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Abstract In this paper, we deal with a per-user power allocation
problem in multiuser multiple input multiple output (MU-MIMO)
systems based on minimum mean square error (MMSE) precod-ing. The MMSE-precoding technique provides a reasonable per-formance due to minimization of a composite interfer-ence-plus-noise power through allowing inter-user interference aswell as the higher capacity with multiuser spatial multiplexing.The technique also has a reasonable computational complexitywith linear transmit processing. The problem of optimizingper-user power allocation under inter-stream interference is not
convex. Hence, we invoke some iterative approaches. In this paper,we propose a simplified iterative water-filling (SIWF) algorithm
for per-user optimum power allocation in multiuserMMSE-precoded MIMO systems, in order to maximize thedownlink sum capacity. This technique can reduce greatly thecomputational complexity for the iterative water-filling processthrough accelerating the iteration process, as compared with the
existing modified iterative water-filling (MIWF) algorithm [1],without any performance loss. In the proposed algorithm, boththe taxation and interference terms are updated at every iterationof the inner loop of iterative water-filling. In addition, per-userpower levels at every iteration for the inner loop are normalizedso that the total transmit power constraint could be satisfied, priorto the next iteration. From computer simulations and complexity
analyses, we show that the proposed algorithm has much lower
complexity but the same capacity, as compared with the originalMIWF algorithm.
KeywordsMultiuser MIMO; MMSE-precoding; Iterative wa-
ter-filling
I. INTRODUCTION
During the last decade, MIMO techniques have drawn a lot
of attentions, because of their promising improvement in terms
of both spectral efficiency and performance. Among many
MIMO issues, MU-MIMO precoding is currently one of hottest
topics, since the MU-MIMO precoding exploits synergically
the high capacity achievable with MIMO multiplexing and the benefit of space division multiple access (SDMA). It also
makes mobile terminals simple, since a lot of complex signal
processing tasks are done in advance at the transmitter [2].
So far, various MU-MIMO precoding techniques includingzero-forcing (ZF) technique, block diagonalization (BD) tech-
nique [3], successive MMSE (SMMSE) techniques [4]-[5],
Tomlinson- Harashima precoding (THP) techniques [6], suc-cessive optimization THP (SO-THP) technique [7] have been
developed. Among them, the MMSE precoding technique [8]
has not only the benefits of SDMA, but also provides a rea-
sonable performance when all user terminals are equipped with
a single receive antenna. The technique has relatively low
computational complexity due to linear transmit processing as
compared with non-linear precoding techniques. Compared
with the ZF technique that enhances the noise, the MMSEtechnique improves the system performance due to minimizing
composite interference-plus-noise power through allowing a
certain amount of inter-stream interference especially at low
SNR.Proper power allocation can result in significant performance
improvement. However, the problem of optimizing per-user
power allocation under inter-stream interference is difficult tosolve, since it is not convex. Recently, the MIWF algorithm [1]
was proposed for per-user power allocation to maximize the
sum capacity under inter-stream interference. The algorithm is
capable of finding sub-optimal solutions due to a taxation
scheme that takes into account the interacting effect of in-ter-stream interference in an iterative manner. In the MIWF
algorithm, while the interference term is updated at every it-
eration for the inner loop of iterative water-filling, the taxationterm is updated at every iteration for an outer loop of iterative
water-filling. In addition, per-user power levels at every itera-
tion for the inner loop cannot be satisfied the total power con-
straint. Hence, it can delay the iterative process to converge into
the solutionsIn this paper, we propose a SIWF algorithm for per-user
power allocation in multiuser MMSE-precoded MIMO systems,
to maximize the sum capacity. In the proposed algorithm, both
the taxation and interference terms are updated at every itera-
tion of the inner loop of iterative water-filling. In addition,
per-user power levels at every iteration for the inner loop are
normalized so that the total transmit power constraint could be
satisfied, prior to the next iteration. This normalization canmake the interference and taxation terms computed more ac-
curately. Hence, the above two modifications in the proposedalgorithm can reduce greatly the overall computational com-
plexity for the iterative water-filling process as compared with
the MIWF algorithm, without any performance loss.
We present the system model of the MU-MIMO downlinksystem with precoding in Section 2, and introduce the simpli-
fied iterative water-filling algorithm for the MU-MIMO sys-tems using MMSE precoding in Section 3. After presenting the
simulation results in Section 4, we analyze the complexity in
Section 5.
II. SYSTEM MODEL
The block diagram of MU-MIMO downlink system is illus-
trated in Fig. 1. In what follows, we assume channel state in-
A Simplified Iterative Water-filling Algorithm for Per-user Power
Allocation in Multiuser MMSE-Precoded MIMO Systems
Min Lee and Seong Keun Oh
School of Electrical and Computer Engineering
Ajou University, Korea
E-mails: {minishow, oskn}@ajou.ac.kr
978-1-4244-1645-5/08/$25.00 2008 IEEE 744
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formation (CSI) knowledge at both the base station (BS) and
mobile stations (MSs) for downlink precoding. Mand Kare
used to represent the number of transmit antennas and receiveantennas, i.e. there areKusers quipped with a single antenna.
s P F1
h
Kh
1n
Kn
1r
Kr
Fig. 1. MU-MIMO downlink system with precoding.
The received signal vector at MSs is
= +r HFPs n , (1)
where r = (r1,r2,,rK)T
is a column vector of received signals at
KMSs. H is aKM composite channel matrix. F is an MK
precoding matrix, P is aKKdiagonal matrix for power allo-
cation, s is an 1Kcolumn vector for transmit signals, and n is
an 1Kcolumn vector for zero-mean additive Gaussian noises
at MSs.Here, H is the MU-MIMO downlink channel matrix made up
of per-user row channel vectors as follows:
1
K
=
h
H
h
, (2)
where hi (i=1,2,,K) denotes an 1Mper-user MISO channel
vector made up of channel gains from transmit antennas to
receive antenna for the i -th user. The precoding matrix F again
consists of combining per-user precoding vectors as
[ ]1 K=f f f , (3)
where fi denotes an M1 column vector of per-user precoding
vector for the i -th user. The power allocation matrix is made
up of square root power allocated users signals as
1 0
0 K
p
=
P
, (4)
whereip is a square root power allocated the signal of the
i-th user. The transmitted signal vectors can be decomposed as
[ ]1T
Ks s=s , (5)
where si is a signal transmitted to the i -th user. The noise
vectorn is decomposed as
[ ]1T
Kn n=n , (6)
where ni is an additive white Gaussian noise at the mobile
receiver of the i -th user.
From (2) through (6), we can rewrite (1) as
[ ]1 1 1 1 1
1
0
0
K
K K K KK
r p s n
r s np
= +
h
f f
h
. (7)
Hence, the received signal of the i -th user can then be ex-
pressed as
1 , 1, 2, ,
K
i i k k k ikr p s n i K =
= + =
h f
. (8)
III. SIMPLIFIED ITERATIVE WATER-FILLINGALGORITHM
In this section, we present the SIWF algorithm for multiuserMMSE-precoded MIMO systems with inter-stream interfer-
ence. To find an effective algorithm of optimizing per-user
power allocation under inter-stream interference, we invoke
some iterative approaches to solve the Karush-Kuhn-Tucker
(KKT) system of the non-convex optimization problem.
When the inter-stream interference is regarded as noise, the
K-user optimization problem to maximize the sum capacity canbe expressed as
2
,
22
21
,
1( )
max log 1K
i i i
Ki
n j i j
j j i
p h
p h=
=
+
+
1
s.t.K
i T
i
P=
0 1, ,i
i K = , (9)
where,i j
h is an effective channel gain experienced by the
signal of thej-th user that interferes with the i-th user, andPTdenotes the total power constraint. Then, the effective channel
matrix can be expressed as
1,1 1,2 1,
2,1 2,2 2,
,1 ,2 ,
K
K
eff
K K K K
h h h
h h h
h h h
= =
H H F
. (10)
Solving the Lagrangian optimization problem of (9), weobtain the following solution as:
{ }( )
2
,
22
21
,
1( )
, log 1K i i i
i Ki
n j i jj j i
p hL p
p h
=
=
= +
+
1
K
i T
i
P=
. (11)
By taking the derivative of the above with respect topi, we can
obtain the KKT system of the optimization problem:
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2 22
, ,
1( )
1iK
i i i j i j n
j j i
t
h p p h
=
= +
+ + , (12)
where ti is defined as follow [9]-[10]:
2
,
2 2 21( )
, ,1( )
Kj j j
i K
j j i
j j k j k nk k j
p ht
p h p h =
=
=
+ +
2
,
2 2,
1( )
j i
K
k j k n
k k j
h
p h
=
+
. (13)
Solving the optimization problem (9) can now be thought of as
solving the KKT system (12)-(13) along with the power con-
straint and positivity constraints onpi, and the equality in (12)
needs to be replaced by less than or equal to, whenpi = 0.The ti term stands for the effect of interference caused the i-th
user towards other users and is called as the taxation term. To
solve this non-convex optimization problem, we fix both the titerm and the interference term from other users, denoted byIi,
where
2
,
1( )
K
i j i j
j j i
I p h=
= . (14)
Here, one of our main ideas is to update both the ti andIi termsaccording to the newly obtained per-user power levels pi
(i=1,,K) at every iteration for the inner loop of iterative wa-
ter-filling. The other is to normalize the newly obtained pi so
that the total power constraint could be satisfied, prior to the
next iteration. These make the algorithm accelerate the iteration
process. Then, we repeat the process until convergence.
This strategy is numerically efficient because (12) is essen-tially a modified water-filling step, which can be rewritten as:
2
2,
2 21( )
, ,
1K j i j ni
j j i ii i i i
p hp
th h
=
+ + =+
. (15)
Fixing ti andIi, the iterative water-filling algorithm can find
the optimalpi. The first step is to recognize that given,pi canbe obtained from (15) as follows:
2
2
,
1 i ni
ii i
Ip
t h
+
+
= +
. (16)
wherex+ := max(x,0). Here, is essentially the inverse of the
water-filling level, but modified by ti term. Next, we get
2
21
,
1K i nT
i ii i
IP
t h
+
=
+
= +
. (17)
With ti andIi fixed, this is now an equation of a single variable.
Further, the right-hand side of (17) is a monotonic function of.
Thus, (17) can be solved through only a one-dimensionalsearch (e.g. using bisection). After has been found,pi can be
obtained from (16).
Once the optimum ({pi }, ) is obtained, we may then iterate
the water-filling loop over total users, taking into account up-dating the taxation and interference terms at every iteration forthe inner loop of iterative water-filling, until the process con-
verges. Finally, repeat the process until the entire KKT systemis solved. The algorithm is summarized as follows:
Algorithm Consider aK-user system. Assume the total powerconstraintPT. The power allocation algorithm works as fol-
lows:
Initialize(0 )
/i T
P K= , 0n = .
Repeat (outer loop)
1n n= +
Repeat (inner loop)
( 1)n
i ip p
2 2
, ,
2 2 21( ) 2 2
, , ,
1( ) 1( )
K j j j j i
iK K
j j i
j j j k j k n k j k n
k k j k k j
p h ht
p h p h p h =
= =
=
+ + +
2
,
1( )
K
i j i j
j j i
I p h=
=
Obtain via bisection on2
21
,
1K i nT
i ii i
IP
t h
+
=
+
= +
2
2
,
1 i ni
ii i
Ip
t h
+
+
= +
Normalize to satisfy1
K
T i
i
P p=
=
Untili
converges.( )n
i ip
Until( ) ( 1)n n
i ip p <
IV. SIMULATION RESULTS
We performed computer simulations to evaluate the per-
formance of the proposed algorithm and to compare it with thatof the MIWF algorithm [1] for MU-MIMO downlink systemwith MMSE precoding. The channel H was assumed to be
spatially white and flat Rayleigh fading. In order to describe theantenna configuration of the MU-MIMO systems, we use the
notation { }1, ,
KR R T M M M . In all simulations for obtaining
the ergodic sum capacity, we considered {1,1,1,1}4 con-
figuration. In addition, we used the receive signal-to-noise ratio
defined as 2SNR /T n
P = [7].
Fig. 2 shows the ergodic sum capacity of the SIWF and
MIWF algorithms as a function of SNR. From Fig. 2, we see
that both algorithms have equal performance, since both of
them use the iterative water-filling algorithm based on the sametaxation scheme in order to solve an optimization problem.
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0 5 10 15 200
2
4
6
8
10
12
14
16
18
{1,1,1,1} X 4
Ergodicsum
capacity[bits/Hz]
SNR [dB]
MIWFSIWF
Fig. 2. Ergodic sum capacity of the SIWF and MIWF algorithms as afunction of SNR.
V. COMPLEXITY ANALYSES
In this section, we analyzed the overall computational com-
plexity of the proposed SIWF algorithm and compared the
complexity with those of the MIWF algorithm and the MIWFalgorithm based on per-iteration power normalization (PIPN)
[12]. For simplicity, we assumed that the number of transmit
antennas is equal to the number of users equipped with a single
receive antenna, that is, K. To count operations for computa-
tional complexity, we take account of the number of iterationsfor the outer loop of iterative water-filling, since the computa-
tional complexity of iterative water-filling depends heavily on
the number of iterations of the outer loop. We take account of
only the number of multiplications. In counting the number of
multiplications, we first consider the multiplications of four
basic operations such as bisection for finding the water-filling
level, normalization, and updating of taxation and interference
terms. These complexities can be used to compute the overallcomputational complexity in combination with the number of
outer loop iterations, inner loop iterations per outer loop, andbisection loop iterations per inner loop.
Let K be the number of users who are positive-power al-
located in an inner loop of iterative water-filling. The bisectionmethod is used to find the water-filling level through iterative
way. It requires 4 1K+ multiplications for every bisection [11].The normalization term is considered only for the MIWF al-
gorithm based on PIPN and the proposed algorithm. This
makes a water-filling solution satisfy the total power constraint,
and requires K multiplications for every inner loop of iterative
water-filing. The taxation and interference terms, respectively,
require 23K K and2
K K multiplications from (13) and
(14) for every inner loop of iterative water-filing. However, if
the interference term is calculated in advance, the taxation term
requires only22K multiplication. While the taxation term is
updated for every outer loop, and the interference term is up-dated for every inner loop in the MIWF algorithm, both terms
are updated for every inner loop in the proposed SIWF algo-
rithm. We summarize the number of multiplications required
for the bisection process, the normalization process, interfer-
ence updating, and taxation updating in the Table 1.
Table 1. Number of multiplications for four basic operations in aniterative water-filling algorithm.
Function Number of multiplications
Bisection 4 1K+
Normalization K
Taxationupdate
23K K or22K with interference term
Interference
update2
K K
We analyzed the overall computational complexity by com-
puting the total number of multiplications required for the
number of outer loop iterations, inner loop iterations per outer
loop, and bisection loop iterations per inner loop. We set SNR
of 3dB. We counted the average number of multiplicationsfrom 10000 independent channel realizations.
Fig. 3 shows the average number of iterations for the outer
loop in the SIWF and MIWF algorithms as a function of thenumber of transmit antennas (or users). From Fig. 3, we see that
the proposed algorithm requires 20% lower iterations for the
outer loop than the MIWF algorithm, except for two transmit
antennas (or users). We also see that only the normalization can
reduce the number of iterations of the MIWF algorithm by
about 8% as compared with the original MIWF algorithm.
2 3 4 5 6 7
0
2
4
6
8
10
Avergaenumberofiterations
Number of Tx antennas (or users)
MIWF
MIWF with PIPN
SIWF
Fig. 3 Average number of iterations of the SIWF and MIWF algo-rithms.
Fig. 4 shows the average number of multiplications for ob-taining the solutions in the SIWF and MIWF algorithms as a
function of the number of transmit antennas (or users). From
Fig. 4, we see that the proposed algorithm requires fewer mul-
tiplications than the MIWF algorithm by about 28%. We also
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