Multiuser Detection and Statistical Physics

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    Multiuser Detection and Statistical Physics

    Dongning Guo and Sergio Verdú Dept. of Electrical Engineering

    Princeton University

    Princeton, NJ 08544, USA

    email: {dguo,verdu}@princeton.edu August 26, 2002

    Abstract

    We present a framework for analyzing multiuser detectors in the context of statistical physics. A mul-

    tiuser detector is shown to be equivalent to a conditional mean estimator which finds the mean value of the

    stochastic output of a so-called Bayes retrochannel. The Bayes retrochannel is equivalent to a spin glass in the sense that the distribution of its stochastic output conditioned on the received signal is exactly the distri-

    bution of the spin glass at thermal equilibrium. In the large-system limit, the bit-error-rate of the multiuser

    detector is simply determined by the magnetization of the spin glass, which can be obtained using powerful

    tools developed in statistical mechanics. In particular, we derive the large-system uncoded bit-error-rate of 

    the matched filter, the MMSE detector, the decorrelator and the optimal detectors, as well as the spectral

    efficiency of the Gaussian CDMA channel. It is found that all users with different received energies share

    the same multiuser efficiency, which uniquely determines the performance of a multiuser detector. A uni-

    versal interpretation of multiuser detection relates the multiuser efficiency to the mean-square error of the

    conditional mean estimator output in the large-system limit.

    Index Terms:  Multiuser detection, statistical mechanics, code-division multiple access, spin glass, self-

    averaging property, free energy, replica method, multiuser efficiency.

    1 Introduction

    Multiuser detection is central to the fulfillment of the capabilities of code-division multiple access (CDMA),

    which is becoming the ubiquitous air-interface in future generation communication systems. In a CDMA

    system, all frequency and time resources are allocated to all users simultaneously. To distinguish between

    users, each user is assigned a user-specific spreading sequence on which the user’s information symbol is

    modulated before transmission. By selecting mutually orthogonal spreading sequences for all users, each

    user can be separated completely by matched filtering to one’s spreading sequence. It is not very realistic to

    maintain orthogonality in a mobile environment and hence multiple access interference (MAI) arises. The

    problem of demodulating in the presence of the MAI therefore becomes vital for a CDMA system.

    A variety of multiuser detectors [1] have been proposed to mitigate the MAI. The simplest one is the

    single-user matched filter, which totally ignores the existence of the MAI. Its performance is not very sat-

    isfactory and is particularly limited by the near-far problem. In the other extreme, the individually optimal (IO) and the jointly optimal (JO) detectors achieve the minimum probability of error but entail prohibitive

    complexity which is exponential in the number of users. A wide spectrum of multiuser detectors offer per-

    formance in between the matched filter and the optimal detectors with substantially reduced complexity. The

    most popular ones include the MMSE detector and the decorrelator. The performance of multiuser detectors

    has been studied extensively in the literature. A collection of results is found in [1]. In general, the perfor-

    mance is dependent on the spreading factor, the number of users, the received signal-to-noise ratios (SNR),

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    and the instantaneous spreading sequences. The dependence on this many parameters results in very complex

    expressions for all but the simplest case. Not only are these expressions hard to evaluate, but the complica-

    tion allows little useful insight into the detection problem. To eliminate the dependency by averaging over all

    spreading sequences (e.g. [2]) is plausible but usually a prohibitive task.

    Recently, it is found that performance analysis can be greatly simplified for random-spread systems the

    size of which tend to infinity [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Of special interest is the case where the number

    of users and the spreading factor both tend to infinity with their ratio fixed. This is referred to as the large- system limit in the literature. As far as linear multiuser detectors are concerned, an immediate advantage of 

    the large-system setting is that the multiple-access interference, as a sum of contributions from all interfering

    users, becomes Gaussian-like in distribution under mild conditions in the many-user limit [13]. This allows

    the output signal-to-interference ratio (SIR) and the uncoded bit-error-rate (BER) to be easily characterized

    for linear detectors such as the matched filter, the MMSE detector and the decorrelator. The large-system

    treatment also finds its success in deriving the capacity (or the spectral efficiency) of CDMA channels when

    certain multiuser detectors are used. Somewhat surprisingly, in all the above instances, the SIR, the BER

    and the spectral efficiency are independent of the spreading sequence assignment with probability 1. The

    underlying theory is that dependency on the spreading sequences vanishes in the large-system domain. In

    particular, the empirical eigenvalue distribution of a large random sequence correlation matrix converges to

    a deterministic distribution with probability 1 [14, 15, 16]. One may compare this to the concept of typical

    sequences in information theory [17]. We can say that for a sufficiently large system, almost all spreading

    sequence assignments are “typical” and lead to the (same) average performance. Unlike in the above, a more recent but quite different view of large CDMA systems is inspired by the

    successful analysis of certain error-control codes using methods developed in statistical mechanics [18, 19,

    20, 21, 22, 23, 24]. In [25] a CDMA system finds an equivalent spin glass similar to the Hopfield model. Here

    a spin glass is a statistical mechanical system consisting of a large number of interacting spins. In [26, 27],

    well-known multiuser detectors are expressed as Marginal-Posterior-Mode detectors, which can be embedded

    in a spin glass. The large-system performance of a multiuser detector is then found as a thermodynamic

    limit of a certain macroscopic property of the corresponding spin glass, which can be obtained by powerful

    techniques sharpened in statistical mechanics. In [28, 24], Tanaka analyzed the IO and the JO detectors,

    the decorrelator and the MMSE detector under the assumption that all users are received at the same energy

    (perfect power control). In addition to the rederivation of some previously known results, Tanaka found for

    the first time, the large-system BER of the optimal detectors at finite SNRs, assuming perfect power control.

    This new statistical mechanics approach to large systems emerges to be more fundamental. In fact, the

    convergence of the empirical eigenvalue distribution, which underlies many above-mentioned large-system

    results, can be proved in statistical mechanics [29, Chapter 1]. Deeply rooted in statistical physics, the new

    approach brings a fresh look into the decades-old multiuser detection problem. In this work, we present a

    systematic treatment of multiuser detection in the context of statistical mechanics based on [30, 31]. We

    introduce the concept of  Bayes retrochannel, which takes the multiaccess channel output as the input and

    generates a stochastic estimate of the originally transmitted data. The characteristic of the Bayes retrochan-

    nel is the posterior probability distribution under some postulated prior and conditional probability distribu-

    tions. A multiuser detector is equivalent to a  conditional mean estimator  which finds the expected value of 

    the stochastic output of the Bayes retrochannel. By carefully choosing the postulated prior and conditional

    probability distributions of the Bayes retrochannel, we can arrive at different multiuser detection optimality

    criteria. Importantly, the Bayes retrochannel is found to be equivalent to a spin glass with the spreading

    sequence assignment and the received signal as quenched randomness. That is, the conditional output distri-

    bution of the Bayes retrochannel is exactly the same as the distribution of the spin glass system at thermal

    equilibrium. Thus, in the large-system limit, the performance of the detector finds its counterpart as a cer-

    tain macroscopic property of the thermodynamic system, which can be obtained using the replica method

    developed in statistical mechanics.

    In particular, we present analytical results for the large-system BER of the matched filter, the decorrelator,

    the MMSE detector and the optimal detectors. We also find the spectral efficiency of the Gaussian CDMA

    channel, both with and without the constraint of binary inputs. Unlike in [28], we do not assume equal-energy

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    users in any case. It is found that the all users share the same multiuser efficiency, which is the solution to a

    fixed-point equation similar to the Ts

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