arXiv:1201.6548v3 [cs.IT] 8 May 2014 Orthogonal Multiple Access with Correlated Sources: Achievable Region and Pragmatic Schemes A. Abrardo, G. Ferrari, M. Martal` o, M. Franceschini, and R. Raheli Abstract— In this paper, we consider orthogonal multiple access coding schemes, where correlated sources are encoded in a distributed fashion and transmitted, through additive white Gaussian noise (AWGN) channels, to an access point (AP). At the AP, component decoders, associated with the source encoders, iteratively exchange soft information by taking into account the source correlation. The first goal of this paper is to investigate the ultimate achievable performance limits in terms of a multi-dimensional feasible region in the space of channel parameters, deriving insights on the impact of the number of sources. The second goal is the design of pragmatic schemes, where the sources use “off-the-shelf” channel codes. In order to analyze the performance of given coding schemes, we propose an extrinsic information transfer (EXIT)-based approach, which allows to determine the corresponding multi-dimensional feasible regions. On the basis of the proposed analytical framework, the performance of pragmatic coded schemes, based on serially concatenated convolutional codes (SCCCs), is discussed. Index Terms— Correlated sources, orthogonal multiple access, joint channel decoding (JCD), noisy Slepian-Wolf problem, EXIT chart, serially concatenated convolutional code (SCCC). I. I NTRODUCTION The efficient transmission of correlated signals, observed at different nodes, to one or more collectors is one of the main challenges in various networking scenarios, e.g., wireless sensor networks [1]. In the case of one collector node, this problem is often referred to as reach-back channel problem [2]–[4]. In the case of separated additive white Gaussian noise (AWGN) channels, the separation between source (up to the Slepian-Wolf limit) and channel coding is known to be optimal [2], [5]. However, implementing a practical system based on separation, i.e., given by distributed source coding (DSC) followed by channel encoding, is not straightforward [6], [7] and the design of practically good codes is still an open issue [8]. Alternative approaches are represented by cooperative source-channel coding and distributed joint source-channel coding (JSCC). In the JSCC case, no cooperation among sources is required, each source is independently encoded, and the correlation between the sources is exploited at the joint decoder by means of joint channel decoding (JCD) [9]– [12]. In other words, for a given source neither the data transmitted from the other sources nor the correlation model A. Abrardo is with the Department of Information Engineering, University of Siena, Italy. Email: abrardo@dii.unisi.it. G. Ferrari, M. Martal` o and R. Raheli are with the Department of Information Engineering, Univer- sity of Parma, Italy. Email: {gianluigi.ferrari,marco.martalo,raheli}@unipr.it. M. Franceschini is with IBM T.J. Watson Research Center, Yorktown Heights, NY, USA. Email: franceschini@us.ibm.com. This paper was presented in part at the 2009 and 2010 Information Theory and Applications Workshop (ITA), UCSD, San Diego, CA, USA. are available at the encoder. The correlation model between the sources must instead be assumed to be known at the (common) receiver, which aims at the reconstruction of the information streams transmitted by the sources. The problem of designing good codes for this scenario has been, however, only partially addressed. In [12], the authors state that for two orthogonal channels the type of concatenated code utilized for the encoding process is not critical, and good results can be obtained, provided that powerful codes are employed. In [13], recursive non-systematic convolutional encoders are proposed as constituent encoders for heavily biased sources, leading to a signal-to-noise ratio (SNR) penalty between 0.74 dB and 1.17 dB with respect to the Shannon limit. In [14], optimized low-density parity-check (LDPC) codes are designed, by means of puncturing and proper iterative decoding schedule at the access point (AP). Extensions to universal codes (i.e., capacity-achieving codes for all possible channel parameters) through spatial coupling has been also recently considered [14]–[16]. More precisely, the approaches in [14]–[16], relative to a two-source scenario, have the following characteristics: at each source, LDPC coding is used; at the AP, message-passing decoding is carried out on a joint bipartite graph (combining the graphs of the two codes) and the asymptotic performance, for infinite codeword length, is investigated. However, the extension of the proposed joint graph-based approach to an arbitrary number of sources is a challenging research direction. Another interesting approach has been presented in [17], where practical concatenated coded schemes are designed for faded multiple-input multiple-output (MIMO) scenarios. However, the scheme is evaluated only for the case of two sources. In this paper, we consider a generic number of correlated sources which transmit to a common AP through orthogonal AWGN channels. The sources do not explicitly use source codes, but only channel codes. At the AP, a proper itera- tive receiver is used to exploit the source correlation. This extends our previous works for two-source scenarios [18], [19], as well as [20], where practical coding/decoding schemes have been designed in the presence of block faded channels. The first contribution of this paper is to shed light on the characterization of JCD schemes with an arbitrary number of correlated sources, by characterizing the multi-dimensional achievable region in the space of channel parameters for an arbitrary number of sources. It will be shown that a few characteristic points are sufficient to accurately characterize this achievable region. The asymptotic behavior, for a large number of sources, is also investigated. The impacts of the correlation level and of the number of sources, as well as the speed of convergence of the achievable region to the