1
Abstract
In this paper, exact expressions for the level crossing rate (LCR) and average fade duration (AFD) for two-branch selection, equal-gain and maximal-ratio combining systems in a Hoyt fading environment are presented. The expressions apply to unbalanced, non-identical, correlated diversity channels and have been validated by specializing the general results to some particular cases whose solutions are known. In passing, the joint bidimensional envelope-phase Hoyt distribution with arbitrary fading parameters is obtained.
Index Terms
Average fade duration, selection combining, equal-gain combining, maximal-ratio combining, Hoyt fading channels, level crossing rate.
I. I NTRODUCTION
Level crossing rate (LCR) and average fade duration (AFD) are widely-used performance measures of wireless diversity systems. However, although the branch signals may be correlated and non-identically distributed in practical systems [1]–[4], the literature on LCR and AFD of diversity techniques over non-identical correlated fading is rather scarce. Pioneering work on this issue was carried out by Adachi et al. [1] for dual branch selection (SC), equal-gain (EGC), and maximal ratio combining
(MRC) over balanced correlated Rayleigh channels. The unbalanced correlated Rayleigh case was addressed in [2] for two-, three-, and four-branch MRC. More recently, [3] presented a unified treatment for the LCR and the AFD of M -branch SC over unbalanced correlated Rayleigh, Ricean, and Nakagamim channels. In [4], the LCR and AFD for the MRC were derived for a correlated, unbalanced Nakagami environment.
This paper fully generalizes the approach used in [1], and provides expressions for the LCR and AFD of dual-branch SC,
EGC, and MRC operating over non-identical, correlated Hoyt (Nakagamiq ) channels.
This work is organized as follows: Section II derives the Joint Bidimensional Envelope-Phase Hoyt distribution; Section III presents general expressions for LCR and AFD of the combining output; Section IV-A derives the matrices for the conditional joint distribution; Section IV-B computes the means and variances for each diversity system; Section V shows some numerical plots, and finally Section VI draws some conclusions.
G. Fraidenraich was with the State University of Campinas, Campinas SP, 13083-970 Campinas, SP, Brazil. He is now with the Department of Electrical
Engineering at Stanford University (e-mail: gfraiden@stanford.edu). M. D. Yacoub, J. R. Mendes, and J. C. S. Santos Filho are with the Department of
Communications, State University of Campinas, 13083-970 Campinas, SP, Brazil (e-mail:[michel,candido]@decom.fee.unicamp.br).

2
II. T HE J OINT B IDIMENSIONAL E NVELOPE -P HASE H OYT DISTRIBUTION
In a Hoyt fading environment, the received signal at the i -th antenna ( i = 1 , 2 ), can be represented in a complex form as
X i
( t ) + jY i
( t ) = R i
( t ) exp( j Θ i
( t )) (1) where X i
( t ) , Y i
( t ) are zero mean independent Gaussian processes with variance σ 2
X i and σ 2
Y i
, respectively. The variates R i
( t ) and Θ i
( t ) follow the envelope and phase of the Hoyt distribution [5], respectively.
We now proceed to determine the joint distribution of X i
, X i
( t ) , Y i
, Y i
( t ) , ( i = 1 , 2 ). Defining the vector Z =
[ X
1
Y
1
X
2
Y
2
] = [ R
1 cos (Θ
1
) R
1 sin (Θ
1
) R
2 cos (Θ
2
) R
2 sin (Θ
2
)] , the joint Gaussian distribution p
Z
( z ) can be written as
[6] p
Z
( z ) =
1
(2 π )
2
(det b )
1 / 2 exp −
1
2 zb
− 1 z
T
(2) where:
(i) [ · ]
T denotes the transpose matrix, b is the covariance matrix given by

 b = 



σ 2
X
1
0
σ
X
1
σ
X
2
µ
12
− σ
X
1
σ
Y
2
η
12
0
σ 2
Y
1
σ
Y
1
σ
X
2
η
12
σ
Y
1
σ
Y
2
µ
12
σ
X
1
σ
X
2
µ
12
σ
Y
1
σ
X
2
η
12
σ 2
X
2
0

− σ
X
1
σ
Y
2
η
12
σ
Y
1
σ
Y
2
µ
12
0
σ 2
Y
2





(i i) µ ij
= µ ij
(0) and η ij
= η ij
(0) are the correlation coefficients, defined as
µ ij
( τ ) =
Cov ( X i
( t ) , X j
( t + τ )) p Var ( X i
( t )) Var ( X j
( t + τ ))
=
Cov ( Y i
( t ) , Y j
( t + τ )) p Var ( Y i
( t )) Var ( Y j
( t + τ ))
( i 6 j )
(3)
(4) and
η ij
( τ ) = −
Cov( X p
Var ( X i i
( t
( t
))
) , Y
Var j
(
( t
Y j
+
( t
τ ))
+ τ ))
= cov( Y i
( t ) , X j
( t + τ )) p
Var ( Y i
( t )) Var ( X j
( t + τ ))
( i = j ) (5) and
η ii
( τ ) =
Cov( X i
( t ) , Y i
( t + τ )) p
Var ( X i
( t )) Var ( Y i
( t + τ ))
( i = 1 , 2) (6)
Var( · ) and Cov( · ) are the variance and covariance operators, respectively. For Gaussian processes, the following relations are valid: µ
22
= µ
11
, µ
21
= µ
12
, η
21
= − η
12
, and η
22
= η
11
[7]. The joint density p
R
1
,R
2
, Θ
1
, Θ
2
( r
1
, r
2
, θ
1
, θ
2
) can be written as p
R
1
,R
2
, Θ
1
, Θ
2
( r
1
, r
2
, θ
1
, θ
2
) = | J | p
Z
( z ) , where | J | = r
1 r
2 is the Jacobian of the transformation. Accordingly, the joint bidimensional envelope-phase Hoyt distribution, as derived here, is given by p
R
1
,R
2
, Θ
1
, Θ
2
( r
1
, r
2
, θ
1
, θ
2
) = r
1 r
2 e
−
2(1
1
−
ρ
2)
" r
2
1 sin2( θ 1)
σ
2
Y
1
+ cos2( θ 1)
σ
2
X
1
!
+ r
2
2 sin2( θ 2)
σ
2
Y
2
+ cos2( θ 2)
σ
2
X
2
!#
× e
4 π 2 (1 − ρ 2 ) σ
X
1
σ
Y
1
σ
X
2
σ
Y
2
( r
1
1
− r
2
ρ
2 )
»
µ
12
„ sin( θ
1) sin(
σY
1
σY
2
θ
2) + cos( θ
1) cos(
θ
σX
1
σX
2
2)
«
+ η
12
„ sin( θ
1) cos(
θ
2)
σY
1
σX
2
+ cos( θ
1) sin(
θ
σX
1
σY
2
2)
«–
(7)

3 where
ρ 2 = µ 2
12
+ η 2
12
(8)
III. LCR AND AFD
The LCR n
R
( r ) and AFD T
R
( r ) of a random signal are defined, respectively, as the average number of upward (or downward) crossings per second at a given level and as the mean time the signal remains below this level after crossing it in the downward direction. The LCR and AFD of the combiner output R = R ( t ) at level r are, respectively, given by [8] n
R
( r ) =
T
R
( r ) =
Z
∞
0
P
R
( r ) n
R
( r )
R,
˙
( r, ˙ ) d r (9)
(10) where p
R,
˙
( · , · ) is the joint probability density function (JPDF) of R and its time derivative
˙
, and P
R
( · ) is the cumulative distribution function (CDF) of R . In the following, (9) and (10) shall be calculated for the dual-branch, correlated, non-identical
Hoyt fading environment using the SC, EGC and MRC techniques.
A. Diversity Systems
The output envelope and its time derivative for the SC, EGC and MRC combining systems are given, respectively, by

R =









 max { R
R
1
+
2
R
2
1
, R p R 2
1
+ R 2
2
2
} SC
EGC
MRC

˙
=
















˙
1
R
1
> R
2
R
2
R
1
< R
2
˙
1 √
2
R
2
R √ 1
+ R
2
R 2
1
R
2
+ R 2
2
SC
EGC
MRC
(11)
Given Θ i
, it is known that i is zero-mean Gaussian [9], therefore it is clear from (11) that the conditional density p
˙ | R
1
,R
2
, Θ
1
, Θ
2
( ˙ | r
1
, r
2
, θ
1
, θ
2
) is also Gaussian with means m
˙
( r
1
, r
2
, θ
1
, θ
2
) and variance σ 2
R
( r
1
, r
2
, θ
1
, θ
2
) . Of course, these quantities depend on the combining scheme and shall be determined latter. Now p
˙
1
,R
2
, Θ
1
, Θ
2 r, r
1
, r
2
, θ
1
, θ
2
) = p
˙
| R
1
,R
2
, Θ
1
, Θ
2
( ˙ | r
1
, r
2
, θ
1
, θ
2
) p
R
1
,R
2
, Θ
1
, Θ
2
( r
1
, r
2
, θ
1
, θ
2
) (12)
Knowing p
˙
1
,R
2
, Θ
1
, Θ
2
( ˙
1
, r
2
, θ
1
, θ
2
) , as in (12), and the relations of R
1
, R
2
,
˙
1
,
˙
2
, as in (11), the joint density p
˙ r, r ) can be obtained to be used in (9). The kernel of the problem now turns out to be the estimation of p
˙ | R
1
,R
2
, Θ
1
, Θ
2
( ˙ | r
1
, r
2
, θ
1
, θ
2
) = √
2 π σ
˙
1
( r
1
, r
2
, θ
1
, θ
2
) exp −
( ˙ − m
˙
2 σ 2
R
(
( r r
1
1
, r
, r
2
2
, θ
, θ
1
1
, θ
, θ
2
2
)
))
2 !
(13)
More specifically, the tricky part of the problem is the determination of m
˙
( r
1
, r
2
, θ
1
, θ
2
) and σ 2
R
( r
1
, r
2
, θ
1
, θ
2
) for each combining scheme. For the moment, assume that these quantities are known. Then, by means of [1, Eq. 8] for SC and of [9,

4
Eqs. (12) and (17)] for EGC and MRC, respectively,
 n
R
( r ) =
























R
2 π
R
2 π
0 0
R r
0
ϑ ( r
1
, r ) p
R
1
,R
2
, Θ
1
, Θ
2
( r
1
, r, θ
1
, θ
2
) dr
1 dθ
1 dθ
2
+ R
2 π
0
R
2 π
R r
0 0
ϑ ( r, r
2
) p
R
1
,R
2
, Θ
1
, Θ
2
( r, r
2
, θ
1
, θ
2
) dr
2 dθ
1 dθ
2
R
2 π
R
2 π
R
√
2 r
0 0 0
√
2 ϑ ( r
1
,
√
2 r
− r
1
) p
R
1
,R
2
, Θ
1
, Θ
2
` r
1
,
√
2 r
− r
1
, θ
1
, θ
2
´ dr
1 dθ
1 dθ
2
R
2 π
0
R
2 π
0
R r
0
√ r
2 r
− r
2
1
ϑ ( r
1
, p r 2
− r 2
1
) p
R
1
,R
2
, Θ
1
, Θ
2
“ r
1
, p r 2
− r 2
1
, θ
1
, θ
2
” dr
1 dθ
1 dθ
2
SC
EGC
MRC in which
ϑ ( r
1
, r
2
) ,
Z
∞
0
˙
˙
| R
1
,R
2
, Θ
1
, Θ
2
( ˙ | r
1
, r
2
, θ
1
, θ
2
) d r
Now, with (13) in (15)
ϑ ( r
1
, r
2
) =
σ
˙
( r
1 2
, θ
1
, θ
2
)
2 π exp − m 2
R
2 σ 2
R
( r
1
, r
2
, θ
1
, θ
2
)
( r
1
, r
2
, θ
1
, θ
2
)
!
+ m
˙
( r
1
, r
2
, θ
1
, θ
2
)
1 + erf
2 m
2 σ
˙
( r
1
, r
2
, θ
1
, θ
2
)
( r
1
, r
2
, θ
1
, θ
2
)
!!
where erf ( · ) is the error function. The CDF P
R
( r ) can be obtained as [10]
(14)
(15)
P
R
( r ) =
Z
γ
1
Z
γ
2
Z
2 π
Z
2 π p
R
1
,R
2
, Θ
1
, Θ
2
( r
1
, r
2
, θ
1
, θ
2
) dθ
1 dθ
2 dr
2 dr
1
0 0 0 0 with






γ
1
γ
1
= γ
2
√
=
= r
2 r γ
2 for SC
=
√
2 r − r
1 for EGC





γ
1
= r γ
2
= p r 2 − r 2
1 for MRC
The AFD follows directly from (10), (14), and (16).
(16)
(17)
IV. C ONDITIONAL S TATISTICS OF
The aim of this section is to find the mean m
˙
( r
1
, r
2
, θ
1
, θ
2
) and the variance σ
˙
( r
1
, r
2
, θ
1
, θ
2
) of the conditional Gaussian distribution p
˙
| R
1
,R
2
, Θ
1
, Θ
2
( ˙ | r
1
, r
2
, θ
1
, θ
2
) for each combining technique.
A. Preliminaries
From (1), it follows that R i
= cos( θ i
X i
+ sin( θ i
Y i
. Then
σ
˙ m
˙ i
( r
1
, r
2
, θ
1
, θ
2
) ,
E h ˙ i
| Z i
= cos ( θ i
) E h ˙ i
| Z i
+ sin ( θ i
) E h ˙ i
| Z i
σ
2
˙ i
( r
1
, r
2
, θ
1
, θ
2
) , Var i
| Z = cos
2
( θ i
) Var h ˙ i
| Z i
+ sin
2
( θ i
) Var h ˙ i
| Z i
+ sin (2 θ i
) Cov h ˙ i
, Y i
| Z i
1
, ˙
2
( r
1
, r
2
, θ
1
, θ
2
) , Cov
˙
1
,
˙
2
| Z = E h ˙
1
˙
2
| Z i
− E h ˙
1
| Z i
E h ˙
2
| Z i
=
= cos ( θ
1
) h cos ( θ
2
) Cov X
1
, X
2
| Z + sin ( θ
2
) Cov
1
, Y
2
| Z i
+
+ sin ( θ
1
) h cos ( θ
2
) Cov Y
1
, X
2
| Z + sin ( θ
2
) Cov Y
1
, Y
2
| Z i
(18)
(19)
(20)

5
Note that from (18) and (19) the mean and the variance of
˙ i given Z can be computed as a function of the mean, the variance, and the covariance of X i and Y i given Z . In order to obtain these statistics, the following will be carried out: 1) Defining
˙ = [ ˙
1
Y
1
X
2
Y
2
] , the multivariate Gaussian distribution p
{ ˙ }
( { ˙ } ) 1 will be determined; 2) Using the result of [6], the mean matrix M and the covariance matrix ∆ of the conditional Gaussian distribution p
˙
| Z
( ˙ | z ) will be obtained; 3) Using these matrices, the mean, the variance and the covariance of X i and Y i given Z will be attained; 4) Finally, using (18), (19), and (20) the mean and the variance of
˙ given Z will be found.
In order to determine p
{ ˙ }
( { ˙ } ) , the joint multidimensional Gaussian distribution given in (2) will be used with the covariance matrix given by

Λ = E h
{
˙
}
T
{
˙
} i
− E h
{
˙
}
T i
E h
{
˙
} i
= 
 a c
T
 c b


From [11, Eq. 9.106], the following relations are valid: E [P( t ) ˙P( t + τ )] = d E [P( t )P( t + τ )] d τ
, E [ ˙P( t )P( t + τ )] = − d E [P( t )P( t + τ )] d τ
,
E [ ˙P( t ) ˙P( t + τ )] = − d
2
E [P( t )P( t + τ )] d τ 2
. Now we define ˙ ij d
2
η ij d τ 2
( τ )
τ =0
, where µ ij
( τ ) and η ij
( τ )
= d µ ij
( τ ) d τ
τ =0
, ¨ ij
= d
2
µ ij d τ 2
( τ ) are given by (4), (5), and (6), respectively. Then
τ =0
, ˙ ij
= d η ij d τ
( τ )
τ =0
, ¨ ij
=

¨
11
σ
2
X
1 a =








0
¨
12
σ
X
1
σ
X
2
12
σ
X
1
σ
Y
2
0
¨
11
σ 2
Y
1
− ¨
12
σ
Y
1
σ
X
2
− ¨
12
σ
Y
1
σ
Y
2
-
0
¨
12
σ
X
1
− ¨
12
σ
Y
1
¨
11
σ 2
X
2
σ
X
2
σ
X
2
¨
12
σ
X
1
σ
Y
2

− ¨
12
σ
Y
1
σ
Y
2
0








¨
11
σ 2
Y
2
(21)

0 c =








11
σ
X
1
σ
Y
1
µ
12
σ
X
1
σ
X
2
− ˙
12
σ
X
1
σ
Y
2
˙
11
σ
X
1
σ
Y
1
0
12
σ
Y
1
σ
X
2
µ
12
σ
Y
1
σ
Y
2
− ˙
12
σ
X
1
− ˙
12
σ
Y
1
0
11
σ
X
2
σ
X
2
σ
X
2
σ
Y
2
12
σ
X
1
σ
Y
2

− ˙
12
σ
Y
1
σ
Y
2
˙
11
σ
X
2
σ
Y
2








0
(22) and the matrix b is given in (3). Note that the diagonal elements in the matrix c are all null, because for a stationary process the correlation coefficient between the process and its time derivative is always null at τ = 0 ( ˙
11
= 0 ) [11].
Using the results from [6, pp. 495-496], the conditional distribution of
˙ given Z 2
, p
˙ | Z
( ˙ | z ) , is Gaussian distributed with the mean matrix M and the covariance matrix ∆ , respectively, given by

M =








E h
X
1
| Z i
E h
Y
1
| Z i
E h ˙
2
| Z i
E h
Y
2
| Z i








 = cb
− 1
Z (23)
1 { ˙ } = h
˙
1
Y
˙
1
˙
2
˙
2
X
1
Y
1
X
2
Y
2 i
2
Note that to provide the information about the variates Z = [ X
1
Y
1
X
2
Y
2
] is the same as to provide the information about the variates [ R
1
Θ
1
R
2
Θ
2
] .

6

Var
∆ =








Cov
Cov
Cov
1
˙
,
1
| Z
Y
1
| Z
Cov
Var
˙
1
,
˙
2
| Z Cov
1
, Y
2
| Z Cov
˙
˙
1
1
,
˙
1
|
Y
1
| Z
,
Y
1
,
˙
2
|
Z Cov
Z
Y
2
| Z
Cov
Var
Cov
˙
1
Y
1
,
˙
,
˙
2
| Z Cov
X
2
| Z
2
| Z
X
2
, Y
2
| Z
Cov
Cov
Var
˙
1
Y
1
,
˙
2
,
,
Y
2
| Z
˙
˙
2
|
2
Y
2
| Z
|
Z
Z








 = a − cb
− 1 c
T
Using this and after a tedious procedure, the matrices obtained are given by
(24)

M =







 m
1 x
1
+ m
2
σ
X
1
σ
Y 1 y
1
+ m
3
σ
X
1
σ
X 2 x
2
+ m
4
σ
X
1
σ
Y 2 y
2
m
2
σ
Y
1
σ
X
1 x
1
m
3
σ
σ
X
2
X
1 x
1
+
+ m m
1
4 y
1
σ
σ
X
2
Y
1
− m
4
σ
σ
Y
1
X
2 x
2
+ m
3
σ
σ
Y
1
Y
2 y
1
− m
1 x
2
+ m
2
σ
σ
X
2
Y
2 y
2 y
2
m
4
σ
σ
Y
2
X
1 x
1
− m
3
σ
σ
Y
2
Y
1 y
1
− m
2
σ
Y
2
σ
X
2 x
2
− m
1 y
2










σ 2
X
1
∆
1


∆ = −






0
σ
X
1
σ
X
2
∆
2
σ
X
1
σ
Y
2
∆
3
0
σ 2
Y
1
∆
1
σ
X
2
σ
Y
1
∆
3
σ
Y
1
σ
Y
2
∆
2
σ
0
X
1
σ
X
2
σ 2
X
2
σ
X
2
∆
2
σ
Y
1
∆
3
∆
1
σ
X
1
σ
0
Y
1
σ
Y
2
∆
3
σ
Y
2
σ 2
Y
2
∆
1
∆
2








 where m
∆
∆
1
3
= ¨
12
1
=
+
µ
12
=
µ
12
µ
12
+ η
1 − ρ 2
12
η
12 , m
2
=
η
12
µ
12
− µ
12
1 − ρ 2
η
12
− ˙
11
µ
11
+
η
11
+ µ
12
2
12
η
12
)+ η
1 − ρ 2
2
12
12
(
˙
η
2
11 11
( µ
12
2
12
− ˙
1 − ρ 2
2
11
− ˙
2
12
)
η
12
− ˙
12
, and ρ
η
12
)
, ∆
2 is given in (8).
, m
3
=
(25)
=
µ
12
η
11
η
12
−
1 − ρ 2
˙
12
+ 12
( η
12
, m
4
=
µ
12
− ˙
11
) − µ
12
1 − ρ 2
( ˙
2
12
12
1 −
11
ρ 2
µ
12
− ˙
2
11
− ˙
2
12
)
,
,
B. Mean and variance of R i s
Using (11), the conditional means and variances for each combining scheme can be obtained as:
1) Selection Combining:
• If R
1
> R
2 m ( r
1
, r
2
, θ
1
, θ
2
) = m
R
1
( r
1
, r
2
, θ
1
, θ
2
)
σ
2
R
( r
1
, r
2
, θ
1
, θ
2
) = σ
2
˙
1
( r
1
, r
2
, θ
1
, θ
2
)
•
If R
1
< R
2 m
˙
( r
1
, r
2
, θ
1
, θ
2
) = m
˙
2
( r
1
, r
2
, θ
1
, θ
2
)
σ
2
R
( r
1
, r
2
, θ
1
, θ
2
) = σ
2
˙
2
( r
1
, r
2
, θ
1
, θ
2
)
2) Equal Gain Combining: m
˙
( r
1
, r
2
, θ
1
, θ
2
) =
σ 2
R
( r
1
, r
2
, θ
1
, θ
2
) = m
˙
1
σ 2
R
1
(
( r
1
, r
2 r
1
, r
2
, θ
1
, θ
2
, θ
1
, θ
2
) +
√
2 m
˙
2
) + σ 2
R
2
( r
1
, r
( r
1
, r
2
2
, θ
1
, θ
, θ
1
, θ
2
2
)
) + 2 σ
R
1
, ˙
2
( r
1
, r
2
, θ
1
, θ
2
)
2
(26)
(27)
(28)
(29)
(30)
(31)

7
3) Maximal Ratio Combining: m ( r
1
, r
2
, θ
1
, θ
2
) =
σ
2
R
( r
1
, r
2
, θ
1
, θ
2
) = r
1 m
R
1 r 2
1
σ 2
˙
1
( r
1
, r
2
, θ
1
, θ
2
) + r
2 m
R
2 p r 2
1
+ r 2
2
( r
1
, r
2
, θ
1
, θ
2
) + r 2
2
σ 2
˙
2
( r
1
, r
2
, θ
1
, θ
2
)
( r
1
, r
2
, θ
1
, θ
2
) + 2 r
1 r
2
σ
˙
1
,
˙
2
( r
1
, r
2
, θ
1
, θ
2
) r 2
1
+ r 2
2
(32)
(33)
C. Special Cases
For the Rayleigh case, σ
X i
= σ
Y i
= σ ( i = 1 , 2 ), then equations from (26) to (33) reduce in a exact manner to those of
[1, Eqs. 26 and 27]. In the case of branch independence (e.g. large separation between the antennas) the mean and variance are not functions of R
1
, R
2
, Θ
1
, and Θ
2 because µ
12
µ
12
µ
12
= η
12 with those of [9] for EGC and MRC with the number of branches M = 2 .
η
11 12
= ¨
12
= 0 . Then the results coincide
V. N UMERICAL R ESULTS
The expressions obtained for the LCR and AFD are general and can be applied to any type of diversity (space, frequency or time). In this section, we assume space diversity at the mobile station as [1]. For incoming multipath waves having equal amplitude and independent phases, the crosscorrelation functions are given by [7], [12]
µ
µ
11
12
(
(
τ
τ
) =
) =
J
0
(2 πf m
τ )
2
1 + ∆ ωT
J
0
2 π q
( f m
τ )
2
+ ( d/λ )
2
− 2 ( f m
τ ) ( d/λ ) cos ( α )
1 + ∆ ωT
2
η
11
( τ ) = ∆ ωT µ
11
( τ )
η
12
( τ ) = ∆ ωT µ
12
( τ )
(34)
(35)
(36)
(37)
(38) where J
0
( · ) is the zero-order Bessel function, λ is the carrier wavelength, f m is the maximum Doppler shift in Hz, d is the antenna spacing, ∆ ω is the angular frequency separation, T is the time delay spread, and α ∈ [0 , 2 π ] is the angle between the antenna axis and the direction of the vehicle motion in radians.
For a nil frequency separation, then η
11
( τ ) = 0 and η
12
( τ ) = 0 . The corresponding correlation coefficients can be calculated as
µ
11
= 1
µ
12
= J
0
(2 πd/λ )
µ
11
= 0
µ
12
= 2 πf m cos ( α ) J
1
(2 πd/λ )
µ
µ
11
12
= − 2 ( πf m
)
2
= (2 πf m
)
2
J
1
(2 πd/λ )
2 πd/λ cos (2 α ) − cos
2
( α ) J
0
(2 πd/λ )
(39)
(40)
(41)
(42)
(43)
(44)

8 where J
1
( · ) is the first-order Bessel function. And this is the case explored here (as well as in [1]).
In the illustrations that follow we use the Hoyt parameter [5] b i
,
σ
σ
2
Xi
2
Xi
− σ
+ σ
2
Yi
2
Yi and the individual power branches Ω i
= σ 2
X i
+ σ 2
Y i
Figs. 1 and 2 show the normalized LCR (left vertical axis), N
R
/f m
, and AFD (right vertical axis), T
R f m
, for α = 0 ◦ and
.
α = 90 ◦
, respectively, as a function of the envelope level, for SC, EGC, and MRC. The following arbitrary parameters have been used: d/λ = 0 .
06 , b i
= 0 .
5 .
Figs. 3 and 4 show the normalized LCR and AFD for two different antenna angles α = 0 ◦ and α = 90 ◦
, respectively, as a function of the parameter d/λ , for the SC, EGC, and MRC. An envelope level at r/ q
Ω
1
+Ω
2
2
= − 20 dB, identical fading parameters b i
= 0 .
5 , and balanced channels have been used. It can be seen that as the antenna spacing becomes larger, the
LCR decreases, becoming oscillatory and convergent. Fig. 3 also shows that the MRC has the smaller LCR in both cases of antenna angles. It can be seen in Fig. 4 that the shape of the AFD curves for the SC, EGC, and MRC are loosely dependent on the antenna spacing when α = π/ 2 .
Also note in Fig. 1 that the LCR for EGC is larger than the LCR for SC for the specific value d/λ = 0 .
06 . Observe in
Fig. 3 that the LCR for EGC is larger than the LCR for SC in the range ( 0 6 d/λ 6 0 .
18 ), but for d/λ > 0 .
18 , we have the result approaching the independence case, in which case the LCR for EGC is smaller than that for SC. The interesting point to observe is that the conclusions drawn for the independence case, which are well known in the literature, do not hold for the dependence case any longer. The comparative performance now must take into consideration the level of correlation between the branches.
Figs. 5 and 6 show the effect of power imbalance and the use of non-identical parameters in the normalized LCR as a function of the envelope level and as a function of the parameter d/λ , respectively, for SC, EGC, and MRC. Fig. 5 shows the balanced ( Ω
1
= 1 , Ω
2
= 1 ) and unbalanced cases ( Ω
1
= 1 .
8 , Ω
2
= 0 .
2 ) for antennas arranged parallel to the direction of the vehicle motion ( α = 0 ◦
), using Hoyt parameters ( b
1
= 0 .
5 , b
2
= 0 .
2 ), and d/λ = 0 .
4 . It can be seen that, for this specific correlation value between the branches, the effect of the power imbalance deteriorates the performance of the LCR for SC, EGC, and MRC cases. Fig.6 shows the balanced ( Ω
1
= 1 , Ω
2
= 1 ) and unbalanced cases ( Ω
1
= 1 .
8 , Ω
2
= 0 .
2 ) for
( α = 0 ◦
), using Hoyt parameters ( b
1
= 0 .
5 , b
2
= 0 .
2 ), and r/ q
Ω
1
+Ω
2
2
= − 20 dB. It also can be seen that the effect of the power imbalance deteriorates the performance of the LCR for SC, EGC, and MRC cases.
Fig. 7 shows the effect of power imbalance and the use of non-identical parameters in the normalized AFD as a function of the parameter d/λ for SC, EGC, and MRC. This figure shows the balanced ( Ω
1
= 1 , Ω
2
= 1 ) and unbalanced cases ( Ω
1
= 1 .
8 ,
Ω
2
= 0 .
2 ) for antennas arranged parallel to the direction of the vehicle motion ( α = 0 ◦
), using Hoyt parameters ( b
1
= 0 .
5 , b
2
= 0 .
2 ), and r/ q
Ω
1
+Ω
2
2
= − 20 dB. It can be seen in Fig. 7 that the shape of the AFD curves for the SC, EGC, and MRC are extremely dependent on the antenna spacing when α = 0 ◦
.
VI. C ONCLUSIONS
Exact formulas for level crossing rate and average fade duration of the dual branch SC, EGC and MRC techniques in a unbalanced, non-identical, and correlated Hoyt fading environment have been presented. In passing, this paper derives the joint
Hoyt bidimensional envelope-phase distribution. These formulas have been validated by specializing the general results to some

9
10
0
10
-1
10
-2
No Diversity
SC
EGC
MRC
α
=0
°
10
-3
-30 -20 -10
20 Log(r/ √( ( Ω
1
+ Ω
1
)/2))
0 10
10
-3
Fig. 1.
Normalized LCR and AFD for d/λ = 0 .
06 , α = 0 ◦ , and identical Hoyt fading parameter b i
= 0 .
5 .
10
0
No Diversity
SC
EGC
MRC
10
3
10
2
10
-1
10
1
α
=90
°
10
0
10
-2
10
-1
10
-2
10
-3
-30 -20 -10
20 Log(r/ √( ( Ω
1
+ Ω
2
)/2))
0
Fig. 2.
Normalized LCR and AFD for d/λ = 0 .
06 , α = 90 ◦ , and identical Hoyt fading parameter b i
= 0 .
5 .
10
10
-3
10
3
10
2
10
1
10
0
10
-1
10
-2 particular cases whose solutions are known. Sample numerical results were presented by specializing the general expressions to a space-diversity system with horizontally spaced omnidirectional antennas at the mobile station.
R EFERENCES
[1] F. Adachi, M. T. Feeney, and J. D. Parsons. Effects of correlated fading on level crossing rates and average fade durations with predectection diversity reception. In Proc. Inst. Elect. Eng., pages 11–17, Feb. 1988.
[2] X. Dong and N. C. Beaulieu. Average level crossing rate and fade duration of maximal-ratio diversity in unbalanced and correlated channels. In WCNC
- IEEE Wireless Communications and Networking Conference, pages 762–767, Mar. 2002.
[3] L. Yang and M. S. Alouini. An exact analysis of the impact of fading correlation on the average level crossing rate and average fade duration of selection combining. In IEEE Vehicular Technology Conference, volume 1, pages 241–245, Apr. 2003.
[4] D. Li and V. K. Prabhu. Average Level Crossing Rates and Average Fade Durations for Maximal-ratio Combining in Correlated Nakagami Channels.
In WCNC - IEEE Wireless Communications and Networking Conference, pages 339–344, Mar. 2004.

10
10
0
10
0
α =0 °
LCR
10
-1
No Diversity
SC
EGC
MRC
10
-2
10
-3
10
-1
10
-4
AFD
10
-5
0 0.2
0.4
d/ λ
0.6
0.8
10
-2
Fig. 3.
Normalized LCR and AFD for r/ q
Ω
1
+Ω
2
2
= − 20dB , α = 0 ◦ , and identical Hoyt fading parameter b i
= 0 .
5 .
10
0
α
=90
°
10
0
10
-1
LCR No Diversity
SC
EGC
MRC
10
-2
10
-1
10
-3
AFD
10
-4
10
-5
0 0.2
0.4
d/ λ
0.6
0.8
10
-2
Fig. 4.
Normalized LCR and AFD for r/ q
Ω
1
+Ω
2
2
= − 20dB , α = 90 ◦ , and identical Hoyt fading parameter b i
= 0 .
5 .
[5] R. S. Hoyt. Probability Functions for the Modulus and Angle of the Normal Complex Variate. Bell System Technical Journal, 26:318–359, Apr. 1947.
[6] M. Schwartz, W. R. Bennett, and S. Stein. Communication Systems and Techniques. New York, 1966.
[7] W. C. Jakes. Microwave Mobile Communications. New York: Wiley, 1974.
[8] S. O. Rice. Mathematical analysis of random noise. Bell System Technical Journal, 23:282–332, Jul. 1944.
[9] G. Fraidenraich, J. C. S. Santos Filho, and M. D. Yacoub. Second-Order Statistics of Maximal-Ratio and Equal-Gain Combining in Hoyt Fading. IEEE
Commun. Lett., 9(1), Jan 2005.
[10] D. G. Brennan. Linear diversity combining techniques. In IRE, volume 47, pages 1075–1102, Jun. 1959.
[11] A. Papoulis. Probability, Random Variables, and Stochastic Processes. Fourth edition, 2002.
[12] W. C. Y. Lee. An extended correlation function of two random variables applied to mobile radio transmission. Bell Syst. Tech. J., pages pp. 3423–3440,
1969.

10
-3
10
-4
10
-5
-30
10
0
10
-1
Unbalanced
(
1
=1.8,
2
=0.2)
=0
10
-2
Balanced (
1
=1,
2
=1)
-20 -10
20 Log(r/ (
1
+
2
)/2))
0
- SC
- EGC
- MRC
10
Fig. 5.
Normalized LCR for d/λ = 0 .
4 , balanced and unbalanced channels, α = 0 ◦ , and Hoyt fading parameters b
1
= 0 .
5 and b
2
= 0 .
2 .
10
0
=0
- SC
- EGC
- MRC
10
-1
Unbalanced (
1
=1.8,
2
=0.2)
10
-2
10
-3
0 0.2
0.4
d/
Balanced (
1
=1,
2
=1)
0.6
0.8
Fig. 6.
Normalized LCR for r/ q
Ω
1
+Ω
2
2
= − 20dB , α = 0 ◦ , balanced and unbalanced channels, Hoyt fading parameters b
1
= 0 .
5 and b
2
= 0 .
2 .
11

10
-1
Unbalanced
(
1
=1.8,
2
=0.2)
=0
- SC
- EGC
- MRC
10
-2
0
Balanced
(
1
=1,
2
=1)
0.2
0.4
d/
0.6
0.8
Fig. 7.
Normalized AFD for r/ q
Ω
1
+Ω
2
2
= − 20dB , α = 0 ◦ , balanced and unbalanced channels, Hoyt fading parameters b
1
= 0 .
5 and b
2
= 0 .
2 .
12