Second-order description of complex random vectors - Statistical Signal Processing of Complex-Valued Data - page 82

ξ y

x ω

A − 1

M A

yy

xx

R xx R yy

K K

A −∗ M ∗ A yy

xx

ξ ∗

x ∗ y ∗ ω ∗

Figure 3.6 Two-channel model for complex ICA. The vertical arrows are labeled with the
cross-covariance matrix between the upper and lower lines (i.e., the complementary
covariance).

rather than general linear transformations wastes a considerable degree of freedom in
designing the blind inverse M # .
In this section, we demonstrate that, in the complex case, it can be possible to determine
M # using second-order information only. This was first shown by DeLathauwer and
DeMoor (2002 ) and independently discovered by Eriksson and Koivunen (2006 ). The
key insight in our demonstration is that the independence of the components of x means
that, up to simple scaling and permutation, x is already given in canonical coordinates.
The idea is then to exploit the invariance of circularity coefficients of x under the linear
mixing transformation M .
Theassumptionof independent components x implies that the covariance matrix R xx
and the complementary covariance matrix R xx are both diagonal. It is therefore easy to
compute canonical coordinates between x and x ∗ , denoted by

=

A xx x . In the strong

uncorrelating transform A xx =

xx is a diagonal scaling matrix, and F xx is
a permutation matrix that rearranges the canonical coordinates

F xx R − 1 / 2 , R − 1 / 2

xx

such that ξ 1 corresponds

to the largest circularity coefficient k 1 , ξ 2 to the second largest coefficient k 2 , and so on.

This makes the strong uncorrelating transform A xx monomial. As a consequence,

also

has independent components .
The mixture y has covariance matrix R yy =

MR xx M H and complementary covari-

ance matrix R yy = MR xx M T . The canonical coordinates of y and y ∗ are computed as

A yy y ∗ . The strong uncorrelating transform A yy is
determined as explained in Section 3.2.2 .
Figure 3.6 shows the connection between the different coordinate systems. The impor-
tant observation is that

F yy R − 1 / 2 ∗ =

= A yy y = y , and

yy

are both in canonical coordinates with the same circularity
coefficients k i . In the next paragraph, we will show that

and

and are related as = D

by a diagonal matrix D with diagonal entries ± 1, provided that all circularity coefficients

are distinct. Since has independent components, so does . Hence, we have a solution

to the ICA problem.

Result 3.9. The strong uncorrelating transform A yy is a separating matrix for the com-
plex linear ICA problem if all circularity coefficients are distinct.

The only thing left to show is that D = A yy MA − 1 is indeed diagonal with diagonal

xx

elements ± 1. Since and are both in canonical coordinates with the same diagonal