Supplementary Mathematical Arguments - Electromagnetic Brain Imaging - page 245

Note that proper complexGaussian randomvectorswith zeromean are called circular.
In this section, we assume that the circularity holds for z . Under the assumption that
z is a proper complex random vector, we derive

E

T

2 ʣ xx + ʣ yx ,

ʣ zz = ( x + i y )( x − i y ) = i (C.14)

ʣ − 1
zz

is found to be 4

and

2 − 1 I

xx

1

ʣ − 1
zz

ʣ yx ʣ − 1

= − i , (C.15)

where

= ʣ xx + ʣ yx ʣ − 1

xx ʣ yx .

(C.16)

| ʣ zz | | ʣ ˆˆ |

We first derive the relationship between the determinants and .Using

Eq. ( C.14 ), we have

2

yx

2 ʣ xx − ʣ yx ,

T T

ʣ

zz =

ʣ xx + i ʣ = i

ʣ − 1
zz

and is expressed as

2 − 1 ʣ xx −

ʣ yx ʣ − 1

1 1

4 − 1

ʣ − 1
zz

T

zz ʣ − 1

= i = ʣ

xx . xx

Taking the determinant of both sides of the equation above, we get

1
4 N | | − 1

| ʣ zz | − 1 | ʣ zz | | ʣ xx | − 1 = ,

and finally,

2 N | ʣ zz |= | ʣ xx || | . (C.17)

On the other hand, using the determinant identity in Eq. ( C.94 ), | ʣ ˆˆ | is expressed

as

=| ʣ xx || ʣ yy − ʣ yx ʣ − 1

T
yx
ʣ yx ʣ yy

ʣ xx ʣ

T

| ʣ ˆˆ |=

xx ʣ yx |=| ʣ xx || | ,

(C.18)

where the relationships in Eq. ( C.13 ) are used to obtain the right-most expression.
Comparing Eqs. ( C.17 ) and ( C.18 ), we finally obtain the relationship,

2 N | ʣ ˆˆ | .

| ʣ zz |= (C.19)

4 It is easy to see that

ʣ − 1
zz

ʣ = I holds. zz