adaptive equalization of time-varying mimo channels

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doi:10.1016/j.sig

�CorrespondiE-mail addre

marius@wooste

(V. Koivunen).

Signal Processing 85 (2005) 81–93

www.elsevier.com/locate/sigpro

Adaptive equalization of time-varying MIMO channels

Mihai Enescu�, Marius Sirbu, Visa Koivunen

Signal Processing Laboratory, SMARAD CoE, Helsinki University of Technology, PO Box 3000, FIN-02015 HUT, Finland

Received 17 May 2001; received in revised form 6 September 2004

Abstract

This paper addresses the problem of adaptive multiple-input multiple-output (MIMO) equalization of time-varying

channels (TVC).

The proposed technique is based on Kalman filtering and decision feedback equalization. The time-varying channels can

be estimated and tracked using a Kalman filter type of algorithm. An equalization technique for MIMO channels is derived

based on a novel decision feedback equalizer structure. Moreover, we present a method to estimate the measurement and

state noise variances used by the Kalman filter because they are crucial parameters in obtaining a reliable performance.

The performance of the algorithm is investigated in simulations using realistic channels generated based on the COST 207

model. The results show that the algorithm performs well even in low SNR conditions or in difficult channel conditions.

r 2004 Elsevier B.V. All rights reserved.

Keywords: Adaptive equalization; MIMO systems; Time-varying channels; COST 207

1. Introduction

Multiple-input multiple-output (MIMO) channelswith intersymbol interference (ISI) and inter-userinterference (IUI) arise in many applications includ-ing wireless communications. In addition, the time-varying nature of the wireless channels makes theequalization even more difficult to achieve [9]. Whenthe mobile station speed is high, deep fades occur inthe transmitted signal as well as rapid changes in thechannel. In this case the traditional equalization

e front matter r 2004 Elsevier B.V. All rights reserve

pro.2004.09.003

ng author. Tel.: +358 9 460 224.

sses: [email protected] (M. Enescu),

r.hut.fi (M. Sirbu), [email protected]

schemes based on channel identification and deci-sion-directed (DD) equalization cannot track thechannel variations. Moreover, these techniquesrequire that a significant percent of the transmittedsymbols are training symbols used for adapting theequalizer [13]. An ideal equalizer would not need anytraining sequence and would be capable of trackingfast variations of the channel.In this paper we derive an adaptive MIMO1

decision feedback equalizer (DFE) algorithmcapable of identifying, tracking and equalizingtime-varying channels (TVCs). The algorithm is

d.

1In this paper, multiple inputs may correspond to multiple

independent users, where each is equipped with a single antenna

or spatial multiplexing system with independent data streams.

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M. Enescu et al. / Signal Processing 85 (2005) 81–9382

derived using the state-variable model. The ad-vantages of the estimation and tracking stages canbe summarized as follows: the estimator is akin tothe conventional Kalman filter (KF), it is thus anexact solution to the estimation problem. KF hasbeen used in the literature in order to track thetime-varying channels in both single-input single-output (SISO) [14] and MIMO [7] cases. Theassumption that the process noise and measure-ment noise variances in the state-space model areknown is commonly used in various trackingapplications [14,7] but it is not necessarily valid.It has been recognized that in many practicalimplementations the erroneous values of themeasurement and process noise covariances de-grade significantly the performance of the KF [10].The goal in this paper is to have a channelestimation stage that does not require addi-tional knowledge of the state-space modelparameters in order to estimate the state. Conse-quently, the noise estimation problem needs tobe addressed. The method presented here forestimating the noise statistics stems from [8] andrelies on covariance matching between the theore-tical and estimated covariances of measurementand observation noises. A recursive formula isderived and the quality of the noise statisticsestimates are assessed by performing a non-parametric whiteness test on the innovationsequence of the KF. Whiteness of innovations isnecessary for optimal performance of KF-basedchannel estimators.The main contribution of the paper is a novel

MIMO DFE which belongs to the family ofnon-connected (NC) MIMO DFEs [11]. Thisimplies that individual DFEs are applied on thereceived signals. This type of DFE is used incombination with the KF in order to achieveequalization of time-varying channels. The DFEstructure is derived using the MMSE criterion.The MIMO DFE cost function takes into accountthe cross channels, hence the equalizer is able tocancel both the IUI and the ISI. The mainassumption is that the input signals and thenoise are uncorrelated. Combining the Kalmanfiltering and the MIMO DFE we get a truereal-time algorithm in the sense that it is recursivein time and the storage space needed to evaluate

the estimates remains constant, as time progressesand the amount of received data increases.Consequently, the method can cope with a largenumber of parameters. Some other MIMO DFEstructures have been derived in the literature [1,16].They belong to the category of fully connected(FC) DFEs (see, [11] for a MIMO DFE classifica-tion), meaning that block matrices of feedforwardand feedback filters are applied on the receivedsignals in order to perform equalization. The NCDFE proposed in this paper achieves goodperformance at lower complexity compared toFC DFEs.The rest of the paper is organized as follows.

The system model is presented first. Then adescription of the proposed algorithm is given.Two methods for estimating noise statisticsare introduced. In Section 5, simulation re-sults of equalization for MIMO channels arepresented.

2. System model

Let us consider a MIMO system with m

transmitters and n receivers. The observationsfrom receiver j (with j ¼ 1; . . . ; nÞ at time t aregiven by

yjðtÞ ¼Xm

i¼1

XLij�1

l¼0

hijðlÞxiðt � lÞ þ vjðtÞ; (1)

where xiðt � lÞ is the complex symbol drawn froma constellation X of the ith user at time t � l; hijðlÞ

is the impulse response of the TVC, yjðtÞ is thereceived signal, and vjðtÞ is the additive Gaussiannoise at receiver j: Setting L ¼ max Lij ; thechannel length of hij ; we obtain the followingvector form:

yðtÞ ¼XL�1l¼0

HlðtÞxðt � lÞ þ vðtÞ; (2)

where y is a column vector of n received signals, xis a column vector of m transmitted signals, HlðtÞ

is an n � m matrix containing the channel taps andv is an additive noise vector. An m � n model is

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+

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.

x1

xi

xm

h11

h1j

hij

hmj

hmn

v1

vj

vn

y

yj

yn

Fig. 1. The ðm � nÞ MIMO system model.

0500

10001500

2000

0

5

10

15

200

0.2

0.4

0.6

0.8

1

t/T

|h|

τ

Fig. 2. Typical HT channel.

M. Enescu et al. / Signal Processing 85 (2005) 81–93 83

presented in Fig. 1. Let us assume that the jthreceived signal is a superposition of Np paths. Theresulting channel impulse response can then bedescribed using the Gaussian distributed wide-

sense stationary with uncorrelated scattering

(WSSUS) model [5]

hijðt; tÞ ¼1ffiffiffiffiffiffiNp

p XNp

p¼1

ejð2pf d;ptþypÞ hRFðt� tpÞ; (3)

where f d;p is the Doppler spread of the path p dueto the receiver motion. The maximum Dopplerspread can be expressed as f d;max ¼ v=l; where v isthe mobile station speed and l is the signalwavelength. yp is the angular spread of the pathp: It is a random variable with uniform distribu-tion in the interval ½0; 2p�: tp is the delay spread ofthe path p; which is a random variable withprobability density function proportional to themean power delay spectrum of the propagationenvironment. hRFðtÞ is the impulse response of thereceiver filter.Four propagation environments are widely used

for simulating receiver performance: typical urban(TU), bad urban (BU), hilly terrain (HT) and ruralarea (RA), each of them having specific parametervalues. This model is suitable for many channels ofpractical interest in mobile wireless communica-tions, which is our concern in this paper. As anexample, an HT channel for a mobile user having aspeed of 90 km/h is presented in Fig. 2.Assuming that the input sequence is a white

sequence drawn from a known constellation andthat the output signal is sampled at symbol rate we

may write the following discrete-time model:

yðkÞ ¼ XðkÞhðkÞ þ vðkÞ; (4)

where X is an n � nmL data matrix defined as

XðkÞ ¼ ½x1ðkÞIm . . . xnðkÞIm . . .

x1ðk � L þ 1ÞIm . . . xnðk � L þ 1ÞIm�;

Im is an m � m identity matrix and

hðkÞ ¼ ½h011ðkÞ . . . h01nðkÞ . . . h0m1ðkÞ . . . h0mnðkÞ . . .

hL�111 ðkÞ . . . hL�1

1n ðkÞ . . . hL�1m1 ðkÞ . . .

hL�1mn ðkÞ�T ð5Þ

is a vector of length nmL containing the MIMOchannel coefficients.

3. Recursive algorithm

The actual algorithm consists of two stages: firstwe estimate and track the channel and in the secondstage we perform the equalization. This type ofstructure lends itself to real-time implementation.

3.1. Estimation and tracking of channel coefficients

In this section we are interested in derivingestimators for the channel coefficients hijðl; kÞ: Ouralgorithm is based on the well-known Kalmanfilter (KF) [4]. KF has been used in order to trackchannel variations both in SISO [14] and MIMO

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[7] cases. The main advantage of our proposedmethod is the acquisition of the noise covariancesprior to performing equalization. Typically thisstatistics is supposed to be known [7], however,this might not be the case in practical applications.Details of the noise statistics estimation methodfollow in Section 4. In matrix notation we have thefollowing state-space equations:

yðkÞ ¼ XðkÞhðkÞ þ vðkÞ; (6)

hðkÞ ¼ Ahðk � 1Þ þ wðkÞ; (7)

where XðkÞ contains transmitted symbols, hðkÞ arethe channel taps at time k and A is the statetransition matrix, in our case a matrix close toidentity ðA ¼ aI; a ¼ 0:99Þ: The state transitionmatrix describes the dynamics of the state vector,i.e. its time autocorrelation properties. Low orderauto-regressive models are widely used in literature[6,7,15]. A relationship between the elements of Aand Doppler frequency exists and it has beenexploited, e.g. in [6,7,14]. Noises v and w aremutually uncorrelated, white noise sequences withcovariance matrices R and Q: They may beestimated prior to performing the equalization orrecursively while we do the equalization.During the training period the transmitted

symbols are known to the receiver. Let us denotethem by Xtraining according to (5). The KFequations [4] can then be summarized as follows:

hðkjk � 1Þ ¼ Ahðk � 1jk � 1Þ;

Pðkjk � 1Þ ¼ APðk � 1jk � 1ÞAT þQ;

KðkÞ ¼ Pðkjk � 1ÞXHtrainingðkÞ

�½XtrainingðkÞPðkjk � 1ÞXHtrainingðkÞ þ R��1;

rðkÞ ¼ yðkÞ � XtrainingðkÞhðkjk � 1Þ;

hðkjkÞ ¼ hðkjk � 1Þ þ KðkÞrðkÞ;

PðkjkÞ ¼ Pðkjk � 1Þ � KðkÞXtrainingðkÞPðkjk � 1Þ;

ð8Þ

where Pðkjk � 1Þ is the prediction error covariancematrix, PðkjkÞ is the filtering error covariancematrix, KðkÞ is the Kalman gain and rðkÞ are theinnovations. Thus, the estimated channel taps at

time instance k are given by the filtered stateestimate hðkjkÞ:

3.2. Equalization

DFE equalizers can be grouped in two categories:one group that uses the channel in order to find thefeedforward (FF) and feedback (FB) coefficientsand the second group which adapts the equalizertaps either blindly or using DD techniques. Sincewe estimate and track the channel taps, we derivean equalizer from the former category [13].Another type of DFE classification was pre-

sented in [11]. DFE categories are introducedconsidering whether the FF and FB filters aremutually connected or operate independently.Based on these definitions, two main categoriescan be considered. The first one is of non-connected (NC)-FF/NC-FB MIMO DFE andthe second one is of (FC)-FF/FC-FB MIMODFE. The method presented in this paper belongsto the first category, the one of NC methods.MIMO DFE structures were also proposed in[1,3]. They belong to the category of FC MIMODFEs.In our derivation, assuming that the input and

noise processes are uncorrelated we compute pairsof FF–FB filters for the received signal at eachantenna based on the MMSE criterion. Let us startby defining the channel convolution matrices Hij

of dimension Nch � N f ; where Nch ¼ L þ N f � 1and N f is the FF filter length.

HijðkÞ ¼

hijð0;kÞ 0 . . . 0

hijð1;kÞ hijð0;kÞ. .. ..

.

hijð2;kÞ hijð1;kÞ 0

..

.hijð2;kÞ hijð0;kÞ

hijðLh�1;kÞ...

hijð1;kÞ

0 hijðLh�1;kÞ hijð2;kÞ

..

.0 ..

.

..

. ... . .

. ...

0 0 hijðLh�1;kÞ

0BBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCA

:

(9)

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The DFE equalizer has N f FF coefficients and Nb

FB coefficients. This will be written asDFEðN f ;NbÞ: Applying the FF filter to the pastN f received observations and the FB filter to thepast Nd estimated symbols for each output we getthe soft estimate:

zjðkÞ ¼XN f

q¼1

f jqyjðk � qÞ �XNd

q¼1

djqxjðk � qÞ: (10)

Equalization is achieved via FF f j ¼

ðf j1; . . . ; f jN fÞT; and FB dj ¼ ðdj1; . . . ; djNd

ÞT filters.

These filters are obtained by minimizing thefollowing cost functions with respect to f j and dj:

Jj ¼ Efjxjðk � dÞ � zjðkÞj2g; (11)

where d is the equalization delay. In Eq. (11) weassume that past decisions are correct. Moreover,the input sequences are assumed to be uncorre-lated with each other or with the noise.Let us consider the observation vector received

at receiver j: When we calculate the pair ðf j ; djÞ;index j is fixed and i is running from 1; . . . ;m: Weuse the notation Hijji¼j for the case when theindexes i and j are equal, hence, Hii ¼ Hijji¼j

corresponds to the direct path channel.Under the above assumptions and notation, for

an m � n MIMO system we obtain

f j ¼Xm

i¼1iaj

HHij Hij þHHij ji¼jP

jDFEHijji¼j � lI

264

375�1

�HHij ji¼jed;

dj ¼MjHijji¼jf j ; (12)

where Mj ¼ ð0Nd�d INd�Nd0Nd�Nch�Nd�dÞ; and

PjDFE ¼ ðI�MT

j MjÞ: Furthermore l ¼ s2x=s2v and

ed ¼ ð0; . . . ; 0; 1; 0; . . . ; 0ÞT is the standard basisvector, with one at the position d; 0pdpN f : Thederivation of the above two equations is presentedin the Appendix. We note that even though theDFEs are non-connected, each of the FF filters istaking into account the cross channels of theMIMO model. Hence, when performing equaliza-tion is also able to cancel the IUI. The noisestatistics needed by the FF filters is estimated

using the noise estimation stage (as explained inSection 4).Finally, the symbol estimate xi at time k is

obtained by

xiðkÞ ¼ arg mina2X

ja� ziðkÞj; (13)

where X is a finite alphabet.The non-connected DFE presented here has the

complexity of OðnN3f Þ: Computational savings are

made with respect to FC DFEs, for example theone used in [7] has the complexity Oðn3N3

f Þ:

3.3. Implementation of the algorithm

The algorithm operates in two modes:Training mode: In the training mode only the

KF is running.Step 1: Obtain the observations yðkÞ and

generate the local training data sequenceXðkÞtraining; k ¼ 0; . . . ;N train � 1; where N train isthe length of the training sequence.

Step 2: Estimate the channel coefficients hðkÞ byrunning the Kalman algorithm described by set ofEqs. (8).Operating mode: In the operating mode both

DFE and Kalman algorithms are running in analternating manner. We assume that the hðkÞ hasbeen estimated during the training period andXtraining ¼ X:

Step 1: Run DFE algorithm and estimate XðkÞ:Step 2: Having XðkÞ run Kalman algorithm and

obtain hðkÞ: The channel estimate hðkÞ is used atnext step k þ 1 by the DFE.

4. Noise covariances estimation

Several equalization algorithms using Kalmanfiltering for channel tracking have been proposed[7,14]. In these algorithms measurement andprocess noise variances are assumed to be known.The proposed algorithm for noise statistics estima-tion stems from the work introduced in [8] and hastwo stages: covariances estimation and optimalitytesting. An iterative batch technique is presentedfor estimating the noise statistics. A recursiveversion of the algorithm is also introduced. The

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main steps of the batch algorithm may besummarized as follows:

Step 1: Run the KF using some random initialvalues for R and Q; and store rðkÞ;Pðkjk � 1Þ andPðkjkÞ for all k: The data matrix XðkÞ is known tothe KF.

Step 2: Estimate the covariance matrices R andQ as described in Section 4.1.

Step 3: Perform the whiteness test as in Section4.2. If the test fails, perform the previous steps (1and 2) again by initializing the KF with the lastestimated values of the noise covariances. If thetest does not fail it means that the filtering processwas close to optimal (i.e., the innovation sequencesare white).

Step 4: Keep the values of R and Q obtained instep 3 and start the equalization.

4.1. Batch covariance estimation

During the first stage we estimate the measure-ment and process noise covariances. At this pointwe know the innovation sequence rðkÞ and thematrices Pðkjk � 1Þ and PðkjkÞ for each k: Sincethis is a batch processing separate from theKalman filtering we will use the index l insteadof k in order to avoid confusion. By using (6) wemay rewrite the innovation as

rðlÞ ¼ XðlÞ½hðljlÞ � hðljl � 1Þ� þ vðlÞ; (14)

where l ¼ 1; . . . ;N and N is the length of theinnovation process. The theoretical covariance ofthe innovation is given by

Sr ¼ XðlÞPðljl � 1ÞXHðlÞ þ R: (15)

The estimated covariance matrix of the innovationis

Sr ¼1

N � 1

XN

l¼1

½rðlÞ � r�½rðlÞ � r�H; (16)

where r ¼ 1=NPN

l¼1 rðlÞ is the mean of theinnovation. From (15) and (16) an estimate ofthe measurement noise covariance matrix is

obtained

R ¼1

N � 1

XN

l¼1

½rðlÞ � r�½rðlÞ � r�H�

�N � 1

NXðlÞPðljl � 1ÞXHðlÞ

�: ð17Þ

The same type of approach can be used in order toobtain the observation noise covariance matrix.We start by defining the residual process as

qðlÞ ¼ hðl þ 1Þ � AhðljlÞ; (18)

qðlÞ ¼ A½hðlÞ � hðljlÞ� þ wðlÞ: (19)

The theoretical covariance of the residual processis given by

Sq ¼ APðljlÞAH þQ: (20)

The estimate of the residual process covariancematrix can be computed as

Sq ¼1

N � 1

XN

l¼1

½qðlÞ � q�½qðlÞ � q�H; (21)

where q ¼ 1=NPN

l¼1 qðlÞ is the mean of theresidual process. Combining Eqs. (20) and (21)an estimate of the process noise covariance matrixis obtained

Q ¼1

N � 1

XN

l¼1

½qðlÞ � q�½qðlÞ � q�H�

�N � 1

NAPðljlÞAH

�: ð22Þ

4.2. Optimality testing

Now we have the estimates of covariances R andQ:We have to determine if the estimates are goodenough in order to ensure reliable tracking. Thiscan be checked by testing the whiteness of theinnovation process. The whiteness test is based onthe autocorrelation.An estimate of the autocorrelation of the

innovation process is given by

CðpÞ ¼1

N

XN

i¼p

rðiÞrði � pÞH; (23)

where N is large compared with p:

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The elements of the autocorrelation functioncijðpÞ are normalized by the zero lag value yieldingelements rijðpÞ in the range ½�1;þ1�: Assumingnegligible cross-correlations, the whiteness test canbe done for the diagonal elements

riiðpÞ ¼ciiðpÞ

ciið0Þ: (24)

Let us define [8] the 95% confidence interval onriiðpÞ for pa0:

jriiðpÞjp1:96ffiffiffiffiffi

Np : (25)

It has been recognized that in many practicalimplementations erroneous estimation of themeasurement and process noise variances degradessignificantly the performance of the filter [10].Consequently, if less than 5% of the riiðpÞ exceedthe threshold introduced in (25) then the innova-tion sequence is considered white and no signifi-cant performance loss is experienced.

4.3. Recursive formulation

From the set of Eqs. (14)–(22) we observe thatonly a few equations have to be rewritten in arecursive manner in order to obtain an on-linemethod for covariance estimation. To achieve thiswe need recursive updates only for two covariancematrices and two means. The updates for the meanof the innovation sequence, r; and for the mean ofthe residual process, q; can be written as follows:

rðkÞ ¼k � 1

krðk � 1Þ þ

1

krðkÞ;

qðkÞ ¼k � 1

kqðk � 1Þ þ

1

kqðkÞ: (26)

Eqs. (16) and (21) can be written in the followingrecursive manner:

SrðkÞ ¼k � 1

kSrðk � 1Þ

þ1

k½rðkÞ � rðkÞ�½rðkÞ � rðkÞ�H;

SqðkÞ ¼k � 1

kSqðk � 1Þ

þ1

k½qðkÞ � qðkÞ�½qðkÞ � qðkÞ�H: ð27Þ

Using the previous recursive expressions,the update of the estimated noise covariancematrix is

RðkÞ ¼ SrðkÞ

�1

k

Xk

l¼1

XðlÞPðljl � 1ÞXHðlÞ ð28Þ

and the update of the process noise covariancematrix is

QðkÞ ¼ SqðkÞ �1

kAPðkjkÞAH: (29)

In the case when the statistics of the channels areslowly time varying compared to the convergencetime of the covariance estimator, the updateweights ðk � 1Þ=k and 1=k have to be replacedwith aðkÞ and bðkÞ: These two parameters controlthe memory length of the algorithm, hence, thebetter these weights are chosen the closer thecovariances will be to the true ones. This case isnot considered in this paper.

4.4. Non-parametric optimality testing

In the case of recursive covariance estimation anon-parametric test is suitable for verifying thewhiteness of the innovation sequences. A suitablesolution for our recursive algorithm is the runs test[12]. New sequences are formed from the innova-tion processes taking the sign of their samples. Arun is defined as a set of identical signs containedbetween two different signs. A sequence would beconsidered non-random if there are either toomany or too few runs and random otherwise.We collect the number of positive sign values in

N1; the number of negative sign values in N2 andthe number of runs in V : Then we have a samplingdistribution of the statistic V : It can be shown [12]that this sampling distribution has the mean and

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variance given by

mV ðkÞ ¼2N1ðkÞN2ðkÞ

N1ðkÞ þ N2ðkÞþ 1

s2V ðkÞ

¼2N1ðkÞN2ðkÞ½2N1ðkÞN2ðkÞ � N1ðkÞ � N2ðkÞ�

½N1ðkÞ þ N2ðkÞ�2½N1ðkÞ þ N2ðkÞ � 1�

:

ð30Þ

By using (30), we can test the hypothesis ofrandomness at appropriate levels of significance.If N1 and N2 are both larger than 8, then thesampling distribution of V is nearly a normaldistribution and then

gðkÞ ¼V ðkÞ � mV ðkÞ

sV ðkÞ(31)

is normally distributed with zero mean andvariance 1. At each time instance k the absolutevalue of the g-score obtained with (31) is comparedwith the 5% level of significance. The hypothesisof randomness is rejected if the threshold isexceeded.

1 2 3 4 5 6 7 8 9 1010-1

100

101

number of iterations

user1 user2 real value

Fig. 3. The estimated measurement noise variance, batch

method.

5. Simulations

In this section we present simulation resultsillustrating the performance of the algorithm. Inthe simulations we use linearized GMSK signals[2]. The pulse shape of this modulation is used asthe receiver filter impulse response. The inputsignal is a binary sequence of N ¼ 1000 symbols.A two-transmit two-receive type of scenario isconsidered. The length of each channel is L ¼ 3:For each simulation we consider 100 Monte Carlorealizations. The downlink connection in a cellularcommunication system with carrier frequency of900 MHz is considered.

5.1. Noise statistics estimation

We have performed simulations using bothbatch and on-line covariance estimation methods.In this experiments we considered an HT channelwith a mobile speed of 100 km/h and 1000 symbolswere transmitted at SNR=10 dB. The data matrixX is known to the receiver. In the batch case,

estimates of R andQ are obtained at each iterationand the whiteness test is performed according to(25). These estimates are used as initial values forthe next iteration. The estimates of R and Q arepresented in Figs. 3 and 4. We note that Qconverges to a steady state value even if the truevalue is not known. The results of the whitenesstest are presented in Fig. 5. It can be observed thatafter 5 iterations the parameter acquisition is doneand equalization can start.Simulation results using the on-line estimation

method are presented next. Estimates of R and Qare presented in Figs. 6 and 7. The results of thenon-parametric whiteness test are presented in Fig.8 where formula (31) has been applied. The resultsindicate that the innovation sequences are whiteand hence the filter performs reliably.

5.2. Equalization of MIMO channels

In the simulations, a training sequence of 50symbols is used for the algorithm initialization. Weassume that the acquisition of R and Q has beenperformed in the previous stage. One may alsoperform the training of the equalizer during theparameter acquisition stage. This may lead to theelimination of the training sequence during the

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1 2 3 4 5 6 7 8 9 1010-3

10-2

10-1


Fig. 4. The estimated process noise variance, batch method.

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25


whi

tene

ss te

st r

esul

t [%

]

1st channel2nd channel

Fig. 5. The whiteness test results, batch method.

0 200 400 600 800 100010-2

10-1

100

101

number of samples

Fig. 6. The on-line estimated diagonal elements of measure-

ment noise variance, where dotted line is the true variance. Each

line represents the time evolution of the diagonal elements of R

corresponding to each user.

0 200 400 600 800 100010-3

10-2

10-1

number of samples

Fig. 7. The on-line estimated process noise variance. Each line

represents the trace of the elements of Q corresponding to each

user.


equalization stage. Since we illustrate the conver-gence and the tracking performance of the algo-rithm, we prefer to use the training sequenceduring the rest of the simulations.The simulations are done for ‘HT’ propagation

environment with the receiver speed of 100 km/h(HT 100). The corresponding maximum Dopplershift is 83.3Hz. The estimated channel magnitudes

for HT 100 direct channels are presented in Figs. 9and 10, respectively, where a DFE(3,3) was used.The length of the channels, L ¼ 3; is assumed to beknown.

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100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

# of samples

Fig. 8. Results of whiteness test. g-score values and 5%

significance level.

0 250 500 750 1000

1

|h11

(1,k

)|

0 250 500 750 1000

1

|h11

(2,k

)|

0 250 500 750 1000

0.6

|h11

(3,k

)|

k=t/T

Fig. 9. The estimated channel magnitude (dashed line) and the

true channel magnitude (solid line) for the main path h11; HT at100km/h.

0 250 500 750 10000

1

|h22

(1,k

)|

0 250 500 750 10000

1

|h22

(2,k

)|

0 250 500 750 10000

1

k=t/T

|h22

(3,k

)|

Fig. 10. The estimated channel phase (dashed line) and the true

channel phase (solid line) for the main path h22; HT at 100km/h.


The symbol error rate (SER) for each user atdifferent SNR is shown in Fig. 11. In order to havean accurate analysis we also compute the co-channel interferer powers as follows:

IUI1ðdBÞ ¼ 10 log

PLi¼1

PNj¼1 jh21ði; jÞj

2PLi¼1


2

!; (32)

IUI2ðdBÞ ¼ 10 log

PLi¼1


2PLi¼1


2

!; (33)

where IUI1 is the power of the co-channelinterferer due to user 2 affecting user 1, and IUI2is the power of the co-channel interferer due touser 1 affecting user 2. We note that the quality ofreception is different for the two users. This is dueto the fact that we have set different IUIs for bothusers.The equalizer performance depending on the co-

channel interferer power was also investigated.Different power levels of the co-channels wereused. The SER of the first user at different SNRsare illustrated in Fig. 12. It is apparent from thesecurves that the equalizer performance degrades asthe co-channel interferer power increases. Theanalysis is highly dependent on channel type anduser speed.The influence of error propagation was also

investigated. In the DFE structure we have fedback the estimated and the correct symbols. Theresult of the simulation for the first user is

presented in Fig. 13. Since the computationof DFE filter coefficients does not dependone time index to another, the problem oferror propagation is reduced if we can havegood estimates of the channel taps. However,errors in channel estimation will lead to signi-ficant errors in DFE filtering process. This mayhappen especially when deep fades occur in thechannel.

ARTICLE IN PRESS

8 9 10 11 12 1310-4

10-3

10-2

10-1

100

SNR[dB]

SE

R IUI= 18dB

IUI= 14dB

user 1user 2

Fig. 11. SER vs. SNR for different powers of the co-channel

interferers.

8 9 10 11 1210-5

10-4

10-3

10-2

10-1

100

SNR[dB]

SE

R

single user 21 dB 15 dB 11 dB 9 dB

Fig. 12. SER vs. SNR depending on the power of the co-

channel interferer.

8 9 10 11 1210-4

10-3

10-2

10-1

100

SNR[dB]S

ER

without error propagationwith error propagation

Fig. 13. Error propagation influence for DFE.


6. Conclusions

This paper focused on real-time equalization oftime-variant MIMO channels. The channel track-ing is performed using a KF and the transmittedsymbol estimation is done by using a novel MIMODFE structure. Two methods for estimating thenoise statistics needed by the KF were presented.

The batch method is based on a whitening testwhich ensures a good performance of the KF, therecursive method allows for real-time implementa-tion, making the algorithm fully on-line. Thechannel is a fast TVC with the coherence timeequal to the symbol period. The channel model fitsvery well to the wireless communication problem,when the signal arrives to the receiver fromdifferent paths with different delay spread, angularspread and Doppler spread. The simulation resultsshow that the algorithm achieves a good perfor-mance even in very demanding conditions.

Appendix A. MIMO–DFE

Let us define the following notations:xiðkÞ ¼ ½xiðkÞ . . . xiðk � Nch � 1Þ�

T is a vector oftransmitted symbols from user i:xjðkÞ ¼ ½xjðk � 1Þ . . . xjðk � NdÞ�

T is a vector ofestimated symbols from receiver j;Nd is the lengthof the feedback filter dj :vjðkÞ ¼ ½vjðk � 1Þ . . . vjðk � N f Þ�

T is the noisevector at the receiver j;N f is the length of thefeedforward filter f j (Fig. 14).Let us start by defining the channel convolution

matrices Hij of dimension Nch � N f ; where Nch ¼

ARTICLE IN PRESS


L þ N f � 1:

HijðkÞ ¼

hijð0;kÞ 0 . . . 0

hijð1;kÞ hijð0;kÞ. .. ..

.

hijð2;kÞ hijð1;kÞ 0

..

.hijð2;kÞ hijð0;kÞ

hijðLh�1;kÞ...

hijð1;kÞ

0 hijðLh�1;kÞ hijð2;kÞ

..

.0 ..

.

..

. ... . .

. ...

0 0 hijðLh�1;kÞ

0BBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCA

;

(34)

where hij are the channel estimates obtained fromthe KF part.Equalization is achieved via FF f i ¼

ðf i1; . . . ; f iN fÞT; and FB di ¼ ðdi1; . . . ; diNd

ÞT filters.

In the rest of the derivation computations areperformed at step k: The soft estimate at sensor j

at time k is

zj ¼ xT1H1jf j þ � � � þ xTi Hijf j þ � � �

þ xTmHmjf j þ vTj f j � x

Tj dj : ð35Þ

We have to compute a pair ðf j ; djÞ such thatJj isminimized.

Jj ¼ Efjxjðk � dÞ � zjðkÞj2g: (36)

Assuming that past decision are correct, xjðkÞ ¼

xjðk � dÞ; we have xjðkÞ ¼ ½xjðk � d� 1Þ . . . xjðk �

d� NdÞ�T:

Fig. 14. DFE structure.

The cost function becomes

Jj ¼ Efj xjðk � dÞ|fflfflfflfflffl{zfflfflfflfflffl}desired symbol

� xT1H1jf j � � � � � xTi Hijf j � � � � � xTmHmjf j � vTj f j þ x

Tj dj|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

equalized symbol

j2g:

ð37Þ

The desired symbol xjðk � dÞ can be also writ-ten as xjðk � dÞ ¼ xTj ed; where ed ¼ ð0; . . . ; 0;1; 0; . . . ; 0ÞT is the standard basis vector, with oneat the position d; 0pdpN f : The cost function canbe further written as

Jj ¼ EfjxTj ed � xT1H1jf j � � � � � xTi Hijf j

� � � � � xTmHmjf j � vTj f j þ x

Tj djj

2g: ð38Þ

Which leads to

Jj ¼ EfðxTj ed � xT1H1jf j � � � � � xTi Hijf j


Tj djÞ

H

�ðxTj ed � xT1H1jf j � � � � � xTi Hijf j


Tj djÞg: ð39Þ

Considering now that the input and noisesequences are uncorrelated we have the follow-ing assumptions Efx

�j xTj g ¼ s2xI; Efv�j v

Tj g ¼ s2vI;

Efx�i xTj gji¼j ¼ s2xM

Tj ; where Mj ¼ ð0Nd�d INd�Nd

0Nd�Nch�Nd�dÞ; Efxjxjðk � dÞg ¼ 0; EfxjxTi g ¼ 0;

EfxjvTj g ¼ 0; e

Hd M

Tj ¼ 0: Let us consider the ob-

servations vector received at receiver j: When wecalculate the pair ðf j ; djÞ; index j is fixed and i isrunning from 1; . . . ;m:We use the notation Hijji¼j

for the case when the indexes i and j are equal,hence, Hii ¼ Hijji¼j corresponds to the direct pathchannel.Considering the previous assumptions, the cost

function can be written as

Jj ¼ eHd s2xed � s2xe

Hd Hijji¼jf j

� s2xfHj Hijji¼jed � s2xf

Hj Hijji¼jM

Tj dj

þ s2xXm

i¼1

fHj HHij Hijf j þ f

Hj s

2vIf j

� s2xdHj MjHijji¼jf j þ s2xd

Hj dj : ð40Þ

The minimum of Jj is found by setting thegradient of Jj with respect to f j and dj to zero.

ARTICLE IN PRESS


A.1. Feedback filter

Based on the previous assumptions and on thefinal form (40), the gradient of Jj with respect todj is given by

rdjJj ¼ �s2xMjH

�ijji¼jf

�j þ s2xd

�j ; (41)

where � denotes complex conjugate. By setting thisto zero and applying complex conjugate, we getthe FB filter

dj ¼MjHijji¼jf j : (42)

A.2. Feedforward filter

The FF filter is found by first taking the gradientof Jj with respect to f j:

rf jJj ¼ s2x

Xm

i¼1

HTijH�ijf

�j þ s2vf

�j

� s2xHTij ji¼jM

Tj d

�j � s2xH

Tij ji¼jed:

Setting this result to zero and replacing the FBfilter with the result from (42) we obtain

s2xXm

i¼1

HTijH�ijf

�j � s2xH

Tij ji¼jM

Tj MjH

�ijji¼jf

�j

� s2vIf�j � s2xH

Tij ji¼jed ¼ 0; ð43Þ

which after applying complex conjugate andmaking the notation l ¼ s2x=s

2v leads to

Xm

i¼1iaj

HHij Hij þHHij ji¼jPDFEHijji¼j � lI

264

375f j

¼ HHij ji¼jed ð44Þ

with PjDFE ¼ I�MT

j Mj :The FF filter is given by

f j ¼Xm

i¼1iaj

HHij Hij þHHij ji¼jPDFEHijji¼j � lI

264

375�1

�HHij ji¼jed: ð45Þ

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adaptive equalization of time-varying mimo channels

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