conSistent EStimation and lArge random MatricEs – SESAME

The project addresses estimation problems in the fields of signal processing and wireless communications.
We consider the frequent case where the observed signal yn corresponds to a multivariate time series
of dimension M and where one needs a certain function of the covariance matrix RM = E(ynyH
n ) of
the received signal. The standard estimators are generally based on the fact that the “true” unknown
covariance matrix RM can be replaced without much error with the empirical covariance matrix defined
as
ˆR
M =
1
N
XN
n=1
ynyH
n ,
where N is the number of available observations. Unfortunately, this error is small only in the case where
the dimension M of the observed vector is small compared to the number N of available observations.
Frequently, this assumption happens to be restrictive. In many practical cases, M and N are indeed of
the same order. Our aim is to develop robust estimation tools in this context. We furthermore focus on
the case where the dimensions M and N are “large”. This assumption is realistic in a large number of
situations in wireless communications such as multiple access techniques with a large number of users,
antenna arrays, etc. In these cases, it is possible to develop relevant estimators based on Random Matrix
Theory (RMT).
When M and N are both large and of the same order of magnitude, it is pertinent to consider the
asymptotic regime where N ! +1, M ! +1, et M/N ! where is a positive constant. In this
regime, classical estimators are not consistent. An approach initiated by Girko and developed further by
Mestre and Rubio provides consistent estimators of functionals of the eigenvalues of RM from ˆRM, as
well as functions related to the eigenvectors of RM. Nevertheless, an important theoretical and practical
work is still needed to build estimators based on this approach for more and more general random matrix
models, and to analyze their performance.
In a first step, one has to explore completely the existing RMT mathematical tools needed for building
these estimators and conducting their performance analysis.
In a second step, many specific problems related to the Code Division Multiple Access (CDMA)
and Multi-Carrier CDMA techniques will be explored. In the framework of applications related to
metrology or to the so called cognitive radio, a receiver has to probe the radio network and to passively
estimate certain parameters, without any knowledge of the number of active users, neither their spreading
sequences, nor their powers. In such a context, the aim is to analyze i) estimators of the number of active
users, ii) estimators of their power distribution, or iii) estimators of performance indexes of the receiver
in case he connects to the network.
In a third step, we tackle certain estimation problems related to the “Information plus Noise” models.
In these cases, matrix Y = (y1, . . . , yN) is assumed to be written as Y = X + V, where X is a random
or deterministic matrix, and V is a random perturbation matrix independent of X. Our purpose is
to build and analyze consistent estimators of certain parameters related to matrix 1
NXXH by relying
on the observation Y only. This type of problems has been rarely considered in the literature. Various
applications motivate the study of this problem. Among these, one can cite the estimation of Directions of
Arrivals by subspace methods in the case of correlated source signals, the estimation of certain functionals
of the radio channel when the latter is partially known, or the implementation of linear receivers robust
to errors in channel knowledge.
Finally, we propose to study problems mathematically close to the ones described above but specifically
related to time series analysis. Two kinds of problems will be considered:
? Analysis of performance of estimated longWiener filters: it is assumed that the observation (yn)n2Z
is a scalar (or low dimension multivariate) time series defined as a noisy filtered version of some
scalar (or low dimension multivariate) white noise sequence (sn)n2Z, that is
yn = [h(z)]sn + vn
where v is a white Gaussian noise independent of s, and [h(z)]sn represents the output of filter h(z)
driven by s at time n. Signals y and s are observed over the interval [1,M +N], and the addressed
problem is the estimation of the finite impulse response (FIR) f = [f0 f1 · · · fM-1] minimizing the
mean-square error of E|sn -fyn|2 in the case where M and N are of the same order of magnitude.
? Analysis of subspace methods concerning time series: subspace frequency estimation of noisy harmonic
signals, and subspace channel estimation of noisy single input / multiple outputs systems.