ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 111

is a weight function that depends on the density generator g ( ) of the underlying cir-
cular CES distribution. For the CN distribution (i.e., when g ( d ) ¼ exp ( d )), we
have that w ml ; 1, which yields the SCM C as the MLE of S . The MLE for T k , n dis-
tribution (cf. Example 2.2), labeled MLT( n ), is obtained with

2 k þ n
nþ 2 x :

w ml ( x ) ¼ (2 : 24)

Note that MLT(1) is the highly robust estimator corresponding to MLE of S for the
complex circular Cauchy distribution, and that MLT( n ) !C (as T k , n !F k )as
n !1 , thus the robustness of MLT( n ) estimators decrease with increasing values
of n .
We generalize (2.22), by defining the M -estimator of scatter, denoted by C w , as the
choice of C [ PDH( k ) that solves the estimating equation

n X

n

1

w ( z i C 1 z i ) z i z i

C ¼ (2 : 25)

i¼ 1

where w is any real-valued weight function on [0, 1 ). Hence M -estimators constitute a
wide class of scatter matrix estimators that include the MLE's for circular CES distri-
butions as important special cases. M -estimators can be calculated by a simple iterative
algorithm described later in this section.
The theoretical (population) counterpart, the M -functional of scatter, denoted by
C w ( z ), is defined analogously as the solution of an implicit equation

C w ( z ) ¼ E [ w ( z H C w ( z ) z ) zz H ] :

(2 : 26)

Observe that (2.26) reduces to (2.25) when F is the empirical distribution F n , that is,
the solution C w of (2.25) is the natural plug-in estimator C w ( F n ). It is easy to show that
the M -functional of scatter is equivariant under invertible linear transformation of the
data in the sense required from a scatter matrix. Due to equivariance, C w ( F S ) ¼ s w S ,
that is, the M -functional is proportional to the parameter S at F S , where the positive
real-valued scalar factor s w ¼ s w ( d ) may be found by solving

E [ w ( d=s w ) d=s w ] ¼ k

(2 : 27)

where d has density (2.12). Often s w need to be solved numerically from (2.27) but in
some cases an analytic expression can be derived. Since parameter S is proportional to
underlying covariance matrix C ( F S ), we conclude that the M -functional of scatter is
also proportional to the covariance matrix provided it exists ( i.e. , g [ G

1 ). In many
applications in sensor array processing, covariance matrix is required only up to a
constant scalar (see e.g. Section 2.7), and hence M -functionals can be used to
define a robust class of array processors.