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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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