Second-order description of complex random vectors - Statistical Signal Processing of Complex-Valued Data - page 81

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

P FA P FA

(a) (b)

Figure 3.5 Receiver operating characteristics of the GLRT detector. In (a), the SNR is fixed at
0 dB. From northwest to southeast, the curves correspond to phase tracking-error variance of
0.7, 0.95, 1.2, and 1.5. In (b), the phase tracking-error variance is fixed at 1. From northwest to
southeast, the curves correspond to SNR of 5 dB, 0 dB, and − 5 dB. In all cases, the number of
samples was M = 1000.

3.5 Independent component analysis

An interesting application of the invariance property of the circularity coefficients is
independent component analysis (ICA). In ICA, we observe a linear mixture y of
independent complex components (sources) x , as described by

y = Mx . (3.100)

We will make a few simplifying assumptions in this section. The dimensions of y
and x are assumed to be equal, and the mixing matrix M is assumed to be non-
singular. The objective is to blindly recover the sources x from the observations y ,
without knowledge of M , using a linear transformation M # . This transformation M #
can be regarded as a blind inverse of M , which is usually called a separating matrix .
Note that, since the model ( 3.100 ) is linear, it is unnecessary to consider widely linear
transformations.
ICA seeks to determine independent components. Arbitrary scaling of x , i.e., multi-
plication by a diagonal matrix, and reordering the components of x , i.e., multiplication
by a permutation matrix, preserves the independence of its components. The product
of a diagonal and a permutation matrix is a monomial matrix, which has exactly one
nonzero entry in each column and row. Hence, we can determine M # up to multiplication
with a monomial matrix.
Standard ICA requires the use of higher-order statistical information, and the blind
recovery of x cannot work if more than one source x i is Gaussian. If only second-order
information is available, the best possible solution is to decorrelate the components,
rather than to make them independent. This is done by determining the principal com-
ponents U H y using the EVD R yy =

E yy H UΛU H . However, the restriction to unitary

=