Introduction to complex random vectors and processes - Statistical Signal Processing of Complex-Valued Data - page 33

can be calculated as

H 2 H − 1 H 2 ) − 1 H 2 H − 1 H 2 ) − 1 H 2 H −∗ ( H 1 − − ( H 1 −

H − 1

1

=

H 2 H − 1 H 2 ) − 1 H 2 H − 1 H 2 H − 1 H 2 ) − 1

( H 1 − ( H 1 −

−

1

N 1 N 2
N 2 N 1

= = N ,

which has the block structure ( 2.9 ).

we often require that all factors
in matrix factorizations represent widely linear transformations. If that is the case, we
need to ensure that all factors satisfy the block pattern ( 2.9 ). If a factor H

When working with the augmented matrix algebra W

,it
does not represent a widely linear transformation, since applying H to an augmented
vector x would yield a vector whose last n entries are not the conjugate of the first n
entries.

∈ W

n × n
into a lower-triangular and upper-triangular factor. If we require that the Cholesky factors
be widely linear transformations, then H

Example 2.2. Consider the Cholesky factorization of a positive definite matrix H ∈ W

XX H with X lower triangular and X H upper

=

n × n . Instead, we determine the Cholesky

triangular will not work since generally X ∈ W

factorization of the equivalent real matrix

1

2 T H H T

LL H

= , (2.12)

1

2 TLT H . Then

and transform L into the augmented complex notation as N =

H = NN H

(2.13)

is the augmented complex representation of the Cholesky factorization of H with N ∈

n × n but N
itself is not generally lower triangular, and neither are its blocks N 1 and N 2 . An exception
is the block-diagonal case: if H is block-diagonal, N is block-diagonal and the diagonal
block N 1 (and N 1 ) is lower triangular.

n × n . Note that ( 2.13 ) simply reexpresses ( 2.12 ) in the augmented algebra

W W

2.1.2 Inner products and quadratic forms

[ a T b T ] T

Consider the two 2 n -dimensional real composite vectors w = , and z =

[ u T v T ] T , the corresponding n -dimensional complex vectors y

, = a + j b and x = u + j v ,

and their complex augmented descriptions y = Tw and x = Tz . We may now relate the

inner products defined on IR 2 n ,C 2 n
∗

, and C n as

Re y H x . w T z

1

2 y H x

= = (2.14)

Thus, the usual inner product w T z defined on IR 2 n

equals (up to a factor of 1 / 2) the

inner product y H x defined on C 2 n
∗

, and also the real part of the usual inner prod-
uct y H x defined on C n . In this topic, we will compute inner products on C 2 n
∗

as