8/3/2019 03-09-06

1/5

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

8/3/2019 03-09-06

2/5

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:

745

8/3/2019 03-09-06

3/5

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.

746

8/3/2019 03-09-06

4/5

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

747

8/3/2019 03-09-06

5/5