SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 191053-5888/03/$17.00©2003IEEE

Recent developments have demonstrated that particle filter-ing is an emerging and powerful methodology for sequen-tial signal processing with a wide range of applications inscience and engineering. It has captured the attention of

many researchers in various communities including those of signal pro-cessing, statistics, and econometrics, and this interest stems from its po-tential for coping with difficult nonlinear and/or non-Gaussianproblems. Based on the concept of sequential importance sampling andthe use of Bayesian theory, particle filtering is particularly useful in deal-ing with nonlinear and non-Gaussian problems. The underlying princi-ple of the methodology is the approximation of relevant distributionswith random measures composed of particles (samples from the space ofthe unknowns) and their associated weights.

In this article, first we present a brief review of the particle filtering the-ory, and then we show how it can be used for resolving many problems inwireless communications. We demonstrate its application to blind equal-ization, blind detection over flat fading channels, multiuser detection,and estimation and detection of space-time codes in fading channels.

IntroductionParticle filtering is a sequential Monte Carlo methodology where the ba-sic idea is the recursive computation of relevant probability distributionsusing the concepts of importance sampling and approximation of proba-bility distributions with discrete random measures. The earliest applica-tions of sequential Monte Carlo methods were in the area of growingpolymers [19], [49], and later they expanded to other fields includingphysics and engineering. Sequential Monte Carlo methods found limiteduse in the past, except for the last decade, primarily due to their very highcomputational complexity and the lack of adequate computing resourcesof the time. The fast advances of computers in the last several years andthe outstanding potential of particle filters have made them recently avery active area of research. Their potential for parallel implementation

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represents additional impetus for their development. Thecurrent interest in particle filtering for signal processingapplications was brought on by [17]. Recent reviews andaccounts of new developments on the subject can befound in [3], [10], and [11].

A large portion of the theory on sequential signal pro-cessing is about signals and systems that are representedby state-space and observation equations, that is, equa-tions of the form

x f x u

y g x v

t t t t

t t t t

==

−( 1 , )( , )

(1)

where y t is a vector of observations, x t is a state vector,g t ()⋅ is a measurement function, f t ()⋅ is a system transitionfunction, u t and vt are noise vectors, and the subscript tdenotes time index. The first equation is known as stateequation, and the second, as measurement equation. Thestandard assumptions are that the analytical forms of thefunctions and the distributions of the two noises areknown. Based on the observations y t and the assump-tions, the objective is to estimate x t recursively.

The method that has been investigated the most andthat has been most frequently applied in practice is theKalman filter [1]. The Kalman filter is optimal in the im-portant case when the equations are linear and the noisesare independent, additive, and Gaussian. In this situation,the distributions of interest (filtering, predictive, orsmoothing) are also Gaussian and the Kalman filter cancompute them exactly without approximations. For sce-narios where the models are nonlinear or the noise isnon-Gaussian, various approximate methods have beenproposed of which the extended Kalman filter is perhapsthe most prominent of all [1].

The particle filtering method has become an impor-tant alternative to the extended Kalman filter. With parti-cle filtering, continuous distributions are approximatedby discrete random measures, which are composed ofweighted particles, where the particles are samples of theunknown states from the state space, and the particleweights are “probability masses” computed by usingBayes theory. In the implementation of particle filtering,importance sampling plays a crucial role and, since theprocedure is designed for sequential use, the method isalso called sequential importance sampling. The advan-tage of particle filtering over other methods is in that theexploited approximation does not involve linearizationsaround current estimates but rather approximations inthe representation of the desired distributions by discreterandom measures.

In this article we discuss the use of particle filtering inseveral important problems in wireless communications.Figure 1 presents a diagram that sorts out the addressedproblems into two groups, one related to single-user sys-tems, and the other to multiple access systems. For singleuser systems, the interest revolves around detection in flatfading and equalization, where the emphasis of the latteris on time-invariant, time-variant channels, and orthogo-nal frequency division multiplexing (OFDM) systems.For multiple access systems, the focus is on particle filter-ing for multiuser detection in code division multiple ac-cess (CDMA) systems and space-time decoding in fading

20 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

Notation

ak kth coefficient of an autoregressive (AR) processbk kth coefficient of a moving-average (MA) processC chip-rate (processing) gainC code and modulation matrixc()⋅ code and modulation functionft()⋅ system transition function at time tg t()⋅ measurement function at time th t communication channel at time tm running index for particles or trajectoriesK number of users in a CDMA systemL order of communication channel symbol alphabetM total number of particles ( , )µ σ2 Gaussiandistributionwithmeanµ andvarianceσ2

c ( , )µ σ2 complex Gaussian distribution with meanµ andvariance σ2

NT number of transmit antennasN R number of receive antennasπ()⋅ importance sampling functionp ()⋅ probability distribution functionra order of an AR processrb order of an MA processR correlation matrix channel covariance matrixs t symbol transmitted at time tσ2 noise variancet discrete time index, where t ∈

T size of a frame of symbolsTs symbol durationτ continuous time indexu t system noise vector at time tvt observation noise vector at time twt

m( ) weighting coefficient of particle m at time twt

m*( ) nonnormalized weighting coefficient of particlem at time t

x t system state vector at time tx0:t , , , ,x x x0 1 K t a trajectory of statesx t

m( ) mth particle at time tx0:

( )t

m mth trajectory of particlesξ k signature of kth user in CDMA transmissiony t observation vector at time ty0:t , , , ,y y y0 1 K t a sequence of observations

channels. In all these cases, the first step is de-fining the problem with state-space represen-tation. For example, a general basebandcommunications model for a fading channelcan be written as

x f x u

s h

t t t t

t t t ty v

=

= +−( , )1

(2)

where yt is the discrete time signal receivedat the receiver, and x t is the state of the sys-tem composed of vectors of transmittedsymbols s t and fading channel coefficientsh t . The state varies in time according to aknown function f t which describes aMarkov process driven by the noise u t . Fi-nally, vt is additive channel noise. The pri-mary objective is to sequentially detect the transmittedsymbols and/or estimate the channel as the observa-tions arrive. From a Bayesian point of view, this impliesobta in ing es t imates o f p s yt t t( , | ):h 0 , wherey y y yt t0 0 1: , , , = K , which is exactly what particle fil-ters are designed for. Many other problems in wirelesscommunications can be described similarly as by (2),some of which are presented in the sequel.

Fundamentals of Particle FilteringConsider a system/signal with a state-space representa-tion given by (1). As already pointed out, the main task ofsequential signal processing is the estimation of the statex t recursively from the observations y t . In general, thereare three probability distribution functions of interest,and they are the filtering distribution p t t( | ):x y 0 ; the pre-dictive distribution p t l t( | ):x y+ 0 , l ≥1; and the smoothingdistribution p t T( | ):x y 0 , where T t> . All the informationabout x t regarding filtering, prediction, or smoothing iscaptured by these distributions, respectively, and so themain goal is their tracking, which is obtaining p t t( | ):x y 0from p t t( | ):x y− −1 0 1 , p t l t( | ):x y+ 0 from p t l t( | ):x y+ −1 0 , orp t T( | ):x y 0 from p t T( | ):x y+1 0 . The algorithms that exactlytrack these distributions are known as optimal algo-rithms. In many practical situations, however, the opti-mal algorithms are impossible to implement, primarilybecause the distribution updates require integrations thatcannot be performed analytically or summations that areimpossible to carry out due to the number of terms in thesummations.

For the joint a posteriori distribution of x 0 , x1 , K, x t ,in case of independent noise samples which are assumedthroughout the article, we can write

p p p pt tk

t

k k k k( | ) ( | ) ( | ) ( | ): :x y x y y x x x0 0 0 01

1∝=

−∏ .(3)

It is straightforward to show that a recursive formula forobtaining p t t( | ): :x y0 0 from p t t( | ): :x y0 1 0 1− − is given by

pp p

ppt t

t t t t

t tt( | )

( | ) ( | )( | )

(: ::

:x yy x x x

y yx0 0

1

0 10= −

−−1 0 1| ):y t − .

(4)

Since the transition from p t t( | ): :x y0 1 0 1− − to p t t( | ): :x y0 0 isoften analytically intractable, we resort to methods thatare based on approximations.

In particle filtering, the distributions are approximatedby discrete random measures defined by particles andweights assigned to the particles. If the distribution of in-terest is p x( ) and its approximating random measure is

χ ==

x wm m

m

M( ) ( ),1 (5)

where x m( ) are the particles, w m( ) are their weights, andM is the number of particles used in the approximation, χapproximates the distribution p x( ) by

( )p x w x xm

Mm m( ) ( ) ( )≈ −

=∑

1

δ(6)

where δ()⋅ is the Dirac delta function. With this approxi-mation, computations of expectations (which involvecomplicated integrations) are simplified to summations,that is, for example,

E g X g x p x dx( ( )) ( ) ( )=∫ (7)

is approximated by

( )E g X w g xm

Mm m( ( )) ( ) ( )≈

=∑

1

.(8)

The next important concept used in particle filtering isthe principle of importance sampling. Suppose we wantto approximate a distribution p x( )with a discrete randommeasure. If we can generate the particles from p x( ), eachof them will be assigned a weight equal to 1/ M. Whendirect sampling from p x( ) is intractable, one can generateparticles x m( ) from a distribution π( )x , known also as

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 21

1. Problems in communications addressed in the article.

importance function, and assign (nonnormalized)weights according to

wp x

xm*( ) ( )

( )=

π (9)

which upon normalization become

w w

w

mm

i

Mi

( )*( )

*( )

=

=∑

1

.

(10)

Suppose now that the posterior distributionp t t( | ): :x y0 1 0 1− − is approximated by the discrete randommeasure χ t t

mt

mmMw− − − ==1 0 1 1 1 , :

( ) ( )x . Note that the trajecto-ries or streams of particles x 0 1:

( )t

m− can be considered parti-

cles of p t t( | ): :x y0 1 0 1− − . Given the discrete randommeasure χ t −1 and the observation y t , the objective is toexploit χ t −1 in obtaining χ t . Sequential importance sam-pling methods achieve this by generating particles x t

m( )

and appending them to x 0 1:( )

tm

− to form x 0:( )

tm , and updating

the weights wtm( ) so that χ t allows for accurate estimates

of the unknowns of interest at time t.If we use an importance function that can be factored

as

π π π( | ) ( | , ) ( | ): : : : : :x y x x y x y0 0 0 1 0 0 1 0 1t t t t t t t= − − − (11)

and if

x x y0 1 0 1 0 1:( )

: :~ ( | )tm

t t− − −π (12)

and

wp

tm t

mt

tm

t−

− −

− −

∝10 1 0 1

0 1 0 1

( ) :( )

:

:( )

:

( | )( | )x yx yπ (13)

we can augment the trajectory x 0 1:( )

tm

− with x tm( ) , where

( )x x x ytm

t tm

t( )

:( )

:~ | ,π 0 1 0− (14)

and easily associate with it an updated weight wtm( ) ob-

tained according to

wp p

tm t t

mt

mt

m

tm

tm

( )( ) ( ) ( )

( ):

( )

( | ) ( | )( |

∝ −

−

y x x xx x

1

0 1π , ):

( )

y 01

tt

mw − .(15)

The sequential importance sampling algorithm canthus be implemented by performing the following twosteps for every t: 1) Draw particles x x x yt

mt t

mt

( ) ( ):~ ( | , ),π −1 0 where

m M=1 2, , ,K . 2) Compute the weights of wt

m( ) according to (15).The importance function plays a very important role in

the performance of the particle filter. This function musthave the same support as the probability distribution thatis being approximated. In general, the closer the impor-tance function to that distribution, the better the approxi-

mation is. In the literature, the two most frequently usedimportance functions are the prior and the optimal im-portance function. The prior importance function isgiven by p t t

m( | )( )x x −1 , and it implies particle weight up-dates by

( )w w ptm

tm

t tm( ) ( ) ( )|∝ −1 y x . (16)

The optimal importance function minimizes the vari-ance of the importance weights conditional on the trajec-tory x 0 1:

( )t

m− and the observations y 0:t and is given by

p t tm

t( | , ):( )

:x x y0 1 0− [11]. When the optimal function isused, the update of the weights is carried out according to

( )w w ptm

tm

t tm( ) ( ) ( )|∝ − −1 1y x . (17)

Note that implementations of particle filters with priorimportance functions are much easier than those with op-timal importance functions. The reason is that the com-putation of p t t

m( | )( )y x −1 requires integration.A major problem with particle filtering is that the dis-

crete random measure degenerates quickly. In otherwords, all the particles except for a very few are assignednegligible weights. The degeneracy implies that the per-formance of the particle filter will deteriorate. Degener-acy, however, can be reduced by using good importancesampling functions and resampling.

Resampling is a scheme that eliminates particles withsmall weights and replicates particles with large weights.In principle, it is implemented as follows: 1) Draw M particles, x t

m*( ) from the discrete distribu-tion χ t . 2) Let x xt

mt

m( ) *( )= , and assign equal weights (1 / M)to the particles.

The idea of resampling is depicted in Figure 2 withM =10 particles. There, the left column of circles repre-sents particles before resampling, where the diameters ofthe circles are proportional to the weights of the particles.The right column of circles are the particles afterresampling. In general the large particles are replicatedand the small particles are removed. For example, the“blue” particle with the largest weight is replicated threetimes and the “yellow” particle, two times, whereas thegreen particles, which have small weights, are removed.Also, after resampling all the circles have equal diameters,that is, all the weights are set to1/ .M In Figure 3, we rep-resent pictorially the random measures and the actualprobability distributions of interest as well as the threesteps of particle filtering: particle generation, weight up-date, and resampling. In the figure, the solid curves repre-sent the distributions of interest, which are approximatedby the discrete measures. The sizes of the particles reflectthe weights that are assigned to them. Finally in Figure 4,we display a flowchart that summarizes the particle filter-ing algorithm. At time t, a new set of particles is gener-ated, and their weights are computed. Thereby we obtainthe random measure χ t , which can be used for estimationof the desired unknowns. Before we proceed with the

22 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

generation of the set of particles for time instant t +1, weestimate the effective particle size (a metric that measuresthe degeneracy of the particles [29], [36]). If the effectiveparticle size is below a predefined threshold, resamplingtakes place; otherwise we proceed with the regular stepsof new particle generation and weight computation.

Recently, a special class of particle filters that approxi-mate the posterior distributions by single Gaussians hasbeen introduced [30]. Although in their derivation it is as-sumed that all the relevant distributions are Gaussian, as isdone with some other filters including the extendedKalman filter and its variants, they are distinguishable inthat the updating of the filtering and predictive distribu-tions is accomplished by propagating particles. This entailsadvantages of easier implementation than is the case withthe standard particle filters and improved performanceover other Gaussian based approximation filters. TheGaussian particle filter has also been used as a buildingblock for more complex filters called Gaussian sum particlefilters [31]. These filters approximate the filtering and pre-dictive distributions by weighted Gaussian mixtures andbasically represent banks of Gaussian particle filters.

Before we continue with the presentation of applica-tions of particle filtering to communication problems, wesummarize the procedure for developing particle filteringalgorithms. The procedure involves the following steps: 1) description of the problem by a discrete state-spacemodel as in (1) 2) selection of a proposal function for particle generation 3) derivation of the equations for the weight update.

Additional issues are the choice of resampling algo-rithm and the schedule for resampling. We proceed withshowing how these steps are applied to resolving theproblem of blind equalization.

Blind EqualizationWhen digital symbols are transmitted over fre-quency-selective channels, inter-symbol interference(ISI) occurs, which has a detrimental effect on the detec-tion at the receiver. To allow for symbol detection withreasonable error, channel equalization is needed to re-verse the effect of ISI. A popular equalization techniqueapplies the principle of maximum-likelihood estimationthat results in the Viterbi algorithm for symbol detec-tion. When the channel parameters are unknown, theyare first estimated, usually from training data whosetransmission contributes to significant overheads andbandwidth-inefficient communication.

Blind equalization involves detection of transmittedsymbols without using training data. This can be accom-plished either without explicit estimation of the channelparameters or by joint symbol detection and channel pa-rameter estimation.

Recently, several researchers have employed particle fil-tering for problems of blind equalization. The flexibility ofparticle filtering has allowed for the application of several

variants of blind equalization including ones that involvetime-invariant channels, [38], [41], time-varying channels[4], [5], [15], [16], additive Gaussian and non-Gaussianchannels [44], as well as OFDM systems [55].

For the convenience of discussing in greater detailsome of the work done in equalization, we adopt the fol-lowing signal model. When digital symbols s t are trans-mitted over a frequency selective channel, the receivedsignal can be represented as

y s h v vt t ll

L

t l t t t t= + = +−=

−

∑0

1

, h s

(18)

where yt is the received signal at time instant t,s t t t t Ls s s = ⋅⋅⋅− − +[ ]1 1 , h t t th h = [ , ,0 1 ⋅⋅⋅ ht L, ]−1 are the co-efficients of the unknown finite impulse response (FIR)channel impulse response, L is the length of the channel,and vt is an additive noise which is usually considered as azero mean Gaussian process with a known variance σ v

2 .The objective is to detect the transmitted symbols by firstobtaining the posterior distribution p s yt t( | ): :0 0 and thenusing it to perform detection.

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 23

*

*

*

*

*

*

*

*

*

*

1

0

1/10

1/10

2. A schematic description of resampling.

Time-Invariant ChannelsWhen the channels are time invariant, we have h ht = . Ifwe assume Gaussian priors for the channel coefficients,we can analytically marginalize them and directly drawsamples from the posterior distribution of the symbolsonly. This allows for performing equalization without ex-plicitly estimating the channel coefficients. In otherwords, as per (15), for the update of the weighting coeffi-cients we would use

wp y s y p s s

stm t t

mt t

mt

m

tm

( ) :( )

:( )

:( )

(

( | , ) ( | )(

∝ − −0 0 1 0 1

π ):

( ):

( )

| , )s yw

tm

tt

m

0 1 01

−− .

(19)

Following [41] and with the assumption that the datasymbols s t ∈ − + , 1 1 are independent identically distrib-uted (i.i.d.) uniform random variables, we can write thestate-space model of the observed data as

s Fs u

h st t t

t t ty v

= +

= +−1

(20)

where F is an L L× state transition matrix given by

F =

0 0 0 0 01 0 0 0 00 1 0 0 0

0 0 0 1 0

L

L

L

M M M O M M

L

(21)

vt v~ ( , ) 0 2σ , and u t tsT = [ ... ]0 0 . If we assume that thechannel coefficients have a Gaussian prior distribution,h h~ ( , ) − −1 1 , it can be shown that the posterior distri-bution of the channel, p s ym

tm

t( | , )( ):

( ):h 0 0 , which corre-

sponds to the mth trajectory of symbols, s tm

0:( ) , is Gaussian,

that is, p s ymt

mt t

mt

m( | , ) ( , )( ):

( ):

( ) ( )h h0 0 = . The mean andthe covariance matrix are recursively updated by

hs

htm

tm t

mt

vt

mt

m

tm

y( ) ( )( )

( ) ( )

(

= +

−

−−

−

σ 2 11

1

1 )( ) ( )

( ) .= + −−s st

mt

m

vt

m

σ 2 11

(22)

In [38] and [41], it is shown that we can obtain an ana-lytical expression for the likelihood function,p y s yt t

mt( | , ):

( ):0 0 1− , by marginalizing out the channel coef-

ficient, i.e.,

24 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

3. A pictorial description of particle filtering.

( )p y s yt tm

tt

m

v tm

| ,| |

( | |)

ex

:( )

:

( ) /

( ) /0 0 1

1 2

21

1 22−−

=

πσ

p ( ) ( ) ( )− +

−

− −−

−12

2

2 1 11

1

2

y

y

t

vt

mt

mt

m

t t

v

σ

σ

h h

s

+

× +

−−

−

−−

−

tm

tm

tm t t

vt

mt

y

11

1

2 11

( ) ( )

( ) ( )

h

sh

σ 1( ) .m

(23)The optimal importance function is proportional to

the likelihood function, or,

( ) ( )π s s y p y s s yt tm

t t t tm

t| , | , ,:( )

: :( )

:0 1 0 0 1 0 1− − −∝ .

It is readily shown that the function has the form

( )π s s y

p y s s yp y s

t tm

t

t t tm

t

t

| ,

( | , , )( |

:( )

:

:( )

:

0 1 0

0 1 0 1

−

− −=t t

mt t t t

mts y p y s s y= + = −− − − −1 10 1 0 1 0 1 0 1, , ) ( | , , ):

( ): :

( ):

(24)and that the updating of the particle weights is carried outby

( )(w w p y s s y

p y s s

tm

tm

t t tm

t

t t

( ) ( ):

( ):| , ,

| ,

∝ =

+ = −

− − −1 0 1 0 11

1( ))0 1 0 1:( )

:, .tm

ty− − (25)

When the particles and weights ofall the trajectories are obtained, sym-bol detection can be carried out usingsmoothing. Smoothing in this con-text has been addressed in [8] and[9], where the emphasis is on themethods for fixed-lag blind equaliza-tion and the interest lies in obtainingthe smoothing distr ibutionp s yt l t( | ):− 0 , where l is the fixed lag,followed up by detection of s t l− .

An interesting extension of theequalization algorithm is its modifica-tion to cope with unknown channelorders. The problem can be resolvedby employing a marginalization strat-egy where the unknown channel orderis marginalized. Another possibility isto estimate the channel order with thetransmitted symbols.

Time-Variant ChannelsFor time-varying channels, such asmobile communication channels,symbol detection and channel esti-

mation can be performed jointly following the concept ofmixture Kalman filtering (MKF) [6], [11]. The problemis again formulated by writing the channel model as astate equation. As before, the channels are represented asFIR filters, but now they have time varying complex coef-ficients whose magnitudes are randomly varying Ray-leigh processes. The coefficient variations can beapproximately modeled as autoregressive-moving aver-age (ARMA) processes given by

h a h b ut l kk

r

t k l kk

r

t k l

a b

, , ,= − +=

−=

−∑ ∑1 0 (26)

where ht l, is the lth coefficient at time t, b0 1= , and ut l, isthe driving noise process of the lth coefficient. The pa-rameters a a1 2, ,K, a ra

, and b b1 2, ,K, brbare functions of

the fading rate of the channels and can be evaluated if theDoppler spread of the channels and the symbol rate areknown. Suppose for clarity of presentation that the varia-tion of the channel is modeled as an autoregressive pro-cess (AR) (all the bk s except b0 in (26) are zero). Then, ifwe define the channel state as x h h ht t t t ra

= ⋅⋅⋅− − +[ ]1 1 ,where h t t t t Lh h h = −[ , , , ], , ,0 1 1K (that is, x t is an r La ×1vector), we obtain the following state-space model:

s Fs zx Ax Du

s x

t t t

t t t

t t t ty v

= += +

= +

−

−

~

.

1

1

(27)

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 25

4. A block diagram of particle filtering.

Here s t i s an r La ×1 vector defined bys t t t t Ls s s = ⋅⋅⋅ ⋅⋅⋅− − +[ ]1 1 0 0 , ~F is an r L r La a× matrixgiven by

~F

F 0 00 0 0

0 0 0

=

⋅⋅⋅

L

L

M M M M

where F is defined by (21), z t is an r La ×1 vector whoseelements are all equal to zero except for the first one whichis equal to s t , and u t is an L×1zero mean Gaussian noisevector whose covariance matrix has diagonal elementsproportional to the power of each lag. The matrix A hasdimensions r L r La a× and is constructed from the coeffi-cients of the AR process according to

A

I I II 0 0

0 0 I 0

=

− − ⋅⋅⋅ ⋅⋅⋅ −⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅

a a a ra1 2

M M M M M

(28)

and D is an r L La × matrix given by

D

I0

0

=

M

where I and 0 in the above expressions are the identity andzero matrices of sizes L L× .

Now the objective is to jointly estimate the state of thechannel x t and to detect the transmitted symbols s t . Itshould be noted that given the transmitted symbols, thestate-space model becomes a linear Gaussian model andthe posterior distribution of the channel p s yt t t( | , ): :x 0 0 isGaussian for all t. This permits recursive estimation of thechannel parameters using Kalman filtering [16].

We consider the importance function

( ) ( )π s s y p y s s y

p s s

t tm

t t t tm

t

t

| , | , ,

|

:( )

: :( )

:

:

0 1 0 0 1 0 1

0

− − −∝

× ( )( )

tm

t

t t tm

t

y

p y s s y

− −

− −∝

1 0 1

0 1 0 1

( ):

:( )

:

,

| , , (29)

where as before we assume that the symbols are i.i.d. uni-form random variables. The last term can be expressed as

( )p y s s y p y s s yt t tm

t t t t tm

t| , , | , , ,:( )

: :( )

:0 1 0 1 0 1 0− − − −= ∫ x( )( )

1

0 1 0 1× − −p s s y dt t tm

t tx x| , ,:( )

:

(30)where the second factor of the integrand is the predictivedistribution of the Kalman filter. Then, the expression forthe importance function can be rewritten as

( ) ( )π σs s y yt tm

t c t tm

t

c t tm

t

| , ; ,

; ,

:( )

:( )

( )

0 1 02

− ∝

×∫

s x

x

( )( )

( )

( ) ( ) ( ) ( ) ( )~ , ~ ~

mt

c tm

tm

v tm

tm

tm

dx

s s s= + σ 2

(31)where, for clar i ty, c t t

mty( ; , )( )s x σ 2 and

c t tm

tm( ; , )( ) ( )x denote that yt and x t have complex

Gaussian distributions, respectively;~ [( ) ( ) ( )stm

t tm

tms s s = − −1 2

⋅⋅⋅ ⋅⋅⋅− +s t Lm

1 0 0( ) ]; and tm( ) and t

m( ) are the predictive meanand covariance of x t of the mth trajectory, respectively.Similarly, we can show that the corresponding weightscan be computed from

( )w w p y s s y

w

tm

tm

st t

mt t

t

t

( ) ( )

:

( ):

(

| , ,∝

=

−∈ ±

− −

−

∑11

0 1 0 1

1 ( )mc

st t

mt

mv t

mt

mt

m

t

y)

( ) ( ) ( ) ( ) ( ); , .∈ ±∑ +

1

2s s s σ

(32)When implementing the algorithm, for each particle

we evaluate the predictive mean and covariance of twoKalman filters, each corresponding to the symbols s t =1and s t = −1, respectively. Then we draw samples using(31) and calculate the weights by (32). Finally, theKalman filters corresponding to the sampled symbols areupdated. Once the particles with their correspondingweights are obtained, the symbol is estimated using themaximum a posteriori (MAP) criterion. If needed, a min-imum mean square error (MMSE) (or other type) esti-mate of the channel state can be obtained from the meanupdates of the posterior distribution of x t .

Unlike other standard methods, particle filtering caneasily be extended to non-Gaussian noises. In [44], theadditive complex noise is modeled as a mixture of J zeromean Gaussians having different variances. There, a la-tent variable λ t is defined to indicate the distribution ofvt . The procedure draws particles from an importancefunction p s s yt t t

mt

mt( , | , , ):

( ):

( ):λ λ1 1 1 1 0− − , and the particles are

used for approximation of the joint posterior distributionp s yt t t( , | ):λ 0 from which the MAP estimates of the sym-bols are obtained. It is reported that the algorithm out-performs existing methods based on Gaussian noise [44].

Recently, a similar treatment has been extended toOFDM systems over frequency selective channels [55].One important difference with the above treatment is thatthe received signal yt is considered to be an observation inthe frequency domain where the index t there representsdifferent subcarriers. In such systems the observed signalsof all the subcarriers are simultaneously received. Thechannel parameters are assumed to have Gaussian a prioridistribution which allows for, as discussed earlier, the per-formance of symbol detection without explicitly deter-mining the channel parameters.

Blind equalization for satellite communications wasconsidered in [33]. There, a state equation was developedfor a nonlinear satellite channel consisting of a cascade oflinear filters and a memoryless nonlinear traveling wave

26 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

tube amplifier. The state equation coupled with the ob-servation, which consists of the state variable embeddedin additive Gaussian noise, provided a dynamicstate-space model for the system. With the assumptionthat all the system parameters and noise variances areknown, a generic particle filtering detector employing theprior importance function was applied to combat thenonlinear distortion of the channel.

SimulationsIn our experiment we have simulated a scenario of atime-invariant channel with an impulse response oflength L=3. We assumed a Gaussian prior for the channelcoefficients, h h~ ( , ) − −1 1 , where

h − −=

=

1 1

100

010000 0 00 024569 00 0 005475

,.

..

.

This choice of prior mean and covariance matrix corre-sponds to an environment with a strong line of sight com-ponent and two weaker, zero-mean paths. This is a fairlyrealistic scenario for an indoor communication system,for instance. The numerical values of the means and vari-ances of the channel taps are selected to yield the delaypower profile in Figure 5.

To estimate the bit error rate (BER) under this channelmodel, we have randomly generated 5,000 signal burstsof duration T = 40 (i.e., 200,000 bits). A new samplechannel is drawn from the above prior distribution foreach burst. At the receiving end, we have simulated threeequalizers: A one-shot linear MMSE (LMMSE) equalizer: Thisequalizer has perfect knowledge of the channel response,and it represents a Wiener matrix-filter that processes allthe observations in a single burst at once [41]. Clearly,this is not a realistic receiver, but it yields a lower boundon the BER of simpler LMMSE equalizers. The maximum likelihood equalizer (MLE): It, too, hasperfect knowledge of the channel impulse response and isimplemented via the Viterbi algorithm. This is the opti-mal sequence detector and, therefore, it yields a lowerbound on the BER. The blind MAP equalizer: It is implemented via the se-quential importance resampling (SIR) algorithm [17].The number of particles is M =300 and resampling iscarried out each time the effective particle size goes belowthe threshold ε =025. M. (The effective sample size is de-fined by M w

t tm

m

M

eff ==∑11

2/ ( )( ) .) The importance dis-tribution is the optimal one.

The estimated BER curves are shown in Figure 6.Each point in the plots results from a trimmed averageover the set of 2,500 signal bursts. Extreme simulationsamples (the highest 0.5% and smallest 0.5%) havebeen discarded.

Blind Detection over Flat Fading ChannelsIn this section, we discuss in detail the applications of par-ticle filters to detection over flat fading channels. Figure 7shows the baseband communications system block dia-gram over a frequency flat fading channel. The input sig-nal to the system is a sequence of symbols s t , transmittedafter bandlimiting the pulses using a pulse shaping filterg( )τ . The symbol period is T s , and the channel is repre-sented by the complex time varying process h( )τ and theadditive noise v( )τ .

Fading, which is the variation of the received complexamplitude, is a result of the multipath nature of the chan-nel. Signals arriving at the receiver via multiple paths havedifferent complex gains and add up, resulting in a fadingchannel. The time variation in the number of paths, am-plitudes, and mainly the phases of the multiple paths pro-duces a fading process with random nature. When themultiple paths arrive roughly within the same symbol pe-riod, the received signal does not undergo distortion infrequency and hence is called frequency flat or

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 27

5. Delay power profile of the three-tap channel.

6. BERs of different equalizers.

nondispersive fading. The fading process, however, ishighly correlated and characterized by its bandwidth orDoppler frequency denoted as f d . The random nature ofthe fading is described by the distribution of the processat each time instant. When the real and complex compo-nents are Gaussian, the resulting amplitude is Rayleighdistributed, while a line-of-sight component results inRicean fading. For a more elaborate discussion on fadingchannels; see, for example, [43] and [50].

It is important to note that the multiple paths arrive inthe same symbol period, and therefore there is no ISI.The receiver observes both a random complex gain forthe transmitted symbol and additive channel noise. Whilethe Jakes model [25] is often used to model the flat fadingchannel, a Markov model is preferred herein to obtain astate-space model. The flat fading process can be gener-ated by filtering complex white noise by a low pass filter,whose spectral characteristics match that of the fadingprocess [27]. As mentioned in the previous section, anAR (or ARMA) model adequately represents the fadingprocess [50], [56], [60]. The AR (ARMA) parametersare chosen to match the spectral characteristics with thoseof the fading process. A simpler method, which uses atwo-path model to build AR(2) and AR(3) processes canbe found in [12] and [56] respectively; the results thereclosely approximate more complex path models.

The state-space representation of the baseband com-munications system can thus be given by

h Ah u

s ht t t

t t t ty v

= +

= +−1

(33)

where s t ts = [ ]0 0 0K , and the channelh t t t t rh h h

a

= − −[ ]1 K is modeled as an autoregressionwith

A =

− − − −

−a a a ar ra a1 2 1 01 0 0 0 0

0 0 0 1 0

L

L

M M M M M

L

where the AR parameters may be known or unknown,and u t tu = [ ]0 0K , where ut is complex white andGaussian with variance σ u

2 . This is a reasonable assump-tion, since the AR parameters depend on the second or-

der statistics of the channel and hence do notchange as rapidly as the channel gain.

Our goal is to estimate the Bayesian poste-rior distribution p s yt t t( , | ):h 0 . For achievingthis, we discuss two groups of algorithms ofwhich the first assumes that the AR coeffi-cients are known and the second that they areunknown.

Known AR CoefficientsDetector IDetector I employs the prior distribution as the impor-tance sampling function, i.e.,

( )( )

π1 1 1

1

( , ) , | ,

| ( )

( ) ( )

( )

h h h

h h

t t t t tm

tm

t tm

t

s p s s

p p s

≡

=

− −

−(34)

where the last equality is due to the Markovian nature ofthe channel and assuming that the transmitted bits are in-dependent of the channel and each other and are identi-cally distributed. The weight update equation can then bewritten as

( )w w p y stm

tm

t tm

tm( ) ( ) ( ) ( )| ,∝ −1 h (35)

( )= −w p y h stm

t tm

tm

1( ) ( ) ( )| , . (36)

This detector can be used for both linear and nonlinearchannel models. It can also be applied to channels withmore general fading characteristics than the ones of Ray-leigh fading, for as long as the fading process can be repre-sented by a Markov model. This detector can also be usedfor non-Gaussian noise channels.

Detector IIHere the importance sampling function is the optimalfunction given by [11]

( )π 2 1 1

1

( , ) , | , ,

| ,

( ) ( )

( )

h h h

h h

t t t t tm

tm

t

t tm

t

s p s s y

p s

=

=

− −

−( ) ( ), | , .( )y p s yt t tm

th −1

(37)

Thus, a sample s tm( ) is first obtained from

( ) ( )p s y p y s p s

s h

t tm

t t t tm

t

t tm

v

| , | , ( )

,

( ) ( )

( )

h h− −

−

∝

= +

1 1

12

σ( )σ u tp s2 ( )(38)

followed by a sample from p s yt tm

tm

t( | , , )( ) ( )h h − =1 ( , )( ) ( ) t

mt

m , where

28 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

7. Continuous-time model of the communication system.

tm

u

tm

tm

v

tm

tm t

( )( ) ( )

( ) ( )

= +

=

−

−

12 2

1

1

σ σI

s s

Ah

( ) ( )

.m

u

tm

t

v

yσ σ2 2

+

s

(39)

The weight update equation is given by

( )w w p y stm

tm

t tm

tm( ) ( ) ( ) ( )| ,∝ − − −1 1 1h (40)

where

( )p y s p s y s ht tm

tm

ts

t t tm

v ut

| , ( ) ; ,( ) ( ) ( )h − −∈

−= +∑1 1 12

σ σ( )2 .(41)

In the last expression, ( ; , )( ) ( )y s ht tm

tm

v u− +12 2σ σ are

Gaussian probability distribution functions with means ht

mt

m( ) ( )−1 and variance σ σv u

2 2+ and computed at yt . Thisdetector is optimal in the sense described earlier. The gen-eralization to non-Rayleigh fading channels andnon-Gaussian channel noises is straightforward.

Detector IIIThis detector combines a bank of weighted Kalman filtersand a particle filtering algorithm. It obtains the posteriordistribution of the transmitted symbols by marginalizingthe channel [7], [44]. Given the transmitted symbols, thestate-space model in (33) is linear, and hence a Kalman fil-ter can be used to track the channel. For the posterior ofthe symbol s t , we can write

( ) ( )p s s y p y s s y p st tm

t t tm

t t t| , | , , ( ):( )

: :( )

:0 1 0 0 1 0 1− − −∝ (42)

where

( )p y s s y p y s

p s

t tm

t t t t t

t tm

| , , ( | , )

|

:( )

:

:( )

0 1 0 1

0 1

− −

−

=

×∫ h

h( ), :y dt t0 1− h (43)

and p s yt tm

t( | , ):( )

:h 0 1 0 1− − is obtained from the Kalman filter.The weight update is given by

( )w w p y s y

w p y

tm

tm

t tm

t

tm

st

t

( ) ( ):

( ):

( )

| ,

|

∝

=

− − −

−∈∑

1 0 1 0 1

1

( )s s y p st tm

t t, , ( ).:( )

:0 1 0 1− −(44)

In [7], extensions to non-Gaussian channel noise and codedsymbols are provided. Non-Rayleigh fading channels canalso be tracked using Gaussian sum particle filters [32].

Unknown AR Coefficients—Blind DetectionIn the previous subsection we have assumed that the AR(or ARMA) coefficients of the channel model are known.In practice, however, the channel statistics are unknown,which implies that the channel model coefficients must beestimated. One approach to their estimation is by using

pilot signals. For accurate estimation, long sequences ofpilot signals may be required, especially for slow fadingchannels. Further, for nonstationary channels, pilot sig-nals must be constantly retransmitted. An attractive alter-native is to build a receiver for joint channel coefficientsestimation and symbol detection.

The presence of unknown model coefficients does notallow for a direct extension of the above proposed particlefilters. In particular, there are three related difficulties.First, the use of the prior importance function results ininefficient implementations, whereas the employment ofthe posterior importance function is prohibited. Second,ambiguities arise between the symbols and the model co-efficients (for phase-shift keying (PSK) modulated sig-nals, there are also phase ambiguities). Third, if the modelcoefficients are static parameters, they may create prob-lems because the diversity of their representations impov-erishes after resampling. To provide particle filters withmore diversity, rejuvenation procedures are required.

Detector I (RLS Based)A blind algorithm for joint channel estimation and detec-tion was first presented in [34]. The algorithm is a hybridmethod which updates the unknown AR coefficients us-ing the recursive least squares (RLS) method, while thechannel and the symbols undergo particle filtering up-dates. The prior is used as importance sampling function,and the AR coefficients are marginalized as follows:

( )π h h h ht t tm

tm

t t tms s y p y, | , , | ,:

( ):

( ): :

( )0 1 0 1 0 1 0 1 0− − − −≡ ( )

( )( )

:

( )

:( )

( )

( ) | ,

|

t t

t t tm

tm

p s

p s p

p d

−

−

−

=

×∫

1

1

0 1

h h a

a h a (45)

where a = [ , , ]a a ra1 K . This results in a hybrid algo-rithm, which is a weighted bank of RLS filters that up-date a, and whose weights are computed recursively

( )w w p y h stm

tm

t tm

tm( ) ( ) ( ) ( )| ,∝ −1 . (46)

Detector IIAnother blind particle filtering detector was reported in[21]. The detector adopts an AR(2) model for the fadingchannel

h a h a h ut t t t= − − +− −1 1 2 2 . (47)

The parameters a1 and a 2 are obtained from

( )a r a rd d d1 222 2 2= − =cos /πΩ and (48)

where rd is the pole radius of the AR(2) model, andΩ d df T= is the normalized maximum Doppler fre-quency. The expressions described by (48) impose an in-teresting and important relationship between theunderlying fading channel and the model coefficients. By

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 29

considering the physical communication system, the rela-tionship enables us to combat the ambiguity between thesymbols and the coefficients and to implement a fullyblind detector. Furthermore, to achieve efficient imple-mentation, the detector first marginalizes out ht usingMKF, and then employs a hybrid importance function fors t and the model coefficients a1 and a 2 [22]. The hybridimportance function is expressed as

( )π s s y

p s s

t t tm

tm

t

t t tm

, | , ,

| , ,

:( )

:( )

:

:( )

a a

a a

0 1 0 1 0

0 1

− −

−= ( ) ( )0 1 0 1

0 1 0

:( )

:( )

( ):

( )

, |

| , ,:

tm

t t tm

t tm

tm

y p

p st

− −

−=−

a a

a a s( )( ) ( )

1 0

1 1 1 2 2 1

( ):

, :( )

, :( )

,mt

t tm

t tm

y

a a a a× − −− −δ δ (49)

where a t t ta a= [ , ]1 2 , and the last equality is obtained

based on the state equations a at t1 1 1, ,= − and a at t2 2 1, ,= − .The corresponding weights are computed by

( )w w p y y

w p

tm

tm

t tm m

t

tm

t

( ) ( ):

( ) ( ):

( )

| , ,:

∝

=

− −

−

−1 0 0 1

1

0 1a s

( )s

t t tm

tm

tt

y s y∈

− −∑

| , , , .:( )

:( )

:a s0 0 1 0 1(50)

Note that, as suggested by its name, the hybrid impor-tance function (49) is a combination of the posterior andthe prior importance functions. As opposed to the poste-rior importance function, it is easily implementable be-cause the sampling from (49) and the computation of theweight in (50) can be readily carried out. Finally, to over-come the impoverishment of the particle representationin a generic implementation, an auxiliary particle filterwith a smoothing kernel can be applied.

Detector IIIIn addition to linear models, alternative modeling of fad-ing channels may be preferred, especially if one wants tocapture the nonlinearities of channels. A wavelet-basednonparametric modeling of fading channels was used in[18] where the fading process is decomposed using wave-let expansions, i.e.,

ht t t= (51)

where t is the wavelet basis vector at time t, and t de-notes the wavelet coefficients, where the two vectors areof size k. A blind receiver employing MKF was proposedfor joint estimation of the wavelet coefficients and symboldetection. What is more, the receiver treats kas a static un-known parameter and evolves according to k kt t= −1 . As aresult, the blind receiver updates the number of waveletsdynamically and requires no channel statistics.

Some ExamplesIn this subsection, we present results of two sets of experi-ments. In all the simulations, it is assumed that data aretransmitted continuously without pilot symbols or anyform of reinitialization bits in between. Hence, the dataare transmitted even during harsh conditions when theinstantaneous signal-to-noise ratio (SNR) is very low,which happens during zero fading.

Rayleigh Fading with Gaussian ChannelNoise—AR Parameters KnownThe first experiment considered a Rayleigh fading channelwith additive Gaussian noise. A binary PSK (BPSK) mod-ulation scheme was used for data transmission, with thesymbols s t = −1 and s t =1 being equally likely. Data were

differentially encoded to mitigate the phaseambiguity problem. All three detectors wereimplemented for a channel with normalizedDoppler spreads set to f d =0001. , which corre-sponds to slow fading, and f d =001. , which is afast fading scenario. An AR(3) process wasused to model the channel, where the AR coef-ficients are a function of the Doppler spreadand are obtained from the method suggested in[56]. The particle filter detectors were com-pared with the clairvoyant detector, which per-forms matched filtering and detectionassuming that the channel is known exactly atthe receiver. Thus, it serves as an unachievableupper performance bound. The number of par-ticles chosen for Detectors I and II wasM =1 000, , while for Detector III was M =50.

In Figure 8, BERs as functions of SNR forthe slow fading case ( f d =0001. ) are plotted.The ARcoefficients are given by( , , )a a a1 2 3 ≡( . , . , . )− −29916 29833 09917 . From the figure,it can be seen that Detectors II and III per-form similarly. However, it was observed that

30 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

8. BER performance of Detectors I, II, and III over flat Rayleigh fading withfd = 0001. for differentially encoded BPSK signaling.

as the SNR was increased, Detector II degraded in per-formance. Detector I does not work well in slow fadingscompared to the other two detectors.

Figure 9 shows the BERs as functions of SNR for thefast fading case, where f d =001. . The AR coefficients aregiven by ( , , ) ( . , . , . )a a a1 2 3 29145 28344 09197≡ − − . De-tectors I and II have equally good performance and areonly slightly better than Detector III.

Rayleigh Fading with Gaussian ChannelNoise—AR Parameters UnknownFigure 10 displays the BER performance versus SNR ofthe blind Detector II described earlier. The fading processwas generated using the Jakes’ method witheight oscillators and Ω d =003. . We can see thatDetector II clearly outperforms the differen-tial detector and performs closely to the lowerbound achieved by the pilot aided MKF. Forthe pilot aided MKF, 1,000 pilot symbols werefirst used to estimate the AR coefficients (themodified covariance method [20], [26] wasemployed for the estimation), and then MKFwas implemented with the estimated coeffi-cients set as true coefficients.

Particle Filteringfor Multiuser DetectionMultiuser detection (MUD) has received agreat deal of attention since the 1980s due toits potential for increasing CDMA system ca-pacity. A specific feature of MUD is that itdoes not treat the multiple access interference(MAI) present in CDMA systems as noise butas information. Since the optimum MUD isexponential in complexity, numerous approxi-mate detectors have been developed to reducethat complexity. However, the performance ofthese detectors is suboptimal since they use in-terim hard decisions.

Application of particle filtering to MUDrequires a representation of the system by a dy-namic state-space model. Since the symbols ofa CDMA system are uncorrelated across dif-ferent time slots, it is not obvious how to con-struct such representation.

The earliest application of particle filters toMUD appeared in [2] and subsequently in[45]. The state-space model is constructed us-ing time dynamics of the fading channels, andthe symbols of all the users at a time slot aretreated as one super symbol. As a result, thesuper symbol has a large alphabet whose sizegrows exponentially with the number of users.Recently in [58] and [59], an alternativestate-space representation of CDMA systemswas proposed. It is based on whitened matched

filter (WMF) outputs, where the dynamics of the systemevolves with user index. This representation allows for anefficient application of particle filtering. Here we also notethat MUD methods based on Markov chain Monte Carlosampling were also proposed [53], [54].

State-Space RepresentationsConsider a synchronous CDMA system with chip-rate(processing gain) C and K users. Let T s denote symbolduration and ξ τk ( )the normalized deterministic signaturewaveform assigned to the kth user. Here τ∈[ , ]0 T s andk K∈ , , 1 K . Let s k ( ) , τ ∈ − +1 1 be the symbol transmit-ted by the kth user, hk ( )τ the fading coefficient of the kth

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 31

9. BER performance of Detectors I, II and III over flat Rayleigh fading withfd = 001. for differentially encoded BPSK signaling.

10. BER performances of Detector II, the pilot aided MKF, and the differentialdetector. The fading channel was generated by Jakes’ method andΩ d = 003. .

user, and v( )τ the received zero mean complex whiteGaussian noise with variance σ v

2 . We can express the re-ceived signal, y( )τ , as

y h s v t Tk

K

k k k s( ) ( ) ( ) ( ) ( ) [ , ]τ τ τ ξ τ τ= + ∈=∑

1

0 .(52)

After sampling at the system chip rate and modeling theRayleigh flat fading channels of all users as ARMA pro-cesses as in (26), the ARMA coefficients are chosen to fitthe spectra of the fading processes. Here, without loss ofgenerality, we assume that r r ra b= = . We can write forthe state-space representation

t t t= +−

~A Euρ 1 (53)

h Bt t= (54)

wheret t t t r t K t r = ⋅⋅⋅ ⋅⋅⋅− − −[ ], , , , ,ρ ρ ρ ρ ρ1 1 1 1 2 is an auxiliary

state vector of size ( )r K+ ×1 1introduced to facilitate therepresentation, and h t t K th h= ⋅⋅⋅[ ], ,1

is the fading coef-ficient vector of all users. The matrices~A and Bare knownand of sizes ( ) ( )r K r K+ × +1 1 and K r K× +( )1 , respec-tively, where ~A is a block diagonal matrix, i.e.,~ , , , A A A A= diag K , with A being an( ) ( )r r+ × +1 1 ma-trix defined by (28), and

B

b 0 00 b 0

0 0 b

=

L

L

M M M M

L

where b = ⋅⋅⋅[ ]b b br0 1 is a vector of the MA parametersof the ARMA process. Finally, E is an ( )r K K+ ×1 matrixwith zero elements except for ones at positions(( ) , )k r k− +1 1 , where k K=1 2, , ,K , and u t is a K ×1noisevector. The observations can now be written as

y vt t t t= +s B (55)

where s t = [ , , ], ,s st K t1 K , and is a K K× diagonal ma-

trix of spreading codes.In the algorithm from [2] and [45], (53) and (55)

form the representation of the system that evolves basedon the chip duration, whereas the parameter of interest s tis static within each symbol time. Samples of s t are takenfrom an alphabet of size 2 K , and the channel state is inte-grated out. This is equivalent to the MKF algorithm pro-posed in [6] and [11]. With large number of users, thealphabet of s t grows exponentially and the calculation ofthe importance weight becomes computationally very ex-pensive. In [45] and [46], it was found that deterministicmethods which preserve the most likely particles were themost efficient.

In the algorithm from [58] and [59], WMF outputswere utilized for the system representation because they

are sufficient statistics of the transmitted symbols [52].We can express the matched filter output as

y RH s vt t t t= + (56)

where R is the crosscorrelation matrix whose ijth elementis def ined by R ij i=< ξ , ξ ξ τ ξ τ τj i

T

js d>=∫0

( ) ( ) ,Ht t K th h= ⋅⋅⋅diag , , , ,1 is the diagonal matrix of the chan-nel fading coefficients which are considered static withineach symbol interval, s t t K ts s = [ , , ], ,1 K is the user sym-bol vector, and v t is a complex-valued Gaussian vectorwith independent real and imaginary components andcovariance matrix equal to σ v

2 R.The cross-correlation matrix is positive definite, and

Cholesky factorization can be employed. There exists aunique lower triangular matrix F such that R F F= .When we applyF F− −= ( ) 1 to the matched filter output,we obtain

y F yt t= −( ) 1 (57)

= +FH s vt t t (58)

= +FS h vt t t (59)

where S t t t K ts s s= diag( , , , ), , ,1 2 K is the user symbol ma-trix. It can be verified that the covariance matrix of v t isσ v

2 I, where I is the identity matrix. Since the noise be-comes i.i.d. white Gaussian, y t is called the whitenedmatched filter output. Component wise, it can be writtenas

y F h s vk ti

k

k i i t i t k t, , , , ,= +=∑

1

.(60)

Particle Filtering Implementation

MUD When the Fading Coefficients Are KnownIn this case the unknowns are the user symbols which areuncorrelated across different time slots. In the followingpresentation, we drop the time subscript t. First, note thats k is independent of y k1 1: − . Therefore, the posterior distri-bution of the symbols of the first k users can be factoredaccording to

p s yp y s y p s y

p y yk kk k k k k

k

( | )( | , ) ( | )

( |: :: : : :

0 10 1 1 0 1 1= − −

1 1

0 0 1 1 1

:

: : :

)( | ) ( ) ( | ).

k

k k k k kp y s p s p s y−

− −∝ (61)

Now if we choose an importance function of the form

π( | ) ( | , ) ( | ,: : : : : :s y p s s y p s s yk k k k k k k k1 1 1 1 1 1 1 2 1 1= − − − − ) ( | )( | , ) ( | ): : : :

⋅⋅⋅= − − −

p s yp s s y s yk k k k k

1 1

1 1 1 1 1 1 1π (62)

32 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

we can create trajectories from p s yK K( | ): :1 1 using impor-tance sampling recursively and the standard particle filter-ing procedure.

During the kth recursion, the trajectory s km

1:( ) is

weighted with respect to p s ykm

k( | ):( )

:1 1 according to

wp s y

s y

w p y s

km k

mk

km

k

km

k

( ) :( )

:

:( )

:

( )

( | )( | )

( |

=

∝ −

1 1

1 1

1

π

1 1 1 1

1

:( )

:

( ) ( )

, )

.km

k

km

km

y

w− −

−= η (63)

Observe that at recursion k, the η( )km s are all that we need

to have for calculating the weights wkm( ) .

We note that drawing particles from the importancefunction π( | , ) ( | , ):

( ): :

( ):s s y p s s yk k

mk k k

mk1 1 1 1 1 1− −= is easy since

s k ∈ + − , 1 1 . In particular, we can write

( ) ( )p s s y p y s s y

p sk k

mk k k k

mk= ∝ =

×− − −1 11 1 1 1 1 1 1| , | , ,

(:

( ): :

( ):

( )k k k

k k km

k

s y

p y s s p s

=

= = =− −

−

1

1 11 1 1 1

1 1

| , )

| , ( ): :

:( )

(64)

where p y s sk k km( | , ):

( )1 1− and p s k( )are the likelihood function

and the prior distribution at recursion k, respectively, andthey can be easily computed. An analogous expressioncan be written for p s s yk k

mk( | , ):

( ):= − −1 1 1 1 . Next, observe that

ηkm

s k k km

kp y s s p sk

( ):

( )( | , ) ( )=∑ −1 1 , and thus the incrementalweight is proportional to the sum of the importance func-tion from (64), which is also readily obtained.

As in other applications of particle filtering,resampling is required in the algorithm. However, theweight must be clearly associated with all the particles inthe trajectory at all times (that is, for all users). This is un-like in other applications where only the present particlesin the trajectories are retained and therefore afterresampling, the connection between the present weightsand previous particles is lost.

The complexity of the algorithm is O KM( ), i.e., pro-portional to the product of the number of particles andthe number of users. If the number of particles is fixed,the complexity is only linear with respect to the numberof users.

MUD When the Fading Coefficients Are UnknownWhen considering the joint channel estimation andMUD problem, we have to incorporate the state-spacerepresentation of the Rayleigh fading channel in (46) and(47) in the final state-space representation. If we assumethat the channel is static within each symbol durationT s ,the WMF output at time t can be written as

y FS B vt t t t= + . (65)

With f k representing the kth row of F, the WMF output

corresponding to the kth user can be expressed asy vk t k t t k t, ,= +f S B . If we process a data block ofT sym-

bols, we can organize the WMF outputs into a sequenceof KT observations, i.e., y 1 1 1 1 1: , , ,KT K Ty y y= ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅y K T, . We introduce now a new index i t K k= − +( )1 , andwe summarize the state-space representation by

~ , ( , )~ ,

,

SXS X

A Eu

ii

i i

ii i

i K=

=+

=+

−

−

if modelse

if

1

1

1 mod

else( , )

,~

i K

y vi

i k i i i

=

= +−

1

1

f S B

(66)

where X i is a K K× matrix whose only nonzero elementis ( ) ,X i kk k is= . As we will see later, such representation fa-cilitates the application of particle filtering.

The posterior distribution can be factored based on thestate-space equations in (66),

( ) ( ) ( )( )

p p p

p y p

i i i i i i

i i i

~ | ~|~ ~ |

|~ ,

: : : :S y S S S y

S

1 1 1 1 1 1 1∝ − − −

∫ ( )i i i id|~ ,: :S y1 1 1 1− − (67)

and if we choose the importance function in the form of

( ) ( ) ( )π π π~ | ~|~ , ~ |: : : : : :S y S S y S y1 1 1 1 1 1 1 1 1i i i i i i i= − − −

we can obtain particles using the MKF algorithm. Specif-ically, the optimal importance function is p i i

mi(~|~ , ):

( ):S S y1 1 1−

which can be evaluated from

( ) ( ) ( )p p p y

p

i im

i i im

i i

i

~|~ , ~|~ |~ ,:( )

: :( )S S y S S S1 1 1 1 1− −∝

×

∫

( )|~ , .:( )

:S y1 1 1 1im

i id− − (68)

Note that given~:( )S1 1i

m− , the state-space model in (66) is lin-

ear in i and is Gaussian, and the term p i( |~ , ):

( ):S y1 1 1 1i

mi− −

can be obtained via the prediction step of the Kalman fil-ter. In turn, the integration in (68) can be evaluated ana-lytically. The term p i i

m(~|~ ):( )S S1 1− is equal to zero if its first

k−1elements on the diagonal do not match those of~Si−1 ;otherwise, it is equal to 1/2.

The importance weight becomes

( )( )

wp

pi

m im

i

im

im

i i

( ) :( )

:

( ):( )

: :

~ |~ |~ , ~=

− −

S y

S S y S

1 1

1 1 1 1π( )( )

1 1 1

1 1 1 1 1

1

( ):

( ): :

( )

~

|

|~ ,

mi

im

i i i

im

w p y

w pi

y

S y

S

−

− − −

−

∝

= ∑ ( )( ) ( )

~|~

|~ , |~ ,

:( )

:( )

:

S S

S S y

i im

i i i i im

i ip y p d

1 1

1 1 1 1

−

− −×∫ .(69)

Further details of the algorithm can be found in [59].

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 33

SimulationsFor the algorithm based on WMF outputs, first we simu-lated a synchronous CDMA system with K =15users and achip-rate of C =30. The spreading codes were generatedrandomly and the same spreading code was used in all simu-lated detectors. Residual resampling [37] was performed af-ter every five users. In Figure 11 we present a performancecomparison for the equal power case with other popularCDMA MUD detectors: the three-stage successive interfer-ence cancellation detector with decorrelating first stage(3-stage), the Gibbs sampler, and the decorrelating decisionfeedback detector (DDF). (In our simulations we includethe Gibbs sampler because it is another Monte Carlo basedtechnique where estimates and detection are based on gen-

erated samples from desired a posteriori distributions.) Forthe Gibbs sampler, we experimented with two detectorswhose difference was in the length of the burn-in periods(periods until convergence). The first detector drew 100samples of which the first 50 were discarded (Gibbs-50).The second detector generated 150 samples, and the first100 samples were discarded (Gibbs-100). As a reference,the Breadth-first tree-search algorithm which is optimumas described in [47] was simulated to provide a lowerbound. In the simulation of the particle filtering method,we used 50 particles (PF-50) and 100 particles (PF-100).The performance curve in Figure 11 was obtained by aver-aging the BER of all 15users. We can see that in the equalpower case, particle filtering can provide near-optimum

performance. It seems that the performancegain by increasing the number of particles from50 to 100 is only marginal.

In the near-far case, the targeted user (thefirst user), had an SNR of 9 dB. The signalstrength of the remaining 14 users was identi-cal and compared with that of the targeteduser, and was varied from −10 dB to 10 dB.The results of the experiment are shown inFigure 12. It is clear that particle filtering per-forms almost always better than thethree-stage detector and although it performsworse than the Gibbs sampler with weaker in-terferers, it is more consistent than the Gibbssampler. Also, it is near optimum in the rangefrom −4 dB to 10 dB.

Then we simulated the performance of the al-gorithm based on WMF outputs in the case ofjoint channel estimation and MUD. We simu-lated a quasi-static Rayleigh flat fading casewhere the channel was considered static within ablock of L l+ 0 symbol times. The first l0 sym-bols were known to the receiver and they weresent as pilot signals. The fading coefficientwithin each block for each user was a complexGaussian random variable with unit varianceand zero mean. The fading coefficients of differ-ent blocks were considered independent fromeach other. We simulated blocks of L=10 andl0 1= . For comparison, we also simulated theSAGE-JDE(↑) algorithm from [28]. For everysimulated point, at least 300 bit errors were ac-cumulated. In Figure 13, we see that the particlefiltering algorithm consistently outperformedSAGE-JDE(↑).

Estimation and Detection ofSpace-Time Codes inFading ChannelsSpace-time coding (STC), originally intro-duced in [13] and further developed in [51],provides a framework for exploiting spatialand temporal diversity to increase data rate in

34 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

11. Performance comparison of various detectors for C K= =30 15, , equalpowers, and known channels.

12. Performance comparison of various detectors for C K= =30 15, , knownchannels and near-far resistance of 9 dB.

wireless communications. Although space-time trelliscodes (STTCs) are deemed to possess the best coding ef-ficiency among space-time codes, they are hard for detec-tion especially when unknown time varying fadingcoefficients are involved. As demonstrated in [7] and[44], it is quite straightforward to represent binary orM-ary convolutional (trellis) coded systems in fadingchannels using state-space models, and in [57] particle fil-tering was considered for this problem.

Suppose that a communication system employs N Ttransmit and N R receive antennas and that a sequence ofuser data symbols, s s t0 ,..., , where s t ∈, is put through atrellis space-time encoder. The new state vector of the trel-lis space-time encoder at time t is determined accordingto the state transition equation s st t tf s= −( , )1 , where s t −1is the previous state, and s t is the new user state. Based onthe current state vector, the space-time encoder then gen-erates a set of N T symbols, c s s s ( ) [ ( ),..., ( )]t t N tc c

T= 1 ,

to be transmitted by the N T antennas, where ci ()⋅ denotesthe code and modulation function of the ith antenna.Suppose hn n tT R , is the fading coefficient from the nT thtransmit antenna to the nR th receive antenna at time t.Let h n t n t N n tR R T R

h h, , ,[ ] = ⋅⋅⋅1 represent the set of channelstates from all transmit antennas to the nR th receive an-tenna. If we arrange all the channel states at time t into asingle N NT R ×1 vector h h ht t N tR

= ⋅⋅⋅[ ], ,1 , all the re-ceived signals at time t can be written in vector form as

y C s h vt t t t= +( ) (70)

where y t t N ty yR

= ⋅⋅⋅[ ], ,1 is the received signal vector, andv t t N tv v

R

= ⋅⋅⋅[ ], ,1 is the noise vector. The code and mod-ulation matrix C s( )t is an N N NT T R× matrix of theform

C s

c s 0 00 c s 0

0 0 c s

( )

( )( )

( )

t

t

t

t

=

L

L

M M M M

L

.

Note that here 0 is an N T ×1 zero vector.This somewhat odd matrix representation isselected to simplify the description of the jointestimation and decoding algorithm.

The fading coefficients from the N T th tothe N R th antenna are modeled in the sameway as in (53) and (54) except that the sub-script n nT R is added, i.e.,

n n t n n n n t n n n n tT R T R T R T R T R, , ,= +−A E u1 (71)

h Bn n t n n n n tT R T R T R, ,= (72)

where A n nR T, E n nT R

, and B n nT Rhave the

same definition as before.We can represent the whole system in a

compact state-space form as follows:

t t t= +−

~A Eu1 (73)

h Bt t= (74)

where t t n t n t n n tT R T R

= [ , , , , , , ], , , ,ρ ρ ρ ρ11 1 1K K K i sthe extended state vector, and the matrices ~A, B, andEareconstructed accordingly from the matrices A n nR T

,E n nT R,

and B n nT R, n NT T=1 2, , ,K , n NR R=1 2, , ,K , respec-

tively. Note that in the algorithm described above, it is notrequired that the fading coefficients or the noise vector atthe receive antennas be independent as required in mostother algorithms.

Since the space-time code is trellis coded, besides the cur-rent and previous received signals y 0:t , future observationsalso hold information about the current user state. Hence itis appropriate to use the delayed importance function as wellas the delay weight method in evaluating the posterior dis-tribution function. The number of delays can be chosen ac-cording to the constraint length of the trellis code.

SimulationWe simulated an STTC system with two transmit and onereceive antenna. The ARMA model for the fading coeffi-cient is described in [7], and it was of order ( , )3 3 . Thismodel corresponds to a fast fading scenario with a nor-malized Doppler frequency (with respect to the symbolrate 1 / T s ) f dT s

=005. .For our simulations, we designed a special STTC

based on the Tarokh eight-state STTC [51] that can com-bat phase ambiguity utilizing the time correlation of thechannels. The details of the design can be found in [57].We compared the performance of the system using ourSTTC with that of a system that uses Tarokh’s STTC andpilot signals. The SNR was varied between 20 and 25 dB.During our simulations, we found that we had to applyresampling frequently, which is essential for good perfor-mance of the algorithm. We used the same residual

SEPTEMBER 2003 IEEE SIGNAL PROCESSING MAGAZINE 35

13. C K= =17 31 10/ , equal power in Rayleigh quasi-static flat fading.

resampling process as described in [7] and resamplingwas performed every five steps. Because the constraintlength is one in the STTC, the number of delayed weightsand delayed samples was one. For every simulated point,at least 100 symbol errors were accumulated.

The comparison results are shown in Figure 14. Therewe also represent the performance of a detector that has ex-act information about the channel states. The simulationsshow that it is viable to use particle filtering for decodingSTTC with unknown fading channels. In addition, a sig-nificant performance improvement can be obtained by us-ing STTCs that can combat phase ambiguity [57].

Summary of Additional WorkBesides the presented applications, there has been re-search in applying particle filtering to other problems incommunications such as synchronization of communica-tion systems [14], [40] and detection of signals inBLAST systems [23]. (BLAST stands for Bell Laborato-ries Layered Space-Time.) In this section, we briefly sum-marize some of this work.

It is broadly recognized that many practical communi-cation channels present a high degree of structure andthat they can be accurately characterized through a set ofreference parameters with a clear physical meaning. Sincethe observed signals collected by the receiver are affectedby these parameters, they should be estimated and com-pensated prior to data detection in order to achieve opti-mal or close-to-optimal performance. The generalizedsynchronization problem consists of the recovery of a setof such physical parameters, namely the symbol timing,phase offset and carrier frequency error. Up to date, manydifferent techniques [39], [48] have been proposed in or-der to solve the synchronization problem, but they arebased on approximate and heuristic methods, as optimal

estimation of the parameters of interest seemsto be analytically intractable [39].

Before particle filters are applied for syn-chronization, again, the observed signals areexpressed in a state-space form. Synchroniza-tion, however, poses the additional difficultyin that the parameters of interest are usuallymodeled as fixed [39], while standard particlefiltering algorithms are aimed at trackingtime-varying unknowns.

To address the optimal recovery of the refer-ence parameters and data detection, two differ-ent approaches are considered. In the firstapproach [40], the idea of rejuvenation [35] isapplied in deriving a recursive algorithm aimedat approximating the joint smoothing proba-bility distribution of the transmitted symbolsand their (fixed) synchronization parameters.The other approach, [14], allows the synchro-nization parameters to evolve in a random wayusing artificial evolution mechanisms (the pa-

rameters are modeled as autoregressive stochastic pro-cesses [24], [42]) and therefore, they can be estimatedusing the traditional particle filtering approach. Accordingto computer simulations, the two methods appear promis-ing for joint data detection and synchronization.

In BLAST systems, spatial diversity is exploited to im-prove efficient transmission in broadband wireless com-munications. It is achieved by using multiple transmittingand receiving antennas, and it provides significant capacitygains. An optimum solution to the detection problem inBLAST systems is based on the maximum likelihood(ML) principle. Its complexity, however, is exponentialwith the number of transmitting antennas, and thereforeits use in practice is prohibitive. A reasonable tradeoff be-tween complexity and performance was proposed wherethe detection proceeds along the signal layers in a decreas-ing order of their signal-to-noise ratio and where the detec-tion in each layer is a two-step scheme of cancellation andnulling [13]. A weakness of this approach, known also as aV-BLAST receiver, is the propagation of error.

As an alternative, one can derive a dynamic state-spacemodel of the BLAST system. The states evolve in spacerather than in time, and the construction of the model isbased on QR decomposition and the output of thefeedforward filters. The proposed particle filters for theconsidered state-space model do not suffer from errorpropagation, and the simulations show that they greatlyoutperform the V-BLAST method and have near opti-mum performance [23].

AcknowledgmentsThis work was supported by the National Science Foun-dation under Awards CCR-9903120 andCCR-0082607. Joaquín Míguez acknowledges the sup-port of Xunta de Galicia, the Ministry of Science and

36 IEEE SIGNAL PROCESSING MAGAZINE SEPTEMBER 2003

14. Comparison results of a genie aided detector and particle filter detectorsthat use Tarokh’s and a newly proposed STTCs.

Technology of Spain and the FEDER funds of the Euro-pean Union (project TIC2001-0751-C04-01).

Petar M. Djuric received his B.S. and M.S. degrees fromthe University of Belgrade, Yugoslavia, in 1981 and1986, respectively, and his Ph.D. degree from the Univer-sity of Rhode Island in 1990, all in electrical engineering.From 1981 to 1986 he was a research associate with theInstitute of Nuclear Sciences, Vinca, Belgrade. Since1990 he has been with Stony Brook University, NewYork, where he is a professor in the Department of Elec-trical and Computer Engineering. He works in the area ofstatistical signal processing and has interests in the theoryof modeling, detection, estimation, and time series analy-sis and its application to a wide variety of disciplines, in-cluding wireless communications and biomedicine.

Jayesh H. Kotecha received a B.E. in electronics and tele-communications from the College of Engineering, Pune,India, in 1995, and M.S and Ph.D. degrees in electricalengineering from the Stony Brook University, New York,in 1996 and 2001, respectively. Since January 2002, hehas been with the University of Wisconsin at Madison as apost-doctoral researcher. His research interests are chieflyin the fields of communications, signal processing, andinformation theory. In particular, he works on problemsrelated to statistical signal processing, adaptive signal pro-cessing and particle filters, physical layer aspects in com-munications like channel modeling, estimation,detection, and coding in the contexts of single antennaand multi-input, multi-output systems.

Jianqiu Zhang received her B.S. and M.S. degree in elec-trical engineering from Zhejiang University, Hangzhou,and Zhongshan University, Guangzhou, China, in 1992and 1995, respectively. She received her Ph.D. degreefrom the Department of Electrical and Computer Engi-neering at Stony Brook University, New York, in 2002.From 1995 to 1997, she was a software engineer with theResearch and Development of Guangdong Nortel,Guangzhou, China. She is now with the Department ofElectrical and Computer Engineering, University of NewHampshire, Durham, as an assistant professor. Her re-search interests include combined error protection andchannel estimation coding, information theory, statisticalsignal processing, turbo coding and decoding, space-timecoding, and multiuser detection.

Yufei Huang received the B.S. degree in applied electron-ics from Northwestern Polytechnic University, Xi’an,China, in 1995 and the M.S. and Ph.D. degrees in electri-cal engineering from Stony Brook University, New York,in 1997 and 2001, respectively. From September 2001 toAugust 2002, he was with the Department of Electricaland Computer Engineering at Stony Brook University asa postdoctoral researcher. He is now an assistant profes-sor with the Department of Electrical Engineering, Uni-versity of Texas at San Antonio. His research interests are

in the general area of statistical signal processing, MonteCarlo sampling methods, and digital communications,and particular the theory of particle filtering and its appli-cations to problems in wireless communications.

Tadesse Ghirmai received his B.Sc. degree from AddisAbaba University, Ethiopia, in 1988 and his M.S.E.E. de-gree from University of South Florida, Tampa, in 1995.He is currently pursuing the Ph.D. degree at Stony BrookUniversity, New York. He worked at the University ofAsmara from 1988 to 1993 and from 1995 to 1999 as anassistant lecturer and as a lecturer, respectively. His re-search interests are in statistical signal processing withemphasis on theory of Monte Carlo methods and theirapplication to communications.

Mónica F. Bugallo received the Ph.D. degree in computerengineering from the University of A Coruña, Spain, in2001. From 1998 to 2000 she was with theDepartamento de Electrónica y Sistemas at theUniversidade da Coruña, Spain, where she worked on in-terference cancellation applied to multiuser communica-tion systems. In 2001, she joined the Department ofElectrical and Computer Engineering at Stony BrookUniversity, where she is currently a lecturer and teachescourses in digital communications, and information the-ory. Her research interests lie in the area of statistical sig-nal processing and its applications to different disciplinesincluding communications and biology.

Joaquín Míguez obtained the Licenciado en Informatica(M.Sc.) and Doctor en Informatica (Ph.D.) degrees fromUniversidade da Coruña, Spain, in 1997 and 2000, re-spectively. Late in 2000, he joined Departamento deElectrónica e Sistemas, Universidade da Coruña, as an as-sistant professor and, from April 2001 through Decem-ber 2001, he was a visiting scholar in the Department ofElectrical and Computer Engineering, at Stony BrookUniversity. His research interests are in the field of statisti-cal signal processing and its applications to communica-tions, with particular emphasis on multiple access,space-time coded and smart-antenna systems.

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