Particle swarm optimization for SAGE maximization step in channel parameter estimation

PARTICLE SWARM OPTIMIZATION FOR SAGE MAXIMIZATION STEP IN CHANNEL

PARAMETER

ESTIMATION

Harun Bodur, Celal Alp Tunc, Defne Aktas, Vakur B. Erturk, and Ayhan Altintas

Dept. ofElectrical and ElectronicsEngineering, BilkentUniversity, Ankara, Turkey, e-mail: harun@ee.bilkent. edu.tr

Keywords: MIMO; channel estimation; SAGE; particle swarmoptimization.

Abstract

Thispaper presents anapplication of Particle Swarm Op-timization (PSO) in Space Alternating Generalized Ex-pectation Maximization (SAGE)algorithm. SAGE algo-rithm is apowerful tool for estimating channel parame-terslikedelay, angles (azimuth and elevation) of arrival and departure, Doppler frequency and polarization. To demonstrate theimprovementinprocessing time by uti-lizing PSOin SAGE algorithm, the channel parameters areestimated fromasynthetic data and thecomputational expenseof SAGEalgorithm with PSO is discussed. 1 Introduction

Recentworks have shown thatappropriate coding using multiple antennas atthe transmitter and receivercan in-crease the capacity of mobile systems

[5].

Such sys-tems arecalled

Multiple

Input

Multiple

Output (MIMO) systems. Design and optimization of MIMO systems require realistic model of the propagation channel. In otherwords, model needs tocharacterize the properties of eachpropagating path such as

delay, angles

(azimuth and

elevation)

of arrival (AoA) and departure (AoD), Doppler frequency and polarization [6]. To generate ac-curatechannel

models,

extensivechannelmeasurements andhigh resolution estimation toolsarerequired. Various estimation tools have been usedto estimate the channel parameters such as

Multiple Signal

Classifica-tion(MUSIC), Estimation of

Signal

Parametervia Rota-tional Invariance

Techniques (ESPRIT)

and maximum-likelihood (ML) methods like Expectation Maximiza-tion (EM) and Space

Alternating

Generalized Expecta-tion MaximizaExpecta-tion (SAGE)

algorithms

[2]. MLmethods

yield

more accurateresults andprovide higher resolution than othermethods, but

computational complexity

ishigh dueto the brute force search requiredto find the likeli-hoodmaximizingparameters.

The SAGE algorithm is apopular technique for

param-eter estimation. It

updates

eachparameter

sequentially

and

successively,

soit hasalower

complexity

andafaster convergence

[1].

The

algorithm

hastwosteps:

expecta-tion(E) and maximization (M)steps.

The mostcomputationallyintensive part of the SAGE al-gorithm is inthe M-step, and hence, fast search proce-dures arerequiredtoreduce thecomputational complex-ity. In this paper, we intend to acceleratethe SAGE algo-rithm using PSO for the search procedure. PSO is a pop-ularevolutionary computation technique which is based onintelligence andmovementofswarms [4].Itis shown that PSO isaneffectivealgorithm and hasalow computa-tionalcomplexity for solving optimization problems. By using PSO, SAGE algorithm converges rapidly and this convergence will be showninSection6.

The signal model for MIMO channels is summarizedin Section 2. Section 3 provides a brief introduction to SAGE algorithm. Section 4 explains how PSO can be utilized in SAGEalgorithm. Section 5presents the per-formance of theproposedalgorithm forasynthetic data. Weconclude withsomeremarksinSection7.

2 MIMOSignal Model

Inatypical MIMO channel environment showninFigure 1 there existsatransmitterantennaarray(Tx),areceiver antennaarray(Rx) and the propagation paths of the trans-mitted signal. Obstacles thatarelocatedinthe environ-ment causereflection, diffraction and refraction. There-fore, the transmitted signal propagates to the receiver through a certain number of paths (1,2...L). When the transmitted signal travels from Tx to Rx on a path, it is delayed, its amplitude is attenuated and its phase is changed. With thisknowledgeathand the receivedsignal vectorcoming from any oneof thepathscanbe written as;

s(t;

e1

)=

exp{j2wvlt}CRj

(QRX,l,)Al

CT.

(QTX

r

1)u(t-

TO)

whereu(t) denotes the transmitted signalvector,e) given by

(1)

f--N

Path#1

Pat utX

Figure1. MIMO channel environment.

is a vectorthat consists of channel parameters such as

angles (azimuth andelevation)of arrival

(QR,

,il),

angles (azimuth andelevation)ofdeparture

(QT

,i), delay (Ti), Dopplerfrequency (vi) and thepolarizationmatrix

(Al).

Qisaunitvectorthat describesthe direction froma ref-erencepoint and givenas

Q =

[cosQ()

sin(0),

sin(X)

sin(0),

cos(0)IT,

(3)

with 0 and X denoting the elevation and azimuth

an-gles, respectively. Finallyinequation (1),

CTX,P(Q)

and

CRX,P(Q), the steering vectors for the transmit and

re-ceivearrays,aregiven by

CTX

,P(Q)=

LfT ,m,p(Q)exp{j

2wAo

m = 1,....

M]T(p:

1(Q.rT.m)}

=

v,h).

CRX,P(Q)=

fRT

,n,p(Q)exp{j2Ao

1

(Q.rR

n = 1, ...,

N]T(p

=v,

h).

wherepdenotes the

polarization

type

(either

vertical

(v)

orhorizontal

(h)),f

is thefieldpatternofantennas,ris

an-tenna

location,

M is number transmitterantennas and N is number receiverantennas.

Equation

(1)

models

signal

ofonepath foran antennaandtotal

signal

atthe receiver isasfollows:

Figure2. MIMOchannelenvironment.

Figure2is theflowgraph of the SAGE

algorithm.

The al-gorithm hastwosteps,

namely, expectation

(E)

and

max-imization (M) steps. In anE-step, the

expected

value of onepathat areceiverantennaoutputis

computed.

Inan

M-step, thelikelihood function is maximized for param-eters ofone

path

sequentially.

For each

iteration,

these steps are performed for all

paths

and SAGE

algorithm

stopsuntil allparametersconverge.

(4)

The formulas of SAGE

algorithm depend

on two

con-cepts: unobservable admissible data and observable

in-complete

data.

Y(t)

describedin

equation

(6)

is the

in-complete

data and theadmissible data forone

path

is

L,,,,n)

(;

(5)

Xl(t)

=

s(t;

E1)

+

W(t).

₍₇₎

3.1 E-step

Since

Xl

is an unobservable

function,

estimate of this function is basedontheobservable datavector

Y(t)

by

L

Y(t)

=

E

s(t;

El)

+

W(t).

1=1

(6)

Xl(t)

=

Y(t)

L E

s(t;

E,,)

i'=1,1'7zl

Here W is the additive white Gaussian noise which is

assumedtobeindependent identically distributed

Gaus-sian.

3 SAGEAlgorithm

SAGE algorithm is an ML estimation method that has

beenused for differentapplicationslikechannel

parame-terestimation.

where

E1,

is the estimated parameters for l'thpathfrom

previous

iteration(s)[2].

Thus, estimation of admissible data ofonepath (Xi (t)) canbe foundby subtractingestimatedsignalsof all other

pathsfrom observed dataY(t). NO K

No

SAGE +

outpu.i

a

4 Particle Swarm Algorithmfor SAGE The computation ofparameters is done from the

like-lihood function and this function is maximized for the parametersindividually. The SAGE coordinate-wise up-dates for theparameters are derivedin [6] and given as follows,

Ti

= arg

maxT,

Z(q5T.

1,

T

,IqR,,

O

RX

X,I,

T1,

V1

;

xi)

(9)

Vi=

argmax,j

IZ(q5T.,l,O:T.,l,q$R,,l,ORX,ll,Vl

;xi)

(10)

O¢)i

I=

arg

maxoT,,I

Z()T.,l,

OT.,l,

¢)RX,l

OR,,,l

-l,

I,)

;

:x)

(11)

OT,I=

arg

maxoTx I

|Z(OTl,I

OTx,l,

ORx,I:

,l,

-l,T, VI)

;

xi)

(12)

ORxll=

arg

maxx,

I

|Z(OTx,l

Txl,

OR,lR

(13,

)l,

I)l

;

--h)

(13)

As we see from theformulas, thealgorithm should search for theparametervalues maximizing the likelihood func-tion. Since the search domain is continuous this is a te-dious operation requiring efficient searchprocedures. To this endwepropose to usePSOtoperform the optimiza-tion.

PSO is one of the evolutionary computation techniques developed for non-linear optimization problems with continuousvaluedparametersthough itcanalso be used with discrete variables [4]. The procedure is based on researches on swarmslike bees and birdflocking and in-spired by social behavior ofswarms [4]. This procedure is simple and its computation time is short.

According to thebiological research, swarms find their food collectively notindividually; information is shared within the members ofswarm. Intheswarm,each mem-ber's position and its information are known, so each member'sposition andvelocityaremodified.

Swarmmovementoptimizesacertainobjective function. Each member knows its bestvalue

(pbest)

and its posi-tion. Moreover, each member knows the best value in theswarm

(gbest)

among

pbests.

Each memberwants to change its position accordingto its

velocity

and the dis-tance to

pbest

and

gbest. Velocity

andposition of each membercanbecalculated by thefollowing equations:

v= w*v+cl* rand* (pbest

-present)

+c2* rand* (gbest -present).

present =present+v *t.

Al=

(|CTX,P

(Q)|CRX,P(Q) PTSC)

Z(Tl,I,STl:XRX),l SRl,-Tl,Vl;--h) (15)

whereargmaxstands for theargumentof themaximum,

Pdenotes signal power,

T,c

denotes the sensing period

ofareceiverantennaand zis likelihood function which

isgivenby

Z(ei; Xi) =

CRX

(QRXm,i)HXl (t;Ti,vli)CTX (QTX,l)

(16) InEquation (16) (.)* denotes theconjugate, (.)H denotes the Hermitianoperators and(.)denotes normalization.

For the initialization procedure the successive

inter-ference cancelation is used and for the details of the derivation readermayreferto[6].

Here v is the member velocity, w is a scaling factor,

presentis thecurrentmember'sposition,rand isa

ran-dom number between (0,1). cl,c2 are learning factors,

usuallytakenas cl = c2= 2.

6 Performance of SAGEwith PSO for aSynthetic

Channel Data

To investigate the performance of SAGE algorithm

to-gether with PSO, asynthetic datais created. In the

cre-atedscenario, double polarizedsixantennasareused both

Txand Rx. L = 5waves arepropagating betweenRxand

Tx, and theparameters arechosenasinTable 1. The

an-tennainput signal consists of 55 rectangular pulses with

10 nanoseconds duration, the received signal (y(t)) is

sampledat1 GHz andsignaltonoise ratio(SNR)is taken

as40 dB.

The channel parameters are estimated by SAGE

algo-rithmusing three different search procedures: PSO,

ran-dom search anddogleg. Random Searchmeanssampling

the cost function within the reference domain and then

;xi)

(14)

(17)

(18) 3.2

M-step

2.5 a) 1.5 a)C' X 1 C,) 0u Cn 0.5-_ PSO with Nb=5

-Trust Region Dogleg Algorithm

+ Random Search

2 3 4 5 6 7 8

Initialization(0) and SAGE iterations (1-10)

O0

_{0 9 10}

Figure3. MSE ofanangleofdeparture

Table]1. Createdsyntheticdataparameters.

choosing maximum of that function and dogleg isawell

knownalgorithm [3].

We observed that the SAGEalgorithm estimated the pa-rameters correctly and rapidly. Figure 3 shows MSE of

anangle of departureversusinitialization of SAGE

algo-rithm(0) and SAGE iteration numbers (1-10). From

Fig-ure 3, SAGE algorithm converges in 3 iterations for all

three algorithms. However, computation time for three approachesaresignificantly different. Table 2 shows the

computation time of search procedures both at initializa-tion and SAGE iterainitializa-tions. Table2illustrates that PSO's computation time for each iteration is significantly better than other methods.

7 Conclusion

In this paper we presented an application of particle swarmoptimization in SAGEalgorithm for channel pa-rameterestimation. PSOprovidesanefficientwayto

per-form optimization of the likelihood function for the

max-imizationstepof the SAGEalgorithm.

Table 2. Computational expense ofthe algorithm

com-pared with other methods. The algorithm works in Mat-labenvironmenton anAMDAthlon 3800processor

rate in terms of number of iterations, the computation time for each iteration is significantly improved by PSO comparedtoother searchprocedures. Therefore PSOcan

also beappliedtochannelparameterestimationproblem

toprovideanefficientoptimization.

References

[1] J. A. Fessler and A. 0. Hero, "Space-alternating

generalized expectation maximization algorithm," IEEE Trans. on Signal Processing, vol. 42, no. 10, pp.

26642677, Oct. 1994.

[2] B. H. Fleury, D. Dahlhaus, R. Heddergott, and M.

Tschudin,

"Wideband angle of arrival estimation using the SAGEalgorithm," in Proc. IEEE Fourth Int. Symp. Spread Spectrum Techniques and Applications (ISSSTA '96),pp. 79-85, Mainz, Germany, Sept. 1996.

[3] M.J.D. Powell, "A Fortran subroutine for solving sys-temsof nonlinearalgebraic equations," Numerical Meth-ods for Nonlinear Algebraic Equations, (P. Rabinowitz, ed.), Ch.7, 1970.

[4] J. Robinson and Y. Rahmat-Samii, "Particle swarm

optimizationinelectromagnetics,"IEEETrans. Antennas Propag., vol. 52,no. 2,pp. 397-407,Feb. 2004.

[5] I. E, Telatar. "Capacity of multi-antenna gaussian

channels," European Transactions on

Telecommunica-tion, vol. 10,pp. 585595, Nov/Dec. 1999.

[6] X. Yin, B. H. Fleury, P. Jourdan, and A. Stucki, "Po-larization estimation of individual propagation paths

us-ing the SAGE algorithm," in Proceedus-ings of the IEEE In-ternational Symposium onPersonal, Indoor and Mobile

Radio Communications (PIMRC), Beijing, China, Sept. 2003.

To demonstrate thegains in computational complexity by utilizing PSOinSAGEalgorithm,weestimated the

chan-nel parameters from a synthetically generated data. We

observed that although there is no gain in convergence

Computation time of one

Estimation of iteration

onepath with (seconds)

Initialization 120 Random Search Sage iteration 1.67 Initialization 1.9 Dogleg Sage iteration 1.5 Initialization 1.75 PSO Sage iteration 0.025 channel parameters 1 =2 f= 3 £=4 f =5 Ti (nsecs) 25 35 45 55 65 AoA azimuth(°) 100 120 140 160 200 AoA elevation(°) 20 40 60 80 100 AoD azimuth(°) 300 330 20 50 70 AoD elevation

(')

10 30 50 70 80