Communications and Network, 2013, 5, 319-326
http://dx.doi.org/10.4236/cn.2013.53B2059 Published Online September 2013 (http://www.scirp.org/journal/cn)
Diversity–Multiplexing Tradeoff and Outage Performance
for 2×2 Dual-Polarized Uncorrelated Rice MIMO
Channels*
Yanping Huang, Guangliang Ren
State Key Lab. of Integrated Services Networks, Xidian University, Xi’an, China
Received July, 2013
ABSTRACT
In this paper, diversity-multiplexing tradeoff (DMT) curve for 2×2 Dual-Polarized uncorrelated Rice MIMO chan nels is
studied. Exact expressions for statistic information of mutual information exponent are derived. Impacts of channel pa-
rameters such as signal to noise ratio (SNR), k-factor and cross polarization discrimination (XPD) on mutual informa-
tion exponent are analyzed. Compared to conventional single-polarized (SP) Rice MIMO systems, a qualitatively dif-
ferent behavior is observed for DP Rice systems. The work in this paper, allows to identify quantitatively for which
channels (k-factor) and SNR levels the use of dual polarization becomes beneficial. Gamma or lognormal distribution
are used to describe mutual information component, and a theoretical formulation for finite-SNR DMT curve in 2×2 DP
uncorrelated Rice channels is presented for the first time, which is valid in low and medium SNRs when multiplexing
gain is larger than 0.75.
Keywords: DMT curve; Dual-Polarized; Uncorrelated Rice Channel; Mutual Information Exponent; k-factor; Outage
Probablity Approximation
1. Introduction
Due to the space-cost of the conventional Single-polar-
ized (SP) multiple-input multiple-output (MIMO) sys-
tems, dual-polarized (DP) MIMO has been receiving
much attention as an attractive alternative for realizing
MIMO architectures in compact devices [1-7]. Co mpared
with SP MIMO, DP MIMO exhibits many different
characteristics. For instance , in [1] it has been clearly
illustrated that in Rice fading, after some k-factor (de-
fined as the ratio of the power in the fixed exponent to
the power in the variable exponent), error probability of
zero-forcing detection method for polarization multi-
plexing starts to decrease with increasing k-factor, while
SP systems perform the opposite. Moreover, various lit-
eratures, such as in [1-4], an idea has been well devel-
oped that polarization diversity wo rks well only in corre-
lated Rayleigh fading or Rice fading channels with LOS
components. It is necessary to note that measurements
have been done to get real parameters of DP channels,
which helps in getting more accurate polarized channel
model [6]. To go further, channel correlation and capac-
ity are discussed in these literatures, proving that such
dual polarization has de-correlation effect on correlated
channels from a practical aspect. Nevertheless, this result
does not extend to diversity systems, such as Almouti
coded MIMO, where polarization confronts performance
loss [2]. In conv entional MIMO systems, it is known that
there exists a fundamental tradeoff between achievable
diversity and multiplexing gains of any transmission over
tr
nn

MIMO channel, i.e., diversity-multiplexing tradeoff
(DMT), as has been clearly illustrated in [8] for sign al to
noise ratio (SNR) approaching infinity. Moreover, it is
also pointed out that DMT curve at finite SNR is quite
different [9-13]. Under realistic propagation conditions,
since SNR cannot reach infinity, it would be meaningful
to study DMT behavior at finite SNRs that are practical
in operating regimes. Up to now there are no literatures
that investigate finite-SNR DMT for dual-polarized sys-
tems.
In previous literatures [9-13], DMT curve is discussed
based on the assumption that, elements of
H
H
H follow
Wishart distribution. However, for polarized MIMO,
because of the asymmetric properties of the generalized
channel matrix, random matrix theory results fo r Wishart
matrices cannot be leveraged. Inspired by the idea pro-
posed in [14], which used gamma, lognormal or weibull
*This work was supported in part by the State Natural Science Founda-
tion of China, Grant No.61072102 and National Major Specialized
Project of Science and Technology, Grant No.2011ZX03001-0 0 7 -01.
C
opyright © 2013 SciRes. CN

Y. P. HUANG, G. L. REN
320
distribution to approximate outage capacity for dual-
polarized MIMO in high SNR regime, by approximating
mutual information exponent, we get theoretical DMT
curve for DP in low and medium SNR regimes in 2×2
uncorrelated Rice channels.
The rest of this paper is organized as follow. Section2
describes the channel model developed for a 2×2
Dual-Polarized uncorrelated Rice fading channels. Sec-
tion 3 discusses statistic characteristic of mutual infor-
mation exponen t, outage probability and DMT curve and
their approximations. Section4 shows the simulation re-
sults. Finally, section5 is the conclusion.
In this paper, and

Ex


Dx represents the ex-
pectation and variation of random variable x, respectively,
* stands for the element-wise conjugation, H for conju-
gate transpose,


det
A
is the determinant of matrix A.
2. System Model and Definitions
Consider a system with one dual-polarized transmit and
one dual-polarized receive antenna. The channel is as-
sumed frequency-flat over the band of interest. The chan-
nel matrix is given by
11 21
12 22
hh
Hhh




(1)
Assume that both transmitter and receiver employ the
same polarization scheme, i.e. both of them employ
horizontal/vertical or slanted polarization. Decomposing
the channel matrix into the sum of a fixed exponent and a
variable exponen t as

1
11
k
H
H
kk


H (2)
The elements of the matrix
H
do not vary and satisfy
22
1122 1hh, 22 12 21
f
hh

. The elements

ij
h
of the matrix

H
are complex random variables, which
satisfy
 
22 22
11 221221
1,Eh EhEhEh

  
 
  


(3)








111212 2212 211122
0; 0Ehh EhhEhh Ehh 
(4)
where 01
f


, 0
1
,,;,1,2
11
~1
,,;,
11
ij
f
k
Nijij
kk
hk
Nij
kk















1,2ij
(5)
In [8], conventional asymptotic definitions of multi-
plexing and diversity gains for a MIMO channel are
given by:


*log
lim log
R
r




 (6)
*log
lim log out
P
d



 (7)
where and represent the asymptotic multiplex-
ing and diversity gain respectively,
*
r* d

is the average
SNR per receive antenna, R is the system data rate and
out is the outage probability. Assuming that no CSI is
available at the transmitter, is defined by
P
out
P



P2

R
out PI RPW  (8)
where I is the mutual information between received and
transmitted signals over the MIMO channels, and is
the mutual information ex ponent satisfy W log
I
W.
The asymptotic DMT is given by the piece-wise linear
function connecting the points , where


*
,id i



*
di
is given by [8]:






*, 0,..,min,
rt r
dinin iinn 

t
n

(9)
r, t are numbers of receive and transmit antennas,
respectively. Note that the asymptotic DMT describes
situation where SNR approches infinity. However, for
practical system design, it is desirable to characterize the
diversity-multiplexing tradeoff at operational SNRs. The
finite-SNR definitions for diversity and multiplexing
gains can provide useful tool to characterize the DMT at
real environment. The finite-SNR multiplexing gain r is
defined as the ratio of to the capacity of an AWGN
channel at SNR with array gain [11],
n n
RG

min ,
rt
Gn

log 1
R
rG

 (10)
The finite-SNR outage probability

,
out
Pr


for a
given and
r

is given
1

 are related to the XPD for
the fixed and variable exponent of the channel, respec-
tively. Good discrimination of orthogonal polarizations
amounts to small values of

and
f

, and vice versa.
Clearly, when f1, 1

 
the model becomes the
conventional SP (single polarization) channel. For 2 × 2
Dual-Polarized uncorrelated Rice MIMO channels ij
are complex Gaussian random variables whose parame-
ters are:
h



1
P, 2r
G
out rPW



 (11)
The finite SNR diversity gain

,dr


is defined by
the negative slope of the plot

,Pr
out

versus log

:


,
,
out
out
Pr
dPr




  (12)

Y. P. HUANG, G. L. REN 321
3. Computation of DP finite-SNR DMT
In this section, the DMT for 2×2 Dual-polarized Rice
channels is examined. First, we derive an exact expres-
sion for the mean and variation of mutual information
exponent, based on which some discussions on channel
parameters are proposed to have a deeper insight into
dual-polarization system. Second, using the expressions
of statistic information derived in the first step, approxi-
mation equations of outage probability are presented.
Finally, DMT for both asymptotic and finite-SNR in 2×
2 Dual-polarized Rice channels at are investigated.
3.1. Statistic Information of Mutual Information
Exponent
Consider that channel state information (CSI) is perfectly
known at the receiver. The MIMO mutual information I
conditioned on the channel realization is given by


1
logdet
log1
r
H
H
n
rank HH
i
it
IIHH
N









(13)
where

11
H
rank HH
i
it
Wn







,
and i

denotes the eigenvalues of
H
H
H. For the case
of or MIMO, mean and variances of W
as a function of k-factor and
2
t
n2r n,
f

 are expressed be-
low:

2
1ii EW EE
nn



 

 
  

 

(14)


24
3
2,
i
ii
WD WDD
nn
R
n

i




 

 

  
 

 
 


(15)
where , and

min ,
rt
nn

n

i
E

,


i
D


,

i
E

,

,
i
R

i


 are given in the Appendix 1,
for the sake of space saving.
The distribution of the mutual information exponent
provides information about the available diversity in the
system. describes the ergodic mutual informa-
tion exponent, which can be used to get upper bound of
mutual information I. And presents some in-
formation about outage probability, i.e., the smaller the
variance, the lower the probability of the outage error is
when transmitting at a fixed rate [8].

EW

DW
From the analytical expression of and

EW


DW
given in (14)-(15), we find that both of them are influ-
enced by k-factor and SNR. With the existences of po-
larization indicators
f

and

, the influence are dif-
ferent. Let


SP
EW,


W
DP be mean of information
exponent of SP and DP, we get
E
 

2
2n
 1
1
SP
EWAk BkC
k 
DP
EW (16)
Then k-factor fo r


E

DP
EW W

SP is divered:
24
2
BB AC
k
A
 
 (17)
where


 

2
22

2
1221
2
121
2
f
ff
f
An
Bn
Cn

f










 



 (18)
In suburban area, where XPD is measured to be
8-15dB range, let =0.4, 0.3
f

 . We find that in SP,
mean of mutual information exponent decrease fast with
the increase in k-factor, while for DP, declension is less.
At 0dB


, required k-factor to fill the gap between
DP and SP is k= -0.5754 or -2.4376; 10dB

, required
k = 11.3118 or -0.4546; when

, k = 4.0184 or
-0.4266.
3.2. Approximating of Outage Probability
Motivated by the work [14], in this section, we derive the
approximation curve for outage probability at finite SNR
for 2×2 dual-polarized uncorrelated Rice channels.
The steps begin with the approximation of statistical
information of mutual information exponent W.
If we assume gamma distribution for W, i.e.
 
/w
p



1,0
Wp
e
fw w

p
w

(19) 







 



22
,
,,
,
,,
DWDW k
kEWEW k
EWEW k
pk DWDW k

 




 

 

(20)
Then outage probablity at given multiplexing gain and
SNR is



1
,p

,
r
out
G
Pr p








 (21)
where is the incomplete gamma
function.

1
0
,xkt kxte dt 


If we assume lognormal distribution for W, i.e.

Y. P. HUANG, G. L. REN
322

 
2
2
ln
2
2
1,
2
wu
W
fwe w
w




0 (22)

2
1
lnlnln 1
2
DW
uE WEWEW





(23)
 
2
2
lnln 1DW DW EW 







(24)
Then
 
ln 1
11
P, 22 2
out
rGu
rerf



 
 
(25)
where both and are given in section2.
Note that format parameters of W are directly related to
polarization parameters

EW

DW
,
f


 as well as k-factor, SNR.
Corollary: When , for outage probability of
DP, k


P0,0 min,
outr t
rN N (26)
In contrast, outage probability in SP is given by [12]
01
P1, 1
out
r
r



 (27)
Proof: As , for conventional SP, the Rice fad-
ing channel approaches a rank-one AWGN channel, such
that the outage probability is 1 for , and 0 for
; However, for DP Rice fading channels, as
, thanks to polarization orthogonality, channel
matrix remains full rank. Thus, as k increases, channel
approach two rank-one AWGN channels. Therefore as
, outage probability for both
k
1r
0
1r
k
k DP P
out 1r

and . 1r
3.3. Asymptotic DMT for Rice Dual
Polarized Channels
Theorem: The asymptotic DMT curve for dual-polarized
channels is independen t of

,
f

, which is identical to
conventional asymptotic DMT in SP channels as de-
scribed in (9) [8 ].
Proof: The proof is given in appendix 2.
3.4. Diversity and Multiplexing Trade-off at
Finite SNR
Simulated by the method in [11], we get finite-SNR
DMT using (11).
 

 
1
0
12
P, ,,
,,
r
G
out r
f
wkdw
Ar Ar








 


 (28)
where for gamma approximation,

 
  

/,
,1
,
,, ,,
wk
pk
p
k
e
fw kwpk k






 (29)
 

 

1
1
1
20
,1 1,
,,,
r
rr
G
,
A
rrGGfG
Arfw kdw






 



k 
(30)
For lognormal approximation, calculation step is
similiar, which is omitted for the sake of space.
4. Simulation
4.1. Impact of k-factor and SNR on Mutual
Information Exponent
As it has been known that DP and SP systems perform
rather diffident in Rice channels. In order to study how
such a difference occurs, Figure 1 plots


EW and


DW as a function of the k-factor at 0dB


and
10dB


. Assume that for DP, 0.4

, 0.3
f


, and
without loss of generality ,take 1
ij
h for ,1,ij 2

.
The theory curves are identical to the ones by Monte
Carlo simulations, which validate the derived expression
of


EW and


DW in section 2. As expected, a
quantitatively differen t behavior is observed for DP Rice
system. Although either in SP or DP case, expectation
and standard deviation of the mutual information expo-
nent drop dramatically with increasing k-factor, espe-
cially in low k-factor regime, where k-factor manifests
the variation of W. It is clear that in DP, the drop is far
less than that in SP both for and

EW


DW
dB
, since
polarization can reduce the channel correlation brought
by LOS component. Moreover, at 0

, no cross
points for




Wk
SPDP at are found. But
at EW E0 10dB


, 11 k

, a cross point appears ,matching
the previous results from (17). Such a phenomenon can
Figure 1. Comparison of mean and variation of W for SP
and DP.

Y. P. HUANG, G. L. REN 323
be explained by the effect of eigenvalue of
H
H
H. At
medium SNR, minimum eigenvalue begins to affect
channel information exponent. For conventional SP 2×2
Rice systems, when k-factor increases, channel matrix
tends to be a rank-deficient matrix, leading the minimum
eigenvalue to be smaller even approaching zero. In con-
trast, eigenvalues of DP systems nearly stays constant,
without being hugely affected by varying k-factor. Hence,
channel matrix does not become ill conditioned, i.e., not
badly affected by LOS component. Thus, in seminars
with strong LOS component, we suggested DP be used.
To illustrate the different eigenvalues we can see Figure
2.
4.2. Impact of XPD on Mutual Information
Exponent
As a final parameter dependency study, we examine on
mutual information exponent as a function of the XPD in
LOS component. Using the analytical formation in sec-
tion2, Figure 3 plots plots and as a
function of the

EW

DW
f

at 0dB

 and 10dB

, with
fixed , 10 k0.4

.
Figure 2. Eigenvalues of SP and DP for 2×2 uncorrelated
Rice fading.
Figure 3. XPD influence on mutual information exponent in
2×2 Rice uncorrelated channels with k=10.
From Figure 3, it is clear that at low SNRs in-
creases with EW
f

. However, at moderate SNR,
starts to drop with improving EW
f

. Conclusions can be
made that XPD and SNR have impacts on the mutual
information at the same time. It is then meaningful to
find the optimal SNRs for different DP systems for opti-
mal code design.
4.3. Outage Probability in Finite SNR
In this part, we study some plots of outage probability
versus SNR in uncorrelated Rice fading with rt nn


2G

.
In Figure 4, given a fixed multiplexing gain 1r

,
outage probability versus SNR curves are plotted for SP
and DP at 5,12k

. It is seen that contrary to SP, outage
probability of DP always drops as k-factor improves. At
some SNR, negative gap of outage probability between
SP and DP turns into positive, coinciding with previous
analysis.
In Figure 5, gamma or lognormal approximation are
Figure 4. Outage probability SP VS DP for k =5 and k =12.
Figure 5. Outage probability approximation for different
multiplexing gains and k-factor DP.

Y. P. HUANG, G. L. REN
324
plotted as well as results of Monte Carlo simulation for
outage probability in various multiplexing gain. The dash
curves represent the approximation value, while the cir-
cle, square symbol represent the DP systems for
respectively. When (in the
plot), the gamma approximation matches the simulation
well. At (), we can use the lognormal
distribution instead, which works well in medium SNR
0-15dB. Note that the higher multiplexing gain, the more
accuracy of the gamma approximation. By using this
approximation method, it becomes simple to estimate the
outage probability of DP Rice channels at or me-
dium SNRs without time-cost simulation.
5, 10kk
r
1 r1.5r
1r
10.75r
4.4. Diversity Gain at Finite SNR
Figure 6 is a plot of diversity and multiplexing gain
tradeoff in finite SNR for DP Rice channels, 10 k

and
.
[0.75,2]r
Obviously, the approximation curve agrees with the
Monte Carlo simulation. For , as it has been indi-
cated in [11], the diversity gain in SP Rice fading chan-
nels approaches zero rapidly since the rank-one LOS
matrix limits the effective degrees of freedom in the
channels. However, for DP Rice fading, a relatively high
diversity gain can still be observed in . At
1r
1
r
10dB

, , diversity gain can be as high as one.
Explanations can be found from minimum eigenvalue of
DP systems, that thanks to the de-correlation effect,
minimum eigenvalue of DP do not approach zero despite
of the exists of strong LOS component.
10 k
For very high k-factor, the channel matrix only de-
pends on the Rice exponent. As , the channels
tend to be AWGN, and the capacity increases only with
SNR. For DP, asymptotic diversity gain becomes infi-
nite.
k
Figure 6. Finite-SNR DMT for DP Rice channels.
5. Conclusions
In this paper, outage probablity and DMT for asymotic
and finite-SNR are studied in 2×2 dual-polarized uncor-
related Rice fading channels. Exact expression mean and
varition of mutual information in DP Rice channels are
derived, based on which how channel paremeters as
k-factors, SNR or XPD influence channel infromation
exponent are discussed. Results show that in subran en-
vironments where 0.4


, 0.3
f

, at 10dB


, a
11k

is required to fill the gap between erogotic mean
of mutual information exponent of SP and DP. Outage
probablity as well as asympotic and finite-SNR DMT are
compared between of SP and DP. Using the gamma or
lognormal distribution, their appromaxition curves for
2×2 dual-polarized uncorrelated Rice channels at 10k

are given. The result in this paper, helps in finding the
inner difference between DP and SP channels. And the
appromaxiton approach for DMT in this paper, alough
not so accurate in low multiplexing gain, can provide
references in pratical code design in dual-polarized Rice
systems, expecially in systems with large amouts of an-
tennas.
REFERENCES
[1] C. Degen and W. Keusgen, “Performance of Polarisation
Multiplexing in Mobile Radio Systems,” Electronics Let-
ters, Vol. 38, No. 25, 2002, pp. 1730-1732.
doi:10.1049/el:20021118
[2] R. U. Nabar, H. Bolskei, V. Erceg, D. Gesbert and A. J.
Paulraj, “Performance of Multiantenna Signaling Tech-
nique in the Presence of Polarization Diversity,” IEEE
Trans. Signal Process., Vol. 50, No. 10, 2002, pp.
2553-2562. doi:10.1109/TSP.2002.803322
[3] Y. Deng, A. Burr and G. White, “Performance of MIMO
Systems with Combined Polarization Multiplexing and
Transmit Diversity,” in proc.2005 Conf. Vehicular Tech-
nol. VTC 2005-Spring. IEEE 61st, Vol. 2, pp. 869 -873.
[4] Mathini Sellathurai, Paul Guinand, and John Lodge
“Space-Time Coding in Mobile Satellite Communications
Using Dual-Polarized Channels”, IEEE Trans, vol. 55,
NO. 1, Jan 2006
[5] Claude Oestges, Bruno Clerckx, Maxime Guillaud, and
M´erouane Debbah, “Dual-Polarized Wireless Commu-
nications:From Propagation Models to System Perform-
ance Evaluation”, IEEE Trans. Wireless, vol. 7, no. 10,
oct, 2008
[6] M. Shafi, M. Zhang, A. Moustakas, P. Smith, A. Molisch,
F. Tufvesson, and S. Simon, “Polarized MIMO channels
in 3-D: Models, measurements and mutual information,”
IEEE Journal on Seleted Areas Commun. , vol. 24,no. 3,
pp. 514–527, Mar. 2006. doi:10.1109/JSAC.2005.862398
[7] Y. B. Li, H. Q. Wang and X.-G. Xia, “On
Quasi-Orthogonal Space-Time Block Codes for
Dual-Polarized MIMO Channels,” IEEE Transactions,

Y. P. HUANG, G. L. REN
325
Wireless Commun., Vol. 11, No. 1, 2012.
[8] L. Zheng and D. Tse, “Diversity and Multiplexing: A Fun-
Damental Tradeoff in Multiple Antenna Channels,” IEEE
Trans Info. Theory, Vol. 49, No. 5, 2003, pp. 1073-1096.
[9] H. Yao and Wornell and W. Gregory, “Structured
Space-Time Block Codes With Optimal Diver-
sity-Multi plexing Tradeo ff and Minimum Delay, ” in proc.
Conf. Global Telecommunications (GLOBECOM), 2003,
Vol. 4, pp. 1941-1945.
[10] R. Narasimhan, A. Ekbal and J. Cioffi, “Finite-SNR Di-
versity-Multiplexing Tradeoff of Space-time Codes,” in
Proc. 2005 , Int. Conf. Commun., 2005, pp. 458-462.
[11] R. Narasimhan, “Finite-SNRdiversity-multiplexing Trade-
off for Correlated Rayleigh and Rician MIMO Channels,”
IEEE Trans. Inf. Theory, Vol. 52, No. 9, pp. 3965-3979,
2006. doi:10.1109/TIT.2006.880057
[12] W.-Y. Shin, S.-Y. Chung and Y. H. Lee, “Diver-
sity-Multiplexing Tradeoff and Outage Performance for
Rician MIMO Channels,” IEEE Transactions Informa-
tions Theory, Vol. 54, 2008, pp. 1186-1196.
doi:10.1109/TIT.2007.915884
[13] C. Ammar El, Falou, W. Hamouda, C. Langlais, C. Abdel
Nour and C. Douillard, “Finite-SNR Diver-
sity-Multiplexing Tradeoff for Rayleigh MIMO,” IEEE,
Communications Letters, Vol. 17, 2013, pp. 753-756.
doi:10.1109/LCOMM.2013.022213.130007
[14] F. Talebi and T. G. Pratt, “Approximating the Outage
Capacity of Asymmetric 2×2 Dual-polarized MIMO at
High SNR,” in proce Int. Conf. (ICNC), 2013, pp. 290
-294.
Appendix 1

,
ii
ii i
R
EEE

i












According to the distribution of channel elements (5), an

(33)
2
H
i
F
HH  

which follows non-central chi-square distribution. Mean we get
and variance of i

 can be derived as :
 
 

2
2
2
2
2
2
2
12 11
1
2
21+
11
1
1
21
11
1
ii
f
fff
ff
E
k
k
k
k
kkk
kk
k
kk
kk
k


 
 









 




  


















  






 











 








 















(34)

2
2
21
1
22
21
f
i
f
i
k
Ek
kk
Dk









 






 

1

(31)
As for i

，by

112 2
11 221221
rjsrjs
H
hhhhr js


，

i
E


 and

i

D

 calculated as in [14] section
3 (11)-(14).


22
2
2
11
f
f
i
kk
Ek







 
 (32) Finally, using (15), exact expression of


DW is
given.
According to (14),is derived. For, as

EW

DW Appendix 2
Using the method prosed in [9] we derive the proof for
asymptotic DMT curve in dual-polarized uncorrelated

Y. P. HUANG, G. L. REN
326
Rice channels.The proof begins with Rayleigh fading
cases.
According to [9], let

2
logR

. Firstly ,we de-
compose H as:
11 1211 12
22 22
,00
rr rr
HQRXR X
rr



 



 (35)
Since

2
11 :, 1rH, and is approximated as
2
11
r
2
11 4
r2

, similarly，22
12 2
r

， 2
22 2
r2

.Thus, for
，keep quadratic term
1r22

and neglect the other
lower terms, we have


2
22
11 22
r
out t
PP rr
n











(36)
As is a variable with a higher order than , and
that small is mainly due to small , i.e., the
main event that causes
2
11
r2
22
r
22
11 22
rr 2
22
r


2
22
11 22
r
t
rr
n


 

 is

 
22
11 22
1r rr 

2
2
occurring.
Thus
 

222
11 22
1rr
out
PPr r


 
12r . When , diversity gain is derived as

,drr   2
1
(37)
And when , neglecting the constant term, we 0r
get



2
222 2
1112 22
2
22
11 22
t
out
r
t
rrr
N
PP
rr
N














 (38)
Here, main events are




 
2
2122 12
1112 22
21 134
1
22
rr
rr r
rrr
1

 
 


 







1
(39)
So that for 0r

 the diversity gain is derived as


,dr   3r 4，without any relationship with

.
Eventually, as LOS component do not affect the
high-SNR diversity gain [12], the asymptotic DMT
analysis here hold on for Rice channels. Therefore, as-
ymptotic DMT curve for DP Rice channels at infinite
SNR is the same as the conventional conclusion(9) in
[8].