Optimal precoder designs for sum rate maximization in MIMO multiuser multicells

e system is maximized. 
Since the MBS can exploit the spectrum 
frequency without awareness of the existence of 
the FBSs, we first derive the MBS transmission 
strategy. Then, the FBSs transmission scheme is 
introduced to handle both intra-tier and cross-tier 
interference. 
3.1 MBS transmission strategies 
In practice, the FBS operates in the plug and 
play mode. Thus, the MBS transmission strategy 
is oblivious the existence of the FBSs and the FBS 
must guarantee the harmless interference levels to 
MUs [6]. With the exclusion of cross-tier 
interference, the received signal of the i-th MU in 
the MBS is given by 
 1
( 1) ( 1) ,( 1) ( 1) ( 1)
( 1) ,( 1) ( 1) ( 1) ( 1)
1,
.
i i i i
K
i j j i
K K K K K
I
K K K K K
j j i

    
    
 
 

y H V s
H V s n
 (7) 
To remove intra-cell interference at the i-th 
user in the macro cell, we adopt the BD scheme 
[14]. Form Eq. (7), the conditions for zero-
interference are given by 
( 1 ) ,( 1) ( 1) 10, ; , 1, ...,i jK K K Kj i i j I      H V
(8) 
It immediately implies that the channel 
capacity of the i-th user in the macro-cell is given 
by 
 ( 1)( 1) 2 ( 1) ,( 1)2
( 1)
( 1) ( 1) ( 1) ,( 1)
1log
.
i K ii
i
i i i
K N K K
K
H H
K K K K
C
  
   
 I H
V V H
 (9) 
Define ( 1)jKH as the 
1
( 1) 1
1,
xM
K
j
I
K K
j j i
N

 
 
 
 
 
 
channel matrix for all users other than the j-th user 
in the MBS 
 1 ( 1) ( 1)
(K 1)
( 1) ( 1) ( 1) ( 1)
( 1)
... ...
... .
j j j
T T T
K K K K
TT
K
 

   

 


H H H
H
H
 (10) 
Condition (8) is equivalent to 
( 1) ( 1) 10 , 1, ..., .j jK K Kj I   VH
 (11) 
In other words, ( 1) jKV is the null space of 
matrix ( 1)jKH . Accordingly, the null space 
condition (11) imposes 1
1 ( 1)
1,
K
j
I
K K
j j i
M N

 
 
  
( 1) jK
d  . Applying the SVD to ( 1)jKH yields 
 ( 1) ( 1) ( 1) ( 1) ,j j j j
H
K K K KU V    H (12) 
where ( 1)jKU  and ( 1)jKV  are the left and right 
singular matrices of ( 1)jKH . The diagonal matrix 
( 1) jK 
 contains the decreasing ordered 
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 
Trang 96 
singular values on its diagonal. To satisfy (11), we 
chose 
 ( 1) ( 1) 1 ( 1) 1 ( 1):, 1 :j j j jK K K K K KM d M       V V T (13) 
where ( 1) ( 1)x ( 1)
K Kj j
j
d d
K C
 
 T can be an 
arbitrary matrix subject to the power constraint at 
the MBS. From (2), the transmitted power 
constraint at the MBS is rewritten as 
 
 
1
1
1 ( 1) ( 1)
1
max
( 1) ( 1) 1
1
trace
trace .
K
i i
K
i i
I
H
K K K
i
I
H
K K K
i
P
P


  

  




= V V = 
T T
 (14) 
Then, from (9) the sum rate at the MBS can 
be calculated as 


1
( 1)
( 1)1 2 2
1 ( 1)
( 1)( 1) ( 1)
1log
,
K
iK i
i
ii i
I
KK N
i K
HH
KK K
C




 
 
  I H
T T H
 (15) 
where we define 
( 1) ( 1) ,( 1) ( 1 )i i iK K K K   
H H V 
 1 ( 1) 1:, 1:jK K KM d M    for simplicity. 
The design problem of interest is to find the 
precoders at the MBS to maximize the total sum 
rate. Thus, the optimal design of precoders can be 
mathematically posed as 
1
( 1)
( 1)
2 ( 1) ( 1) ( 1)2
1 ( 1)
1logmax
K
K i i iiK i
i
I
H
N K K K
i K



  
 
Q I H Q H
(16a) 
  
1
max
( 1) 1
1
traces.t.
K
i
I
K K
i
P

 

 Q (16b) 
where ( 1) ( 1) ( 1)i i i
H
K K K  Q T T . It is clear that 
the objective function and constraints of (16) are 
convex optimization. It is well known that in such 
a convex optimization problem, a local optimum 
is also a global optimum. Thus, problem (16) can 
be efficiently solved by standard optimization 
software packages, e.g., CVX [15]. After 
obtaining the optimal solution 
( 1) iK 
Q to problem 
(16), we calculate the singular value 
decomposition ( 1) ( 1) ( 1) ( 1)i i i i
H
K K K K    Q U V and 
obtain the optimal 
( 1 ) iK 
T by 
 1/2( 1) ( 1) ( 1) .i i iK K K   T U (17) 
We summarize the design steps of the 
precoders at MBS in Algorithm 1. 
Algorithm 1 : MBS transmission strategies 
1: Input: m ax
1 ( 1) 1, , , C S I, .iK K KK M N P  
2: Output: 
( 1) ( 1) 1and , .i iK K KC  T V
3: Compute ( 1)jKV from (12). 
4: Solve the convex optimization (16) to 
obtain 
( 1 ) .iK Q Then, applying the SVD 
( 1) ( 1) ( 1) ( 1) ,i i i i
H
K K K K     Q U we have 
1/2
( 1) ( 1) ( 1) .i i iK K K   T U 
5: Obtain 
( 1)iK
V by (13). 
6: Evaluate the sum rate in (15). 
3.2 FBS transmission strategies 
Note that the FBSs can operate in the same 
frequency with the MBS if they do not cause 
harmful interference to MUs. In addition, the 
intra-tier interference should be mitigated to 
enhance the sum rate of the femtocells. This 
means that the FBSs should be deal with both 
cross-tier and intra-tier interference. The received 
signal of the i-th user in the k-th cell, defined by 
(4), yields the channel capacity of the i-th user in 
k-th femtocell given by 
 1
2 , ,log ,i k i i i i ii
H H
k N k k k k k k kC
 I H V V H  (18) 
where 2
,l ,
( , ) ( , )
i i k i j j ii
H H
k k N k l l k l
l j k i


  I H V V H . 
The sum-rate of the femtocells is 
   12 , ,
1 1
log .
k
i k i i i ii
IK
H
k N k k k k k k
k i
R 
 
 Q I H Q H 
(19) 
TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 
 Trang 97 
It is highly desired that the femtocell 
deployment still guarantees the quality of service 
of the MUs. The inter-cell interference from K 
femtocells caused to the i-th user in the MBS must 
by constrained by an acceptable threshold 
( 1)iK
  
  ( 1) ( 1) , ( 1) , ( 1)
1 1
trace
k
i i j i i
IK
H
K K k k K k K
k j
   
 
 I H Q H (20) 
where we define 
w i th a n d = 1 , . . . = 1 , . . . 
j j j k
H
k k k k K j IQ V V (21) 
For simplicity of notation, we define 
 
 =1,..., and =1,..., j k
k k K j I
Q Q . To investigate on 
the efficiency of the precoder design, we consider 
the scenario in which perfect CSI is exchanged by 
the BSs via the backhaul links [10] and global CSI 
is perfectly known at all BSs. The problem of 
interest is to find the precoders at FBSs to 
maximize the total sum rate of K femtocells. Thus, 
the optimal design of precoders can be written as 
  max R
Q
Q (22a) 
   max
1
s.t. trace 1,...,
k
i
I
k k
i
k KP

 Q (22b) 
  
1
( 1) , ( 1) ,
1 1
( 1)
trace
1, ...,,
k
i j i
i K
IK
H
K k k K k
k j
K i I 
 
 
 
 H Q H (22c) 
where constraint (22b) is imposed on the 
transmitted power per FBS while constraint (22c) 
guarantees that the interference power at the i-th 
MU receiver is less than an allowable threshold 
( 1)iK
  . 
It is obvious that the constraints of (22) are 
convex while the objective function is 
nonconcave. Thus, problem (22) is nonconvex 
which renders the mathematical challenges to find 
the optimal solutions of (22). Our approach is to 
recast problem (22) into a d.c. optimization and, 
then, develop an iterative d.c. programming for 
finding the precoders. To this end, we rewrite (22) 
as 
2 2 , ,
1 1
min log log
k
i i i i i
IK
H
k k k k k k k
k i 
   Q H Q H  
(23a) 
  max
1
s.t. trace 1,...,
k
i
I
k k
i
k KP

 Q (23b) 
  
1
( 1) , ( 1) ,
1 1
( 1)
trace
1, ...,,
k
i j i
i K
IK
H
K k k K k
k j
K i I 
 
 
 
 H Q H (23c) 
where 2
,l ,
( , ) ( , )
i i k i j ii
H
k k N k l k l
l j k i


  I H Q H . This 
minimization problem is still nonconvex because 
2log ik
 concave. Nevertheless, it can be solved 
by applied the local d.c. programming [16], [17]. 
Since 
2log ik
 is concave, at the th  iteration, 
one can has 
   
2 2
( )
1( ) ( )
, ,
( , ) ( , )
log log
trace
i i
i i i j j
k k
H
k l k k l l l
l j k i

 

 
    H H Q Q
 

(24) 
Replacing (24) into (23) yields the following 
optimization problem 
   
2
2
( )
1 1
1( ) ( )
, ,
( , ) ( , )
, ,
min log
trace
log
k
i
i i i j j
i i i i
IK
k
k i
H
k l k k l l l
l j k i
H
k k k k k k

 
 


 
   
  


Q
H H Q Q
H Q H



(25a) 
  max
1
s.t. trace 1,...,
k
i
I
k k
i
k KP

 Q (25b) 
  
1
( 1) , ( 1) ,
1 1
( 1)
trace
1, ...,,
k
i j i
i K
IK
H
K k k K k
k j
K i I 
 
 
 
  H Q H (25c) 
which is a convex optimization and, thus, it 
can be efficiently solved. As a result, the iterative 
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 
Trang 98 
procedure to solve problem (23) is summarized in 
Algorithm 2 where ϵ is an acceptable accuracy. 
Note that problem (23) is of d.c. programming and 
thus, it can be proved that the convergence of 
Algorithm 2 is guaranteed [16], [17]. 
Algorithm 2 : Interative algorithm for sum 
rate maximization of FBSs 
1: Initialization: Set 0  , choose  (0 )
ik
Q
, and calculate   (0 )ikR Q . 
2: th  iteration: Solve the convex 
optimization problem (25) to obtain the solution 
*
ik
Q and set 1   , ( ) *
i ik k
 Q Q and 
calculate   ( )ikR Q . 
3: Ending iteration: If 
        ( ) ( 1) ( )/i i ik k kR R R         Q Q Q ϵ, 
then stop; else go to step 2. 
4. ILLUSTRATIVE RESULTS 
In this section, we evaluate the performance 
of the proposed method by numerical simulations. 
We consider the HetNet with 1 MBS and K = 2 
FBSs. Each MBS or FBS is equipped with Mk = 
M = 2 antennas. Each base station serves Ik = I = 
2 users, each equipped with Nk = N = 2 antennas. 
Each base station transmits d
ik
= d = 1 data 
stream to its intended user. The MIMO Rayleigh 
channels are randomly generated with zero mean 
and unit variance entries. All noise variances are 
normalized 2
ik
 = 2 = 1, k = 1, , K+1 and i 
= 1, , Ik. We assume that all FBSs have the same 
maximum allowable transmitted power 
m a x
m a xkP P while the MBS has 
m a x
1 m a x2KP P  . We investigate the sum rate of 
the system for 
m a xP from 0 to 30dB. We set the 
acceptable interference threshold at all MUs 
( 1 )K i   . 
Firstly, we study on the convergence 
characteristic of the proposed Algorithm 2. We set 
= 0.1 and ϵ = 10-9. Figure 2 illustrates the 
evolution of the objective function (22) over 
iterations. It can observed that the objective 
function is monotonically increased over 
iterations and it quickly converges in less than 20 
iterations. In addition, as the maximum allowable 
transmitted power increases, the sum rate of the 
FBSs also increases. 
Figure 2. Convergence characteristic of the proposed 
algorithm. 
Next, we investigate the sum rate 
performance loss of the macrocell for different 
interference powers caused by FBSs. As can seen 
from Fig. 3 that for a small fixed value of  = 0.1 
the reduction in the sum-rate of the macrocell 
when there is the presence of FBSs is negligible. 
When we increase the allowable interference 
threshold  = 1, the sum-rate performance loss of 
the marocell increases. When the allowable 
interference threshold varies with respect to the 
transmitted power at base stations, max0.1P or 
max0.5P , the sum-rate of the macrocell does not 
increase when the transmitted power is large 
enough due to an increasing interference power at 
MUs. 
0 5 10 15 20
8.8
9
9.2
9.4
9.6
9.8
10
Iterations
Su
m
 ra
te
 o
f F
B
Ss
 (b
ps
/H
z)
Pmax=10dB
Pmax=12dB
Pmax=14dB
Pmax=16dB
TAÏP CHÍ PHAÙT TRIEÅN KH&CN, TAÄP 18, SOÁ K6- 2015 
 Trang 99 
Figure 3. The sum rate loss of the macrocell for 
difference interference threshold. 
Figure 4. The sum rate of the femtocells for difference 
interference threshold. 
On the other hand, the sum rate of femtocells 
is given in Fig. 4. In contrast to the macrocell, the 
average sum-rate of femtocells increases when the 
interference constraints at the MUs are more 
relaxed, as shown in Fig. 4. It is observed from 
Figs. 3 and 4, the average sum rate of the 
macrocell is higher than that of the femtocells. 
The reasons are that the MBS has higher 
transmitted power than the FBSs and the 
transmission strategies of FBSs must guarantee 
the harmless interference to MUs. 
Now, we compare the total sum-rate 
performance of our proposed method with that of 
time division multiple access (TDMA) in [1] and 
selfish approach in [12]. In TDMA, each base 
station transmits the signals in different time slots 
so that there is no inter-cell interference [1]. In 
selfish approach, each base station only cares 
about its signals in its own cell and does not care 
about interference to users in other cells [12]. As 
observed from Fig. 5, our proposed method 
outperforms the other methods for all interested 
region 
maxP . 
Figure 5. The sum-rate of the HetNet. 
Especially, when
m a xP is large, the sum-rate 
performance gap between our method and the 
selfish approach is significant. This is because the 
inter-cell interference is dominant for large
maxP 
while selfish approach is not aware to inter-cell 
interference. On the other hand, the TDMA 
approach can handle inter-cell interference but 
each cell can use only a part of time for 
transmission. By choosing an appropriate 
interference threshold, our method can provide 
an improved sum-rate since it can efficiently 
handle both inter and intra-cell interference. 
5. CONCLUSION 
This paper has presented the transmission 
strategies for the downlink of multicell multiuser 
MIMO HetNets. The block diagonalization 
scheme is used for the macrocell. The precoders 
at the FBSs are designed to maximize the total 
sum-rate while keeping interference to the MUs 
below the acceptable threshold. The convex 
optimization is exploited to find the MBS 
precoders while the d.c. programming is used to 
find the precoders at the FBSs. The simulation 
0 5 10 15 20 25 30
0
5
10
15
20
25
P
max
 (dB)
A
ve
ra
ge
 su
m
 ra
te
 o
f M
B
S 
(b
ps
/H
z)
=0.1
=1
=0.1Pmax
=0.5Pmax
without FBS
0 5 10 15 20 25 30
4
6
8
10
12
14
16
Pmax (dB)
A
ve
ra
ge
 su
m
 ra
te
 o
f F
B
Ss
 (b
ps
/H
z)
=0.1
=1
=0.1Pmax
=0.5Pmax
0 5 10 15 20 25 30
0
5
10
15
20
25
30
35
Pmax (dB)
A
ve
ra
ge
 su
m
 ra
te
 o
f H
et
N
et
 (b
ps
/H
z)
=0.1
=1
=0.1Pmax
=0.5Pmax
TDMA
Selfish 
SCIENCE & TECHNOLOGY DEVELOPMENT, Vol.18, No.K6 - 2015 
Trang 100 
results show the effectiveness of the proposed 
method as compared with the typical TDMA 
scheme and selfish transmission strategy in terms 
of the sum rate. 
ACKNOWLEDGEMENT 
This research is funded by Vietnam National 
Foundation for Science and Technology 
Development (NAFOSTED) under grant number 
102.04-2013.46.
Thiết kế bộ tiền mã hóa tối ưu để cực đại 
tổng tốc độ bit trong hệ thống MIMO nhiều 
cell nhiều người sử dụng 
 Hà Hoàng Kha 
 Nguyễn Đình Long 
 Đỗ Hồng Tuấn 
Trường Đại học Bách Khoa, ĐHQG-HCM, Việt Nam 
TÓM TẮT 
Bài báo nghiên cứu về vấn đề thiết kế các 
bộ tiền mã hóa tuyến tính cho hệ thống mạng 
không đồng nhất bao gồm nhiều người sử 
dụng có nhiều antenna phát và nhiều 
antenna thu. Mô hình mạng không đồng nhất 
bao gồm nhiều trạm phát femto hoạt động 
đồng thời trong vùng phủ sóng của một trạm 
phát macro. Để giải quyết vấn đề can nhiễu 
giữa các người sử dụng trong macrocell, 
chúng tôi sử dụng kỹ thuật khối chéo hóa và 
kỹ thuật tối ưu lồi để cực đại hóa tổng tốc độ 
bit của người sử dụng trong macrocell. Kỹ 
thuật truyền của các trạm femto được thiết kế 
để tối đa hóa tổng dung lượng của người sử 
dụng trong femtocell với ràng buộc về công 
suất phát và mức can nhiễu gây ra cho người 
sử dụng trong macrocell. Vấn đề thiết kế này 
tổng quát là bài toán tối ưu không lồi, và việc 
tìm lời giải tối ưu là một thách thức. Giải pháp 
của chúng tôi là biến đổi vấn đề thiết kế các 
bộ tiền mã hóa thành bài toán tối ưu hiệu của 
hai hàm lồi, và chúng tôi phát triển một giải 
thuật lặp hiệu quả để tìm các bộ mã hóa tối 
ưu. Các mô phỏng số đã chỉ ra rằng giải pháp 
đề xuất cung cấp tổng tốc độ bit cao hơn các 
phương pháp khác cho mạng không đồng 
nhất. 
Từ khóa: Tiền mã hóa tuyến tính, kênh can nhiễu MIMO, nhiều cell, mạng không đồng nhất, 
tối ưu D.C. 
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Clerckx, C. Shin, and K. Jang, Hierarchical 
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