Channel capacity - Signal Processing and Optimization for Transceiver Systems - page 232

For the special case of a scalar complex Gaussian x ,if x − m x is circularly

symmetric, the entropy of x is therefore

( f x )=ln( πeσ x ) , H (6 . 67)

where σ x m x | 2 ] . In this case x = x re + jx im , where the real and

imaginary parts are independent Gaussians with variance σ x / 2. The real and
imaginary parts each have an entropy [Cover and Thomas, 1991]

= E [ | x −

1

2 ln(2 πeσ re )= 1 2 ln( πeσ x ) .

(6 . 68)

Thus the entropy of the complex circularly symmetric Gaussian x is the sum of
the entropies of the real and imaginary parts. We can also use log 2 instead of ln
in the above expressions. If we use log

the entropy is in bits, whereas if we use

2

ln then the entropy is in nats.

6.6.5 Relation to other definitions

The definition for circularly symmetric complex random vectors is given by Eqs.
(6.35)-(6.36). Let us refer to this as Definition 1 . This is different from the
definition given in Tse and Viswanath [2005] which says that x is circularly

symmetric if e jθ x has the same pdf as x ; let us refer to this as Definition 2 .We

now make a number of observations.

1. Definition 2 implies in particular that E [ e jθ

x ]= E [ x ] for all θ. That is,
e jθ E [ x ]= E [ x ] for all θ, which implies E [ x ]= 0 . But Definition 1 (used in
this topic) does not imply zero mean, as seen from Example 6.4 below. So
the two definitions are not equivalent. Even for zero-mean random vectors,
the definitions are not equivalent because Definition 2 restricts the entire
pdf instead of just the second-order moment.

2. If Definition 2 holds then so does Definition 1 because if the pdf of x is iden-
tical to that of e jθ x then the pseudocorrelations ought to be unchanged,

that is,

T ]= e 2 jθ E [ xx T ]

E [ xx (6 . 69)

T ]= 0 , which in turn is equivalent to Definition
1 (Sec. 6.6.1). Thus Definition 2 implies Definition 1.

for any θ. This implies E [ xx

3. For the case of zero-mean Gaussian x , it can be shown that the converse
is also true, that is, Definition 1 does imply Definition 2. This is because
a zero-mean Gaussian satisfying Definition 1 has a pdf of the form

1

det( π C xx ) e − x † C − 1

x

f ( x )= (6 . 70)

xx

as explained in Sec. 6.6.3. If we replace x with e jθ

x then the covariance
C xx does not change, nor does the quadratic form x † C − 1

xx x . So f ( x )is
unchanged as demanded by Definition 2. So in the zero-mean Gaussian
case the two definitions are equivalent.