Int. J. Communications, Network and System Sciences, 2010, 3, 266-272
doi:10.4236/ijcns.2010.33034 blished Online March 2010 (http://www.SciRP.org/journal/ijcns/).
Pu
Performance of Block Diagonalization Broadcasting
Scheme for Multiuser MIMO System Operating in
Presence of Spatial Correlation and Mutual Coupling
Feng Wang, Marek E. Bialkowski, Xia Liu
School of Information Technology & Electrical Enginnering, University of Queensland, Brisbane, Australia
Received December 31, 2009; revised January 30, 2010; accepted February 27, 2010
Abstract
In this paper, the capacity of a multiuser Multiple Input Multiple Output (MIMO) system employing the
block diagonalization broadcasting scheme in presence of spatial correlation and mutual coupling is investi-
gated. It is shown by computer simulations that, in general, the presence of spatial correlation decreases the
capacity of a multiuser MIMO system. However, for some particular antenna element spacing mutual cou-
pling decreases the spatial correlation rendering an increased capacity. The optimized diagonalization
broadcasting technique with a two-stage power allocation scheme is proposed and verified. The presented
simulations results confirm the advantage of the proposed broadcasting scheme.
Keywords: Multiuser MIMO, Block Diagonalizaiton, Mutual Coupling, Dirty Paper Coding
1. Introduction
It has been shown via theoretical derivations as well as
by experiment that using multiple element antennas with
a suitable signal transmission scheme in a rich scattering
propagation environment can enhance peer-to-peer com-
munication without the use of extra frequency bandwidth
[1,2]. This potential of multiple input multiple output
(MIMO) communication systems can be used to advan-
tage using two alternative approaches. In one approach,
the signal transmission quality via diversity can be im-
proved. Alternatively for a chosen quality factor such as
bit error rate (BER), the data rate can be increased by a
stream multiplexing transmission.
Most recent studies on MIMO focus on multiuser sys-
tems. For a multiuser MIMO system, allocation of the
channel resources among independent users either in the
form of multiple accesses (uplink) or broadcasting (dow-
nlink) is considered. The information theory hints that
the broadcasting case is by far the most challenging. In
this case, an inter-user interference occurs due to the
spatially multiplexed transmitted signals at a base station
(BS). For the Gaussian MIMO broadcasting channels, it
has been proved that dirty paper coding (DPC) [3] can
achieve the available capacity [4]. However, to deploy
DPC in a real system is challenging due to the high com-
plexity and computational burden on successive encod-
ing and decoding. An alternative strategy is the block
diagonalization (BD) [5,6]. Compared with DPC sche-
me, BD is a suboptimal technique with much reduced
complexity. Using this technique, signals are transmitted
only to desired users. In turn, null steering is applied to
other users by decomposing the multiuser channel into a
group of parallel single user MIMO channels. To achieve
such decomposition, BS needs to select a suitable beam-
forming matrix for each user. The matrix is vertical to
the space spanned by other users’ channels matrices. If
the channel matrices of all the scheduled users are per-
fectly known at the transmitter, the inter-user interfer-
ence can be eliminated by BD, rendering a simple re-
ceiver structure.
Because of its simplicity and good performance, BD is
under constant research. In [7], the imperfect channel
state information (CSI) assumption was used while in-
vestigating BD. The effect of outdated CSI at transmitter
on multiuser MIMO system with BD was reported. In [8],
a BD algorithm that accounts for the presence of other-
cell interferences was proposed under the assumption
that the transmitter has full CSI and the information
about the interference plus noise covariance matrix for
in-cell users. Most of the research on BD for multiuser
MIMO systems was done by neglecting interactions
within the transmitting and receiving array antennas and
between the array antennas and scatterers. When a

F. WANG ET AL. 267

MIMO transceiver has to be of compact size interactions
within the transmitting or receiving array antennas have
to be taken into account. The small inter-element spacing
in the antenna array in such transceivers renders mutual
coupling. The effect of mutual coupling on a point-
to-point MIMO system has been investigated in many
works, such as [9,10,11]. In this paper, a BD algorithm
that accounts for the effect of spatial correlation and
mutual coupling in array antennas is presented and its
performance is evaluated with respect to the overall sys-
tem capacity.
The paper is organized as follows. Section 2 describes
a multiuser MIMO system model including the channel
model with spatial correlation. Section 3 describes inter-
actions between the array elements and scatterers in the
propagation environment in which mutual coupling ef-
fects cannot be neglected. Section 4 gives details of the
BD algorithm that accounts for the effect of mutual cou-
pling. Section 5 quantifies the effect of mutual coupling
by presenting numerical results. Section 6 summarizes
the findings of the undertaken research.
2. System Model
2.1. Signal Model
A narrowband multiuser system is assumed. It is postu-
lated that it is created around a base station (BS) with L
downlink mobile users. The base station includes N
transmitting antennas. At time t, K mobile stations (MS)
from L available users are scheduled to be serviced by
BS. The k-th mobile station (MS) employs Mk antennas.
The transmitted signal intended for the k-th mobile sta-
tion is denoted by a Qk × 1 dimensional vector xk which
is weighted by an N × Qk pre-processing matrix Wk be-
fore transmission. Qk is the number of parallel data
symbols transmitted simultaneously to the k-th MS. The
MIMO channel between the BS the k-th MS is described
by the complex matrix Hk, whose (i,j)th entries represent
the complex gain between the j-th transmit antenna at BS
and i-th antenna at k-th MS. It is assumed that different
MS experience independent fading. The received signal
at k-th MS can be presented by
1
1,
K
kk kkkk
k
K
kkkkk jjjk
jjk






yH WExn
HWExHWEx n
(1)
where trace(EkE†
k) = pk is the power transmitted to the kth
MS. nk is the additive Gaussian white noise (AWGN)
vector, whose elements are independent identical distri-
bution (i.i.d.) zero-mean circularly symmetric complex
Gaussian random variables with variance σn
2.
2.2. Channel Model
The channel matrix Hk describing the channel properties
between BS and the kth MS is influence by the transmit-
ting and receiving antenna array configurations and a
signal propagation environment. It is assumed that the
BS and MSs are equipped with wire dipoles arranged in
liner arrays. The length of each dipole element is as-
sumed to be half wavelength. Also, the links between BS
and different MSs do not share the same scattering envi-
ronment. This assumption confirms the earlier assumed
independent signal fading for different MSs. For each
link, the Kronecker channel model [5,12] is assumed. In
this model, the correlations at transmitter and receiver
sides are independent and the channel matrix Hk is rep-
resented as


1/2 1/2k
kMS
H HR GR
BS
(2)
where GH is a matrix with i.i.d. Gaussian entries with
zero mean and unit variance and k
M
S
R and RBS are spa-
tial correlation matrices at the kth MS and BS, respec-
tively. In a rich scattering environment, the correlation
for any pair of dipole element with spacing dm,n can be
obtained using Clark’s model and are given by a Bessel
function
,0,
(
mn mn
Jkd )


(3)
Using (3), the correlation matrix for the kth MS can be
generated as
1,1 1,
,1 ,
k
kk
M
k
MS
MM


k
M











R

 

(4)
In turn, the correlation matrix for BS can be obtained
from
1,1 1,
,1 ,
N
BS
NN


N











R

 

(5)
3. Mutual Coupling
For the array formed by linear parallel wire dipoles, the
mutual coupling matrix can be expressed using electro-
magnetic and circular theory described in [9]
1
(ΖΖ)( )
AT TM

 ZCI (6)
where ZA = 73 + j42.5[Ω] is the element impedance in
isolation and ZT is impedance of the receiver at each ele-
ment. It is chosen to be the complex conjugate of ZA to
obtain the impedance match. Z is the mutual impedance
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F. WANG ET AL. 268
matrix with all the diagonal elements equal to ZA + ZT, its
non-diagonal elements Znm are decided by the physical
parameters including dipole length, the horizontal dis-
tance between the two dipoles. For a side-by-side array
configuration and dipole length l equals to 0.5λ, Znm is
given by [9,10]





012
012
300.5722ln(2 )(2 )
30(2) ,
30 2()()()
302() () ()
i
i
mn
iii
iii
lCl mn
jS l
ZCuCuCumn
jSuSuSu


 






(7)
where β=2π⁄λ is the wave number and Ci(u) and Si(u) are
the cosine and sine integral, respectively, given as


0
cos( )
()
u
i
u
i
x
Cu dx
x
sinx
Su dx
x





(8)
and the constants are given by [10]


0
22
1
22
2
h
h
h
ud
udl
udl






l
l
(9)
where dh is the horizontal distance between the two di-
pole antennas.
4. Block Diagonalizaiton with Mutual
Coupling
We assume at time t, K mobile stations (MS) from L
available users are scheduled to be serviced by BS. To
ensure the sufficient freedom for BS to perform BD over
the K scheduled MSs, it is assumed that
1
K
k
k
M
N


 (10)
With spatial correlation and mutual coupling taken
into account, the received signal at kth MS described by
(1) can be rewritten as



1/ 21/2
1
1/ 21/ 2
1/ 21/2
1,
K
kk
kMSMSBSBSkkkk
k
kk
MSMSBSBSkk k
K
kk
M
SMSBSBSjjj k
jjk







H
H
H
yCRGRCWExn
CR GRCWEx
CR GRCWExn


(11)
where, CBS and k
M
S
C are the mutual coupling matrices
for the dipole element array at BS and the kth MS, re-
spectively. Wk is the beamforming matrix at BS for the
kth MS. To eliminate the interference from the signals
transmitted to other MSs, the key idea in the block di-
agonalization is to zero-force the interference by impos-
ing the following condition


1/ 21/2(,1,
kk
MSMSBS BSjjk kjK H
CR GRCW0)
(12)
when the mutual coupling and correlation is taken into
account,


1/ 21/ 2
12
12
[, ,,]
[, ,,]
kk
kMSMSBSB
T
K
K



H
HCR GRC
HHH H
WWW W


S
(13)
where (•)T donates the matrix transpose operation. By
including the condition given by (12), the effective chan-
nel matrix for the multiuser MIMO system with K MSs
can be represented by a ∑Mk × ∑Mk matrix, given as
11121
21 222
1
K
K
KK

K















DHW
HW HWHW
HW HWHW
HW HW





(14)
By using (12), Equation (14) can be rewritten as
11
22
KK















HW0 00
0HW0 0
D00 0
000HW
 (15)
At this point it is important to comment whether the
condition (12) can be met in practice. From the theory of
antennas it is known that an N-element array antenna is
capable of forming N-1 nulls. This means that in the
strict sense, the BS having an N-element array is able
only to null up to N-1 MSs. For a larger number of MSs,
the condition (12) has to be compromised. In such a case,
the BS can direct low sidelobes instead of nulls towards
undesired users. In further considerations, it is assumed
that the number of MSs served by BS is such that the
condition (12) is met.
4.1. Calculation of Beamforming Matrices
In order to transmit a signal only to the desired MS while
steering nulls to the remaining MSs, the beamforming
matrix for the desired MS should be orthogonal to the
space spanned by the channel matrices of the undesired
MSs. We define the channel matrix as
111
[,,,
T
kkk
HHHH H
]
K
(16)

F. WANG ET AL. 269
which is obtained by removing the channel matrix for the
kth MS from . Performing the eigenvalue decomposi-
tion (EVD) over the N × N non-negative Hermitian Ma-
trix, one obtains
H

†
†
†
[] k
kk kk
k






Σ0V
HHVV 00V

 
(17)
where (•) † denotes the conjugate transpose operation.
It can bee seen that Vk is a matrix with the dimension
of N × Mk. Its columns correspond to those zero eigen-
values. By letting Wk = Vk, a perfect null steering to all
the undesired K-1 MSs can be achieved. By repeating the
steps represented by Equation (15) and (16), all the K
beamforming matrices can be obtained. In this way, as
shown in Equation (14), the multiuser MIMO downlink
system is decomposed into K independent single-user
MIMO systems.
4.2. Overall Capacity of Multiuser MIMO
Broadcasting with Block Diagonalization
For the case of a multiuser MIMO downlink system
which is decomposed into K independent single-user
MIMO systems by block diagonalization, the overall
capacity can be obtained as a sum of individual links
capacities, as expressed by
†††
22
1
1
log det
K
k
s
umk kkk kk
kn
C






xx
IEWHRHWE

(18)
With mutual coupling and spatial correlations taken
into account, (18) can be rewritten as
†††
22
1
1
log det
K
k
s
umk kkk kk
kn
C






xx
IEWHRHWE



(19)
where is the kth MS’s input covariance matrix.
The capacity for kth MS is
k
xx
R
†††
22
1
log detk
kkkk
n
C





xx
IEWHRHWE
kkk (20)
We assume that the signal intended for the kth MS is a
Gaussian signal. As a result, (20) can be simplified to
††
22
†† ††
22
log det
log det
k
kkkkk
n
k
kBSkMSMSkBSk
n
p
C
p










IWHHW
IWCHCCHCW
For high SNRs, (21) can be further simplified and
given by



†† ††
22
††
22
††
†† †
22
2
†
log det
log det
det det
log detlog det
det
k
kkBSk MSMSk BSk
n
k
kkkk
n
MS MSBS BS
k
kkkkMSMS
n
BS BS
p
C
p
p












WC HCCHCW
WHHW
CC CC
WHHWC C
CC



(22)
The last part of Equation (22) shows three terms con-
tributing to the capacity. The first term represents the
broadcasting capacity for the kth MS without the effect
of mutual coupling at BS and MS. The second and third
terms represent the mutual coupling at kth MS and BS.
The effect of these terms on capacity depends on the
coupling matrices at the MS and BS ends. If the product
of the determinants of the mutual coupling matrices is
larger than one, the effect of mutual coupling on capacity
is positive. Otherwise, it is negative.
4.3. Power Allocation Scheme
The most straightforward power allocation scheme from
BS to different MSs is accomplished by transmitting
equal power to each MS. That is
†
trace() T
kkk
P
p
K
EE  (23)
where PT is the total transmitted power at BS and pk is
the power allocated to the kth MS.
This power allocation scheme is simple to realize in
practice. However, it does not always provide the best
performance with respect to capacity. To maximize the
capacity, a two-stage power allocation scheme is pre-
sented. At the first stage, the power allocation is accom-
plished according to the objective function at the users’
level, as expressed by


12
12
,,,
1
Max,, ,
K
sum K
pp p
K
kT
k
Cpp p
Subject topP





(24)



(21)
The result of (24) is the optimized power allocation for
different users under service. This is the capacity-greedy
power allocation scheme and is non-linear. The solution
can be obtained by applying a Lagrange method.
At the second stage, the transmit power for each user
can be optimized at an antenna level by using a wa-
ter-filling scheme. At this stage, the power is allocated to
C opyright © 2010 SciRes. IJCNS

F. WANG ET AL. 270
different transmit antennas according to the objective
function, which is described by
2
1
,1,2,,
in
kk
i
k
r
i
kk
i
pi
Subject topp





 




r
(25)
where (z)+=max(0,z) and μk is chosen to obey the power
constraint for the kth MS and r is the rank of the effec-
tive channel matrix between BS and the kth MS



1/2 1/2
rank kk
M
SMS BSBSk
rH CR GRCW (26)
By applying the water-filling scheme, the capacity for
kth MS is

2
2
1
1
log 1
r
i
kkk
in
C  




 




n
(27)
5. Numerical Results
Using the presented theory, computer simulations are
performed for a multi-user MIMO system with 8 trans-
mit antennas at BS and 3 MSs each equipped with 2 re-
ceive antennas. It is assumed that the three MSs are
scheduled and served by BS at the same time. As a result,
this system is referred to as a 3 × (2 × 8) system.
Figure 1 presents the possible impact of spatial corre-
lation and mutual coupling on the broadcasting through-
put. In simulations, the dipole spacing at BS and MS is
assumed to be fixed at 1.0λ and 0.5λ, respectively.
As observed from the results presented in Figure 1,
Dirty Paper Coding, where effects of spatial correlation
and mutual coupling are neglected, offers the largest
-5 05 10 15 20
0
5
10
15
20
25
30
35
40
45
SNR[dB]
Sum Throughput[bps/Hz]
Dirty Paper Coding
BD with Rayleigh Channel
BD with Spatial Correlation
BD with Mu tu a l Coup li n g
Figure 1. Broadcasting throughput for a 3 × (2 × 8) system.
scheme in which spatial correlation and mutual coupling
are neglected shows a reduced throughput. The through-
put for BD with mutual coupling or spatial correlation
included in calculations further reduces the system throu-
ghput. The differences are most pronounced at larger
levels of SNR.
Figure 2 shows the effect of spatial correlation and
mutual coupling on the broadcasting throughput for a 3×
(2×8) system. The SNR is set to 10 dB and the unit for
dipole spacing is the wavelength, represented by λ. The
solid lines represent CDF of broadcasting throughput
with spatial correlation only and the dotted lines are for
the CDF of broadcasting throughput with spatial correla-
tion and mutual coupling combined. It can bee seen form
Figure 2 that the presence of spatial correlation and mu-
tual coupling results in a degraded broadcasting throug-
hput in comparison with an idealized Rayleigh channel. .
In general, spatial correlation is regarded as a negative
factor in a MIMO communication system. However,
mutual coupling can be seen as a positive factor at some
dipole spacing. As observed in Figure 2, for the dipole
spacing of 0.2λ and 0.3λ, the existence of mutual cou-
pling results in a higher capacity. It is interesting to note
that the curve of the capacity with and without mutual
coupling merge at the point of dipole spacing equal to
0.4λ. When the spacing is increased to 0.6λ, the plot rep-
resenting the capacity with mutual coupling is on the left
side of the curve for the capacity with correlation only.
This is the case for which the presence of mutual cou-
pling leads to a lower capacity.
Figures 3 and 4 show comparisons between capacity
with spatial correlation only, and with spatial correlation
plus mutual coupling, as a function of antenna element
spacing.
In the presented simulation results, the SNR is set to
67 8 91011 12 13 1415
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
0. 9
1
Sum Throughput[bps/Hz]
CDF
Dipole Spacing =0.2 No M C
Dipole Spacing =0.2 Wit h M C
Dipole Spacing =0.3 No M C
Dipole Spacing =0.3 Wit h M C
Dipole Spacing =0.4 No M C
Dipole Spacing =0.4 Wit h M C
Dipole Spacing =0.6 No M C
Dipole Spacing =0.6 Wit h M C
Rayl ei gh Channel
Dipole Spaci ng =0.2
Dipole Spacing =0.3
Dipole S pacing =0.6
Dipole S pac i ng = 0. 4
Figure 2. Broadcasting throughput CDF for a 3 × (2 × 8)
system.

F. WANG ET AL. 271
0.10.2 0.30.4 0.50.60.7 0.8 0.91
6
6. 5
7
7. 5
8
8. 5
9
9. 5
Dipole spacing
Sum Throughput[bps/Hz]
BD with Spatial Correlation
BD with Mutual Coupling
Figure 3. Broadcasting throughput vs. MS array interele-
ment spacing for a 3 × (2 × 8) system.
00.2 0.4 0.6 0.8 1
-5
0
5
10
15
20
0
5
10
15
20
25
MS Int er-elem ent spaci ng
SNR(dB)
Sum T hroughput(bps/Hz )
5
10
15
20
BD with Spatial Correlation
BD with Mutual Coupling
Figure 4. Broadcasting throughput vs. MS array interele-
ment spacing and SNR for a 3 × (2 × 8) system.
10 dB and the unit for dipole spacing is the wavelength,
as represented by λ. The dipole spacing ranges from 0.0λ
to 1.0λ. We can see that the curves for BD with spatial
correlation only and BD with spatial correlation plus mu-
tual coupling cross at 0.4λ and 0.95λ. For the dipole
spacing range from 0.4λ to 0.95λ, mutual coupling in-
creases the spatial correlation level and results in a de-
creased capacity. In turn, when the dipole spacing ranges
from 0.1λ to 0.4λ, mutual coupling decreases the spatial
correlation level and renders an increased capacity.
The results presented in Figures 5 and 6 verify the
two-stage power allocation scheme described in Section 6.
One can see from results presented in Figures 5 and 6
that with or without mutual coupling, the optimized
power allocation scheme leads to a higher capacity than
the non-optimized one over the SNR range from 5 dB to
20 dB and the antenna spacing from 0.1λ to 1λ. The
0
0. 2
0. 4
0.6
0.8
1
-5 0510 15 20
0
5
10
15
20
25
30
MS Inter-el ement spac ing
SNR(d B)
Sum T hroughput(bps/Hz)
5
10
15
20
25
Opt im i zed Broadc asting Throughput
Non-optim i zed Broadc asting Throughtput
Figure 5. Comparison of optimized and non-optimized
broadcasting throughput vs. MS array interelement spacing
and SNR for a 3 × (2 × 8) system in the presence of spatial
correlation only.
0
0. 2
0. 4
0. 6
0. 8
1
-5
0
5
10
15
20
0
10
20
30
S NR(dB)
MS Int er-elem ent spac i ng
S um T hroughput(bps/ Hz)
5
10
15
20
25
Opt i m i zed Broadc asti ng Throughut
Non-opti m i zed Broadc asti ng Throughut
Figure 6. Comparison of optimized and non-optimized
broadcasting throughput vs. MS array interelement spac-
ing and SNR for a 3 × (2 × 8) system in the presence of spa-
tial correlation and mutual coupling.
optimized scheme improves capacity in the presence of
spatial correlation and mutual coupling. This achieve-
ment is more apparent at higher values of SNR and lar-
ger inter-element antenna spacing.
6. Conclusions
In this paper, investigations into the capacity of a multi-
user MIMO system with block diagonalization broad-
casting scheme in the presence of spatial correlation and
mutual coupling have been presented. The effect of spa-
tial correlation and mutual coupling on the broadcasting
throughput for block diagonalization broadcasting has
been analyzed. It has been shown by the performed
computer simulations that the presence of spatial correla-
C opyright © 2010 SciRes. IJCNS

F. WANG ET AL. 272
tion leads to a decreased capacity. However, mutual cou-
pling may have negative or positive influence of capacity.
For some particular dipole spacing range, mutual cou-
pling decreases the spatial correlation level, rendering an
increased capacity. The optimized diagonalization broad-
casting technique with a two-stage power allocation
scheme has been proposed and verified. The presented
simulations results have demonstrated a positive impact
of this optimized BD scheme.
7. Acknowledgment
One of the authors (F. Wang) acknowledges the support
of the University of Queensland in the form of Interna-
tional Postgraduate Research Scholarship (IPRS).
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