the fact that a Gaussian source under the Mean-Squared Error (MSE) distortion measure is “matched” to an Additive White Gaussian Noise (AWGN) channel [1] when the source bandwidth and the channel bandwidth are equal, i.e., when the ratio of source sample per channel use, denoted by b in this work, is equal to 1. Unfortunately, this fortuitous matching condition does not hold in general for b ̸=1 and/or in the case of more general source/channel pairs and distortion measures, as those arising from practically relevant sources and commu- nication networks including heterogeneous wired and wireless communication links, including forward error correction at the physical layer. A lossy multicast problem with L users is charac- terized by the distortion region D(b), i.e., the region of achievable distortion vectors (D 1 ,...,D L ) ∈ R L + , for a given bandwidth ratio b, an underlying one-to-L broadcast channel [2], and a given distortion measure. While D(b) is unknown in general, in [3] it is shown that for a Gaussian source over a Gaussian broadcast channel and MSE distortion, the optimal distortion region can be approached within a fixed gap, independent of the channel signal to noise ratios, by the concatenation of successive refinement source coding [4] and superpo- sition coding [2]. Although the approximate optimality result of [3] and [5] does not extend to general sources, broadcast channels and distortion measures, even if the underlying broadcast channel is degraded and the source is successively refinable, the concatenated source- channel coding approach is very appealing in practice and yields a layered coding scheme where the distortion at which any user can reconstruct the source depends on the number of successive refinement layers successfully decoded. Given its practical relevance, without further ado in this paper we focus on such concatenated scheme. In the case of practical codes and finite block length,