Int. J. Communications, Network and System Sciences, 2010, 3, 668-673
doi:10.4236/ijcns.2010.38089 Published Online August 2010 (http://www.SciRP.org/journal/ijcns)
Copyright © 2010 SciRes. IJCNS
Transmit Diversity versus Beamforming System in
Correlated Fading Channel
Abdul W. Umrani1, Yong L. Guan2, Fahim A. Umrani1
1Institute of Information and Communication Technologies, Mehran University of Engineering & Technology,
Jamshoro, Pakistan
2Division of Communication Engineering, School of EEE, Nanyang Technological University, Singapore City, Singapore
E-mail: waheed.umrani@faculty.muet.edu.pk
Received May 25, 2010; revised June 27, 2010; accepted July 31, 2010
Abstract
The performance analysis, using outage capacity for a forward link cellular communications system is pre-
sented. The effects of correlated fading, the distribution of angle of arrivals, and the array configurations are
considered. The objective is to examine the effect of angle of arrival (AOA) energy distributions on two sys-
tems (transmit diversity and beamforming) with angle spread. We do so by comparing the performance of
transmit diversity with a system that uses beamforming to point a relatively narrow beam at the mobile sta-
tion. Analytical results show that the transmit diversity system using uniform linear arrays (ULA) and uni-
form circular arrays (UCA) with truncated Laplacian AOA, performs better even at smaller angle spreads as
compared to other energy distributions. The ULA geometry is a preferable configuration for transmit diver-
sity system as compared to UCA.
Keywords: Antenna Arrays, Angle Spread, Fading Correlation, Diversity Performance, Fading Channels, Cellular
Radio
1. Introduction
The use of multiple antennas at the base station to create
spatial diversity provides an effective technique for red-
ucing the effects of fading. It is known that Transmit
diversity system provides the benefit of diversity with no
array gain, while the transmit beamforming provides
array gain but no diversity [1]. The beamforming system
is normally designed so that the fading at the antennas is
highly correlated for wide range of angular spreads,
while transmit diversity is designed so that the fading
will be decorrelated for small angle spreads. Transmit
diversity systems are typically optimal in rich scattering
environments existing in indoor environments, which
means that the fading between antenna elements are in-
dependent. However, in low scattering environments (e.g.
in line-of-sight) where the fading is correlated, these
systems do not perform well. The authors are interested
in the performance of both systems as the fading correla-
tion changes from perfectly correlated to completely in-
dependent with AOA energy distribution under consid-
eration (i.e., Uniform, truncated Gaussian, and truncated
Laplacian).
Some of the research works using antenna arrays in
the downlink for particular environments have been pro-
posed in [2-6] for correlated fading channels. In these
papers, the authors use combination of space-time block
codes (STBC) with beamforming. In [4,5], the authors,
have studied the impact of fading correlation on the per-
formance of STBC. In [6], it was shown that the SNR for
both the transmit diversity method and beamformer is a
function of the sum of eigenvalues of fading correlation
matrix. Since fading correlation matrix is also a function
of angle of arrival, with some distribution, depending on
the particular environment. Some measurement results
show that the AOA energy distribution in general has a
shape which more closely resembles Gaussian or Lapla-
cian distribution [7], while other experiments suggest the
use of truncated Gaussian AOA distribution, when the
base station is near to the mobile station and truncated
Laplacian AOA distribution for micro-cellular radio en-
vironments.
The authors in [8], presented a comparative analysis
and the tradeoffs between transmit diversity systems and
beamforming for different number of antennas for the
forward link cellular communication system with and
without hand-off. However, the effect of angle of arrival
A. W. UMRANI ET AL.
669
(AOA) distributions and the array configuration (i.e.,
linear, circular, etc.) were not taken into account in [8].
In this paper, we study the transmit diversity and trans-
mit beamforming for multi-input-single-output (MISO)
system, and present a comparative analysis/results of the
two systems by taking into account the various AOA
energy distributions, and array geometry. The objective
is to examine the effect of angle of arrival (AOA) distri-
butions on two systems with angle spread, which is de-
fined as standard deviation of the distribution under con-
sideration (i.e., Uniform, truncated Gaussian, and trun-
cated Laplacian). We also want to see the performance of
uniform linear arrays (ULA) and uniform circular arrays
(UCA) with these distributions. In this paper, frequency
division duplex (FDD) system is considered, where the
up and downlink channels fade independently. Therefore,
the base station (BS) is unable to estimate the downlink
channel. This is also referred to as open-loop transmit
diversity. Certainly, the closed-loop transmit diversity,
which uses feedback, has the potential to provide diver-
sity as well as array gain advantage, but that is beyond
the scope of this paper. We also assume that the receiver
knows the channel by using pilot or training signal. The
part of this work is also reported in [9,10].
In Section 2, we present a detail description of system
model. Then a brief discussion of spatial correlation and
outage capacity is presented in Section 3. Section 4 de-
tails the results and some discussion on it. Finally, con-
clusion is presented in Section 5.
Notation: Lower case boldface letters are used to de-
note vectors and upper case boldface letters to denote
matrices, (.)T denotes the transpose, and (.)H denotes the
Hermitian transpose. In addition, (A) means the real
part of A, (A) means the imaginary part of A.
2. System Model
A wireless communication system with M array antennas
(ULA or UCA) at the base station and a single antenna at
the mobile station is considered. Specifically, the ULA
consists of four elements located on the x direction while
the UCA consists of four elements lying on a circle about
the origin as shown in the Figure 1 below. This configu-
ration is similar to as considered by [11] (Figure 1).
Consider an array receiving a narrowband signal from
a point source. Let the array response vector be denoted
by a(θ) = [a1,…, aM]T, where θ is the direction of the so-
urce relative to array. In the case of ULA, it is given by


21/sinjmd
m
ae

(1)
where am(θ) is the mth entry a(θ), d is the spacing be-
tween the antennas, is the wavelength. Without loss of
generality, we assumed here that the phase of first ele-
ment is zero. Note that adding the same fixed phase to all
1
2
3
4
Figure 1. Four element ULA and UCA antenna array con-
figuration at the base station. R is radius of circular array
and d is the spacing between array elements, d = 2R/3.
of the elements of the array does not change any of the
results. Similarly, in the case of UCA, it is given by,


21/sincos
m
jmR
m
ae

(2)
where R is the circular radius of the array,
is the eleva-
tion angle of arrival. For simplicity, only azimuth angles
are considered in propagation geometry (i.e.,
= 90),
and
m is the angle that each element location makes
with horizontal axis as shown in [11] Figure 1. For
four-element UCA configuration,
m’s are 0, 90, 270
and 360, respectively.
Let SNR denote the average signal-to-noise ratio at the
mobile station, when the base station uses a single an-
tenna. Then, the instantaneous signal-to-noise ratio (snr)
observed by ideal transmit diversity system is given by
[8],
2
SNR
snr
M
(3)
where
2
1
222
MHH
m
m
a

aa uRu (4)
where u is vector of independent Gaussian random vari-
ables with zero mean and unit variance, R is the spatial
covariance matrix of a. Using singular value decomposi-
tion of R,
can finally be written as
2
1
2
M
mm
m
u

(5)
where
m are singular values of R. In general, the pdf of
is given by [12]
 
1
M
mm
m
fp

(6)
where

1,
1,&
m
M
m
mm
iim
mm
pe

i



(7)
Copyright © 2010 SciRes. IJCNS
670 A. W. UMRANI ET AL.
3. Spatial Correlation and Outage
Probability
In [13], a detail analysis on the spatial fading correlation
for uniform AOA on linear arrays was carefully studied.
Further research on spatial fading correlation has been
carried out for various AOA energy distributions in [11,
14-18], which include Gaussian and Laplacian distribu-
tion etc. In [11], the bit error rate performance is com-
puted for ULA and UCA for truncated Gaussian AOA
distribution, and in [14], a correct version of closed form
equation for spatial fading correlation matrix is reported.
Similarly, for truncated Laplacian distribution the fading
correlation equations are reported in [16,17]. Some me-
asurement results suggest that the AOA energy distribu-
tion in general has a shape which more closely resembles
Gaussian or Laplacian distribution [7], while other ex-
periments suggest the use of truncated Gaussian AOA
distribution, when the base station is near to the mobile
and truncated Laplacian AOA distribution for micro-
cellular radio environments.
For the sake of completeness we write the truncated
Gaussian AOA distribution [11], but corrected in [14] as,


2
2
2
s
a
sg s
fCe




(8)
where
1
2
2
g
a
a
C
erf




(9)
Similarly truncated Laplacian AOA distribution is
written as [16],

s
l
fCe




 (10)
where

21
la
s
Ce
(11)
In (8)-(11), θ is the mean AOA, a
is the angle spr-
ead of truncated Gaussian distribution. Similarly, s is the
decay factor, which is related to the angle spread. Spe-
cifically, as s increases, the angle spread decreases. The
closed form expressions for the real and imaginary parts
of spatial covariance matrix R, for uniform linear array
having uniform AOA, can easily be computed from [13].
For the case of truncated Gaussian AOA, the real and
imaginary parts of spatial covariance matrix for ULA,
and UCA are given [11], and modified as in [14] by (12)
-(15), respectively.




22
0
2
2
1
(,) 22
cos2
a
lga
k
kl
k
mnJ ZC
eJZk

 


R
(12)




22
21
2
21
0
(,) 22
sin21
a
k
gak l
k
mnCeJ Z
k




R (13)


22
0
2
2
1
(,) 22
cos2
a
cga
k
kc
k
mnJ ZC
eJZk

 


R
(14)




22
21
2
21
0
(,) 22
sin21
a
k
gak c
k
mnCeJ Z
k






R (15)
Similarly, for the case of truncated Laplacian AOA,
these are given by (16)-(19), for ULA and UCA, respec-
tively [16,17],

 

02
1
2
22
(,) 2
cos2
4
lk
k
mnJ ZJZ
sk
sk
l

R
(16)






2
21 2
2
1
1
(,) 2
21 1
sin21
s
kl s
k
se
mnJ Z
sk e
k
 

 



R
(17)

 

02
1
2
22
(,) 2
cos2
4
ck
k
mnJ ZJZ
sk
sk
c

R
(18)






2
21 2
2
1
1
(,) 221 1
sin 21
s
kc s
k
se
JZ
mn sk e
k


  




R
(19)
where

2
ld
Zmn
 and Zc and
are defined in
[11]. And Jn(.) is the Bessel function of first kind and of
order n.
A convenient approach to comparing the two systems
is to calculate their respective channel capacities. This
provides a comparison that is independent of the specific
system details (such as modulation, coding, frame/block
size, etc.). The Shannon capacity bound is very well-kn-
own. However, it does not always provide a good meas-
ure of performance of practical systems, because of the
underlying assumptions on infinite block lengths (which
imply infinite delay) and error probabilities approaching
zero. Practical wireless systems operate with finite, often
Copyright © 2010 SciRes. IJCNS
A. W. UMRANI ET AL.
671
quite short, block lengths and are designed for small but
non-vanishing error probabilities. Their performance,
especially in the presence of correlated fading, is often
worse than predicted by the Shannon bound.
Outage capacity has been used by number of authors,
for example in [8] and the references therein, in an atte-
mpt to provide a more realistic characterization of per-
formance of wireless communication systems than is
provided by Shannon capacity. In these works, the cap-
acity is treated as a random variable, being a function of
randomly varying signal-to-noise ratio. They define the
outage capacity Cout(p) with probability p. This capacity
has the intuitive interpretation as the highest transmission
rate that can be sustained with probability (1 – p). The
random capacity of a transmit diversity system is given
by,
2
log 12
SNR
CM


(20)
The pdf f(C) of the capacity can be computed from the
pdf of
using standard random variable transformation
and can be given by [8],


1
2log2 2
21
C
C
M
M
fC f
SNR SNR



(21)
The outage capacity Cout(p) for probability p is given
by


out
Cp
p
fCdC

(22)
For the case of a beamformer with weight vector W =
[w1,…, w
M], assumed to have unit-norm. The snr at the
beamformer output is
2
2
1
M
mm
m
s
nrSNRxSNRwa

(23)
where x is a zero-mean complex Gaussian-random vari-
able with variance g = WHRW, with R being covariance
matrix of the channel response vector a. In the fully cor-
related case, we can write (23), as
2
;where, 2
2
SNR
s
nr Ma

 (24)
where a is a complex Gaussian variable with zero-mean
and unit variance. Therefore, the pdf of the capacity of
the beamforming system is given by [8],


1
2log2 2
21
C
C
M
fC f
M
SNR MSNR




(25)
4. Simulation Results
The notation in the following figures are given as, TD-U,
TD-G, and TD-L which stands for transmit diversity sys-
tem with uniform AOA, transmit diversity with truncated
Gaussian AOA, and transmit diversity with truncated
Laplacian AOA, respectively.
In Figure 2, a comparison is shown at angle spread (σ
= 3°). At 2% outage probability the performance of TD-
U and BF (fully correlated case) is same as concluded by
[8] (see Figure 8), but that is not the case with TD-G and
TD-L as can be seen from Figure 2 [9,10]. A similar co-
mparison is also plotted when the base station uses UCA.
In this case, the performance of TD-U and BF (fully corr-
elated case) is almost same, but inferior to the case when
ULA used at the base station. It is clear from the Figure
2 and Figure 3 that the transmit diversity systems do not
perform well at smaller angle spread, especially for trun-
cated Gaussian AOA.
05 10 15 20 2530
10
-3
10
-2
10
-1
10
0
SNR (dB)
Outage Probability
=3
o
BF-cor
TD-uncor
TD -G
TD -U
TD -L
Figure 2. Outage probability for Transmit diversity and
beamforming systems for ULA; mean AOA θ = 0°, σ = 3°,
Cout(p) = 3, M = 4.
05 1015 20 25 30
10
-3
10
-2
10
-1
10
0
SNR (dB)
Outage Probability
= 3
o
Bf-cor
TD, Unco
TD -G
TD -U
TD -L
Figure 3. Outage probability for Transmit diversity and
beamforming systems for UCA; mean AOA θ = 0°, σ = 3°,
Cout(p) = 3, M = 4.
Copyright © 2010 SciRes. IJCNS
672 A. W. UMRANI ET AL.
To see the effect for wider range of angle spreads, the
plot of probability distribution function (pdf) for several
values of angle spread using (6), and (12)-(13) are shown
in Figures 4 and 5 of [10], where it can be observed that,
for a small angle spread of (σ = 1°), the TD-G curve
tends to reach the χ2
2 distribution curve, while for mod-
erate angle spread (σ = 7°), it tends to approach χ2
2M.
This is due to reason that, in the case of uncorrelated
fading R = I, λm = 1, in which case γ has a χ2
2M distribu-
tion, while in the fully correlated case, R has rank one
and λ1 = M, while λm = 0 for m > 1, in that case, γ/M has
χ2
2 distribution as can be observed from Figure 5 of [10].
Figures 4 and 5, show the outage capacity curve as a
function of SNR of transmit diversity system at p = 2%,
for several values of angle spread. In Figure 4, a comp-
arison for TD-U is given for ULA and UCA for two val-
ues of angle spread. We can see that, base station with
ULA configuration provides higher capacity than that of
UCA. Specifically, at σ = 7° and C = 3 bits/s/Hz, UCA is
2 dB inferior to ULA. Similarly for TD-L and TD-G a co-
mparison is also plotted in Figure 5, for ULA and UCA
for σ = 7°. It can be noted that, TD-L provides higher
capacity than the other two energy distributions.
Figure 6, depicts a plot of spatial correlation as func-
tion of antenna separation for ULA for two values of an-
gle spread that is for σ = 3° and σ = 7°. Figure 6 also
gives sufficient justification that the truncated Laplacian
AOA energy distribution gives lower correlation as com-
pared to other energy distributions.
Figure 7, shows the outage capacity curve as a func-
tion of SNR for TD-U for several values of central AOA
at p = 2%, and σ = 3°. It can be seen from the Figure 7
that the central AOA can have significant impact on the
performance. Specifically, at θ = 0° and 90° TD-U provi-
des lower capacity values than other central AOA. Only
UCA and uniform AOA energy distribution is considered.
Similar conclusion can be made for other two energy dis-
05 10 1520 25 30
0
1
2
3
4
5
6
7
8
9
10
SNR (dB)
Capac ity
TD-U
UCA,
= 3
o
UCA,
= 7
o
ULA,
= 3
o
ULA,
= 7
o
Figure 4. Outage capacity for Transmit diversity for ULA/
UCA with uniform AOA; θ = 0°, M = 4, p = 2%.
tributions.
Based on the above results, we can conclude that the
transmit diversity system having ULA or UCA at the ba-
se station with truncated Laplacian AOA, perform better
than uniform and truncated Gaussian AOA, even at sma-
ller angle spreads. We also show that ULA provide hi-
gher capacity than UCA for transmit diversity systems.
This conclusion is in consistent with the results reported
in [19] for the indoor clustered channels for MIMO com-
munications.
5. Conclusions
In this paper, we presented a comparative analysis of tra-
nsmit diversity system and transmit beamforming for the
downlink of wireless communication system, using out-
age capacity as performance measure. We examined the
0510 15 20 25 30
0
1
2
3
4
5
6
7
8
9
10
SNR (dB)
Capaci t y
= 3
o
UCA, TD-G
UCA, TD-L
ULA, TD-G
ULA, TD-L
Figure 5. Outage capacity for Transmit diversity for ULA/
UCA; mean AOA θ = 0°, σ = 3°, M = 4, p = 2%.
00.5 11.5 22.5 33.5 44.5 5
0
0. 2
0. 4
0. 6
0. 8
1
Antenna seperation, D/
|R(1,2)|
U-
= 3
o
G-
= 3
o
L-
= 3
o
U-
= 7
o
G-
= 7
o
L-
= 7
o
Figure 6. Spatial envelop correlation between antenna 1
and 2 for ULA; mean AOA θ = 0°, M = 4.
Copyright © 2010 SciRes. IJCNS
A. W. UMRANI ET AL.
Copyright © 2010 SciRes. IJCNS
673
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05 10 15 20 2530
0
1
2
3
4
5
6
7
8
9
10
SNR (dB)
Capaci ty
TD- U,
= 3
o
= 0
o
=22.5
o
= 45
o
= 67.5
o
=90
o
θ
θ
θ
θ
θ
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Figure 7. Outage capacity for Transmit diversity for UCA
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transmit diversity system using linear arrays or circular
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extreme case, where fading is completely correlated, the
transmit diversity performs no better than a single ante-
nna. Results also show that the transmit diversity system
using arrays (ULA or UCA) with truncated Laplacian
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compared to other energy distributions. Further, ULA se-
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