Sparsely-spread CDMA - a statistical mechanics based analysis
📝 Abstract
Sparse Code Division Multiple Access (CDMA), a variation on the standard CDMA method in which the spreading (signature) matrix contains only a relatively small number of non-zero elements, is presented and analysed using methods of statistical physics. The analysis provides results on the performance of maximum likelihood decoding for sparse spreading codes in the large system limit. We present results for both cases of regular and irregular spreading matrices for the binary additive white Gaussian noise channel (BIAWGN) with a comparison to the canonical (dense) random spreading code.
💡 Analysis
Sparse Code Division Multiple Access (CDMA), a variation on the standard CDMA method in which the spreading (signature) matrix contains only a relatively small number of non-zero elements, is presented and analysed using methods of statistical physics. The analysis provides results on the performance of maximum likelihood decoding for sparse spreading codes in the large system limit. We present results for both cases of regular and irregular spreading matrices for the binary additive white Gaussian noise channel (BIAWGN) with a comparison to the canonical (dense) random spreading code.
📄 Content
The area of multiuser communications is one of great interest from both theoretical and engineering perspectives [1]. Code Division Multiple Access (CDMA) is a particular method for allowing multiple users to access channel resources in an efficient and robust manner, and plays an important role in the current preferred standards for allocating channel resources in wireless communications. CDMA utilises channel resources highly efficiently by allowing many users to transmit on much of the bandwidth simultaneously, each transmission being encoded with a user specific signature code. Disentangling the information in the channel is possible by using the properties of these codes and much of the focus in CDMA research is on developing efficient codes and decoding methods.
In this paper we study a variant of the original method, sparse CDMA, where the spreading matrix contains only a relatively small number of non-zero elements as was originally studied and motivated in [2]. While the straightforward application of sparse CDMA techniques to uplink multiple access communication is rather limited, as it is difficult to synchronise the sparse transmissions from the various users, the method can be highly useful for frequency and time hopping. In frequency-hopping code division multiple access (FH-CDMA), one repeatedly switches frequencies during radio transmission, often to minimize the effectiveness of interception or jamming of telecommunications. At any given time step, each user occupies a small (finite) number of the (infinite) M-ary frequency-shift-keying (MFSK) chip/carrier pairs (with gain G, the total number of chipfrequency pairs is MG.) Hops between available frequencies can be either random or preplanned and take place after the transmission of data on a narrow frequency band. In time-hopping (TH-)CDMA, a pseudo-noise sequence defines the transmission moment for the various users, which can be viewed as sparse CDMA when used in an ultra-wideband impulse communication system. In this case the sparse time-hopping sequences reduces collisions between transmissions.
This study follows the seminal paper of Tanaka [3], and other recent extensions [4], in utilising the replica analysis for randomly spread CDMA with discrete inputs, which established many of the properties of random densely-spread CDMA with respect to several different detectors including Maximum A Posteriori (MAP), Marginal Posterior Maximiser (MPM) and minimum mean square-error (MMSE). Sparsely-spread CDMA differs from the conventional CDMA, based on dense spreading sequences, in that any user only transmits to a small number of chips (by comparison to transmission on all chips in the case of dense CDMA). The sparse nature of this model facilitates the use of methods from statistical physics of dilute disordered systems [5,6] for studying the properties of typical cases.
The feasibility of sparse CDMA for transmitting information was recently demonstrated [2] for the case of real (Gaussian distributed) input symbols by employing a Gaussian effective medium approximation; several results have been reported for the case of random transmission patterns. In a separate recent study, based on the belief propagation inference algorithm and a binary input prior distribution, sparse CDMA has also been considered as a route to proving results in the densely spread CDMA [7]. In addition, this study demonstrated the existence of a waterfall phenomenon comparable to the dense code for a subset of ensembles. The waterfall phenomenon is observed in decoding techniques, where there is a dynamical transition between two statistically distinct solutions as the noise parameter is varied. Finally we note a number of pertinent studies concerning the effectiveness of belief propagation as an MPM decoding method [8,9,10,11], and in combining sparse encoding (LDPC) methods with CDMA [12]. Many of these papers however consider the extreme dilution regime -in which the number of chip contributions is large but not O(N).
The theoretical work regarding sparsely spread CDMA remained lacking in certain respects. As pointed out in [2], spreading codes with Poisson distributed number of nonzero elements, per chip and across users, are systematically failing in that each user has some probability of not contributing to any chips (transmitting no information). Even in the “partly regular” code [7] ensemble (where each user transmits on the same number of chips) some chips have no contributors owing to the Poisson distribution in chip connectivity, consequently the bandwidth is not effectively utilised. We circumvent this problem by introducing constraints to prevent this, namely taking regular signature codes constrained such that both the number of users per chip and chips per user take fixed integer values. Furthermore we present analytic and numerical analysis without resort to Gaussian approximations of any quantities. Using new tools from statistical mechanics we are
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