ROBUST ESTIMATION TECHNIQUES FOR COMPLEX-VALUED RANDOM VECTORS - Adaptive Signal Processing: Next Generation Solutions - page 124

where g is the value of g ( )at F . The second part in the above IF expression can be
written in the form

C 1 IF( t ; g , F ) ¼C 1 a @

@1 [ a H

C 1 ( F 1 , t ) a ] 1

j 1¼ 0

¼C 1 a [ a H C 1 a ] 2 a H IF( t ; C 1 , F ) a

¼w C a H IF( t ; C 1 , F ) g:

Thus, the IF of w C ( ) can now be written as

IF( t ; w C , F ) ¼ [ Iw C a H ]IF( t ; C 1 , F ) g:

Using the IF expression (2.36) of the inverse of the covariance matrix shows that

IF( t ; C 1 , F ) g ¼ [ C 1 tt H C 1 þC 1 ] g

¼C 1 tt H w C þw C :

Thus the IF of w C ( ) can be written

IF( t ; w C , F ) ¼ [ Iw C a H ][ C 1 tt H w C þw C ] :

By noting that [ Iw C a H ] w C ¼ 0 (due to the MVDR gain constraint w H

a ¼ 1),

C

shows that

IF( t ; w C , F ) ¼ [ w C a H I ] C 1 tt H w C : (2 : 37)

This is a compact expression for the IF of w C ( ) that also neatly reveals the vulner-
ability of the conventional MVDR weight vector to outliers. Clearly, contamination
at a point t with large norm ktk has an effect proportional to ktk

2

the IF. We may

also rewrite the IF expression (2.37) as

IF( t ; w C , F ) ¼ r 2 [ w C a H I ] C 1 = 2 uu H

1 = 2 w C

C

C 1 t and u ¼C 1 = 2 t=r is a unit vector. This expression shows that the
norm of the IF grows linearly with r (since u remains bounded).
Let us now consider the case that the reference distribution F is a circular CES dis-
tribution F S ¼ CE k ( S , g ). In this case, since C ( F S ) ¼ s C S and w C ( F S ) ¼ w , the IF
expression can be written as follows in Theorem 3.

where r 2

¼ t H

Theorem 3 The IF of the conventional MVDR functional w C ( ) at a circular CES
distribution F S ¼ CE k ( S , g ) is given by

r 2
s C

[ wa H I ] S 1 = 2 uu H S 1 = 2 w IF( t ; w C , F ) ¼

¼ t H S 1 t, u ¼ S 1 = 2 t=r a unit vector and w¼Ga is defined in (2.35) . where r 2