ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 90

property instead of a method that does not. For example, the generalized uncorrelating
transform (GUT) method [47] that is explicitly designed for blind separation of non-
circular sources has, in general, better performance in such cases than a method that
does not exploit the noncircularity aspect of the sources. Uncertainties related to
system matrix, for example, departures from assumed sensor array geometry and
related robust estimation procedures are not considered in this chapter.

2.1.2 Outline of the Chapter

This chapter is organized as follows. First, key statistics that are used in describing
properties of complex-valued random vectors are presented in Section 2.2. Essential
statistics used in this chapter in the characterization of complex random vectors
are the circular symmetry, covariance matrix, pseudo-covariance matrix, the
strong-uncorrelating transform and the circularity coefficients. The information con-
tained in these statistics can be exploited in designing optimal array processors. In
Section 2.3, the class of complex elliptically symmetric (CES) distributions [46] are
reviewed. CES distributions constitute a flexible, broad class of distributions that
can model both circular / noncircular and heavy- / light-tailed complex random
phenomena. It includes the commonly used circular complex normal (CN) distribution
as a special case. We also introduce an adjusted generalized likelihood ratio test
(GLRT) that can be used for testing circularity when sampling from CES distributions
with finite fourth-order moments [40]. This test statistic is shown to be a function of
circularity coefficients.
In Section 2.4, tools to compare statistical robustness and statistical efficiency of the
estimators are discussed. Special emphasis is put on the concept of influence function
(IF) of a statistical functional. IF describes the qualitative robustness of an estimator.
Intuitively, qualitative robustness means that the impact of errors to the performance
of the estimator is bounded and small changes in the data cause only small changes
in the estimates. More explicitly IF measures the sensitivity of the functional to
small amounts of contamination in the distribution. It can also be used to calculate
the asymptotic covariance structure of the estimator. In Section 2.5, the important
concepts of (spatial) scatter matrix and (spatial) pseudo-scatter matrix are defined
and examples of such matrix functionals are given. These matrices will be used in
developing robust array processors and blind separation techniques that work reliably
for both circular / noncircular and Gaussian / non-Gaussian environments. Special
emphasis is put on one particularly important class of scatter matrices, called the
M -estimators of scatter, that generalize the ML-estimators of scatter matrix parameters
of circular CES distributions. Then, in Section 2.6, it is demonstrated how scatter and
pseudo-scatter matrices can be used in designing robust beamforming and subspace
based DOA estimation methods. Also, a subspace DOA estimation method [13]
designed for noncircular sources is discussed. In Section 2.7, we derive the IF of
the conventional minimum variance distortionless response (MVDR) beamformer
and compare it with the IF of MVDR beamformer employing a robust M -estimator
of scatter in place of the conventional covariance matrix. The derived IF of a conven-
tional MVDR beamformer reveals its vulnerability to outliers. IF is further used to
compute the asymptotic variances and statistical efficiencies of

the MVDR