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Space and Path Diversity : Theoretical Matched Filter Bound and Comparison to

Diversity Gain Obtained with Practical MLSE Receivers

L. Husson, A. WautierSUPLEC

Service Radiolectricit et lectronique

3 rue Joliot-Curie, Plateau de Moulon 91192 Gif-sur-Yvette, Francelionel.husson@supelec.fr, armelle.wautier@supelec.fr

ABSTRACT

Performance of space diversity techniques is consideredfor transmissions in mobile radio environments.Optimal performances are derived for any multipathRayleigh fading channel for Nth-order diversitywhenever the signals delivered by the antennas arecorrelated or not. The proposed theoretical study,illustrated for 2PSK, is suitable for any othermodulation schemes (constellation, symbol rate, filtershaping) to evaluate optimal performance of existingor future digital systems including imperfect channelestimation on fast varying channels. Matched filterbound performance is compared to practical receiversbehavior. Simulations have been carried out in the caseof the GSM system for second order diversityconsidering two practical receiver structures based onMLSE (maximum-likelihood sequence-estimator). Thetheoretical results and simulations show that secondorder diversity can be used in practical to considerablyimprove the BER performance or to improve the linkbudget of the transmission.

1. INTRODUCTION

In TDMA mobile radio systems, the propagationchannel causes perturbation effects on the transmissionlink (attenuation, distortions and fluctuations). In orderto get rid of inter-symbol interference (ISI) consecutiveto the multipath propagation, the receiver needs anequalizer. Usually, when its complexity is notprohibitive, a maximum-likelihood sequence-estimation(MLSE) receiver is preferred due to its ability to recoverthe transmitted symbols with an error probability closeto the optimum. However, in the case of deep fading inthe received signal (destructive recombination of signals

coming from several propagation paths), the receivermay not achieve sufficiently low bit error rate (BER).

Space diversity is a way to considerably improve theBER performance. Several antennas are used by thereceiver. When the antennas are sufficiently spaced, thereceived signals can be considered as independent andthe probability that they both fade at the same time isdrastically reduced. Consequently the performance isexpected to be improved.

The paper is structured as follows. Section II describeshow optimal performance is evaluated for transmissionsover any multipath Rayleigh-fading channel. In Section

III space diversity is considered. We evaluate thematched filter bound (MFB) for spatial diversity withNantennas either when the signals delivered by antennas

are uncorrelated or when they are correlated(insufficiently spaced antennas). We apply these resultsin the case of the TU0 channel proposed by the COST207 [3] for mobile radio systems such as GSM. InSection IV we display simulation results fortransmissions over mobile radio channels with twostructures of receivers using the MLSE. For the mostefficient structure, second order diversity is shown toprovide a significant gain in the signal to noise ratio

(SNR). Finally, Section V concludes the paper.

2. OPTIMAL PERFORMANCE FOR ANY

MULTIPATH RAYLEIGH FADING CHANNELS

Let us consider a binary modulated signal, the basebandequivalent signal can be written as:

s(t) =Tsi

sig(tiTs) (1)

where Ts is the symbol period, symbols si take withequal probabilities the values +A or -A, and g(t) is theshaping pulse used in the transmitter.

The propagation channel is assumed to be a mobileradio channel with m+1 Rayleigh fading paths :

c(t) = k=0

m

kzk(t- k) (2)

where k2 and k are the average power and the delay of

path number kand zk is a complex, zero-mean, unitarygaussian variable.

The received signal r(t) is the signal transmittedthrough the channel affected by an additional whitegaussian noise n(t) having a two-sided density N

o(fig.

1).

r(t) = k=0

m

kzk(t- k) *Tsi

si g(tiTs) + n(t) (3)

receiverpulseshaping

g(t)

channelc(t)

+s(t)

n(t)

r(t)

Figure 1. Baseband equivalent model for thetransmission

The instantaneous SNR, b, is expressed by :

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b =A2Ts2No

zHMz (4)

with z=(zi) and Mij =k.i*R(k-i) and R(t) being theautocorrelation function ofg(t)

M is a non-negative-definite Hermitian matrix withpositive eigenvalues (i) which are depending on the

power-delay profile of the channel and on the pulseshaping. A diagonal (D) and an orthonormal (Q)matrices can be found such as QMQH=D, it follows that[7] :

b =A2Ts2No

i=0

m

i |z'i|2 (5)

with z' = Qz , whose elements are zero-mean unitarygaussian variables.

The matched filter bound is the limit of the achievableperformance. The optimal bit error probability isdirectly related to the SNR. It is shown in [5] that theaverage bit error rate PE(b) corresponding to anaverage SNR b can be derived for any multipathRayleigh fading channel and be expressed as a functionof the eigenvalues (i).

3. OPTIMAL PERFORMANCE FOR

DIVERSITY COMBINING

A. Uncorrelated antennas

Let us consider a receiver with N antennas spacedenough such that the received signals are uncorrelated

(fig. 2). Considering that all the noises are independentand have the same density No, we can evaluate theperformance of the spatial diversity with optimalcombining by considering that the global instantaneousSNR is the addition of the instantaneous SNR on eachantenna. b

i and bi denote the instantaneous and

average signal-to-noise ratios on antenna number i andby b and b the instantaneous and average globalsignal-to-noise ratios [8] [2].

Considering all antennas having the same gain, theautocorrelation of the received signal (denoted by M) isidentical for each antenna. The computation of theoptimal average error probability is similar to the one

previously seen, considering the global (N.m by N.m)matrixM :

M =

M

... 0 M M

0 ... M

(6)

The eigenvalues ofM are the ones ofM with an ordermultiplied byN.

channelc1(t) +

n (t)

r (t)

+

+

pulse

shapingg(t)s(t)

.

.

.

.

.

.

.

.

....

Nthorder

spatialdiversityreceiver

channelci(t)

channelcN(t)

r(t)

rN(t)

ni(t)

nN(t)

Figure 2. Baseband equivalent model for atransmission withNth order spatial diversity

Numerical results are presented for transmissions overthe TU0 model (typical urban environment with mobilespeed equal to zero kilometer per hour) proposed by the

COST 207 [3] when using spatial diversity of orderNwith uncorrelated signals. Modulation scheme is 2PSKwith square root raised cosine pulse shape (roll-off=0.5). Transmission rate is 270.833 kbits/s. Figure 3displays the average error probability as a function ofthe average global SNRb.

0 5 10 1510-6

10-5

10-4

10-3

10-2

10-1

100

b

1

2

3

456

7

Figure 3. Optimal performances for spatial diversitywith uncorrelated signals on TU0 channel model.

ParameterNis the order of diversity ;N=1 (no

diversity), 2, 3, 4, 5, 6, 7, Considering the global SNR makes it possible toevaluate the improvement in the performance due todiversity regarding to the power of the received signal,this is called the diversity gain. For the TU0 modelwhen considering an average BER target of 5 10 -3 thediversity gain is of 2.7 dB for second order diversity andof 3.8 dB for third order diversity (asymptotically thisgain would be of 5.9 dB). The additive diversity gain isthe most significant for second order diversity anddecreases with the increase of the order, which means alimited interest of high diversity systems.

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B. Correlated antennas

When antennas are not spaced enough, the receivedsignals are correlated between each other. Let us assumethat all the antennas have the same gain and that thecorrelation rate is the same between all antennas : .The previous computation can be used by replacing inthe formula (6) M by :

M =

M ... M M M

M ... M

(7)

Figure 4 displays the BER performance as a function ofthe average global SNRb for binary transmission overTU0 channel using spatial diversity with order two forcorrelation rate between 0 and 1. The diversity gain,which is obviously null for=1, is close to its maximum

as soon as is lower than 0.4 (this is obtained when theantennas are spaced of approximately 20 [1]; whichcorresponds to 6 meters for GSM 900 ). This behavior isalso illustrated on figure 5 which represents the averageBER at b = 9 dB as a function of which confirms theresults established by simulations in [4].

0 5 10 1510

-6

10-5

10-4

10

-3

10-2

10-1

100

b

0

1

Figure 4. Optimal performances for spatial diversity

with two correlated antennas on TU0 channelmodel; parameter is the rate of correlation betweenthe signals delivered by antennas = 0, 0.1, 0.2, 0.3,

0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-3

10-2

10-1

correlation rate

2

3

Figure 5. BER as a function of correlation rate atb =9 dB on TU0 channel model. Parameter: order ofdiversity N=2 or 3

4. PRACTICAL RECEIVERS

A. Degradation due to imperfect channel estimate

Optimal theoretical performance can also be derivedfrom the matched filter bound including degradationdue to imperfect channel knowledge from the receiver.This degradation has been evaluated for single antenna

equalizers in [11] and can be easily included in thisanalysis.

0 2 4 6 8 10 12 14 16 18 2010-5

10-4

10-3

10-2

10-1

100

b

2

1

3

Figure 6. Optimal performances for spatial diversitywith imperfect channel estimate on TU0 channel

model; spatial diversity orders are 1, 2, 3.

Figure 6 illustrates the channel estimation degradationover a time-invariant TU0 channel ; channel estimate isrealized on a training sequence composed of a CAZAC(Constant amplitude zero-autocorrelation) sequence of

length 16 and of 5 repeated symbols in order to estimate6 samples of the channel response. When channel erroris only due to noise, the gain of diversity remainsidentical to the one derived for perfect channelestimation (i. e., 2.7 dB for second order and 3.8 dB forthird order when considering an average BER target of5 10-3).

0 2 4 6 8 10 12 14 16 18 2010-5

10-4

10-3

10

-2

10-1

100

BER

b

2

1

3

Figure 7. Optimal performances for spatial diversitywith time-varying channel in absence of tracking

estimation on TU300 channel model; spatial diversity

orders are 1, 2, 3

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Figure 7 shows the performance on a time-varyingTU300 channel (fd Ts= 9 10

-4) in absence of trackingchannel estimation over 58 information symbols. As thetime variations of the channel deteriorate the BERperformance for high SNRs, the impact of diversity isreinforced ; the diversity gain is of 3.4 dB for secondorder and of 4.6 dB for third order, when considering anaverage BER target of 5 10-3.

B. MLSE-based receivers

For flat fading channels, selection combining,maximum ratio combining and equal gain combiningare classical diversity structures [8]. In the case offrequency-selective channels, diversity can be appliedeither by the combining of the received signals beforethe equalizer [9] or by the integration of diversity in thestructure of the MLSE equalizer. Practical methods forimplementing two-order spatial diversity in MLSEreceivers are studied in [6].

0 5 10 1510-5

10-4

10-3

10-2

10-1

100

b

MFB

MC

SS

no diversity

Figure 8. BER performance with a MLSE receiver withperfect channel estimation on TU0 in the followingcases: no diversity, SS method, MC method. The MFB

curve is the optimal performance for diversity of order 2

We compare the performances of two implementationmethods. The first method, sample selection (SS),consists in selecting the received signal with the greatestpower and forgetting the others (which is a combiningof the received signals with coefficients being either 1-for the selected channel-or 0-for the others-). In theother method, metric combining (MC), the metric used

in the Viterbi algorithm of the MLSE equalizer is thesum of the metrics corresponding to each diversitychannel. We have chosen the SS method for its very lowcomplexity and MC method for its efficientperformances.

Simulations have been carried out for 3000 GSM burstsover COST 207 channels with two uncorrelatedantennas. As in GSM system, used modulation isGMSK (BT=0.3) with bit rate equal to 270,833 kbits/s.Bursts are composed of 2*58 information bits and a 26-bit training sequence used for the estimation of thechannel response. Performances are compared withderived optimal diversity (matched filter bound with

perfect estimation of channel) and in absence ofdiversity technique.

Figs 8 and 9 display the average BER performance as afunction ofb for diversity in the case of transmissionsover TU0 channel; the channel is either perfectlyestimated (Fig. 8) or estimated with the trainingsequence (Fig. 9). The performance of the chosenstructures for the implementation of second orderdiversity (SS and MC) is compared to each other and tothe boundaries of optimal diversity and no diversity.

0 5 10 1510-5

10-4

10-3

10-2

10-1

100

b

MFB

MC

SS

no diversity

Figure 9. TU0 model : BER performance for MLSEreceiver (channel estimated with the training sequence)in the cases: no diversity, SS method, MC method. TheMFB curve is the optimal performance for diversity of

order 2

Figure 8 shows that, in the case of perfect channelestimation, the MC method leads to BER performanceclose to the optimum (less than 1 dB of loss in SNR) ;the SS method is less efficient, leading to a loss in SNR

of 1.5 dB in comparison with the MC method. Figure 7shows that, due to the estimation of the channelresponse, the performance is affected of a loss in SNR of1.5 dB. For an average BER target of 5 10-3, thediversity gain in SNR is of 1.6 dB for the SS methodand of 3.1 dB for MC method.

It is possible to evaluate the diversity gain in the linkbudget by considering the partial SNR which is relatedto the power of the emitted signal. This overall gain isthe sum of the gain in the received power due to theaddition of antennas (i.e. 10log(N) dB) and the diversitygain. In the TU0 model, the overall second-orderdiversity gain in SNR is of 4.6 dB for the SS method

and of 6.1 dB for MC method, for an average BERtarget of 5.10-3.

5. CONCLUSION

The performance of the receiver in the case oftransmissions over mobile radio channels can beimproved by resorting to spatial diversity. We derivedthe optimum performance achievable by spatial diversityconsidering the order of the diversity N and thecorrelation between the received signals for anymultipath Rayleigh-fading channel. Results applied to2PSK transmissions over the TU0 model (typical urban

environment with mobile speed equal to zero kilometerper hour) proposed by COST 207 can easily be

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extrapolated for other modulation schemes(constellation, symbol rate, filter shaping) [10].

Results of simulations are displayed for GSM-liketransmissions over the TU0 model. The consideredreceivers are based on MLSE adapted to spatial diversityeither by selection of the best signal to process (SSmethod) or by modifying the computation of the metric

used by the Viterbi algorithm (MC method). It is shownthat, in the considered case, an additive antenna (secondorder diversity) can improve the link budget byapproximately 6.1 dB (3.1 dB of diversity gain in SNRand 3 dB due to the increasing in the received power) ifthe antennas are sufficiently spaced.

REFERENCES

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[3] M. Failli (chairman), Digital land mobile radiocommunications, COST 207 final report, CEC Inf.Technol. And Sciences, Brussels, pp. 135-166, 1989

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