MLSE BASED EQUALIZATION AND FADING CHANNEL MODELING FOR GSM

Vipin Pathak – IEEE Member

HUGHES SOFTWARE SYSTEMSDigital Signal Processing GroupElectronic City, Gurgaon - Indiavpathak@hss.hns.com

ABSTRACT

This paper presents an outline on the need ofequalization in case of GSM systems. Themodulation scheme employed in GSM systemsis GMSK, which is both power and spectrumefficient, but at the cost of increased inter-symbol interference. The effect of inter-symbolinterference is more severe in case of multipathfading channel. This obviates the need of nearoptimum equalization. In this paper fadingchannel modeling and trained equalizationtechnique has been discussed, which includesthe channel estimation and timingsynchronization, followed by a trellis basedequalizer (Maximum likelihood sequenceestimation) using Viterbi Algorithm forestimating the transmitted bits from the receivedsignal and estimated channel response. Thepaper concludes with some of the BER results indifferent channel conditions as per the GSMrecommendations.

Index Terms – GSM, GMSK, Equalization,MLSE, trained equalizer, demodulation, ViterbiEqualization, channel estimation.

1. INTRODUCTION

The mobile communication system suffers fromserious problem of inter-symbol interference,due to multipath fading in the channel. Fading iscaused by interference between multiple replicasof the same signal, which arrive at slightlydifferent times at the receiver. These individualmultipath components also undergo Dopplershift due to the relative motion between thetransmitter and receiver. The widely accepted

standard GSM1 uses power and spectrumefficient modulation technique, GaussianMinimum Shift Keying, which is ideally suitedfor a mobile communication system. This paperfocuses on modeling of mobile channels as perGSM specifications [1] and the equalizationtechnique to combat the effect of the mobilechannel. The paper also briefs about the GMSKmodulation.

2. GMSK MODULATION

GMSK (Gaussian Minimum Shift Keying) hasbeen adopted as the modulation scheme for theGSM systems because of its spectral and powerefficiency. GMSK [4] is a modified form ofMSK with a pre-modulation Gaussian Filterhaving bandwidth time-period product BT = 0.3for GSM. This addition of a Gaussian filterlimits the out of band radiations but at the costof introducing inter-symbol interference in acontrolled manner. There is a trade-off onchoosing the value of BT; a lower value of BTmeans compact spectrum but higher inter-symbol interference. MSK (Minimum ShiftKeying) is a special case of CPM [5](Continuous Phase Modulation) withmodulation index h = 0.5. Thus GMSK alsobelongs to CPM class and is spectrum efficientbecause of its continuous phase and constantenvelope. The use of a Gaussian filter makesGSMK modulation scheme to be non-linearwhich appropriates the use of power efficientClass C non-linear amplifiers at the RF end, thusits ideally suited for mobile communicationsystem.

1 The Frequency band used in GSM900 is 890 – 915MHz for Uplink and 935 – 960 MHz for downlink.

The impulse response of a Gaussian lowpass filter is given by

−

Π= 22

2

2exp

21

)(T

tT

thσσ

(1)

where( )BTΠ

=2

2lnσ , BT = 0.3 for GSM

B is the 3dB bandwidth of the filter and T1 is thesymbol time period. The pulse shaping functionis given as

∗=

Tt

rectthtg )()( (2)

where rectangular function is defined by

<=

otherwise

TtforTT

trect

,0

||,1

(3)

Ideally the Gaussian filter has infinite timedomain response, but its truncated to a length ofL=3 symbol periods. The pulse response g(t) is areal function of length LT with normalizedamplitude (i.e. its zero outside [0, LT]) and canbe written as

( )

+Π−

−Π

=

2ln2/

2

2ln2/

2

21

TtBQ

TtBQ

Ttg (4)

where Q(t) is the Q-function defined as

( ) ττ dtQt∫∞

−Π

= 2/exp21

)( 2 (5)

Thus the phase of the modulated signal is givenas

∑ ∫−

∞−

Π=i

iTt

i duughmt )()(θ (6)

where [ ]1,1 −∈im are the differentially encodedNRZ data bits, h = 0.5 is the modulation index

1 The symbol time period in case of GSM systems is3.69µs, which corresponds to the data rate of 270.833kbits/sec.

which results in a maximum phase change of π/2per symbol period.The final GMSK signal is represented as

( )0)(2cos2

)( φθ ++Π= ttfTE

ts cb (7)

where Eb is the energy per modulating bit, fc isthe center frequency and φ0 is a random phasewhich is constant during one burst.

3. FADING CHANNEL MODELING

In case of a wireless radio channel, propagationof radio waves takes place through reflection,diffraction and scattering so a signal travelsfrom the transmitter to receiver over multiplereflective paths. This causes fluctuations in thereceived signal’s amplitude and phase givingrise to the phenomenon of Multipath fading,which introduces inter-symbol interference inthe signal. Multipath fading occurs when thesignals from multiple paths arrive at the receiverwith different delays and attenuation, thesesignals then interfere constructively anddestructively at the receiving antenna. In amobile scenario the situation worsens as theseindividual multipath components also undergoDoppler shift due to the presence of relativemotion between the transmitter and the receiver.The doppler shift is defined as

αλ

cosv

f d = (8)

λv

Bd = (9)

where ν is the speed, λ is the wavelength, α isthe angle at which signal is received at theantenna, Bd is the maximum doppler shift knownas the Doppler spread. The fading can be flatfading or frequency selective fading dependingupon the delay spread. Flat fading occurs whenthe transmitted signal has a bandwidth smallerthan the coherence bandwidth of the channel i.e.the delay spread is much less than the symboltime. However frequency selective fading, amore serious impairment, occurs when thetransmitted signal has a bandwidth larger thanthe coherence bandwidth of the channel i.e. thedelay spread is larger than the symbol time.

Figure1 shows typical signal propagationthrough a mobile radio channel.

Figure1 Signal Propagation Through MobileChannel

The received signal can be written as

)()()()( tNtgthtg sr += (10)

wheregr(t) is the received signalgs(t) is the transmitted signalh(t) is the channel response (fading process)N(t) is Additive White Gaussian noise.In a mobile radio channel, as there is no directline of sight between the transmitter and thereceiver i.e. there is no dominant path, so thefading process is modeled as a Rayleigh fadingprocess. Hence h(t) is a zero mean complexGaussian random process, whose envelope isRayleigh distributed and the phase is uniformlydistributed. The probability density functions ofthe magnitude and phase are given by equationsbelow

)(2

exp)( 2

2

2 aUaa

af a

−=

σσ (11)

Π<≤Π

= 20,21

)( φφφf (12)

Jakes’ Model [2] has been used for modeling ofRayleigh fading channel. It’s based on the factthat a large number of complex sinusoids withuniformly distributed phases and amplitudes canbe added up to give a complex Gaussian noise(by the central limit theorem), whose amplitudehas Rayleigh distribution. The inphase andquadrature components of fading process aregiven as

αωωβ coscos2coscos2)(0

1

tttx m

N

nnn += ∑

= (13)

αωωβ sincos2cossin2)(0

1

ttty m

N

nnn += ∑

= (14)

where βn=πn/N0, ωm=2πν/λN = 2(2N0+1), ωn=ωmcos(2πn/N), α = π/4N0 = 8 is the number of sinusoids used forgenerating the Rayleigh fading.Jakes’ Model generates flat fading, however incase of GSM, the delay spread is more than thesymbol time i.e. frequency selective fading ispresent. The frequency selective fading isgenerated by summing up individually fadedpaths with different delays and relative power asper the tapped delay model given in GSMspecification 5.05 [1].

Figure 2 Rayleigh Distributed PDF of Magnitude

Figure 3 Uniformly Distributed PDF of the Phase

4. EQUALIZATION

Equalizer is used to mitigate the effects ofchannel induced inter-symbol interference. InGSM the delay spread is more than the symboltime so frequency selective fading occurs.However it is assumed that there isn’t any fastfading degradation during one slot of 577µs i.e.the channel response doesn’t varies significantlyduring one burst. This section discusses trained

N(t)h(t)

gr(t)gs(t)X +

equalization technique based on MLSE [3].Figure 4 shows the basic block diagram of theequalizer.

Figure 4 Block Diagram for the Equalizer

4.1. Channel Estimation and Timing Synchronization

The Channel Estimation and timingsynchronization utilize the knowledge of thetraining sequence present in the GSM burst. TheFigure 5 shows the burst structure of a normalGSM burst.

Figure 5 GSM Normal Burst Structure

There are eight training sequences defined forthe normal burst. All the training sequenceshave good autocorrelation properties. Figure 6shows the autocorrelation of the trainingsequence with its middle 16 bits. It can be seenfrom the figure that the autocorrelation Rn isgiven as

{ }

±±±±±==

=5,4,3,2,1......00............................16

nfornfor

Rn (15)

The training sequence part in the received signalcan be expressed as

0* NhTSeqrTSeq += (16)

The received signal r is convolved with theconjugate of the known training sequence. Thus

**TSeqrv = (17)

The energy estimate is given by2)()( nvne = (18)

Now calculating the window energy of lengthL= (Lh + 1) * OSR, where Lh is the length of

Figure 6 Correlation of the Training Sequence

channel estimate in symbol durations, OSR isover-sampling rate.

∑+

==

Lm

mke kemP )()( (19)

The maximum sample kth in Pe containing thehighest energy directly corresponds to the firstsample of the channel response. Thus theestimated channel response is given as

LkOSRknfornv

nh +== ::...16

)()( (20)

From the knowledge of kth sample and themaximum of h(n) the start of the burst iscalculated. Finally the matched filtering of thereceived signal is done using the estimatedchannel response.

** hrY = (21)

Thus the input to the MLSE detector is thematched filtered signal Y and the autocorrelationof channel estimate Rhh.

4.2. MLSE – Viterbi Equalization

The maximum likelihood sequence estimation isan optimum equalization technique implementedusing the Viterbi algorithm. The number ofstates is given by

12 += hLM (22)

A state is given as σ(n)= {I(n),I(n-1),.I(n-(Lh-1)}where I(n) ∈ {1, j, -1, -1}. Thus σ(n) ∈ {S1, S2,… SM} where S1, S2 ...SM are the possible states.

Y(t)r(t)

he(t)

MatchedFiltering

MLSE usingViterbi Algorithm

Channel Estimation &Timing Synchronization

Reference TrainingSequence

Hard decisionData Bits

TrainingSequence Data BitsData Bits

3 57 1 26 1 57 3

Figure 7 shows the possible states in case whenLh = 2.

Figure 7 Possible 8 States for Lh=2

Just like the general Viterbi decoder, theprevious and the next state tables are generated.When a trellis is drawn, two paths lead to eachstate; the survivor path is the one with highermetric. The metric of a path is calculated byadding the previous state metric to thecontribution by the transition from previous tocurrent state. The metric increment by transitionfrom one state to another is the Gain in themetric and is given by

( )

{ }

−−

=

∑−

−=

1** ][][][Re][][Re

,],[n

Lnmhh

ba

h

mnRmnnYn

SSnYGain

σσσ

(23)

Here while calculating the metric incrementfrom one state transition to another, theknowledge of the estimated channel response isused. The trellis is drawn for all the 148 symbolsin the burst, and then traceback is done knowingthe survivor metric for each state, to estimate thetransmitted bit sequence.

5. SIMULATION RESULTS

Figure 8-9 show the uncoded bit error rate graphagainst Eb/No, for static, hilly terrain1 andtypical urban area channel model. The channelhas been simulated using reduced 6-Tap settingsand Jakes’ model, for different fading rates.

6. CONCLUSIONS

The paper has discussed the modeling of fadingchannel and equalization technique to mitigatethe effects of the channel.

1 HTx, x is the vehicle speed in km/hr.

Figure 8 Simulation for Hilly Terrain

Figure 9 Simulation for Typical Urban Area

7. REFERENCES

[1] ETSI “GSM 05.03, 05.04, 05.05” specs.[2] Bernard Sklar, Digital Communication

Pearson Education Asia ‘2001[3] Ungerboeck “Adaptive Maximum

Likelihood Receiver for carrier modulateddata transmission system” IEEETransactions on Comm. Vol. com-22 no.5May 1974.

[4] Murota and Hirade “GMSK Modulation fordigital mobile radio telephony” IEEE Trans.on Comm. Vol. com-29 no.7 July 1981.

[5] Sundberg and Aulin “Continuous PhaseModulation – Part I” IEEE Trans. on Comm.Vol. com-29 no.3 March 1981.

[6] Forney Jr. “Maximum Likelihood SequenceEstimation of digital sequences in presenceof intersymbol interference” IEEE Trans. onInfo. Theory. Vol. IT-18 no.3 May 1972.

[7] Ekstrom and Mikkelsen “Implementation ofGSM Simulation”, Aalborg University.

-1, jj, -1

-1, -j-j, -1

1, -j-j, 1

1, jj, 1

1-1

j

-j