Estimation - Statistical Signal Processing of Complex-Valued Data - page 146

Thus, a general complex-valued quadratic form can be written as

2 y H H y 2 [ y H H 11 y

y T H 11 y ∗ + y T H 21 y y H H 12 y ∗ ]

1 1

x = = + +

y H H 11 y

1

2 [ y T H 21 y

y H H 12 y ∗ ]

= + + . (5.145)

Combining our results so far, the complex-valued output of a WLQ system can be
expressed as

1

2 g H y

1

2 y H H y

x = c + +

1

2 [ g 1 y

g 2 y ∗ ] y H H 11 y

1

2 [ y T H 21 y

y H H 12 y ∗ ]

= c + + + + + , (5.146)

C 2 m constitutes the widely

where c is a complex constant, and g H y with g H [ g 1 , g 2 ]

= ∈

linear part and y H H y the widely quadratic part.
If x is real, then g 1 = C 2 m

∗

g 2 , i.e., g H g 1 = [ g 1 , g 1 ] = ∈ has the structure of an

H 11 , H 21 = H 12 , i.e., H has the structure of an augmented

augmented vector, and H 11 =

matrix (where N 1 = H 11 and N 2 = H 12 ). Thus, for real x ,

1

2 g 1 y

1

2 y H N y

x = u = c + +

Re( g 1 y ) y H N 1 y Re( y H N 2 y ∗ ) = c + + + . (5.147)

H ] to be a vector in a linear space, 8 We may consider a pair [ g , with addition defined by

[ g , H ] g , H ] [ g , H ] + = [ g + H + (5.148)

and multiplication by a complex scalar a by

a [ g , H ] = [ a g , a H ] . (5.149)

The inner product in this space is defined by
[ g

H ] =

[ g , g H g + tr( H H H ) , H ] , . (5.150)

5.7.2 WLQMMSE estimation

We would now like to estimate a scalar complex signal x : −→ C from a measurement

C m , using the WLQ estimator y : −→

g H y y H H y x = c + + . (5.151)

C 2 m For convenience, we have absorbed the factors of 1 / 2in( 5.146 )into g ∈ and

C 2 m × 2 m . We derive the WLQ estimator as a fairly straightforward extension of
Picinbono and Duvaut (1988 ) to the complex noncircular case, which avoids the use of
tensor notation. We assume that both x and y have zero mean. In order to ensure that x
has zero mean as well, we need to choose c

H ∈

=−

tr ( R yy H ) so that

g H y y H H y

x = + − tr ( R yy H ) . (5.152)

Using the definition of the inner product ( 5.150 ), the estimator is

= [ g R yy ] . H H ] y y H

x , , [ y , − (5.153)