Estimation - Statistical Signal Processing of Complex-Valued Data - page 130

Equivalently,

( R xy − R xy R −∗

yy R yy ) P − 1

( R xy − R xy R − 1

yy R yy ) P −∗ yy y ∗ .

x =

yy y +

(5.58)

yy R yy is the error covariance
matrix for linearly estimating y from y ∗ . The augmented error covariance matrix Q of
the error vector e

R yy − R yy R −∗ In this equation, the Schur complement P yy =

= x − x is

E [ e e H ] R xy R − 1

yy R xy .

Q = = R xx − (5.59)

A competing estimator x = W y will produce an augmented error e = x − x with

covariance matrix

E e e H Q = W ) R yy ( W W ) H

= + − − ,

Q ( W (5.60)

Q . As a consequence, all real-valued increasing functions of Q
are minimized, in particular,

which shows that Q ≤

2

tr Q = e

2

E e = tr Q ≤ E , (5.61)

det Q .

det Q ≤ (5.62)

These statements hold for the error vector e as well as the augmented error vector e
because

Q ⇒ Q .

≤ ≤

Q Q (5.63)

= x −

The error covariance matrix Q of the error vector e x is the northwest block of

the augmented error covariance matrix Q , which can be evaluated as

( R xy − R xy R −∗

yy R yy ) P − 1

( R xy −

yy R yy ) P −∗ yy R xy .

(5.64)

E [ ee H ]

yy R xy −

R xy R − 1 Q = = R xx −

A particular choice for a generally suboptimum filter is the LMMSE filter

R xy R − 1

yy 0
0 xy R −∗

W = ⇐⇒ W = R xy R − 1

yy ,

(5.65)

yy

which ignores complementary covariance matrices. We will examine the relation between
LMMSE and WLMMSE filters in the following subsections.

5.4.1 Special cases

If the signal x is real ,wehave R xy = R xy . This leads to the simplified expression

2Re ( R xy − R xy R −∗

yy y yy R yy ) P − 1

x = . (5.66)

While the WLMMSE estimate of a real signal is always real, the LMMSE estimate is
generally complex.

Result 5.4. The WLMMSE and LMMSE estimates are identical if and only if the error
of the LMMSE estimate is orthogonal to y ∗ , i.e.,

yy R yy − R xy =

( W y y ∗ ⇐⇒ R xy R − 1

− x ) ⊥ 0 . (5.67)