© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue International Journal of Electronics and Communications Letter Successive vs. Joint Decoding under Complexity and Performance Constraints ∗ Ralf R. Müller ∗∗ Abstract Helpful guidelines to efficient design of multiuser systems based on successive decoding under a complexity con- straint are given. It is shown that in practical systems com- plexity saved by chosing shorter codes can often be successfully invested in joint-multiuser decoding in order to achieve a better performance with the same overall complexity. Keywords Information theory, Communication theory, Succes- sive decoding 1. Introduction The complexity for decoding a multiuser code employed on the asynchronous multiple-access channel is exponen- tial in the product of the number of users K and the code word length ℓ, in general [1]. Under some circum- stances, i.e. the users’ individual rates form a vertex of the channel’s capacity region, information theory allows for methods to break up the multiuser decoding problem into K independent single-user decoding problems [2, 3]. This reduces the complexity from O(|A| ℓ K ) to O( K ·|A| ℓ ) with A denoting the code’s alphabet, i.e. A = {+1, -1} for binary codes. All such simplifying methods which are known up to now rely on the principle of successive decoding: That user which is decoded currently does not assume any knowledge about the statistics of the signals of the users which are to be decoded subsequently. After being de- coded, the current user’s signal is reconstructed perfectly and eliminated from the signals the users to be decoded subsequently operate with. This principle has been fre- quently used in multiuser information theory in order to prove the achievability of the capacity region of a variety of multiple-access channels [2]. While successive decoding does not affect the individ- ual users’ channel capacities which are achieved only for infinite code word length, it was already pointed out in Received August 9, 2000. Ralf R. Müller, Forschungszentrum Telekommunikation Wien (FTW), Vienna, Austria. E-mail: mueller@ftw.at ∗ This work was supported in part by the German Academic Ex- change Service (DAAD) under grant 332 4 00 510. ∗∗ This work was performed while the author was with Telecom- munications Institute II of the University of Erlangen–Nuremburg, Germany, and the Department of Electrical Engineering, Princeton University, Princeton, U.S.A. [4] using bounds on the error probability of random codes that for finite code word length the situation is different, in general. Thus, multiuser coding should be more powerful than single-user coding combined with successive decod- ing, in general, but it also involves much larger complex- ity. Hereby, the natural question arises whether multiuser decoding can also be favourable if both schemes are com- pared with respect to identical complexity. This question will be answered for communication over the Gaussian multiple-access channel – the multiuser equivalent to the additive white Gaussian noise (AWGN) channel – under some simplifying, but practically accurate assumptions. The presented analysis will also give practical guidelines how to optimally partition a set of users into subsets of users that shall be decoded jointly within the sets and suc- cessively among the sets. 2. Successive vs. joint decoding The information rate to be transmitted arbitrarily reliable over an AWGN channel is well-known [2] to be R ≤ 1 2 log 2  1 + P N  (1) with P and N denoting the power of the user’s signal and the AWGN, respectively. Equality in (1) holds only for in- finite code word length. In practice, the allowable rate at a given target error probability P e ≪ 1 will be smaller than channel capacity or corollary the signal power required to support capacity rate will be higher. Following the second approach to describe the point enables to write P ( N) = VN ( 4 R - 1 ) (2) where V > 1 denotes the power gap between infinite and finite code word length for a given target error probability. For sake of simplicity, the power penalty V is assumed to be identical for all users. Though an approximation, this assumption is well-justified by the results in [5] showing that low-rate codes can be used as constituent codes in multilevel codes in such a way that the power penalty is almost preserved for all rates up to infinity. Successive decoding means to treat the signals of the users to be decoded subsequently as AWGN although code laws provide statistical dependencies between subse- quent symbols of the same user. Neglecting error propa- gation, the sum power of any pair of users which are neighbouring each other in the successive decoding chain satisfies P S = P 1 ( N S + P 2 ( N S ) ) + P 2 ( N S ) (3) where P 1 and P 2 denote the power of the first and the sec- ond user’s signal, respectively, while N S denotes the sum Int. J. Electron. Commun. (AE ¨ U) 55 (2000) No. 2, 1-3 1434-8411/2000/55/2-1 $15.00/0