Detection - Statistical Signal Processing of Complex-Valued Data - page 187

So what performance advantage does widely linear processing offer over linear pro-
cessing? Answering this question proceeds along the lines of Section 5.4.2 . The perfor-
mance criterion we choose here is the deflection d . Schreier et al . (2005 ) computed the
deflection for the improper signal-plus-white-noise detection scenario:

2 n 2

i
λ i +

λ

N 0

i = 1

d =

2 .

(7.36)

2 n

λ i
λ i +

2 N 0

N 0

i = 1

2 n
i

Here,

1 are the eigenvalues of the augmented covariance matrix R xx . In order
to evaluate the maximum performance advantage of widely linear over linear pro-
cessing, we need to maximize the deflection for fixed R xx and varying R xx .Using
Result A3.3 , it can be shown that the deflection is a Schur-convex function of the
eigenvalues

{ λ i }

=

{ λ i } . Therefore, maximizing the deflection requires maximum spread

of the { λ i } in the sense of majorization. According to Result 3.7 , this is achieved

for

λ i = µ i , = ,..., , λ i = , = + ,..., ,

2 i 1 n and 0 i n 1 2 n (7.37)

n

where

i = 1 are the eigenvalues of the Hermitian covariance matrix R xx . By plug-
ging this into ( 7.36 ), we obtain the maximum deflection achieved by widely linear
processing,

{ µ i }

2 n

2

i

µ

2 µ i + N 0

i = 1

max d =

2 .

(7.38)

n

µ i

R xx

N 0

2 µ i + N 0

i = 1

On the other hand, linear processing implicitly assumes that R xx = 0 , in which case the

deflection is

n 2

i
µ i +

µ

N 0

i = 1

d

R xx = 0 =

2 .

(7.39)

n

µ i
µ i +

N 0

N 0

i = 1

Thus, the maximum performance advantage, as measured by deflection, is

max

R xx d

0 =

,

max
N 0

2 (7.40)

d

R xx =

which is attained for N 0 −→ . This bound was derived by Schreier et al .

(2005 ) . We note that, if ( 7.37 ) is not satisfied, the maximum performance advantage,
which is then less than a factor of 2, occurs for some noise level N 0 >

0or N 0 −→ ∞

0.