Channel capacity - Signal Processing and Optimization for Transceiver Systems - page 227

Proof. The pseudocorrelation of y is

T ]= E [ xx T ]+ E [ qq T ]+ E [ xq T ]+ E [ qx T ] .

E [ yy

Since x and q are circularly symmetric, the first two terms on the right-hand
side are zero. Next, the cross-term can be written as E [ xq

T ]
because of statistical independence. Using the further fact that E [ x ]or E [ q ]
is zero, we therefore get E [ yy

T ]= E [ x ] E [ q

T ]= 0 , proving that y is circularly symmetric

indeed.

Lemma 6.3. Algebraic identities from circular symmetry. Let C xx = P + j Q
have the inverse

♠

xx = P + j Q .

C − 1

(6 . 44)

Then the inverse of

C uu =0 . 5 P − Q

QP

is given by

uu =2 P − Q

.

C − 1 (6 . 45)

Q P

Furthermore

det [2 C uu ]=[det C xx ] 2 (6 . 46)

for any covariance C xx = P + j Q . ♦

Proof. To prove Eq. (6.45) observe that the equation ( P + j Q )( P + j Q )= I
implies

PP − QQ = I PQ + QP = 0 . and

Using this it follows that
P − Q
QP

P − Q
Q P

= I ,

which proves Eq. (6.45). To prove Eq. (6.46) first observe the identity
I j I
0I

P − Q
QP

I − = C xx 0

QC xx

.

j I
0I

Using the fact that [det C xx ] is real (because C xx is Hermitian), and the
fact that

det A0
CD

=det A det D

for square matrices A and D , Eq. (6.46) follows.