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8/18/2019 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