Clustered Precoding for Coordinated Multi-Cell Systems Based on Signal-to-Leakage Ratios Mehdi Sadeghzadeh, Hamid Reza Bahrami, Nghi H. Tran ECE Department, The University of Akron, Akron, Ohio Emails:
[email protected],
[email protected],
[email protected] To overcome the above drawback of BD-based systems, we propose in this paper a novel clustered linear precoding scheme applicable to downlink network MIMO. Our approach is based on the signal-to-Ieakage ratio (SLR) maximization, which was originally introduced in [10]-[12] for single-cell multi-user MIMO systems. Different from the well-known BD techniques, our proposed precoding scheme has no restriction on the number of transmit antennas and receive antennas, and therefore, it can support downlink network MIMO with any antenna configuration. Note in [10]-[12], the connection between the SLR and known criteria like signal-to-interference ratio (SIR) or sum-rate is missing. It is not clear why the SLR approaches are good criteria to design precoder base on and can enhance the system's throughput. To this end, we first provide an insightful connection between the SLR and the total sum-rate for downlink network MIMO. An optimal precoding scheme that maximizes the SLR for a given input covariance is then developed. Asymptotically, these precoders also result in the maximum sum-rate. Assuming per-BTS power constraints (PBPCs), optimal power allocation schemes are further developed to optimize the sum-rate for users with single receive antenna. Numerical results show that the proposed method performs very well when there are more receive antennas than transmit antennas and it outperforms the conventional methods based on the SLR maximization.
Abstract-In this paper, we propose a novel clustered linear precoding scheme based on the signal-to-Ieakage ratio (SLR) maximization applicable to downlink network MIMO. Different from conventional techniques, our proposed scheme is capable of supporting a network configuration in which the number of transmit antennas at the BTSs is smaller than the total number of antennas at all users. To this end, we first provide an insightful connection between the SLR and the total sum-rate. An optimal precoding scheme that maximizes the SLR for a given input covariance matrix is then developed. Asymptotically, the precoders obtained based on the SLR criterion also lead to the maximum sum-rate. By further assuming per-BTS power con straints ( PBPCs), optimal power allocation schemes are developed to optimize the sum-rate for users with single receive antenna. Numerical results show that the proposed method outperforms the conventional schemes based on SLR. T he gain is observed in terms of both the sum-rate and the bit error rate ( BER) performance. Index Terms-Clustered precoding, block diagonalization, OFDM, channel state information, null space
I.
INTRODUCTION
Coordinated multiple-point transmission (CoMP), the long term evolution (LTE-Advanced) terminology for network multi-input-multi-output (MIMO) communication using mul tiple antennas, has been recognized as a primary candidate to fulfil the stringent quality-of-service (QoS) requirements in 4G applications [1], [2]. However, in a multi-cell MIMO environ ment, besides multi-user interference [3], [4] that presents in every cell, signals intended for a particular cell also provide adverse effects to other cells. This type of interference is referred to as inter-cell interference (ICI). Both multi-user interference and ICI therefore significantly reduce the through put, especially for the cell-edge users. As such, one of the main challenges in CoMP is to design a precoding technique to mitigate the multi-user interference and ICI simultaneously. In a multi-user system, the block diagonalization (BD) technique originally proposed in [5], [6] has been considered as one of the most efficient schemes to remove the interference and simplifies the detection at the user side. The technique has been recently extended to CoMP in [7]-[9]. In particular, it was demonstrated in [7]-[9] that the BD scheme not only can successfully remove the multi-user interference and ICI but also increase the number of independent data streams, creating a significant improvement in the sum-rate. While the BD technique is able to provide a superior performance, it imposes a restriction on the antenna config uration of CoMPo Specifically, the BD-based systems require that the total number of transmit antennas must be larger or equal to the total number of receive antennas. This condition, however, does not always hold true in practice. For example, in the downlink network MIMO, a few base transmission stations (BTSs) might need to serve a large number of users simultaneously. If it is the case, the use of BD technique shall result in a unsatisfactory performance.
978-1-4799-5051-5/14/$31.00 ©2014 IEEE
II.
SYSTEM MODEL
We consider a downlink MIMO orthogonal frequency di vision multiple access (OFDMA) network with universal frequency reuse. The network is divided into S clusters or super-cells, each consists of N neighboring cells coordinating with each other in the cluster. Also, we assume that there is one BTS per cell (N BTSs in a cluster) and each BTS uses Nt transmit antennas. In addition, each user is equipped with single received antennas. In every time-slot, each BTS applies a precoding vector to its transmitted data vector over a particular subcarrier. Assuming that there are a total of M users served by a cluster simultaneously on each subcarrier, the received signal for the u-th user (u = 1, 2, . . . ,M) on the n-th sub-carrier in the c-th cluster can be written as N
Lh�,b (n)w�,b (n)d� (n)
y� (n)
b=l N
+
M
L h�,b (n) L wf,b (n)d�(n) b=l S
+
i=l,i#u
N
c=l,c,",c b=l
+
66
M
L L h�b (n) L w�,b (n)df(n)
nC (n)
i=l
(1)
a so-called SLR criterion, which was originally considered for a single-cell MIMO downlink system. Even though this simple approach has been considered to be suboptimal [10]-[12], we shall demonstrate shortly for downlink MIMO network, asymptotically, it performs as good as the sum-rate maximiza tion criterion. The SLR can be interpreted as the signal power divided by the power of the interference of which the desired user has caused to the other users. For a given cluster, the SLR associated with a given cluster with M users can be expressed as
where
h�,b (n) is the
1
Nt
channel vector from the b-th BTS in the c-th super-cell to the i-th user
•
x
•
w�,b (n) is the Nt
•
df (n) is the transmitted data for the i-th user in the c-th
x 1 precoding vector of the b-th BTS in the c-th super-cell applied to the data of the i-th user
super-cell nC(n) is the additive white Gaussian noise (AWGN) with
•
zero mean and variance
O"�
It is assumed that the the covariance of the transmitted data = qu for all users. The first term in is qu; i.e. the right hand side (RHS) of (1) is the desired signal, the second term is ICI, the third one is the interference due to other super-cells, and the fourth term is AWGN. Considering coordination among BTSs and by dropping the sub-carrier index for notational convenience, we rewrite (1) as
E{d;d;H}
1
in
x
NtN
To further provide an insight on about the SLR, let consider
M
c-th
h;,l(n), h;;2(n), ..., h;,N(n)
aggregate channel vector from super-cell to the u-th user,
( w;,l(n)H, w;,2 (n)H, ..., w;,N(n)H )H
.
u=l
)
is the all BTSs
w; (n)
the
IS
NtN
x
1
By calculating logarithm of both sides, using the Jensen's Inequality, and note that logarithm is a concave function, (4) can further be written as
aggregate precoder vector from all the BTSs for the u-th user,
and
Zu
=
,\,s hC ,\,M C C L."c=l,c,," C u L."i=lWi di
from other super-cells
Zu
is
(1)
+
nC
· IS
·h
t e mtenerence ·
"
lus noise. We assume the variance of
M
� Note that (.) indicates the Hermitian operator. can be further expressed as
0";.
10g2 M ,\,M WC Hhcu HhuCW,C 1 ""' L."t=l,#u t 10 g2 � ,\,M C CH C C N u=l L."i=l,,,"u wu Hhi hi Wu 1 M L 10g2 N 1 L N u=l
y� h;:w;:d;: + h;:W;: d;: + Zu u 1, 2, ...,M (2) where W; (wi, ...,W;:_l,W;+l, ...,wM ) and d; ( d1cH, ..., dCu-lH, dCu+lH, ... , dCM H ) H. =
=
=
_
=
Before closing this section, it should be emphasized that when M ::; LNtN J, i.e., the total number of receive antennas is less than the total number of transmit antennas, the block diagonalization precoding has been thoroughly discussed and and it is considered to be a very effective precoding scheme [8], [9]. Unfortunately, when the number of users is larger than LNtN J, the BD-based precoding scheme does not provide a satisfactory performance. In the next section, we shall consider an alternative approach based on the SLR criterion to overcome this issue.
L!110g2 (L;110g2 ( 1) Lu=l 10g2
(SIRu)
U =
1, ... ,M
)
L..,i-1 - ,r L """M C HHc HHc C - M
u=l L..,i=l,=lu Wu
Using the optimal solution in
x
2
= 1,
U
hC H hC H
where ul ' ..., u-1 ' u + 1 ' Rayleigh-Ritz quotient [10], we have
= 1, ...,M
CH ..., hM
] H. Usmg . the
A
XHX'
xHx
(16)
It then follows from (8) and (16) that the proposed solution in (13) not only can maximize the SLR but also minimize g in (7). Equivalently, the obtained precoders asymptotically maximize the total sum-rate. As a result, the proposed method in [12], cannot maximize the sum-rate and actually any other kind of precoders cannot maximize the sum-rate. So, our precoder design is optimal. Note that if the number of receive antenna per each user was more than one, we could not use arithmetic-harmonic inequality in (14) and the result in (15) does not hold true for this case. Due to this limitation, we assume single receive antenna per each user.
It then follows that
Hh; Hh;w ; (10) WuC HHuc HHucwuc c AHA HfCwu'c BHB HC HHu' where x BH -lh;Hh;B- , and A and B can be obtained from the Cholesky decomposition of BH -lh;Hh;B-1 and H; HH;, respectively. In the case that Bw; is the eigenvector of AHA, we have w;
V. OPTIMAL POWER ALLOCATION UNDER PER-BTS POWER CONSTRAINT In the previous sections, optimal precoder vectors have been developed under the assumption of an equal power allocation scheme among users. It is therefore possible to further maximize the sum-rate by finding an optimal power allocation to each user under a total power constraint. To this end, in the following, we first formulate the optimization problem. Then, solutions in closed-form are established. Let qi be the covariance corresponding to each user i-tho The sum-rate of our considered system is then given as
U
(11)
cN,Nxl,
(13), it is easy to see that
M """M C HHcu HHcuw,c """ L.., ,=l,=1u W ' � """M C Hc HHicWuC = M u=l L..,i=l,=lu w u H i
xHAHAx :S;A (9) xHx where A is the eigenvalues of AHA and the equality occurs when x is the eigenvectors of AHA. The result in (9) holds for any antenna configuration, as long as H; HH; is invertible.
Since w; E
(15)
iWu
where is the eigenvector. Furthermore, from the eigenvector properties, we obtain = 1. Therefore, for the solution in (13), one has
(8)
subject tollw�11
[ hC H
i
' HHcwc = """ M . :1:u WuC Hhcto Hhct wuc = WuC HHc u u u Dt=l ,
arg
H'C
(14)
By applying this inequality to the second term in the right hand side of (6), one has
arg
SLRu
LM
M
""" Xi >M i=l Xi """M Yi � Yi L..,i=l i=l
IV. OPTIMAL PRECODERS BASED ON SLR MAXIMIZATION
let Bw; be the eigenvector correspond
M 1 L log2 ( 1 + SINR) N u=l
R=
ing to the largest eigenvalue of BH -lh;Hh;B-1. Then by rewriting (11), one has
(12) where Amax u is the largest eigenvalue of the BH -lh;Hh;B-1. Hence, the optimal solutions of (8) can be expressed in closed-form as follows:
Wuc =
(
Under a per-BTS power constraint (PBPC), one has Tr
l Hhc B-1) BH - hc u u
B-1eigenvectorsmax -;'-=--:: - - =- - - - - - - ...:.: ..:..: = =-==- ;-;----, :-: - � ;-:- ---' =-:=--:- )f---:-:II 11 B-l eigenvectorsmax ( BH -lh�Hh� B-1
where eigenvectorsmax eigenvector corresponds l Blf B-1.
h; Hh;
(
U=
l,
...,M
where
(wc,bQWC,bH) :s; pb, b = I, 2, ..., N c ,b ] b = 1 " 2 Wc b = [WIc,b Wc,b ... WM '
Q = diag (q1,
(13)
2
..., qM). Given that
antenna, (17) can be rewritten as
)
BH -lh;Hh;B-1 the is to the largest eigenvalue of
R
68
� 10 � N � g2 u=l
(
(18)
... ,
N
and
each user has one receive max
u qu ' . 2 L..,i=l,=1u \ qt + (J'u
1 + """MA
)
(19)
10 0 ��������������������� """"'7"- [12], N=1 --e- Proposedmethod, N=1
1O-
��'liHrJ T���������HiI' :","",""*,-il8iH!'+-l�"*--'I
a: w co
a: w co
10-4�__---,-___ o 5 10
1O-5�__---,-___ o 5 10
__---,-___ �__---,-__ -----,
�
15
20
SNR
25
Figure 1. Un-coded bit error rate of proposed and the method in [12] for different number of users, N 9, Nt 4, and Nr 1. =
=
u
= w� H h� H h�w�, u � H [ H h h�w[. Note that Amax W
=
u
15
SNR
20
25
30
Figure 2. Un-coded bit error rate comparison of proposed method with the method in [12], M NtN + 1, Nt 4, and Nr 1.
=
=
1, 2, . . . , M and A; A; due to design of our precoders. The optimal power allocation optimization problem considering PBPC can then be formulated as where Amax
r, t �
__---,-___ �__---,-__ -----,
�
30
- �- [12], N=3 - - - Proposedmethod, N=3 - + - [12], N=4 Proposedmethod, N=4 ---+-- >< - [12], N=7 --e-- Proposedmethod, N=7 - --$- [12], N=9 ---A- Proposedmethod, N=9 -4-[12], N=12 + � C: method, N=12 9 �Proposedmethod, N=19
A.
=
»
=
=
Bit error rate
1) Effect of the number of users: We evaluate the performance of our proposed method in terms of BER and compare it with the proposed method in [12]. We consider 9 cells coordinating with each other in a cluster. Each BTS is equipped with 4 antennas and the users are all equipped with R single antennas. We simulate the performance for different number of users, M = 37,38,39,40 and 41. Fig. 1 shows the BER performance of our proposed scheme and [12]. As b b = 1, 2, . . . , N II w 112 qi:S; p , subject to shown, our proposed method reaches to better result in all the qt ?: 0, Z - 1, ..., M. cases and there is a slight enhancement in terms of BER for Using Lagrange multiplier, the optimal solution can then be our proposed method. The performance curves, however, show a floor effect at SNRs above 20 dB. This is because of the fact expressed as that at high SNRs, the performance bottleneck is the residual interference from other users which reduces SINR and thereby A; qi the performance. Note that by increasing the number of users, Amax the amount of interference gets more and the BER results gets (20) worse. 2) Effect of the cluster size: In this part, we simulate the where j..Lb should satisfy the following effect of the number of cells in a super-cell on the BER. We consider each BTS and user are equipped with 4 and 1 M antennas, respectively. For simplicity, we only consider the b II w� 112 case that there is only one additional user than the number i= of users that can be served in the case of the BD system (7 (M = NtN + 1). Fig. 2 shows the BER performance of the A; qi 2 proposed method and the one in [12] for different cluster sizes. Amax Ama As shown, the performance improves by increasing the number (21) of coordinating cells in a cluster. Note that this performance improvement comes at the cost of an increase in the CSI feed back load as the number of cooperating cells increases. As This can be easily solved numerically to obtain j..Lb. shown, our proposed method's result is slightly better than VI. NUMERICAL RESULTS the other method in [12].
{ L�l
t
L�l,#U
L
l
u
,
L�l,#u
u
:u
)
In this section, the performance of the proposed scheme in terms of BER and sum-rate in various scenarios is investigated. We consider different number of users, antennas at BTSs, and different number of cells within a cluster. Although the Monte Carlo Numerical is done assuming binary phase-shift-keying (BPSK) modulation, we expect to see the same trend for higher order modulations.
B. Sum-rate In this part, the sum-rate of our proposed precoding scheme with equal power allocation and with proposed power alloca tion is investigated. 1) Equal power allocation: Fig. 3 shows the achievable sum-rate of the two schemes for a system with 4 and 1
69
2) Proposed power allocation: Finally, we compare the sum-rate versus the number of users for equal power allocation with that of the proposed power allocation in Fig. 4. We assume that the received SNR is 10 dB, NT = 1, Nt = 4, N = 7,9, and 12. The sum-rate improvement due to the use of the proposed power allocation is significant and it increases with the number of users per cluster. In fact, at a reasonably large number of users a 8 and 9 bps/Hz gain in terms of sum-rate is obtained compared to the case of equal power allocation for N = 9 and N = 12, respectively. As shown here, there is just 1 bps/Hz enhancement between the equal power allocation and using the proposed power allocation from N = 9 to N = 12, we confirm our choice (N = 9) as the optimum number of cells in the cluster.
--- '--- '---'---'--- '--� =1�9-' ' 1c���pr=op=os�ed=m�eth=O d�,N - - - Proposedmethod,N=12 --e-Proposed method, N=9 40 -+ Proposedmethod,N=7 ---*-[12],N=19 2 35 �:� �[12],N=7 _30 � Proposedmethod,N=4 ----.J :l:! -4--"----'-'[1"' 2-],"' N c=-= 4'-__ L 45
� i��l:
(;; �25 Ql
� 20 E "
(j) 15
VII. 2
4
6
8
10
12
14
16
In this paper, we proposed a novel clustered precoding scheme based on the SLR maximization for the downlink of network MIMO systems. Unlike BD scheme that only works when the number of transmit antennas is larger that the number of receive antennas, our proposed scheme is capable of supporting a network configuration in which the number of receive antennas is larger. We also showed that the use of the SLR criterion results in an asymptotically optimal solution with regard to the sum-rate maximization problem. By using the optimal precoder vectors obtained from the SLR criterion, it was demonstrated that the proposed precoding scheme greatly outperforms the the method in [12]. Optimal power allocation scheme was also developed to further improve the total sum-rate of the considered system.
20
18
# of users
Figure 3. Sum-rate of the proposed scheme and the method in [12] assuming equal power allocation for different number of cells, Nt 4, and Nr 1. =
CONCLUSION
=
50 ,-,---,--,---,--, 45 40 _35 N
I
REFERENCES
(;; �30
[1] S. Shamai (Shitz), O. Somekh, and B. M, Zaidel, "Multi-cell com munications: an information theoretic perspective, Joint Workshop on Communications and Coding (JWCC), Oct. 2004. [2] S, Shamai (Shitz) and B, M. Zaidel, E " nhancing the cellular downlink capacity via co-processing at the transmitting end, " IEEE Veh. Techno!. Con!, VT C Spring 2001, pp, 1745-1749, May 2001. [3] G. J. Foschini and M. J. Gans, "On limits of wireless communications in a fading environment when using multiple antennas, " Wireless Personal Commun., vol. 6, no. 3, pp. 311-335, Mar. 1998. [4] S, Verdu, Multiuser Detection, Cambridge, UK: Cambridge Univ Press, 1998. [5] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, "Zero-forcing meth ods for downlink spatial multiplexing in multiuser MIMO channels, " IEEE Trans. Sig. Proc., vol. 52, no. 2, pp. 461-471, Feb. 2004. [6] J. Zhang, R. Chen, J. G. Andrews, A. Ghosh, and R. W. Heath, Jr., "Networked MIMO with clustered linear precoding, " IEEE Trans. Wireless Commun , vol.8, no.4, pp.1910-1921, April 2009. [7] L. U. Choi and R. D. Murch, "A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach, " IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 20-24, Jan. 2004. [8] Z, Pan, K. K. Wong, and T. S. Ng, "Generalized multiuser orthogonal space-division multiplexing, " IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 1969-1973, Nov. 2004. [9] J. Zhang, R. Chen, J. G. Andrews, and R. W. Heath, Jr., "Coordinated multi-cell MIMO systems with cellular block diagonalization, " 46th
Ql
� 25 E "
(j) 20
15
- Proposed power allocation,N=12 � Proposed power allocation,N=9 � Proposed power allocation,N=7 � Equal power allocation, N=12 -e-- Equal power allocation, N=9 --e-- Equal power allocation, N=7 2
4
6
8
10
12
14
16
18
20
# of users
Figure 4. Sum-rate comparison of the proposed power allocation and equal power allocation of the proposed scheme for different number of cells, Nt 4, and Nr 1. =
=
antennas at transmitter and receiver sides, respectively, We assume there are various number of cells collaborating with each other and the received SNR for all the users is 18dB. The number of users which is supported for each case is considered to be M = NtN + 1 for N = 7,9,12, and 19 and 20 for N = 4. The results is shown just for 20 users in Fig. 3 and we compare our result with the method in [12] with the same number of users for each case. Observe from Fig. 3, our proposed method outperforms significantly in comparison with the other method. The difference gets large for high number of cells in the cluster. Again this enhancement comes at the cost of an increase in the CSI feed back load as the number of cooperating cells increases. Since form N = 9 to 19 there is just 5 bps/Hz increase in terms of sum-rate, we assume N = 9 is the best choice for the number of cells in the cluster.
Asilomar Signals, Systems and Computers Conference, 2007, ACSSC 2007, pp.1669-1673, 4-7 Nov. 2007.
[l0] A. Tarighat, M. Sadek, and A. H. Sayed, "A multi user beamforming scheme for downlink MIMO channels based on maximizing signal-to leakage ratios, " IEEE ICASSP, vol. 3, pp. 1129-1132, 2005. [11] M, c. H. Lim, M. Ghogho, and D. C. McLernon, "Spatial Multiplexing in the Multi-User MIMO Downlink Based on Signal-to-Leakage Ratios, " GLOBECOM'07, pp. 3634-3638, 26-30 Nov. 2007. [l2] P. Cheng, M. Tao, and W. Zhang, "A New SLNR-Based Linear Pre coding for Downlink Multi-User Multi-Stream MIMO Systems, " IEEE Commun. Lett. , vo1.l4, no.11, pp.1008-101O, Nov. 2010.
70
