Sequential Monte Carlo methods for complexity-constrained MAP equalization of dispersive MIMO channels

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0165-1684/$ - se

doi:10.1016/j.si

$This work

Ciencia of S

TEC2006-13514

this article wer

phia, March 20

2005).�Correspond

fax: +3498116

E-mail addr

[email protected]

(J. Mıguez).1Tel.: +1 6312Tel.: +34 91

Signal Processing 88 (2008) 1017–1034

www.elsevier.com/locate/sigpro

Sequential Monte Carlo methods for complexity-constrainedMAP equalization of dispersive MIMO channels$

Manuel A. Vazqueza,�, Monica F. Bugallob,1, Joaquın Mıguezc,2

aDepartamento de Electronica e Sistemas, Universidade da Coruna, Facultade de Informatica, Campus de Elvina s/n, 15071 A Coruna, SpainbDepartment of Electrical and Computer Engineering, Stony Brook University, Stony Brook, NY 11794, USA

cDepartamento de Teorıa de la Senal y Comunicaciones, Universidad Carlos III de Madrid, Avenida de la Universidad 30,

28911 Leganes, Madrid, Spain

Received 17 October 2006; received in revised form 26 July 2007; accepted 5 November 2007

Available online 22 November 2007

Abstract

The ability to perform nearly optimal equalization of multiple input multiple output (MIMO) wireless channels using

sequential Monte Carlo (SMC) techniques has recently been demonstrated. SMC methods allow to recursively

approximate the a posteriori probabilities of the transmitted symbols, as observations are sequentially collected, using

samples from adequate probability distributions. Hence, they are a class of online (adaptive) algorithms, suitable to handle

the time-varying channels typical of high speed mobile communication applications.

The main drawback of the SMC-based MIMO-channel equalizers so far proposed is that their computational complexity

grows exponentially with the number of input data streams and the length of the channel impulse response, rendering these

methods impractical. In this paper, we introduce novel SMC schemes that overcome this limitation by the adequate design of

proposal probability distribution functions that can be sampled with a lesser computational burden, yet provide a close-to-

optimal performance in terms of the resulting equalizer bit error rate and channel estimation error. We show that the

complexity of the resulting receivers grows polynomially with the number of input data streams and the length of the channel

response, and present computer simulation results that illustrate their performance in some typical scenarios.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Multiple input multiple output (MIMO); Joint channel and data estimation; Particle filtering (PF); Sequential Monte Carlo

(SMC)

e front matter r 2007 Elsevier B.V. All rights reserved

gpro.2007.11.004

has been supported by Ministerio de Educacion y

pain (projects TEC2004-06451-C05-01 and

-C02-01/TCM). Part of the results contained in

e presented at the 30th IEEE ICASSP (Philadel-

05) and the 61st IEEE VTC (Stockholm, May

ing author. Tel.: +34981167000x1350;

7160.

esses: [email protected] (M.A. Vazquez),

nysb.edu (M.F. Bugallo), [email protected]

6328395; fax: +1 6316328494.

6248749; fax: +34 916249430.

1. Introduction

Sequential Monte Carlo (SMC) methods, alsocommonly known as particle filtering (PF) algo-rithms, are simulation-based techniques that aim atapproximating the a posteriori probability distribu-tion function (PDF) of a time-varying signal ofinterest (SOI), given some related observations,using a discrete probability measure with a randomsupport [1–3]. The fundamental principle of PF is to

.

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ARTICLE IN PRESS

3The receiver in [19] also uses PF for estimating the channel

delay parameters given the detected data.

M.A. Vazquez et al. / Signal Processing 88 (2008) 1017–10341018

explore the space of the SOI by generating randomsamples (termed particles) from a proposal distribu-tion. These particles are then assigned properweights [1] and yield the discrete approximation ofthe a posteriori PDF. The main advantages of themethodology are its generality and the sequentialprocessing of the observations. The former enablesits application to a remarkably broad class ofestimation and detection problems. The lattermeans that the discrete approximation of the aposteriori PDF can be updated recursively when-ever new observations become available, henceSMC methods are inherently adaptive.

The application of PF to a number of problems indigital communications has received considerableattention in the last few years (see the tutorial [4] aswell as the recent work in [5–9]) and, in particular,the close-to-optimal demodulation of multiple inputmultiple output (MIMO) wireless channels has beenaddressed in [10–14] among others. The complexityof optimal MIMO equalization grows exponentiallywith the number of transmitters [15], which makes itimpractical in most scenarios. It has been suggestedthat PF can be used to alleviate this complexity witha limited performance degradation. In [10] it isshown that nearly optimal bit error rate (BER) canbe achieved using PF, but the complexity of thesampling scheme employed in that paper still growsexponentially with the number of input datastreams. In [12], the complexity of the SMCalgorithm is reduced by handling together identicalparticles which are represented as paths in a tree.In an alternative approach, a stochastic M tree-search algorithm is proposed in [13] for the BLAST(Bell Labs layered space-time) coding system[16,17].

Although the techniques in [12,13] are successfulin reducing the number of distinct Monte Carlosamples to be processed, the complexity of thereceiver in [12] still grows exponentially with thenumber of data streams, while the decoder in [13] isarchitecture dependent and needs modifications tobe applied in a generic dispersive MIMO channel.A common approach for reducing the complexity ofdata detection in MIMO systems is the triangular-ization of the channel autocorrelation matrix. Earlyexamples in the context of code division multipleaccess systems (CDMA) can be found in [15] and, incombination with PF, in [18]. Some recent papershave also proposed schemes based on the QRdecomposition to perform data detection in ortho-gonal frequency-division multiplexing (OFDM)

MIMO systems3 [19,20]. The combination of matrixtriangularization and PF for data detection ingeneric MIMO channels has also been investigatedrecently. In [11,14], Dong et al. propose an iterative(turbo) method for channel estimation and datadecoding. At each iteration, decoding is carried outby a sequential importance sampling (SIS) algo-rithm, similarly to [18], given the latest availablechannel estimate. Then, given the detected data, thechannel is re-estimated using a conventional quad-ratic estimator. None of these articles addresses thejoint estimation of an arbitrary MIMO channel andthe data sequence with a particle filter. Although thelatter problem can be considerably hard to solve, ithas been argued that PF algorithms are well suitedfor complex communication problems that involvethe estimation of both discrete (symbols) andcontinuous (channel coefficients) state variables, asopposed to ‘‘simple’’ data detection that can bemore efficiently addressed with deterministic orsemi-deterministic methods [21].

In this paper, we propose two novel SMCschemes for MIMO equalization that performchannel estimation and data detection jointly, witha computational load per particle that increasesonly polynomially with the number of input datastreams in an arbitrary frequency-and-time selectivechannel. The first one exploits the well-knownCholesky triangularization of the channel autocor-relation matrix to enable sequential sampling of thetransmitted data. The novelty with respect toprevious work based on the same principle is thatwe explicitly take into account the uncertainty in theknowledge of the channel and our method yieldsboth data and channel (soft) estimates. The secondSMC scheme relies on the use of simple linear filtersto compute soft estimates of the transmittedsymbols, from which data samples are subsequentlyobtained, and it is, to the best of our knowledge, anew approach to the problem. Computer simula-tions that illustrate the performance of the proposedSMC receivers, both in terms of BER and channelestimation mean square error (MSE), are alsopresented.

The remaining of the paper is organized asfollows. In next section, the signal model fortransmission over a MIMO dispersive channel isdescribed. In Section 3, the standard application ofPF to MIMO equalization is illustrated. The basic

ARTICLE IN PRESSM.A. Vazquez et al. / Signal Processing 88 (2008) 1017–1034 1019

ideas behind our constrained-complexity SMCreceivers are explained in Section 4. Then, the twospecific methods are described in detail in Sections 5and 6. Illustrative computer simulations are shownin Section 7 and, finally, a discussion of theresults and a few concluding remarks are made inSection 8.

2. Signal model

Fig. 1 shows a block diagram of a MIMOcommunication system with N input data streamsand L output observation streams. We assumespatial diversity at both the transmitter and thereceiver, hence we will often refer to the N transmitantennas and the L receive antennas. However, thedescribed results can be extended to arbitraryMIMO systems in a straightforward manner. Thediscrete-time, baseband equivalent model of the L

received signals is [22]

xt ¼Xm�1i¼0

HtðiÞst�i þ gt; t ¼ 0; 1; . . . , (1)

where xt is the L� 1 vector of observationscollected at the receiving antennas, HtðiÞ;i ¼ 0; . . . ;m� 1, is the (time-varying) L�N multi-dimensional channel impulse response (CIR) at timet, st ¼ ½stð1Þ; . . . ; stðNÞ�

> is the N � 1 vector ofsymbols transmitted at time t (fstðnÞg, withn 2 f1; . . . ;Ng, is the nth input stream), m is thelength of the CIR, which yields the span of the inter-symbol interference (ISI), and gt is an L� 1 vectorof independent additive white Gaussian noise(AWGN) components with variance s2g;t.

For the purpose of the algorithm design, it iscommon [4] to model the channel variation bymeans of an autoregressive (AR) process driven bywhite Gaussian noise. We consider the first-order

Fig. 1. Block diagram of the

AR model (see, e.g., [23,24])

HtðiÞ ¼ aHt�1ðiÞ þ VtðiÞ; 0pipm� 1, (2)

where VtðiÞ are L�N matrices of independent andidentically distributed (i.i.d.) Gaussian elementswith zero mean and variance s2v . Model parametersa and s2v are selected to fit any desired channelautocorrelation function. Note that model (2) is notprimarily intended to be a physically realisticrepresentation of the channel fading process, butrather a simple approximation of the channeldynamics that turns out advantageous for thederivation of PF-based MIMO equalizers. This is,indeed, a common approach [5,23–25]. Please, referto [24] for a detailed discussion of AR MIMOchannel modeling.

Using the above equations, the MIMO transmis-sion process can be modeled as a dynamic system instate-space form,

st�UðSN Þ;

HtðiÞ ¼ aHt�1ðiÞ þ VtðiÞ; 0pipm� 1;

)

state equations, ð3Þ

xt ¼Pm�1i¼0

HtðiÞst�i þ gt; t ¼ 0; 1; . . . ;

)

observation equation. ð4Þ

The system state at time t consists of the CIR,fHtðiÞg

m�1i¼0 , which is assumed unknown, and the

symbol vectors st�mþ1:t ¼ fst�mþ1; . . . ; stg. It evolvesaccording to the state equations in (3), i.e., thechannel evolution is driven by a first-order ARprocess with known parameters s2v and a, while thesymbols are modeled as discrete uniform randomvariables in the alphabet S, hence st�UðS

NÞ.The dynamic system representation allows to use

PF in order to compute asymptotically optimal joint

N � L MIMO system.

ARTICLE IN PRESSM.A. Vazquez et al. / Signal Processing 88 (2008) 1017–10341020

estimates of the channel response and the trans-mitted data from the collected observations, asdescribed in next section.

3. MIMO channel equalization

3.1. Sequential importance sampling

Most PF methods rely upon the principle ofimportance sampling (IS) [1] for building anempirical approximation of a desired probabilitydensity function (pdf), say pðyÞ, by drawing samplesfrom a different distribution, known as importance

function or proposal pdf, denoted qðyÞ, with adomain that includes that of pðyÞ. These samplesare then assigned appropriate normalized impor-

tance weights, i.e.,

yðiÞ�qðyÞ, ð5Þ

~wðiÞ ¼pðyðiÞÞ

qðyðiÞÞ, ð6Þ

wðiÞ ¼~wðiÞPM

i¼1 ~wðiÞ, ð7Þ

where M is the number of particles, ~wðiÞ,i ¼ 1; . . . ;M, are the unnormalized weights, and

wðiÞ are the normalized ones, so thatPM

i¼1 wðiÞ ¼ 1.

The particles drawn from qðyÞ are said to beproperly weighted with respect to pðyÞ, meaning that

EpðyÞ½f ðyÞ� ¼

Zf ðyÞpðyÞ ¼

Zf ðyÞpðyÞ

qðyÞ

qðyÞ

¼

Zf ðyÞ ~wðyÞqðyÞ ¼ EqðyÞ½f ðyÞ ~wðyÞ�, ð8Þ

where f ðyÞ is an arbitrary integrable function and~wðyÞ ¼ pðyÞ=qðyÞ. Intuitively, (8) says that the meanof a function f ðyÞ with respect to a pdf pðyÞ can becomputed with respect to another pdf qðyÞ if weproperly weight the function using the factor ~wðyÞ.Note that the normalization step in (7) is necessaryin practice because Mo1 and the integrals inEpðyÞ½f ðyÞ� need to be approximated by sums.

In the equalization problem, we are interested indetecting the transmitted symbols, hence we need toapproximate the a posteriori probability massfunction4 (pmf) pðs0:tjx0:tÞ which contains allrelevant statistical information for the optimal(Bayesian) estimation of s0:t, and the importancefunction has the form qðs0:tjx0:tÞ.

4Note that any pmf can be handled as a pdf by representing it

using Dirac delta functions.

One of the most appealing features of the PFapproach is its potential for online processing.Indeed, the IS principle can be sequentially appliedby exploiting the recursive decomposition of theposterior distribution

pðs0:tjx0:tÞ / pðxtjs0:t;x0:t�1Þpðs0:t�1jx0:t�1Þ, (9)

which is readily derived by taking into account the apriori uniform pmf of the symbols.

The recursive algorithm called SIS combines theIS principle, decomposition (9) and an importancepmf that can be factored as

qðs0:tjx0:tÞ ¼ qðstjs0:t�1; x0:tÞqðs0:t�1jx0:t�1Þ (10)

to build a discrete probability measure with randomsupport that approximates the posterior pmf [2]. LetOt ¼ fs

ðiÞ0:t;w

ðiÞt g

Mi¼1 denote the discrete measure at

time t. The desired pmf is approximated as

pðs0:tjx0:tÞ ¼XMi¼1

dðs0:t � sðiÞ0:tÞw

ðiÞt ,

where dð�Þ is the Dirac delta function. When a newobservation is collected at time tþ 1, the SISalgorithm proceeds as follows to recursively com-pute Otþ1:

(1)
IS: sðiÞtþ1�qðstþ1js
ðiÞ0:t;x0:tþ1Þ:

(2)
Weight update: ~wðiÞtþ1 ¼ w
ðiÞt

pðxtþ1jsðiÞ0:tþ1

;x0:tÞ

qðsðiÞtþ1jsðiÞ0:t;x0:tþ1Þ

(3)
Weight normalization: w
ðiÞtþ1 ¼

~wðiÞtþ1PM

k¼1~wðkÞ

tþ1

The asymptotic convergence, for M !1, of theSIS algorithm is proved in [26] under mild assump-tions. It is straightforward to obtain data estimatesfrom the approximate pmf pðs0:tjx0:tÞ. For example,a marginal MAP detector can be implemented as

smapt ¼ argmax

st

XMi¼1

dðst � sðiÞt Þw

ðiÞt

( ), (11)

which amounts to selecting the particle with thehighest accumulated weight (note that some parti-cles can be replicated).

One major problem in the practical implementa-tion of the SIS algorithm is that after a few timesteps most of the particles have importance weightswith negligible values (very close to zero). Thecommon solution to this problem is to resample theparticles. Resampling is an algorithmic step thatstochastically discards particles with small weightswhile replicating those with significant weight. In its

ARTICLE IN PRESS

Table 1

Pseudocode of the optimal SIS algorithm

for each time instant t ¼ 1; 2; . . .for each particle i ¼ 1; 2; . . . ;Mlet Sl ¼ 0

for each st 2SN

compute the likelihood lðstÞ ¼ pðxtjst; sðiÞ0:t�1; x0:t�1Þ

accumulate Sl ¼ Sl þ lðstÞ

draw a new sample sðiÞt �qtðstÞ ¼ lðstÞ=Sl

obtain the unnormalized weight ~wðiÞt ¼ wðiÞt�1Sl

update the ith Kalman Filter using the sample sðiÞt

compute Sw ¼PM

i¼1 ~wðiÞt

for each particle

normalize its weight wðiÞt ¼ ~wðiÞt =Sw

estimate the effective sample size, Meff ¼ ðPM

i¼1wðiÞt

2Þ�1

if MeffogrM

resample

M.A. Vazquez et al. / Signal Processing 88 (2008) 1017–1034 1021

conceptually simplest form, resampling generates M

new particles fsðiÞ0:t; 1=MgMi¼1 by drawing samples

from the discrete pmf presamplingðsðiÞ0:tÞ ¼ w

ðiÞt .

3.2. Optimal importance function

The performance of the SIS algorithm consider-ably depends on the choice of the importancefunction. If we assume that only the observationsup to time t are available to detect symbol vector st,the optimal (in the sense that the variance of theimportance weights is minimized [2]) proposal pmffor the MIMO equalization problem is

qðstjx0:t; s0:t�1Þ ¼ pðstjs0:t�1; x0:tÞ / pðxtjs0:t;x0:t�1Þ,

(12)

which contains all the information available at timet for the sampling of st. The likelihood on the right-hand side of (12) can be obtained in closed-form.Indeed, if we let Ht ¼ ½Htðm� 1Þ � � �Htð0Þ� be theL�Nm overall channel matrix and use ht to denotethe LNm� 1 vector built by taking all elements inHt column-wise, then we can write

pðxtjs0:t;x0:t�1Þ ¼

Zht

pðxtjst�mþ1:t; htÞ

�pðhtjs0:t�1;x0:t�1Þdht. ð13Þ

Both densities in the integrand are Gaussian and,therefore, the integral can be solved (see, e.g., [23]).Specifically, if we let Nðxjl;CÞ denote the Gaussiandensity of x with mean l and covariance matrix C,then we find that pðxtjst�mþ1:t; htÞ ¼Nðxtj

Pm�1i¼0

HtðiÞst�i; s2g;tIÞ while pðhtjs0:t�1;x0:t�1Þ can be com-puted using the Kalman filter (KF) [27]. The latterobservation becomes apparent if we note that, given thesymbols s0:t, the dynamic system (3)–(4) is linear in ht

and Gaussian. It is actually well known that the KFcan be integrated into the SIS algorithm for con-ditionally linear Gaussian systems and the resultingmethod is termed mixture Kalman filter (MKF) [2,28].Also note that the KF requires the a priori distributionof the channel, pðh0Þ, to be known and Gaussian.

The weight update equation for the importancefunction (12) can be easily derived, and the completealgorithm becomes

sðiÞt �qtðstÞ ¼

pðxtjst; sðiÞ0:t�1;x0:t�1ÞP

~st2SN pðxtj~st; s

ðiÞ0:t�1;x0:t�1Þ

, (14)

wðiÞt / w

ðiÞt�1

pðstjs0:t�1;x0:tÞ

qðstjs0:t�1;x0:tÞ¼ w

ðiÞt�1

�X~st2S

N

pðxtj~st; sðiÞ0:t�1; x0:t�1Þ, ð15Þ

with resampling when needed. Notice that theimportance function in Eq. (14) corresponds to anormalization of (12). The pseudocode of thealgorithm is shown in Table 1, including aresampling step each time the effective sample size[2], Meff ¼ ð

PMi¼1 w

ðiÞ2

t Þ�1, falls below a certain

fraction, 0ogro1, of the total number of particles,M. Intuitively, the Meff parameter represents thenumber of independent particles drawn from thetrue posterior pmf that would yield the sameapproximation as the available M weighted particlesfrom the importance pmf.

From (14), (15), and the pseudocode of thealgorithm, it is seen that the MKF algorithm withoptimal importance function requires the computa-tion of jSjN different likelihoods (one for eachpossible value of st) and one KF step for eachparticle in Ot. As a consequence, the complexity ofthe method grows exponentially with the number oftransmit antennas, which renders the algorithmimpractical. Moreover, the procedure outlined inTable 1 yields a poor average performance when theMIMO channel is highly dispersive. Indeed, due tothe channel convolutional effect, the energy of st

is distributed over two or more symbol periodsand decisions made at time t become highlyunreliable [8].

ARTICLE IN PRESS

Table 2

Pseudocode of the DSIS (optimal) algorithm

for each time instant t ¼ 1; 2; . . .for each particle i ¼ 1; . . . ;Mlet Sl ¼ 0

for each symbol vector st 2SN

initialize the likelihood of st: lðstÞ ¼ 0

let P0 ¼ pðxtjs0:t; x0:t�1Þ

for each ~stþ1:tþd 2SNd

let P ¼ P0

for k ¼ 1; . . . ; dcompute P ¼ P� pðxtþkjs0:t; ~stþ1:tþk ;x0:tþk�1Þ

accumulate lðstÞ ¼ lðstÞ þ P

accumulate Sl ¼ Sl þ lðstÞ

draw new sample sðiÞt �qtðstÞ ¼ lðstÞ=Sl

obtain the unnormalized weight ~wðiÞt ¼ wðiÞt�1Sl

compute Sw ¼PM

i¼1 ~wðiÞt

for each particle

normalize its weight wðiÞt ¼ ~wðiÞt =Sw


i¼1wðiÞt

2Þ�1

if MeffogrM

resample


3.3. Delayed sampling

Whatever the approach, detection in dispersivechannels usually requires smoothing, i.e., st must bedetected based on both past and future observationsx0:tþd , where 1pd is a smoothing lag. In the contextof PF, smoothing is also referred to as delayed

sampling [2,23] because particle sðiÞt cannot be drawn

until xtþd is observed. We will refer to SISalgorithms that perform smoothing as delayed SIS(DSIS) methods.

Assume that the discrete measure Odt�1 ¼

fsðiÞ0:t�1;w

ðiÞtþd�1g

Mi¼1 has been obtained by processing

the observations x0:tþd�1 with a DSIS algorithm.The optimal smoothing importance pmf for st is

qðstjs0:t�1; x0:tþd Þ ¼ pðstjs0:t�1;x0:tþdÞ

/X

~stþ1:tþd2SNd

Yd

k¼0

pðxtþkjs0:t; ~stþ1:tþk; x0:tþk�1Þ,

ð16Þ

where factors pðxtþkjs0:t; ~stþ1:tþk; x0:tþk�1Þ can becomputed in the same way as (13). The normalizingconstant for the importance function in Eq. (16) isP

~st:tþd2SNðdþ1Þ

Qdk¼0 pðxtþkjs

ðiÞ0:t�1; ~st:tþk;x0:tþk�1Þ and

thus, the weight update equation becomes

wðiÞtþd ¼ w

ðiÞtþd�1

X~st:tþd2S

Nðdþ1Þ

Yd

k¼0

�pðxtþkjsðiÞ0:t�1; ~st:tþk; x0:tþk�1Þ, ð17Þ

where ~st:tþd 2SNðdþ1Þ are all the possible sequencesof symbol vectors transmitted from time t to timetþ d.

Therefore, sampling and updating a single parti-cle with this method involves the computation ofjSjNðdþ1Þ likelihoods, one for each possible differentsequence ~st:tþd , and each likelihood requires d þ 1KF steps. This means that the complexity of thealgorithm grows exponentially with the number ofantennas and the smoothing parameter, i.e., it isOðjSjNðdþ1ÞÞ. Although the smoothed MKF proce-dure guarantees a close-to-optimal performance, therequired computational effort is prohibitive. Table 2shows the pseudocode of the DSIS algorithm withoptimal importance pmf.

4. Constrained-complexity smoothing

The results in Section 3 indicate that the maindifficulty we need to overcome is the computational

complexity due to the data vector detection, ratherthan the tracking of the CIR.

In order to perform smoothing with a tractablecomplexity, we propose two novel SMC schemesbased on the ideas of sampling in a higherdimension [29] and sequentially on the data space[18]. Specifically, we address the approximation ofthe joint pdf pðs0:tþd ; ht:tþd jx0:tþdÞ using an impor-tance function qðs0:tþd ; ht:tþd jx0:tþdÞ, to be definedbelow.

The target pdf can be decomposed in a straight-forward manner as

pðs0:tþd ; ht:tþd jx0:tþd Þ

/Yd

k¼0

pðxtþkjhtþk; stþk�mþ1:tþkÞ

�Yd

l¼1

pðhtþl jhtþl�1Þpðhtjs0:t�1; x0:t�1Þ

�pðs0:t�1jx0:t�1Þ, ð18Þ

where all the factors are Gaussian and can becomputed (using the KF for obtaining the mean andcovariance of the term pðhtjs0:t�1;x0:t�1Þ). As animportance pdf, we consider the function

qðs0:tþd ; ht:tþd jx0:tþd Þ

¼ qðst:tþd ; ht:tþd jx0:tþd ; s0:t�1Þqðs0:t�1jx0:t�1Þ. ð19Þ


Using (18) and (19), we can design the DSISalgorithm

ðsðiÞt:tþd ; h

ðiÞt:tþdÞ�qðst:tþd ; ht:tþd js0:t�1; x0:tþd Þ,

wðiÞtþd ¼ w

ðiÞtþd�1

Qdk¼0 pðxtþkjh

ðiÞtþk; s

ðiÞtþk�mþ1:tþkÞ

Qdl¼1 pðh

ðiÞtþl jh

ðiÞtþl�1Þ

qðst:tþd ; ht:tþd jx0:tþd ; s0:t�1ÞpðhðiÞt jsðiÞ0:t�1;x0:t�1Þ (20)

with i ¼ 1; . . . ;M, and resampling when needed.The symbols s

ðiÞtþ1:tþd and the channel sequence h

ðiÞt:tþd

are just auxiliary variables [29] which are sampledfor convenience and can be discarded in order toobtain a discrete probability measure Od

t ¼

fsðiÞ0:t;w

ðiÞtþdg

Mi¼1 which is used as an approximation of

pðs0:tjx0:tþdÞ.We will introduce two schemes that make use of

(20) to update the importance weights, but differin the way the proposal function is selected. Inboth cases, however, we will need to build atransformed observation model. Let us begin withthe L� ðd þ 1Þ stacked vector

xt;d ¼ Ht;dst;d þ gt;d , (21)

where xt;d ¼ ½x>t ; . . . ;x

>tþd �>, st;d ¼ ½s

>t�mþ1; . . . ;

s>tþd �> is the Nðmþ dÞ � 1 vector containing all

contributing symbols, gt;d ¼ ½g>t ; . . . ; g

>tþd �> is the

Lðd þ 1Þ � 1 AWGN vector, and

Ht;d ¼

Htðm� 1Þ> 0 � � � 0

Htðm� 2Þ> Htþ1ðm� 1Þ> � � � 0

..

.Htþ1ðm� 2Þ> . .

. ...

Htð0Þ> ..

. . ..

Htþd ðm� 1Þ>

..

.Htþ1ð0Þ

> . ..

Htþd ðm� 2Þ>

..

. ... . .

. ...

0 0 � � � Htþd ð0Þ>

266666666666666664

377777777777777775

>

(22)

is the Lðd þ 1Þ �Nðmþ dÞ stacked channel matrix.Initially, we factor the non-recursive part of theimportance pdf as

qðst:tþd ; ht:tþd jx0:tþd ; s0:t�1Þ

/Yd

k¼1

pðhtþkjhtþk�1Þ

�pðhtjs0:t�1;x0:t�1Þqtþdðst:tþdÞ ð23Þ

so that the channel vector hðiÞt is drawn from the

Gaussian distribution pðhðiÞt jsðiÞ0:t�1; x0:t�1Þ, which is

given by the KF operating on particle i, and hðiÞtþ1:tþd

are predicted using the prior pdf’s pðhðiÞtþkjh

ðiÞtþk�1Þ.

Once we have sampled hðiÞt:tþd , and since s

ðiÞt�mþ1:t�1 are

already available, we can use both of them tosuppress the causal ISI in the observations and obtain

xðiÞt;d ¼ xt;d � �HðiÞt;d �s

ðiÞt;d ,

where �sðiÞt;d ¼ ½sðiÞ>t�mþ1; . . . ; s

ðiÞ>t�1 �> is an Nðm� 1Þ � 1

vector, and

�HðiÞt;d ¼

HðiÞt ðm� 1Þ H

ðiÞt ðm� 2Þ . . . H

ðiÞt ð1Þ

0 HðiÞtþ1ðm� 1Þ . . . H

ðiÞtþ1ð2Þ

..

. ... . .

. ...

0 0 � � � HðiÞtþd�1ðm� 1Þ

0 0 � � � 0

266666666664

377777777775

is an Lðd þ 1Þ �Nðm� 1Þ matrix (note that all the

matrices HðiÞtþkðjÞ with 0pjpm� 1, and 0pkpd, are

contained in hðiÞt:tþd). Conditional on h

ðiÞt:tþd ¼ ht:tþd

and sðiÞt�mþ1:t�1 ¼ st�mþ1:t�1 (that is, assuming the true

symbols and the true channel coefficients), thetransformed model for the observations can bewritten as

xðiÞt;d ¼ H

ðiÞ

t;d st;d þ gt;d , (24)

where st;d ¼ ½s>t ; . . . ; s

>tþd �> and H

ðiÞt;d is an Lðd þ 1Þ �

Nðd þ 1Þ matrix obtained by removing the first

Nðm� 1Þ columns of matrix HðiÞt;d , constructed

according to model (22) using the sample matrices

HðiÞtþkðjÞ; 0pjpm� 1 and 0pkpd,

HðiÞt;d ¼

HðiÞt ð0Þ

> HðiÞtþ1ð1Þ

> . . . HðiÞtþd ðm� 1Þ>

..

.HðiÞtþ1ð0Þ

> . ..

HðiÞtþd ðm� 2Þ>

..

. ... . .

. ...

0 0 � � � HðiÞtþd ð0Þ

>

266666664

377777775

>

.

(25)

The methods to be introduced in the subsequentsections build their proposal functions based on thetransformed observations of (24).


5. Triangularization of the channel autocorrelation

matrix

Any symmetric positive definite square matrix,say A, can be factorized as A ¼ UUH , where U is anupper triangular matrix with the same dimensionsas A. This factorization is known as Choleskydecomposition [30]. The underlying idea of themethod introduced in this section is to use this resultto obtain a convenient decomposition of matrix H

ðiÞt;d

that will allow to sequentially sample all thesymbols transmitted at time t, in a way similar tothe multiuser detection method proposed in [18].

Since HðiÞt;d is not necessarily symmetric, square

and positive definite, we cannot apply this resultdirectly and some pre-processing is in order. Inparticular, we define a new Nðd þ 1Þ �Nðd þ 1Þ

matrix, RðiÞt;d ¼ H

ðiÞH

t;d HðiÞt;d , which is always positive

definite (as long as HðiÞt;d has full column rank) and

then apply the Cholesky decomposition5

RðiÞt;d ¼ H

ðiÞH

t;d HðiÞt;d ¼ U

ðiÞt;dU

ðiÞHt;d , (26)

where UðiÞt;d is an Nðd þ 1Þ �Nðd þ 1Þ upper trian-

gular matrix. Then, we pre-multiply (24) by

UðiÞ�1t;d H

ðiÞH

t;d , which yields

zðiÞt;d ¼ U

ðiÞ�1t;d H

ðiÞH

t;d HðiÞt;d st;d þU

ðiÞ�1t;d H

ðiÞH

t;d gt;d

¼ UðiÞ�1t;d U

ðiÞt;dU

ðiÞHt;d st;d þ �g

ðiÞt;d

¼ UðiÞHt;d st;d þ �g

ðiÞt;d , ð27Þ

where zðiÞt;d ¼ U

ðiÞ�1t;d H

ðiÞH

t;d xðiÞt;d and �gðiÞt;d ¼ U

ðiÞ�1t;d H

ðiÞH

t;d gt;d

are Nðd þ 1Þ � 1 vectors.The mean and covariance

of �gðiÞt;d are easily computable if we know those of

gt;d . Since gt;d ¼ ½g>t ; . . . ; g

>tþd �>, E½gt;d � ¼ 0 and

Rgt;d¼ E½gt;dg

Ht;d � ¼ Dg;t;d , where Dg;t;d is an Lðd þ

1Þ � Lðd þ 1Þ diagonal matrix with s2g;tþbðk�1Þ=Lc in

the ðk; kÞ position (notation b�c indicates the‘‘integer part’’ of a real number), we can write

E½�gðiÞt;d � ¼ E½UðiÞ�1t;d H

ðiÞHt;d gt;d � ¼ U

ðiÞ�1t;d H

ðiÞH

t;d E½gt;d � ¼ 0,

(28)

5UðiÞt;d could also have been obtained by applying the QR-

factorization directly on matrix HðiÞt;d . This would have yielded the

same computational complexity but with smaller constant.

R�gðiÞ

t;d

¼ E½�gðiÞt;d �gðiÞHt;d � ¼ E½U

ðiÞ�1t;d H

ðiÞH

t;d gt;dgHt;dH

ðiÞt;dU

ðiÞ�Ht;d �

¼ UðiÞ�1t;d H

ðiÞH

t;d E½gt;dgHt;d �H

ðiÞt;dU

ðiÞ�Ht;d

¼ UðiÞ�1t;d H

ðiÞH

t;d Dg;t;dHðiÞt;dU

ðiÞ�Ht;d . ð29Þ

Eq. (27) can be written in a visually more appealingway that explicitly shows the triangularization ofthe channel autocorrelation matrix, namely

zðiÞt;d ¼

uðiÞ1;1 0 . . . 0

uðiÞ1;2 u

ðiÞ2;2

. ..

0

..

. ... . .

. ...

uðiÞ1;Nðdþ1Þ u

ðiÞ2;Nðdþ1Þ � � � u

ðiÞNðdþ1Þ;Nðdþ1Þ

26666666664

37777777775

�

stð1Þ

stð2Þ

..

.

stþdðNÞ

26666664

37777775þ �gðiÞt;d , ð30Þ

where uðiÞl;k is the element in the ðl; kÞ position of

matrix UðiÞt;d .

Given (30), the symbols st:tþd can be sequentiallysampled (starting with stð1Þ) in a relatively simplemanner. Let ½Q�i;j denote the element in the ith rowand jth column of matrix Q (with similar notation,½q�i, for vectors). Then, for k ¼ 1; . . . ;Nðd þ 1Þ,we have

½zðiÞt;d �k ¼

Xk

l¼1

uðiÞl;kstþbðl�1Þ=Ncð1þ ððl � 1ÞCNÞÞ þ ½�gðiÞt;d �k,

(31)

where ðxCyÞ is the remainder of x=y. Furthermore,from (31) we can recover a Gaussian randomvariable with known mean and variance, namely

z ¼ ½zðiÞt;d �k �

Xk�1l¼1

uðiÞl;kstþbðl�1Þ=Ncð1þ ððl � 1ÞCNÞÞ

�NðzjuðiÞk;kstþbðk�1Þ=Ncð1þ ððk � 1ÞCNÞ; ½R

�gðiÞt;d

�k;kÞ.

ð32Þ

Using (32) we sample the symbols in st:tþd sequen-tially. Assume, for simplicity, that S ¼ f�1g (theextension to a larger alphabet is straightforward),then symbol s

ðiÞtþkðnÞ is drawn conditional on

ARTICLE IN PRESS

Table 3

Pseudocode of the DSIS algorithm based on the triangularization

of the channel autocorrelation matrix

for each time instant t

for each particle i ¼ 1; . . . ;M

draw hðiÞt:tþd�pðht:tþd js

ðiÞ0:t�1;xð0:t�1Þ

construct �HðiÞ

t;d , HðiÞt;d and s

ðiÞt�mþ1:t�1

compute xðiÞt;d ¼ xt;d � �HðiÞt;d �s

ðiÞt;d

calculate RðiÞt;d ¼ H

ðiÞH

t;d HðiÞt;d

apply Cholesky decomposition on RðiÞt;d :R

ðiÞt;d ¼ U

ðiÞt;dU

ðiÞHt;d

compute the transformed observation zðiÞt;d ¼ U

ðiÞ�1t;d H

ðiÞHt;d x

ðiÞt;d

compute the covariance matrix of R�gðiÞ

t;d

(Eq. (29))

let st;d ¼ 0Nðdþ1Þ�1

let Q ¼ 0

for each symbol, st;d ðkÞ, in vector st;d , k ¼ 1; . . . ;Nðd þ 1Þ

zðiÞt;d ðkÞ ¼ z

ðiÞt;d ðkÞ � ½U

ðiÞHt;d �k;� st;d

let Sp ¼ 0

for each s 2S

ProbðsÞ ¼NðzðiÞt;d ðkÞ; u

ðiÞk;ks; ½R

�gðiÞ

t;d

�k;kÞ

accumulate Sp ¼ Sp þ ProbðsÞ

draw a sample st;d ðkÞ�qðst;d ðkÞÞ ¼ Probðst;d ðkÞÞ=Sp

accumulate Q ¼ Q� qðst;d ðkÞÞ

update st;d with st;d ðkÞ

construct sðiÞt from the sampled symbols in st;d

obtain the unnormalized weight ~wðiÞt from Eqs. (20) to (34)

compute Sw ¼PM

i¼1 ~wðiÞt

for each particle

normalizeits weight wðiÞt ¼ ~wðiÞt =Sw


i¼1wðiÞt

2Þ�1

if MeffogrM

resample


sðiÞt ð1Þ; . . . ; s

ðiÞtþkðn� 1Þ according to the probabilities

ProbtþdfstþkðnÞ ¼ 1g ¼ ProbfgðiÞtþkðnÞ40g,

ProbtþdfstþkðnÞ ¼ �1g ¼ ProbfgðiÞtþkðnÞo0g,

where

gtþkðnÞ ¼½zðiÞt;d �kNþn �

PkNþn�1l¼1 ul;kNþnstþbðl�1Þ=Ncð1þ ððl � 1ÞCNÞÞ

uðiÞkNþn;kNþn

(33)

is a Gaussian random variable with mean stþkðnÞ

and variance uðiÞ�2

kNþn;kNþn½R�gðiÞt;d

�kNþn;kNþn (to find these

parameter values, substitute (32) into (33)).Finally, the proposal pdf for the ith particle can

be evaluated as

qtþd ðhðiÞt:tþd ; s

ðiÞt:tþd js

ðiÞ0:t�1;x0:tþdÞ

/Yd

l¼1

pðhðiÞtþljh

ðiÞtþl�1Þ

�pðhðiÞt jsðiÞ0:t�1;x0:t�1Þ

�Yd

k¼0

YNn¼1

ProbtþdfsðiÞtþkðnÞg ð34Þ

and the weight update equation can be obtainedsubstituting this expression in the denominator of (20).

The propagation of a single particle in theproposed smoothing algorithm requires the evalua-tion of jSjNðd þ 1Þ symbol probabilities, instead ofthe jSjNðdþ1Þ symbol vector likelihoods of theoptimal smoothing algorithm in (16)–(17). Hence,the computational complexity per particle of thisequalizer grows only polynomially with the numberof data streams, N, and the smoothing lag d.

A detailed pseudocode for the proposed algo-rithm is shown in Table 3.

6. Linear filtering

The second proposed method is based on theavailability of some inexpensive technique tocompute soft estimates of the transmitted symbols.Then, these estimates can be used to design anefficient proposal pmf.

Specifically, let us consider the general decom-position (23) for the importance function. Afterdrawing h

ðiÞt:tþd , we can build the matrix H

ðiÞt;d

according to model (25), and use it to obtain alinear filter (e.g., a minimum mean square error(MMSE) linear receiver or a decorrelating receiver[15]), denoted as the Lðd þ 1Þ �Nðd þ 1Þ matrix

filter FðiÞt , for detecting st;d in Eq. (24), and therefore

st (recall st;d ¼ ½s>t ; . . . ; s

>tþd �>). In particular, for the

ith particle, the soft estimate ~sðiÞt;d ðkÞ, of the symbolst;dðkÞ, is computed as

~sðiÞt;dðkÞ ¼ fðiÞH

t;k xt; k ¼ 1; . . . ;Nðd þ 1Þ, (35)

where fðiÞt;k is the kth column of F

ðiÞt . Subsequently,

the corresponding particle is drawn from theimportance pmf

sðiÞt;dðkÞ�qtðst;dðkÞÞ ¼

Nð~sðiÞt;d ðkÞjst;d ðkÞ;s2qÞPs2SNð~sðiÞt;dðkÞjs;s

2qÞ,

i ¼ 1; . . . ;M, ð36Þ

where the variance s2q should be matched to thevariance of the random variable ðst;dðkÞ � ~s

ðiÞt;dðkÞÞ.

Notice that we have simply assumed a Gaussianmodel for the uncertainty in the soft estimate ~sðiÞt;dðkÞ.

ARTICLE IN PRESS

Table 4

Pseudocode of the DSIS algorithm based on a given linear filter

for each time instant t

for each particle i ¼ 1; . . . ;M

draw hðiÞt:tþd�pðht:tþd js

ðiÞ0:t�1; xð0:t�1Þ

construct �HðiÞt;d , HðiÞt;d and s

ðiÞt�mþ1:t�1

compute xðiÞt;d ¼ xt;d � �HðiÞt;d �s

ðiÞt;d

obtain a matrix filter FðiÞHt from H

ðiÞt;d

compute soft estimates ~sðiÞt ¼ FðiÞHt xt

let Q ¼ 0

for each symbol, st;d ðkÞ, in vector st;d ,k ¼ 1; . . . ;Nðd þ 1Þ

let Sp ¼ 0

for each s 2S

ProbðsÞ ¼Nðsj~sðiÞt ðkÞ;s2qÞaccumulate Sp ¼ Sp þ ProbðsÞ

draw st;d ðkÞ�qðst;d ðkÞÞ ¼ Probðst;d ðkÞÞ=Sp

accumulate Q ¼ Q� qðst;d ðkÞÞ

update st;d with st;d ðkÞ

construct sðiÞt from the sampled symbols in st;d

obtain the unnormalized weight ~wðiÞt from Eq. (38)

compute Sw ¼PM

i¼1 ~wðiÞt

for each particle

normalizeits weight wðiÞt ¼ ~wðiÞt =Sw


i¼1wðiÞt

2Þ�1

if MeffogrM

resample


This is, indeed, a model. We do not imply that theestimate is necessarily Gaussian-distributed (as amatter of fact, in general it is not) and other modelscould also be used. We have found, however, thatthe Gaussian approach is simple and effective.

The idea of assigning probabilities to bitsaccording to (36) is actually inspired in theinterference cancellation method of [31], wheresimilar probabilities are calculated (for one useronly) as a step of a pseudomaximum likelihoodprocedure. Furthermore, a connection with beliefpropagation algorithm, where bit marginal prob-abilities are iteratively computed and updated canalso be established [32].

Although at time t we only need the symbolvector st (the first N elements of st;d ), all the symbolsin st;d must be drawn for computing qtþdðst:tþd Þ.They are sampled independently, so the proposalpmf for the entire vector becomes

qtþdðst:tþdÞ ¼YNðdþ1Þ

k¼1

qtðst;dðkÞÞ (37)

and only Nðd þ 1ÞjSj probabilities are computedper particle. The weight update equation is obtainedsubstituting (37) and (23) into (20), to arrive at

wðiÞt / w

ðiÞt�1

Qdk¼0pðxtþkjhtþk; s0:tþkÞQNðdþ1Þ

k¼1 qtðst;dðkÞÞ, (38)

with i ¼ 1; . . . ;M. The complete algorithm is pre-sented in Table 4.

The success of the proposal function depends, toa great extent, on the choice of the linear filter, F

ðiÞt ,

i ¼ 1; 2; . . . ;M. We have considered two differentproposal functions. The first one is based on theMMSE filter and the second one on the decorrelatorreceiver [15].

A linear MMSE filter [33] yields a vector of softestimates with the least MSE with respect to thetransmitted symbols. Hence, the matrix filter iscomputed as the solution to the quadratic optimiza-tion problem

FðiÞt ¼ argmin

FðiÞt

fE½kFðiÞt xðiÞt;d � st;dk

22�g, (39)

which is unique and can be written as

FðiÞHt ¼ ðs2s H

ðiÞt;dH

ðiÞH

t;d þ R�gt;dÞ�1s2s H

ðiÞt;d , (40)

where ss is the power of the transmitted symbols.The decorrelating receiver is simply the pseudoin-

verse of the channel matrix HðiÞt;d , given by

FðiÞt ¼ ðH

ðiÞHt;d H

ðiÞt;d Þ�1H

ðiÞHt;d . (41)

As for the variance of the estimates, s2q in (36), it is

computed as s2q ¼ E½jfðiÞH

t;k xðiÞt;d � skj

2�, which results

in s2q ¼ 1� hðiÞH

t;k fðiÞt;k (where ht;k is the kth column

of HðiÞt ) for the case of the MMSE filter, and

s2q ¼ fðiÞH

t;k fðiÞt;k for the decorrelator-based proposal

function.Note that, in this case, the computational

complexity per particle is dominated by thecomputation of the inverse matrices in (40) and(41), for the MMSE and decorrelator filters,respectively. In both cases, therefore, the numberof operations required is OððNðd þ 1ÞÞ3Þ. Overall,the computational complexity remains constrainedto be polynomial in N.

7. Simulation results

7.1. General simulation setup

For our computer simulations, we have assumed asystem with N ¼ 2 transmitting antennas and L ¼ 3


receiving antennas. The modulation format isBPSK (thus the alphabet is binary, S ¼ f�1g),and transmission is carried out in bursts of 300symbol vectors (i.e., 600 binary symbols), including ashort training sequence of 15 symbol vectors whichare used to obtain an initial (rough) estimate of theCIR.

In order to assess the performance of theproposed algorithms, we have considered twodifferent procedures to simulate the CIR, yieldingfour different channel models overall.

For the first model, we assume that the CIR is amultidimensional AR process, i.e., we use Eq. (2) togenerate the channel response and, as a conse-quence, the channel statistics perfectly match theassumptions made on the derivation of the SMCequalizers. We will refer to this simulation methodas ‘‘AR model’’.

It is also of interest to assess the robustnessof the proposed equalizers when the channeldynamics do not match the assumption of Eq. (2).Therefore, we have also carried out computersimulations where the CIR’s are generated usingthe classical Clarke model [34]. In this case, thesimulated channel is a (multidimensional) Gaussianwide-sense stationary uncorrelated scattering(WSSUS) fading process. Each coefficient ht

i;jðlÞ

within matrix HtðlÞ is modeled as a Gaussianrandom variable with zero mean and variance s2l .The autocovariance function of the discrete-timeprocess fht

i;jðlÞgt2N is

flðkÞ ¼ E½hti;jðlÞh

tþki;j ðlÞ� ¼ s2l J0ð2pF dkTÞ, (42)

where J0ð�Þ is the zero-order Bessel functionof the first kind, F d ¼ ðvm=cÞF c is the maximumDoppler spread, vm ¼ 180Km=h (or 50m/s) is themobile (transmitter) speed, c is the speed of light,and Fc ¼ 2GHz is the carrier frequency. Thevariances of the different channel delays, s2l ; l ¼0; . . . ;m� 1 are given by the delay-power profile ofthe channel. We have considered three differentprofiles:

�
Uniform [35]: all channel coefficients have thesame variance, i.e., s2l is constant for all l. � Urban micro: proposed in [36] to model urban
microcellular environments in beyond 3G com-munications systems.

6It is apparent that the five PF-based techniques examined in
� this paper produce channel estimates as a side-product.
Urban macro: also proposed in [36] to modelurban macrocellular scenarios.

For each channel model, we have numericallyestimated the BER of the three proposed SMC
smoothers, labeled as
�
DSIS (triangularization), described in Section 5; � DSIS (MMSE linear filter), described in Section 6
and using the matrix filter (40); and
� DSIS (decorrelation filter), described in Section 6
and using the matrix filter (41).

For comparison purposes, we have also consideredfour additional receivers,

�
SIS (optimal): the SIS algorithm with the optimalimportance function (without delayed sampling)described in Section 3.2 and Table 1, to illustratethe maximum performance that can be achievedby a SMC algorithm without smoothing; � DSIS (optimal): the DSIS algorithm with the
optimal importance function described inSection 3.3 and Table 2;
� KFþMMSE: this is a linear MMSE detector
analogous to (40) that uses the channel esti-mated by a KF. Its purpose is to allow thecomparison of the proposed SMC equalizerswith decision-directed (DD) algorithm proposedin [24].
� MLSD: the receiver that performs maximum
likelihood sequence detection (MLSD) using aViterbi algorithm. The necessary channel esti-mates are provided by a genie-aided KF alwaysfed with the (true) transmitted symbol vectors.This algorithm provides a good approximationof the optimal receiver.

Besides the BER, we have also evaluated theperformance of the proposed smoothers in termsof the normalized MSE in channel estimation,6

MSEt ¼ðht � htÞ

Hðht � htÞ

hHt ht

, (43)

where ht is the estimate of ht, and, in order toprovide an adequate bound, we have run a genie-aided KF with perfect knowledge of the transmitteddata, and estimated the resulting MSE.

All plots shown in the remaining of this sectionhave been obtained by simulating the transmissionof many data bursts with independently generatedsymbols, CIR and noise processes.


7.2. AR model

For this set of simulations, each channel realiza-tion has been generated using the AR process of (2)with parameters a ¼ 1� 10�5 and s2v ¼ 10�4.

Fig. 2 shows the BER of the different SMCreceivers for several signal-to-noise (SNR) valueswhen the number of particles is M ¼ 200, togetherwith the KFþMMSE algorithm and the optimalresults of the MLSD . It is seen that the threepolynomial-complexity DSIS techniques proposedin this paper clearly outperform both the exponen-tial-complexity SIS (optimal) receiver, which suffersfrom a serious performance degradation becauseof the MIMO channel frequency selectivity, andthe KFþMMSE smoother. It is also observedthat for this number of particles, the differencebetween our proposed complexity-constrainedSMC equalizers and the optimal DSIS (whosecomplexity is exponential on the number of inputstreams and the smoothing lag) is negligible. Wecan also see in Fig. 2 that the performance of theSIS (optimal) receiver is roughly the same as that ofthe DSIS algorithms for low SNR values, while itfails in the higher SNR region. This is just anindication that the AWGN in the channel is themain source of errors for low SNR, while ISI

1e-04

0.001

0.01

0.1

1

4 6 8

BE

R

SN

optimal DSIS optimal SIS

DSIS (triangularization) DSIS (MMSE linear filter)

DSIS (decorrelation filter)MLSD

Kalman Filter + MMSE

Fig. 2. BER for several values of the SNR (dB). The SMC algorithms e

frames. AR channel model.

becomes the major limiting factor as the SNRbecomes higher.

Fig. 3 depicts the time evolution of the channelestimation MSEt when the SNR value is 15 dB. Itcan be observed that the DSIS (optimal) equalizerand the three proposed complexity-constrainedsmoothers perfectly match the curve of the genie-aided KF. On the other side, the channel estimatesof the SIS (optimal) and KFþMMSE algorithmsdegrade with time. This can be interpreted as anerror-propagation phenomenon: since the dataestimates do not achieve a sufficiently low BER,channel estimation worsens, and this, in turn,prevents the BER from improving.

7.3. Clarke model

7.3.1. Uniform profile

We have considered channels generated accordingto the Clarke autocorrelation model described inSection 7.1 with a uniform delay-power profile andCIR length of m ¼ 3.

As shown in Fig. 4, the performance of thecomplexity-constrained SMC equalizers degradesslightly when the assumptions made during thedesign of the algorithms are not met. However, theoptimal DSIS equalizer still shows nearly the same

10 12 14

R (dB)

mploy M ¼ 200 particles. The results are averaged over 1478 data

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DSIS (triangularization) DSIS (MMSE linear filter) DSIS (decorrelation filter)

Kalman Filter (known data) Kalman Filter + MMSE

Fig. 3. Channel estimation MSE for several values of the SNR (dB). The SMC algorithms employ M ¼ 200 particles. The results are

averaged over 1478 data frames. AR channel model.

1e-04

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MLSD Kalman Filter + MMSE

Fig. 4. BER for several values of the SNR (dB). The SMC algorithms employ M ¼ 200 particles. The results are averaged over 568 data

frames. Clarke autocorrelation model (uniform profile).


BER as the MLSD ( 1 dB loss). The polynomial-complexity methods, on the other hand, have a lossof between 1:5 and 2 dB for BER ¼ 10�3, withrespect to the optimal DSIS technique.

The MSEt evolution of the algorithms for thischannel model can be seen in Fig. 5. From thepicture, it is clear that the tracking of this kind ofnon-Markovian channel is a complicated task, so

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averaged over 568 data frames. Clarke autocorrelation model (uniform profile).


that even the KF with perfect knowledge of thetransmitted symbol vectors does not improve theperformance of the four DSIS equalizers. Note,however, that now the KF is not the best possiblechannel estimator since the channel is no longer anAR process. This also explains the slightly growth inthe MSE that all the algorithms suffer at thebeginning of the frame. A straightforward strategyto improve the performance of all methods thatemploy the KF is to approximate the channeldynamics using a higher order AR process. In [24] itis suggested that an order 2 AR process can providea good representation of the channel autocorrela-tion function.

7.3.2. Urban micro-profile

The channel is generated following the Clarkemodel described in Section 7.1, but the assumeddelay-power profile is given by the values specifiedin [36, Table 5], which yields a channel length ofm ¼ 3 according to our simulation parameters.

In Fig. 6 the performance of the three complexity-constrained equalizers is compared with that of theMLSD for this channel model. The exponentialcomplexity of the latter is avoided at the cost ofapproximately 2.5 dB loss for BER ¼ 10�3. The plotalso shows that the BER of the KalmanþMMSE

algorithm is one order of magnitude above that ofour proposed SMC-based algorithms, which attaina nearly identical performance.

Fig. 7 illustrates how the MSEt curves of theproposed smoothers can barely be distinguishedfrom that of the KF with perfect knowledge of the(true) transmitted symbol vectors.

7.3.3. Urban macro-profile

In this case, the delay-power profile is determinedby the specified values for the homonymous profilein [36, Table 5], which, in our case, give a CIRlength of m ¼ 6. To deal with this channel length,we have carried out simulations with M ¼ 500particles.

Fig. 8 illustrates the BER of the three polynomial-complexity equalizers and the MLSD. Although inthis case the performance loss of the proposedcomplexity-constrained methods with respect to theMLSD is apparent (nearly 4 dB at BER ¼ 10�3),the complexity of the latter when m ¼ 6 renders thealgorithm impractical.

The MSEt evolution for this channel model isshown in Fig. 9, where it can be seen, once again,that the channel estimates obtained by the intro-duced SMC equalizers are very close to those of theKF with known symbols.

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frames. Clarke autocorrelation model (urban micro-profile).

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averaged over 1206 data frames. Clarke autocorrelation model (urban micro-profile).


8. Discussion

Existing PF methods for MAP equalization offrequency-selective MIMO channels have been

shown to attain close-to-optimal performance, butalso to suffer from a stringent limitation becausetheir computational complexity per particle growsexponentially with the product of the number of

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frames. Clarke autocorrelation model (urban macro-profile).

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averaged over 1068 data frames. Clarke autocorrelation model (urban macro-profile).


input data streams (N) and the length of thediscrete-time CIR (m). In this paper, we haveintroduced two novel SMC schemes that avoid this

drawback by sampling in a higher dimension (theCIR is used as an auxiliary variable) and sequen-tially in the data space.


Starting from these principles, we have derivedthree SMC equalizers with a computational com-plexity that is polynomial, instead of exponential, inthe product Nm. The first equalizer is based on anadequate triangularization of the channel autocor-relation matrix that enables the sequential samplingof the symbols, while the other two receivers uselow-complexity linear detectors for proposing newdata particles.

Our computer simulation results show that theproposed schemes:

�
Clearly outperform the MAP equalizer based onthe SIS algorithm with optimal importancefunction (but no smoothing). The complexity ofthis receiver grows exponentially on N, while theproposed methods only present polynomialcomplexity, but it makes decisions based on thefiltering distribution of the data, which is veryinefficient for frequency-selective channels. � Attain a BER which is close to that of the
optimal DSIS. The latter method is very effective(we have shown that it can attain a BER veryclose to the optimal detector) but involvesOðjSjNd Þ operations per particle, which rendersit impractical.
� Allow an appealing trade-off between perfor-
mance and computational complexity, since theBER and channel estimation accuracy can beconsistently improved by drawing a largernumber of particles.

Regarding the last item, it may be argued thatincreasing the number of particles too far may leadto computational loads similar to those of expo-nential-complex receivers. However, recent results[37] show that particle filters can be effectivelyparallelized in such a way that the complexity-per-particle is more relevant than the overallcomplexity in order to attain high processingspeeds. Moreover, the design of SMC methods withan adaptive number of particles may furtherimprove the efficiency of the proposed equalizers(e.g., some CIR’s are easier to equalize and can behandled with a smaller number of particles, whilefor ‘‘difficult’’ channels, the computational effortcan be adaptively increased). Unfortunately, this isan open topic of research and it is not clear yet howto use the statistics of the particles (e.g., the effectivesample size [2] to adapt their number).

An alternative to improve the proposed SMCequalizers is to use ‘‘sphere detection’’ (SD) [38].

The basic idea behind SD is to produce a list ofcandidate data vectors, with associated likelihoodvalues. Such a candidate list can be constructedstarting from the output of a simple linear detector(i.e., exploiting again the idea of Section 6), and, foran adequate choice of the sphere radius, it containsthe data vectors with higher likelihoods. Thecandidate list could then be used in two ways toenhance the proposal SMC equalizers:

�
to draw new particles with probabilities propor-tional to the likelihoods (i.e., we substitute thesimple linear filters of Section 6 for a morecomplex, but also more efficient detector). � to approximate the DSIS (optimal) SMC equal-
izer. Specifically, by considering only the subsetof most likely data vectors, we can approximatethe summations in Eq. (16) and (17) using only apolynomial number of terms.

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Sequential Monte Carlo methods for complexity-constrained MAP equalization of dispersive MIMO channels

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