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MIMO GENERALIZED DECORRELATING DISCRETE-TIME RAKE RECEIVER

Tuncer Baykas, Mohamed Siala, Abbas Yongaco glu

School of Information SUPCOM School of InformationTechnology and Engineering Cite Technologique Technology and Engineering

University of Ottawa des Communications University of OttawaOttawa, Canada Ariana,Tunisia Ottawa,Canada

tbaykas@site.uottawa.ca mohamed.siala@supcom.rnu.tn yongacog@site.uottawa.ca

ABSTRACT

In this paper, we introduce a generalized decorrelatingdiscrete-time RAKE receiver for MIMO systems (MIMOGD-DTR). The MIMO GD-DTR system is a combination

of two other advanced MIMO RAKE reception methods:the jointly decoding generalized rake receiver (JD-GRAKE)and the MIMO decorrelating discrete-time RAKE reception(MIMO D-DTR). The JD-GRAKE has been proposed forcorrelated interference suppression in MIMO systems and itis obtained from the single antenna generalized RAKE re-ceiver (G-RAKE). The MIMO D-DTR, which is obtainedfrom the single antenna decorrelating discrete-time RAKE(D-DTR) system, improves the performance in the presenceof channel estimation errors in diffuse channels. The MIMOGD-DTR combines the complementary advantages of boththe JD-GRAKE and the MIMO D-DTR. It suppresses the in-terference, and takes into consideration the imperfect chan-nel state information at the receiver. Our results show that

the performance of the discrete-time version of the MIMOJD-GRAKE could be worse than a conventional RAKE re-ceiver, when there are channel estimation errors in the sys-tem, whereas proposed MIMO GD-DTR provides gains upto 1.7 dB at a bit error rate of 102 in 3 transmit 3 receiveantenna system (33).

1. INTRODUCTION

The conventional RAKE receivers, which are used in wire-less communication to collect energy from multipath chan-nels, have several weaknesses. This paper focuses on twoof these weaknesses with the intention of reducing them formulti antenna (MIMO) systems. The first weakness is due to

colored interference. Colored interference could be a majorsource of degradation in the system performance, since theconventional RAKE receivers assume the interference to beuncorrelated. Bottomley et al. proposed in [1] a generalizedRAKE receiver in order to have colored (correlated) interfer-ence suppression for one transmit and one receive antennasystems. As a generalization of this principle to MIMO sys-tems, Grant et al. have proposed two methods in [2]. TheJD-GRAKE jointly decodes the symbols spread by the samespreading code and transmitted from different antennas. TheMMSE-GRAKE is an alternative system which is less com-plex than the JD-GRAKE. In the MMSE-GRAKE the sym-bols are decoded one by one, by treating the rest of the sym-bols as interference. Such a decoding decreases the com-

plexity at the price of a decrease in performance. The flaw ofall theses G-RAKE systems is the nonrealistic assumption ofhaving perfect channel state information at the receiver.The second weakness arises from the time-varying nature of

the wireless channels. Conventional RAKE receivers use anacquisition system (or searcher) to detect new paths with sig-nificant power, and a tracking system to follow the continu-ously time-varying delays of the paths [3, 4]. The acquisitionand tracking systems have two main drawbacks. First, the

assumption that the multipaths are resolvable is nonrealistic[4]. Secondly, the acquisition and tracking systems can nei-ther distinguishnor follow paths separated by less then a chipperiod [5, 6]. To eliminate the need for complex acquisitionand tracking devices, a discrete-time RAKE receiver (DTR)has been proposed in [7]. It is obtained by a lossless samplingof a transmit filtered version of the channel impulse response.The simulations show that the DTR is sensitive to channelestimation errors. To cope with this sensitivity, several in-tuitive methods have been presented in [7] and the deriva-tion of an optimum structure, complying with the maximuma posteriori (MAP) criterion, has been made in [4]. Thisoptimum structure, referred to as the decorrelating discrete-time RAKE (D-DTR), exploits the covariance matrix of thechannel values at RAKE fingers to obtain robustness againstchannel estimation errors. In [8], the D-DTR system is ex-tended for the MIMO case.In this work we combine the discrete-time version of the JD-GRAKE receiver withthe MIMO D-DTR receiver and obtainthe MIMO generalized discrete-time RAKE receiver. TheMIMO GD-DTR could suppress the interference and takesinto consideration the channel estimation errors. We obtainthe performance of the proposed receiver in a UMTS up-link model as well as the performances of the conventionalMIMO discrete-time RAKE receiver (MIMO C-DTR), theMIMO D-DTR and the discrete-time version of the MIMOJD-GRAKE. From this point on, we will refer the discrete-

time version of the MIMO JD-GRAKE as MIMO general-ized discrete-time RAKE receiver (MIMO G-DTR) to easethe presentation.The outline of the paper is as follows. In the next section weexplain the system model and receivers for single antennasystems. Afterwards, we show how to extend the GD-DTRto the MIMO GD-DTR. Section 4 presents the simulationresults. We finish the paper with the conclusions.

2. SYSTEM MODEL AND DISCRETE-TIME RAKERECEIVERS

In this section we explain the single antenna versions of theRAKE receivers using a basic DS-CDMA transmitter model.

The modulated symbol sk is spread using a long spreadingcode ck = (ck0,ck1...,ckQ1). The spreading factor of thecode is denoted by Q andckq is the qth chip of the spreadingcode ck. We assume that the code ck has the perfect correla-

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tion property to eliminate the interchip interference. This as-sumption is asymptotically met with large spreading factors.The spread symbol is convolved with a pulse shaping fil-ter with impulse response g(t) and transmitted over a quasi-

static diffuse multipath Rayleigh fading channel. Throughoutthis paper we assume that the modulation scheme is BPSK.The channel impulse response changes independently fromone time slot to another, but the multipath intensity profile ofthe channel does not change. Assuming the impulse responseof the channel for the time slot comprising sk to be h(t), thereceived signal r(t) for the symbol sk can be written as

r(t) = h(t)g(t) sk

Q

Q1

q=0

ckq(t qTc)

+ n(t),

where and(t) denote the convolution operator and Diracdelta function, respectively. In the above equation, Tc de-notes the chip period andn(t) denotes the noise plus interfer-ence. All of the receiver types we mention in this paper arediscrete-time receivers. They sample the received symbol ata rate ofTc/2, i.e. at twice the chip rate, and passes it to thecorrelators. The sampling is lossless for square-root raisedcosine filters with rolloff less than 1. In the correlators, thesymbol sk is despread using the oversampled version of ck.Due to the assumption that the spreading sequence is perfectin nature, the output vector of the correlators, y, for the sym-bol sk can be written as

y = fsk+u,

whereu is a zero-mean complex gaussian additive noise plusinterference vector with covariance matrix Ru, and the vec-tor f consists of elements from the sampled version of thefiltered channel response h(t)g(t). Although y, f andudepend on the time index k, to ease the presentation we onlykeep the index in sk.

If the estimate of f, denoted by f, is available, the C-DTR

obtains the decision variable by forming = fHy, wherethe superscript H denotes hermitian transpose. For BPSKmodulation, a hard decision on the symbol is obtained fromsgn(Re()), where Re is the real part operator. As we men-tioned above, the C-DTR does not take into consideration thecorrelation matrix Ru and also the erroneous nature of theestimated channel. The G-DTR uses Ru to obtain a decision

variable forsk, which is equal to

= (Ru1 f)Hy.

Iff is equal to f, i.e. the channel is perfectly known at the re-ceiver, this method provides the maximum likelihood detec-tion of the symbols [1]. The covariance matrix Ru dependson thermal noise and interference. The noise in u is assumedto be AWGN with variance 2 = 2N0/Tc, due to sampling,where N0 is the power spectral density of the complex noise.In the UMTS uplink, the (x,y)th element of the Ru whichcorresponds the covariance between the interference valuesof the correlators with discrete delays dx and dy can be ap-proximated as:

Ru(x,y) = 2x,y +EI

g(dx )g(dy )d,

where EI denotes the power spectral density of the interfer-ence andx,y denotes the Kronecker delta function. The num-ber of interferers and the effectiveness of the power controlcould affect the values of EI. In our simulations, to set the

value of EI, we define a noise rise coefficient , which isequal to

= (EI +N0)/N0.

The noise rise coefficient is also related to the system loadx through = 1/(1x) [9]. The system load shows thepercentage of energy coming from interference sources to thenoise plus interference in the system.

The D-DTR takes into consideration that only an estimate fof the channel vector is available, such that

f= f+e,

where e is the channel estimation error vector. We assume in

this work that f is estimated with Np pilot symbols and theestimated channel vector is equal to:

f=1

NpEp

Np1

m=0

yms

m,

where Ep is the energy per pilot symbol, ym is the receivedvector of the mth pilot symbol and the sms are the pilot sym-bols within a time slot. The D-DTR uses the channel covari-ance matrix Rf, which is defined as

Rf = E[ffH],

to construct the decision variable. The channel covariance

matrix Rf can be estimated using the estimated channel val-ues or calculated if the delay profile of the channel and theimpulse response of the pulse shaping filter are known. Thedecision variable of the D-DTR is equal to:

= (UWUHf)Hy,

where U is the matrix consisting of the eigenvectors of thechannel covariance matrixRf. The weighting matrix W is adiagonal matrix with the lth diagonal entry being a weightingfactorwl. If the data and pilot symbols have common energy

Es, then wl is equal to

wl =1

1 +2

(

Es

2 +

1

l)

,

where 2=2/(NpEs) andl is the eigenvalue correspond-ing to the lth eigenvector ofRf. In [4], it is shown that theabove weights are optimum under the assumptions of esti-mated channels and white noise.

2.1 The GD-DTR System

The GD-DTR uses both covariance matrices Rf andRu tocombine complimentary advantages of the D-DTR and theG-DTR. The GD-RAKE first decorrelates the colored noiseu in y by using the eigenvectors and eigenvalues of Ru

1

such that,

y =Hy = H(fsk+u),

where is the diagonal matrix of the square roots of the

eigenvalues ofR1u and is the eigenvector matrix ofRu1.

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On the other hand, the channel estimate f is also multipliedwith H such that,

f =Hf.

In the above equation f should be understood as the estimateof the equivalent channel vector of a received vector withan equivalent white noise vector. The decorrelation of noisechanges the covariance matrix Rf to R

fas well,

Rf

= HRf.

At this point the system has white gaussian noise sampleswith unit energy (2=1), since we used the eigenvalues of

the Ru1 in the noise decorrelating process. Additionally,

the channel vector has the new covariance matrix Rf

. Sincethis situation matches the assumptions of the D-DTR, thatmethod can be applied directly. Then the decision variable isequal to:

= (Uf)HWUy,

where U consists of the eigenvectors ofRf

andW is a di-agonal matrix with the lth diagonal entry being the weightingfactorwl, which is equal to

wl =1

1 + 1NpEs

(Es +1l

),

where l is the lth eigenvalue ofR

f.

3. THE MIMO GD-DTR SYSTEM

In this section we explain the MIMO generalized decorrelat-ing discrete-time RAKE receiver. For the MIMO GD-DTR,we assume a system with M transmit and N receive anten-nas. For each receive antenna there are L correlators. Thespreading code is the same for the transmit antennas of a userto reduce the consumption of orthogonal codes and preserveorthogonality between different users in the system. The sig-nals from different transmit antennas are synchronized. Thechannels between any transmit and receive antennas are in-dependent and have the same delay profile and therefore thesame covariance matrix Rf. The received signal, after sam-pling and despreading at the nth receive antenna, is equal to

yn = Fnsk+un,

where Fn is the channel matrix between the transmit anten-nas and the nth receive antenna, sk consists of the symbolssent simultaneouslyfrom all the transmit antennas. The noiseplus interference vectorun has the covariance matrix Ru.The MIMO GD-DTR follows the steps of the GD-DTR. TheMIMO GD-DTR starts with the decorrelation process of thenoise plus interference using the matrices and resultingfrom the decomposition of the covariance matrix Ru, suchthat,

yn = Hyn =

H(Fnsk+un),

andFn=HFn, where F is the estimation ofF. Afterwards

the decorrelating process of the channels are applied to y

n,as:

zn = UHyn,

and to Fn, such that

En = UHFn.

In [8], the probability of receiving zn, given the transmitted

symbol vectors, En and2, is given as:

p(yn|s, En) =L1

l=0

1

2

1 + 2l

wl

Mexp(Xnl)

where Xnl is equal to

Xnl =1

2

1 + 2l

wl

MynlM1

m=0

emnlsm

1 + 2

l

2 (1)

where emnl is the lth estimated channel coefficient from the

mth transmit antenna to the receive antenna n. In (1), smandynl denote the symbol from the mth transmit antenna andthe lth element ofyn respectively. For the GD-DTR system,due to the noise decorrelating process, 2=1 and2=1/NpEs.The MIMO GD-DTR chooses the symbol vector s whichmaximizes the following quantity:

N1

n=0

p(yn|s= s, En),

with respect to all possible s vectors. The soft output of thebit from the mth transmit antenna, m, can be obtained fromthe above equation, such that:

m = log

N1n=0 p(yn|sm = 1, En)

N1n=0 p(yn|sm = 1, En)

.

The MIMO D-DTR system does not apply the noise decorre-lating process, whereas the MIMO G-DTR system does notapply the channel decorrelating process and therefore doesnot have the weight matrix W. The MIMO C-DTR systemapplies neither the noise decorrelating process nor the chan-nel decorrelating process. In the next section we present thesimulation results for 2x2 and 3x3 systems.

4. SIMULATION RESULTS

To show the performance of the MIMO GD-DTR, we firstchoose a 2 transmit and 2 receive antenna (2 2) system.The channel has an exponential delay profile with RMS de-lay spread ofTm=0.25Tc, which is observed in UMTS chan-nels in [10] by Foo et al. The pulse shaping filter is a rootraised cosine filter with roll off = 0.22. The number ofcorrelators L is equal to 9. This system collects %99 of thechannel energy according to [4]. The number of pilot sym-bols per channel realization, Np, is equal to 2. To simulatea highly loaded system we set =10. The results are shownin Figure 1. The curves from top to bottom follow the sameorder of the legend. The gain of the MIMO GD-DTR com-pared to MIMO C-DTR is about 1.4 dB at a bit error rateof 102 and employing the MIMO G-DTR actually causes

a performance loss compared to the MIMO C-DTR. Sincethe performance of the MIMO D-DTR is close to the perfor-mance of the MIMO GD-DTR, we can conclude that mostof the gain of the MIMO GD-DTR comes from suppressing

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6 8 10 12 14 16 18 2010

3

102

101

100

Eb/N

0

BER

MIMO GDTR

MIMO CDTR

MIMO DDTR

MIMO GDDTR

Figure 1: The BER performances of (from top to bottom,in that order) the MIMO G-DTR, the MIMO C-DTR, theMIMO D-DTR and the MIMO GD-DTR for a 2 2 system

6 8 10 12 14 16 1810

3

102

101

100

Eb/N

0

BER

MIMO GDTR

MIMO CDTR

MIMO DDTR

MIMO GDDTR

Figure 2: The BER performances of (from top to bottom,in that order) the MIMO G-DTR, the MIMO C-DTR, theMIMO D-DTR and the MIMO GD-DTR for a 3 3 system

noisy channel estimates.We did another set of simulations with a 33 system. Weobserve a slight increase in the gain of the MIMO GD-DTRsystem with respect to the MIMO C-DTR. For example, inFigure 2, which shows the performance of 33 system with

Np=2 andL=9, the gain of the MIMO GD-DTR is around 1.7

dB at a 102 bit error rate compared to the MIMO C-DTR.Although it is not shown, we did other simulations to see theeffect of different parameters on the system performances.

We can conclude that the gain of the MIMO GD-DTR withrespect to the MIMO C-DTR increases with a decrease of thedelay spread and with an increase of the number of correla-tors and the number of antennas. The MIMO G-DTR system

gives the worst results in all simulations with channel esti-mation errors. Even if the perfect channel state informationis available, the gain of the MIMO G-DTR is very limited.The performance of the MIMO D-DTR receiver is always

between the MIMO GD-DTR and the C-DTR.

5. CONCLUSION

In this paper, we have introduced a generalized decorrelatingdiscrete-time RAKE receiver for MIMO systems to improvethe overall performance in the presence of channel estima-tion errors and colored noise plus interference. The systemexploits the covariance matrix of the channel and the covari-ance matrix of the noise plus interference to produce com-bining weights.Our results showed that gains up to 1.7 dB with respect tothe conventional discrete-time RAKE receiver are availablefor 33 systems. We observed that low number of pilot sym-bols, short delay spreads and high number of correlators andantennas increase the gain of the MIMO GD-DTR with re-spect to the conventional MIMO RAKE receiver. We simu-lated the MIMO G-DTR and the MIMO D-DTR systems aswell. The MIMO G-DTR system is not suitable since its per-formance is worse than the MIMO C-DTR if the channel co-efficients are estimated. The MIMO D-DTR performs betterthan the MIMO C-DTR but worse than the MIMO GD-DTR.

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