ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 132

implies (due to (2.41)) that sources have distinct values of circular kurtosis, that is, k 0
( s i ) = k 0 ( s j ) for all i = j [ (1, ... , k ).
For more properties of DOGMA functionals, see [49], where also alternative for-
mulations of the method are derived along with efficient computational approach to
compute the estimator.

2.8.2 The Class of GUT Estimators
Let C ( . ) denote any scatter matrix functional and P ( . ) denote any spatial pseudo-scatter
matrix functional with IC-property (i.e., they reduce to diagonal matrices when F is a
cdf of a random vector with independent components). As already mentioned, the
covariance matrix is an example of a scatter matrix that possesses IC-property.
Pseudo-kurtosis matrix P kur ( . ) defined in (2.21) is an example of a spatial pseudo-
scatter matrix that possesses IC-property. Sign pseudo-covariance matrix P sgn ( . )
for example do not necessarily possess IC-property. However, as mentioned earlier,
for symmetric independent sources, any scatter or spatial pseudo-scatter matrix auto-
matically possesses IC-property. Again if the sources are not symmetric, a symmetri-
cized version of any scatter or spatial pseudo-scatter matrix can be easily constructed
that automatically possesses IC-property, see [47, 49] for details.

kk

k is called the
Generalized Uncorrelating Transform (GUT) if transformed data s ¼ Wz satisfies

Definition 5 Matrix functional W¼W ( z ) [ C of z [ C

C ( s ) ¼ I and P ( s ) ¼ L (2 : 42)

where L¼L ( s ) ¼ diag (l i ) is a real nonnegative diagonal matrix, called the circular-
ity matrix, and l i ¼ [ P ( s )] ii 0 is called the ith circularity coefficient, i ¼ 1 , ..., k .

The GUT matrix with choices C ¼ C and P ¼ P corresponds to the SUT [21, 22]
described in Section 2.2.2. Essentially, GUT matrix W ( . ) is a data transformation
that jointly diagonalizes the selected scatter and spatial pseudo-scatter matrix of the
transformed data s ¼ Wz . Note that the pseudo-covariance matrix employed by
SUT is a pseudo-scatter matrix, whereas in Definition 5, we only require C ( . )tobe
a spatial pseudo-scatter matrix.

GUT algorithm

(a) Calculate the square-root matrix B ( z )of C ( z ) 2 1 ,so B ( z ) H B ( z ) ¼ C ( z ) 2 1 , and
the whitened data v ¼ B ( z ) z (so C ( v ) ¼ I ).
(b) Calculate Takagi's factorization (symmetric SVD) of P ( . ) for the whitened
data v

P ( v ) ¼ ULU T (2 : 43)