Communications and Network, 2013, 5, 181-186
http://dx.doi.org/10.4236/cn.2013.53B2035 Published Online September 2013 (http://www.scirp.org/journal/cn)
A Power Allocation Scheme Using Updated SLNR Value
Based on Perturbation Theory*
Wenwen Cao1, Zi Teng1,2, Jun Wu1
1College of Electronics and Information Engineering, Tongji University, Shanghai, 201804, China
2 School of Mathematics, Physics & Information Engineering, Jiaxing University, Jiaxing, Zhejiang, China.
Received June, 2013
ABSTRACT
The performance of downlink multiple-input multiple-output (MIMO) cellular networks is limited by co-channel inter-
ference (CCI). In this paper, we propose a linear precoding scheme based on signal-to-leakage-and-noise ratio (SLNR)
criteria which can reduce the CCI significantly. Since each user’s SLNR value is corresponding to the largest eigen-
value of the generalized matrix which indicates the channel quality that we propose a scheme to do a dynamic power
allocation as an auxiliary way to improve SLNR precoding scheme. We use the perturbation theory to update each
user’s SLNR value each time step in time-varying channels rather than directly decompose the channel matrix so as to
reduce the amount of calculation. The simulation results show that the proposed scheme offers about 0.3 bps/Hz addi-
tional capacity gain and 0.5 dB BER gain over conventional SLNR precoding method with lower computational com-
plexity. And it also obtains about 0.5 bps/Hz additional capacity gain and 1 dB BER gain compared to the scheme only
update the preceding vectors.
Keywords: Updated SLNR Value; Perturbation Theory; Power Allocation
1. Introduction
In downlink multi-user MIMO system, because of the
limited frequency resources, a base station (BS) has to
communicate with several co-channel user Equipments
(UEs) to achieve high system capacity. This way of
transmission inevitably causes co-channel interference.
In general, there are non-linear and linear MU-MIMO
precoding schemes to solve this problem. Due to the high
complexity, the nonlinear schemes [1,2] are seldom used
in practical. Only linear MU-MIMO precoding schemes
are applied in 3GPP long-term evolution (LTE) [3]. The
conventional linear schemes such as ZF (Zero- Forcing)
[4-6], MMSE (Minimum Mean-Square Error) [4,6] and
BD (block diagonalization) [7] can eliminate or reduce
the interference among users and data streams. But all
these solutions have the drawbacks that the number of
base station antennas must be equal or greater than the
data streams of all users. In [8], the SINR(Signal to In-
terference plus Noise Ratio) precoding has been pro-
posed which is desirable to maximize the average SINR
for designing a robust precoder connected to maximize
the sum rate. But the algorithm contains iterative part
which increases the complexity of the realization. How-
ever, the expression of SINR is coupled for each user that
the solution of precoder vectors is difficult.
In [9,10], SLNR (signal to 1eakage and noise ratio)
has been developed to suppress the CCI. This scheme
takes a comprehensive consideration of the useful signal,
interference signal and channel noise. For each user, the
transmitter only needs to work out a generalized eigen-
value to obtain the optimal precoding matrix. There is no
constraint on the system configuration in terms of the
number of transmit and receive antennas. So the SLNR
precoding achieves a good tradeoff of algorithm com-
plexity and system performance. In practical wireless
communication system, because of the infraction and
scattering, there exists time-delay and multipath-fading.
In [11], the author has mentioned to employ a perturba-
tion theory to update formula of precoding vector design
as an approximate solution with tolerable small per-
formance loss under time-varying channels. This method
gives us an inspiration that since each user’s SLNR value
is corresponding to the largest eigenvalue of the general-
ized vector, we can use the perturbation theory to update
each user’s SLNR value as well as the generalized ei-
genvector corresponding to the slowly varying channel
information.
In the most of existing precoding methods, it is always
*This work is financially supported by NSFC General Program under
contract No.61173041.
C
opyright © 2013 SciRes. CN

W. W. CAO ET AL.
182
assumed that the transmitter distributes equal power to
each user. But in the real wireless communication system,
the users usually have different channel states. In [12,13],
the multi-user iterative water filling algorithm has been
proposed. Although it achieves a good performance
knowing the perfect CSI, its high complexity restricts the
method to be put into practice. According to this classic
power allocation approach, it is very important to note
here that we tend to allocate more power to users in good
channel condition in order to obtain better system per-
formance. Thus, if the transmit power is allocated accord-
ing to the accuracy CSI, some gains may be achieved due
to the effective power allocation. In conventional SLNR
precoding scheme, the SLNR value seems useless, but in
our proposed scheme, we use each user’s updated SLNR
value as the refernece for power allocation to improve
the system performance.
Notation: Throughout this paper, matrices are denoted
by boldface symbols. 
H
 denotes the conjugate transpose,
F
 represents the Frobenius norm,
N
I
is the NN

identity matrix.
M
N
 represents the set of
M
N

matrices in complex field. Besides, means x is dif-
ferentiable.
x

The paper is organized as follows. The system model,
previous works done on SLNR precoding, and the solu-
tion of updated eigenvalue based on perturbation theory
are introduced in Section II. We propose a new power
allocation scheme using the updated SLNR value in Sec-
tion III. Numerical results and conclusion are provided in
Section IV and Section V.
2. System Model
We consider a downlink MU-MIMO system in Figure 1,
where there is a base station communicating with K users
simultaneously over the same time-frequency resource.
The base station has N transmit antennas and each user is
equipped with Mk antenna. We employ a linear precoding
matrix at the transmitter, so the transmit data symbol
vector 1
N

x can be presented as
K
1
kk
k
 xws (1)
Figure 1. A block diagram of downlink MU-MIMO system.
where 1
L
k


s
LM
k
denotes the transmitted data for k-th
user. L is the number of data streams supported for user k
and it is assumed equal for all the users for simplicity.
Here

and . The scalar data is mul-
tiplied by a
NLM
k
k s
N
1

precoding vector before being trans-
mitted over the channel. We can set as the
transmit power constraint. And is the total transmit
power of downlink.
2
Ex0 p
0
p
For a given user, the received signal vector at the
k-th receiver is
yk
K
1,
kkkki
iik
k



y
HxHxn (2)
where the received signal vector of user k is denoted as
1
k
M
k


y. MIMO channel for the k-th user is k

H
k
M
N

. It is modeled as independent and identically dis-
tributed complex Gaussian variables with zero-mean and
unit-variance. The noise is independent com-
plex Gaussian distributed,
1
k
M
k
n
k~(CN 2
0, k
M
k)

i.e nI.
2.1. Signal to Leakage and Noise Ratio
The SLNR precoding design makes a balance between
eliminating the co-channel interference and the noise.
The basic concept in a SLNR system is that it maximizes
the strength of the desired signal relative to the noise and
total interference caused to the other users [9]. This ap-
proach is discussed below.
We can use the linear SLNR precoder in [9] as an op-
timization metric. Since each user’s precoding vector can
be optimized, the optimization problem turns into a com-
pletely decoupled one. So we can obtain a closed-form
solution. Besides, the SLNR precoding scheme does not
need the constraint on the system configuration in terms
of the number of transmit and receive antennas.
From the definition in [9], the k-th user’s SLNR is
2
F
K2
2F
1, k
SLNR1 K
Mσ
kk
k
kik
ii
i




Hw Hw (3)
The SLNR precoding scheme deals with the total in-
terfering power that user i causes on all other users. The
robust precoding vectors can be achieved by solving the
following optimization problem. The precoder design
aims at maximizing the SLNR can be formulated as
arg max1 K
k
opt k
kSLNR k


w
w (4)
Using the criterion of max each user’s own SLNR
value can decouple the precoding vector { in the
objective function of (3), in [9], the optimum solution is
given by
}
k
w
2H H 1
eigvector((M σ)
kkk
kk k
)
k

wHHHH (5)
By generalized eigenvalue decomposition, there exists
an invertible matrix
N
N
k

T,

W. W. CAO ET AL. 183
2H H 11
(M σ)Λdiag( ,,)
kkkkk
kk kN



HH HHT (6)
The maximum SLNR value is given by
max 1
SLNRmax( ,,)
N
k

  (7)
In the conventional SLNR precoding approach, it is
assumed that equal power is allocated to each user. As
the SLNR value is available to the transmitter and it can
indicate the channel quality of each user, we should take
advantage of the knowledge of SLNR value which may
bring further improvement to the system performance.
2.2. Calculate Updated SLNR Value Using
Perturbation theory
We assume that the time-varying channel is slow fading
which suits the practical wireless communication system.
Actually, at two consecutive time steps, the transmit
channels are not independent that we view the current
channel as a slightly updated version of the previous one
[14]. If we do the eigen-decomposition every time step, it
will be a burdensome to calculation. So we should make
full use of the time dependency of the channels between
two consecutive time steps. Here the perturbation theory
will be applied in calculating the current updated version
of eigenvalues and eigenvectors.
The perturbation theory [15] considers the effect of a
small disturbance in the equation and finds an approxi-
mate solution to a problem which cannot be solved ex-
actly. It often provides a better approximate answer to
what the real solution should be. Our approach is to ap-
ply matrix perturbation theory to the SLNR linear pre-
coding where eigenvalues and eigenvectors are computed
at every time step not using the matrix decomposition
which reduces the complexity of computation.
Let , the definition of the generalized ei-
gen equation is Where , is
called a generalized eigenvector and is called a gen-
eralized eigenvalue. When the equation is perturbed, we
bring a parameter to illustrate the change of the equa-
tion. By the definition, we know that
mn
, AB
xλx.
iii
AB
ε
n
xixi
λi

 

εxελεε x
ii


AA BBi (8)
where . Let
ε1



λε
i

and



xε
i

be the set of
generalized eigenvalues and eigenvectors. The problem
is to find the nontrivial solutions of equation (8).
When we solve the equation (8),




λε and xε
ii
are
differentiable about . Set = 0 in the equation and
the initial conditions to derive an initial value problem
which determines the unperturbed solution
ε ε


λ0
i. The
equation (8) can be transformed to be





x0x0
λ0x0λ0x0λ0x0
ii
iiiiii







AA
BBB
(9)
 
λε and xε
ii
can be expressed in the form of Taylor
expansion.
 
2
λε λ0ελ 0Ο(ε)
ii i
 
 (10)






2
xεx0 εx0 Ο(ε)
ii i


 (11)
All of the other terms in the linear equation are of or-
der . By substituting the expansion (10) and (11)
into the differential equation (9), we obtain solutions of
the eigenvalues and eigenvectors to first order
2
Ο(ε)

H
λελ εx(λ)x
ii i i
 

AB
i
(12)


NH
H
1, i
εx( λ)x
ε
xεx1x xx
2λλ
ii j
ii ij
(13)
iij
jj

 


AB
B
i
In (12), the updated eigenvalues are formed by the
unperturbed solution and the first-order perturbation
correction .
λi
H
i
εx(


λε λ)x
ii
 

AB
From the form of the solution (12) and (13), we can
see that the computation of the updated eigenvalues and
eigenvectors do not need to do eigen-decomposition all
the time. Because of the approximate solution, there ex-
ists small loss of the system performance. In our forth-
coming scheme, we applied the unused eigenvalue to be
an indication of the channel quality, so the transmitter
can do the power allocation according to it. The simula-
tion has proved that it is a good scheme to compensate
the performance loss.
3. Proposed Power Allocation Scheme
In the following, we propose a power allocation scheme
which compensates the power loss due to approximated
computation by perturbation theory. The SLNR precod-
ing design in (3) also depends on the amount of power
allocated to each user which takes the CSI into consid-
eration. However in the practical wireless communica-
tion system, each user’s channel fading is not completely
the same. By (6), we can clearly see that the channel
quality directly determine the SLNR at the receiver, it
can significantly affect the system performance. Since
the total transmission power of BS is constrained, if the
transmit power is allocated according to the perfect CSI,
we can get performance gains due to the improved power-
efficiency.
The receivers feedback the liability CSI to the trans-
mitter and the transmitter choose the preferable precod-
ing vector to transmit the signal. We put forward a
straight solution of a simplified power allocation problem.
As the transmitter knows the perfect CSI, it allocates
more power to the user which has better channel quality.
The precoding vector is the eigenvector corresponding to
the largest eigenvalue of , and it is in propor-
tion to the channel power , so that the largest eigenvalue
can be used as an indication to the quality of channel.
Then, an SLNR-based power allocation scheme can be
λHk
k
HH

W. W. CAO ET AL.
184
formulated as
K
0
1
λ
λ
kk
i
i
p
p



(14)
The solution of the updated SLNR value 

λε
i in (12)
substituted in (14), the updated SLNR value is more ac-
curacy to do the power allocation.



K
0
1
(λλε)
(λλε)
kk
k
ii
i
p
p




 (15)
Using the updated SLNR to do power allocation, we
can see the effect of parameters like eigenvalues, trans-
mit power and channel perturbation on the capacity.


K
0
K
2
11
(λλε)p
C log(1)
σ(λλε)
kk
per
kii
i


 
 (16)
When computing , we define λk 


H
[]n [n]
k
k
AnHH
and 



2H
k
kk
HMσn[nBnH ]
k
at time n.
In the computation of updated SLNR value, we also
achieve performance gain benefits by adjusting the up-
dated precoding vectors obtained from solution (13). The
proposed simplified power scheme barely adds any addi-
tional calculation compared to the scheme only updating
the precoding vectors and it has less computation amount
compared to the conventional SLNR precoding scheme
which do the eigen-composition every time step in the
time-varying channel. So the proposed allocation scheme
using updated eigenvalue can improve the system per-
formance considering of less computation complexity.
4. Numerical Results
In this section, we evaluate the performance of the pro-
posed algorithm. We consider a MU-MIMO system with
4 (K = 4) users. Assume the BS has 12 (N = 12) antennas,
each user equipped with 2 () antennas and each
user is transmitted with 2 (L = 2) streams. In this simula-
tion experiment, the number of transmit antenna is more
than the sum of all the receivers. All simulations are
conducted using a QPSK modulation and a frame with
200 bits averaged over 10000 frames. We use the first
order Gauss-Markov model to simulate a time-varying
Rayleigh-fading channel. And the average bit error rate
(BER) of users and system capacity are used as the eval-
uation criterion.
M2
k
In Figure 2, we can clearly see that when we update
the SLNR each time step, the loss of the SLNR can be
notably reduced compared to the non-updated scheme.
We simulate about 100 depth of time steps and it is ob-
vious that calculating the updated SLNR with the pertur-
bation theory, the SLNR loss is negligible. Figure 2
shows that the SLNR value degrades only 0.1 through
the approximate calculation. However the non-update
scheme degrades about 0.6 which demonstrates that the
updated SLNR scheme will have small effects on the
system performance with reduced computational com-
plexity.
For the purpose of matching our hardware implemen
in Figures 3 and 4 , we compare the BER performance
and system capacity of the proposed power allocation
scheme using updated SLNR value (USLNR- PA) with
MMSE algorithm (MMSE), the conventional SLNR
maximization algorithm (CSLNR) and the updated pre-
coding vectors scheme in [12] (SLNR-PERTUBATED).
When the channel variation is not significant, the
SLNR-PERTUBATED scheme only update the precod-
ing vectors causes about 0.3 dB loss in BER and 0.2
bps/Hz loss in capacity compared to the conventional
SLNR precoding scheme. From the result, the proposed
power allocation scheme shows about 0.5 bps/Hz addi-
tional capacity gain and 1 dB BER gain per user over the
SLNR-PERTUBTED scheme. Although the average
achievable BER and capacity of the proposed algorithm
020 40 60 80100
7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8
Tim e Index
S L NR V a l u e
non-updated SLNR
updated-SLNR
Figure 2. SLNR value degrade with time index.
-5 0510152025
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR (dB)
Average BER
BER vs SNR multi
SLNR-PERTUBATED
US LNR-P A
MMSE
CS LNR
Figure 3. BER performance of K = 4, N = 12, Mk = 2.

W. W. CAO ET AL. 185
Figure 4. Sum capacity of K = 4, N = 12, Mk = 2.
is small but the simplified power allocation scheme ob-
tain the performance gain with no additional computation
and it not only makes up the performance loss in [11],
but also obtains the performance gain about 0.5dB in
BER and 0.3bps/HZ in capacity compared with the con-
ventional algorithm.
The results show that the proposed scheme has the best
performance from both BER and system capacity. Using
perturbation theory to obtain the updated SLNR and the
updated precoding vector rather than decomposing the
matrix to get the generalized eigenvalues and eigenvec-
tors is an excellent way to achieve balance between algo-
rithm complexity and system performance. What’s more,
using the updated SLNR value to do a power allocation
can further improve the system performance. From (2),
the SLNR precoding design depends on the amount of
power allocated to each user so that allocating more
power to the user having good channel quality can in-
crease the system performance. This strategy to compen-
sate the performance loss in [12] is feasible.
5. Conclusions
In this paper, we have investigated the power allocation
scheme using the updated SLNR value base on perturba-
tion theory. As the time-varying channel is taken into
consideration, we avoid doing the eigen-decomposition
in two consecutive time step. It leads to relatively less
amount of calculation compared to the conventional
SLNR algorithm and better system performance com-
pared to the scheme only updating precoding vector.
Then the proposed power allocation scheme using up-
dated SLNR value which is more accuracy as an indica-
tor to the channel quality compensates the performance
loss caused by the approximate calculation.
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