Digital Broadcasts Using a FM Transmitter

M A S T E R ' S T H E S I S

Digital Broadcasts Using a FM Transmitter

Robert Selberg

Luleå University of Technology MSc Programmes in Engineering Computer Science and Engineering

Department of Computer Science and Electrical Engineering Division of Signal Processing

2006:269 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--06/269--SE

Robert Selberg

June 2006

Every night, scrambled audio newspapers are broadcasted over the terres-

trial FM network all over Sweden. To be able to listen to these papers,

the listener have to borrow a special receiver from Taltidningsnämnden, a

Swedish government authority.

Rubico ABand Microbit2.0AB have developed anew receiver called

Adelausingdigitalcomponentsandtheyhavewishtoalsomakethebroad-

castsdigital. It is thepurposeof this thesisto investigate suitable modula-

tion, symbolsynchronization, symboldetection and equalization techniques

for digital transmissions over the existing FM network and implement a

proof-of-concept decodersystemableto run onthe Adelareceiver.

Oftheinvestigatedmodulationtechniquespulseamplitudemodulation,

quadrature amplitude modulation and Gaussian minimum shift keying 

pulse amplitude modulation (PAM) was implemented and tested in a lab

environment.

The result was that pulse amplitude modulation is not an appropriate

modulation for high speed data broadcasts. It is sucient for 23 kBaud

2-PAM broadcasts(in mono)but ifhigher speedis required,a carriermod-

ulation like quadrature amplitude modulation is preferable. However, the

resource requirement of the implementation was found to be very low. In

theory,the presentedsystemcanbe implemented ona DSP systemcapable

ofaslowas5-10MIPSand2KBRAM,whenonlycalculatingtheexecution

timeandmemoryneededbythelteringoperationandusingtheparameters

oftheimplementation presentedinthis thesis.

This master's thesis is the nal work for my Master of Science degree in

computer science/signalprocessing at LuleåUniversity of Technology. The

work hasbeen performedat RubicoABduring thewinterand spring 2005-

2006.

Rubico AB hasdeveloped a digital receiver for radio broadcasted audio

newspapers,calledAdela,whichis compatiblewiththeoldreceiversusedin

Sweden. However, the old receivers tend to break too often and therefore

they are to be replaced with Adela, one after another. In time, all analog

receiverswill bereplacedandtherewillbenoneed foranalogbroadcasts. It

isthepurposeofthisthesistoinvestigatesuitablemodulation techniquesfor

digital broadcast over the existing FM network and make proof-of-concept

implementation ableto run on the Adela hardware.

IwouldliketothankmyexaminerDr. FrankSjöbergandmysupervisor

at Rubico AB, PerJohansson, for their invaluable supportthroughout this

thesis. Iwouldalsoliketo thankCamillaIsakssonandNilsCarlsonfortheir

proofreading and mycolleges at Rubico; HansEklund, AndersLarsson and

Mika Kallijärvifor their viewpoints andcompanyunderthese months.

RobertSelberg

Luleå, Sweden

June 2006

1 Introduction 7

1.1 Background . . . 7

1.2 TheCurrent Receiver . . . 8

1.3 Advantages withDigital Broadcasts. . . 8

1.4 ProjectGoal . . . 9

1.5 ProjectLimitations . . . 9

1.6 Method . . . 10

1.7 Outline ofthe Thesis . . . 10

2 Modulation 11 2.1 Basics . . . 11

2.2 PulseModulationTechniques . . . 12

2.2.1 PulseAmplitude Modulation . . . 14

2.2.2 Quadrature AmplitudeModulation . . . 16

2.3 Pulses,Eye Diagramsand Matched Filters . . . 18

2.3.1 PulseProperties . . . 18

2.3.2 Eye Diagrams . . . 21

2.3.3 Matched Filters . . . 21

2.4 Gaussian Minimum ShiftKeying . . . 23

2.5 SpectrumAnalysis . . . 24

2.6 Discussionand Conclusion . . . 25

2.6.1 Reasons forUsing PAM . . . 26

2.6.2 Reasons forUsing QAM . . . 26

2.6.3 Reasons forUsing GMSK . . . 26

2.6.4 Conclusion . . . 26

3 Demodulation 29 3.1 SymbolSynchronization . . . 29

3.2 SymbolDetection . . . 32

3.3 Adaptive Equalization . . . 33

4.2 MATLABImplementation . . . 38

4.3 Real-TimeImplementation inC . . . 39

5 Results 45

5.1 Investigated Modulations . . . 45

5.2 Implementation . . . 46

6 Discussion and Conclusions 49

6.1 Discussion . . . 49

6.2 Conclusions . . . 50

6.3 FutureWork . . . 50

Introduction

Everynight,localnewspapersarebroadcastedonSverigesRadioP1allover

Sweden. Thesenewspapers arefor thevisually impaired, afatics, dyslectics

anddisabledpersonsthatfor somereasoncan notreadprintednewspapers.

The receivers usedfor recordingthese newspapersarenowoldand have

to be replaced by new ones. The new receiver is built upon a digital sig-

nal processor but the broadcasts still remain analog and it is the purpose

of this thesis to investigate suitable techniques to send and receive digital

broadcasts. Finally a complete systemwill be built to evaluate thechosen

techniques.

1.1 Background

Taltidningsnämnden, aSwedish government authority,isresponsibleforthe

publication of newspapers in spoken form. They also distribute funds to

the66 1

papers thatreadintheir news and oerthem asaudionewspapers.

Toprevent eavesdropping, thebroadcastednewspapersarescrambledbefore

transmission and therefore a special receiver has to be used to be able to

listentothenewspapers. Formoreinformationaboutthecurrentscrambling

technique, seeLarsson[13 ].

The newspapers today,are recorded on a compact audiocassette by an

old analog receiver that can only descramble and record the newspapers;

it is not able to replay them. This receiver, which was developed about

20 years ago, is out of date and tends to break down too often because of

the mechanical parts that get worn out with time. It is also expensive to

repair since itishard toget spareparts. Taltidningsnämnden hastherefore

givenRubico ABand Microbit2.0AB acontract to develop anew, digital,

receiver called Adela 2

. This receiver was put in test operation during the

1

AccordingtoTaltidningsnämnden[18 ].

2

Adelaisalsothenameofthewomanwhowasinvolvedinthebeginningandtherst

rstquarter2006. For morenon-technical information about Adela, seethe

Adela website[19].

The Adela receiver usesthe same descrambling technique as theold re-

ceiver,butthe decodingismadeusingadigitalsignalprocessor (DSP).The

reasonfor thisisthattheoldreceiversarereplacedgradually andthebroad-

castsmustbedecodable byboth receivers. However, therearealso plansto

digitize the radiotransmissions and it isthepurposeof this thesisto inves-

tigate and determine a suitable modulation technique to be usedinside the

existingFMsystem.

1.2 The Current Receiver

Today's analog receiver has some severe limitations; it can only record one

paper per night and each paper lls one 90 minutes cassette. This means

thatapersoncan onlysubscribetoonepaperatatimeandifheorshegoes

onajourney,e.g. overaweekend,someonehastochange thecassetteevery

day. Ifnot,theolderpaperswillbeoverwritten. Besidethis,theoldreceiver

isbig, heavy and the soundquality is very dependent on thequalityof the

analogcassette.

The new digital receiver does not have the same limitations as the old

one. The biggest benet for Taltidningsnämnden is that Adela does not

have any moving mechanical parts that getworn out and therefore it does

not need to be repaired often. From a user's point of view, the biggest

benets arethepossibilitytosavefour newspaperson thememory card 3

,it

iseasyto change article,thereceiverissmallanditcanbebattery-powered.

Otheradvantagesareplaybackspeedcontrol,i.e. youcanplaybackanarticle

faster or slower atyour desiredspeed,which isan requested feature andan

recorded help-voice thatgivestheuserfeedback.

Formoretechnicalinformationabouthowthenewdigitalreceiverworks,

seeLarsson[13 ] andPääjärvi[15]

1.3 Advantages with Digital Broadcasts

There aremany benets of digital broadcasts,the increase inaudioquality

is probably the chief benet from a user's point of view. However, there

areother more concealed benets like higher spectraleciency 4

, simplied

rmware updates and enhanced security through encrypted transmissions.

Mostpeoplewillnot thinkaboutthermwareupdatesorthebroader range

of available newspapers, but these features will greatly reduce thesoftware

maintenance costs and bandwidth usage associated with more transmitted

3

Dependingonthesizeofthememorycard.

4

newspapers and in the long term, it will hopefully lower the costs for the

customers.

Thepossibilityofoeringnewservicesisaninteresting featurewithdig-

ital broadcasts. The libraries, for example, canhave a servicewhere people

can borrow their audio books without having to visit the actual library.

When the loan has expired, the audio book will simply be deleted by the

receiver; no physical audiobookhasto be returned.

However,digitalbroadcastdohavesome drawbacks. For example,ifthe

receiver conditions deteriorate, like in a snow storm, the data stream may

collapse and the data is lost. This may be a problem for some users, but

typicallyit isnot abig problem. The benets far outweighthedrawbacks.

1.4 Project Goal

The goal of this thesis has been to investigate and evaluate suitable mod-

ulation techniques for use in data transmissions over a FM channel. It is

preferred to nd a modulation that makes it possible to send compressed

audio, i.e. transmit more than 64kbit/s,but not arequirement.

Whenan appropriatetechnique hasbeenchosen; itwillbeimplemented

inan embedded system, using theAnalog Devices BlacknBF532 DSP,to

demodulate a datastream inreal-time. The goalis to make theimplemen-

tationrun onasystemwithaDSPcapable of200MIPS 5

and 32KBRAM.

To build the decoder part of a communication system; a synchronizer,

detectorandpossiblyanequalizerhasto beused. Thereforethese partswill

also be investigated andsuitable algorithms willbeimplemented.

TheconclusionsdrawnfromthisprojectdoesnotonlyapplytotheAdela

digital newspaperreceiverbut also to other applications which want to use

theexistingFMradio network infrastructurefor databroadcasts.

1.5 Project Limitations

The tests performed have mostly been done in Mathworks MATLAB [7].

No real-world tests have been done, due to practical reasons but the nal

implementationhasbeentested inalabenvironmentusingaFMmodulator

andtheAdelareceiver. Thiswastosimulateareal-worldtest. Unfortunately

theeectsofmulti-path propagationand otherinterfering signalscouldnot

be investigated using thelabequipment.

The MATLABtestsperformedon themodulationsareresistance toad-

ditive white Gaussian noise (AWGN) and timing errors. No timehas been

giventondingtheoptimalparametersofthedierentpartsinthecomplete

system.

5

1.6 Method

In the beginning of this project, a literature study of dierent modulation

techniqueswasdone 6

. Afew ofthe investigated techniqueswasselected for

acloserlookaccordingto the following criteria:

•

^{Well tested technique used inreal} applications.

•

^{Suitable for} transmissionvia aFMtransmitter.

•

^{Feasible to} investigate and implement within the time frame of this thesis.

After the selection was performed, the three selected modulations was

implemented and tested inMATLAB. The testsresulted inthat one mod-

ulationwasselected for real-time implementation; but to be ableto decode

the signal, appropriate symbol synchronization, equalization and detection

techniqueshasto bechosen. Thereforeanother literaturestudy was done.

After the second study, the selected techniques were also implemented

and tested in MATLAB. Thereafter a complete receiver system was built

andtested,inMATLAB,onamodulateddatasignal;whichwastransmitted

throughtheFMmodulator and recorded byAdela.

When theMATLAB implementation was working, itwas rewrittenina

formthatcould be,and was, implementedinreal-time.

1.7 Outline of the Thesis

Asthereader may have noticed,the rst chapter is an introduction to this

master's thesis. The second chapter will introduce the basic concepts of

digitaltransmissions;theoryaboutpulseshapes,eyediagramsandaderiva-

tionofmatched ltersarediscussed. Apartfromthis,thethree modulation

techniques investigated  PAM, QAM and GMSK  are explained and an

account ofwhy the chosen technique waschosenis given.

Chapterthreeisaboutdemodulation ofthereceivedsignal. Inthischap-

ter, the importance of the symbol synchronization, symbol detection and

adaptive equalization are explained. A derivation of how to make an opti-

malmeansquareequalizerandplots ofhowitimproves thesignalisshown.

Chapter four explains theimplementation indepth, some implemented

andunimplemented optimizationsarealso explained.

Chapterveandsixsummarizestheworkbydiscussingtheresults,con-

clusions,problems andfutureimprovements.

6

The tested modulations are: Pulse Amplitude Modulation (PAM), Quadrature Am-

plitudeModulation (QAM), Quadrature Phase ShiftKeying (QPSK), Oset Quadrature

Phase Shift Keying (OQPSK), Minimum Shift Keying (MSK)and Gaussian Minimum

Modulation

Have you ever wondered how digital data is transmitted through a wired,

beropticor wirelesschannel 1

? Ifyouhave,thischapterwillhopefullygive

you,asareader,some understandingofhowdigitalcommunicationsystems

work.

This chapter will introduce the basic concepts of modulation. First a

short introduction of how a communication system works is given; then

two pulse amplitude modulations are discussed, one baseband modulation

calledPulseAmplitudeModulation (PAM)andonecarriermodulationcalled

Quadrature Amplitude Modulation (QAM). After that, theory about pulse

shapes,eyediagramsandmatchedltersispresentedtogetherwiththeGaus-

sianMinimumShiftKeying (GMSK)modulation,whichisusedintheGSM

cellphone system. Finally a short spectrum analysis of the FM system is

doneandamotivation ofthechoiceofmodulationto be usedinthis project

isgiven.

2.1 Basics

So whyis the datamodulated beforeit istransmitted? To make thetrans-

mittedsignaltwithina certainbandwidthlimit,make itresistanttonoise

andeasy to decode.

Figure 2.1 shows a simple model of a communication system where the

data enters the system from the left, becomes modulated and transmitted

by the transmitter. After transmission, the signal will travel through the

channel and get distorted by both the channel and noise at the receiver.

The receiver therefore applies a so called matched lter, to remove noise,

andanequalizer,tocompensate forthechanneldistortions,beforesampling

anddecoding.

1

Thecommunicationchannelisaphysicalmediumthatisusedtosendthesignalfrom

Figure2.1: Simplied system model  without a symbol synchronizer  for a digital

communicationsystem. Ifananaloglterisused,itisimplementedbefore

thesampler.

Figure2.2: Exampleofanonreturntozero (NRZ)pulseandapossiblepulsetrain.

2.2 Pulse Modulation Techniques

There areseveral ways to modulate digital data. Oneof the simplestbase-

band modulations is non return to zero (NRZ)[17], which is a special case

ofPAM.TheNRZmodulationmapsthedigitalbits0and1tothealphabet

[-1,+1], which in its turn is used together with the NRZ square pulse to

create the necessary transmission symbols. The pulse used by NRZ and a

possible pulse trainareshown inFigure2.2.

The transmission symbols, called symbols from here on, are the basic

buildingblocksofasignalandthesmallestunitsacommunicationsystemcan

transmit. Theycontains the information ofone or morebits, dependingon

thesize ofthealphabet. Thenumberofbitsasymbolcontainsiscalculated

accordingto the following formula:

nr bits =

^log

2 (alphabet size).

^(2.1)

The NRZ modulation is easy to synchronize due to the sharp edges of

thepulse,but theyalso makesthepulsetrain verybandwidthinecient. A

pulse train with independent symbols has the same frequency spectra asa

singlepulse and the discontinuities in theNRZ pulse makes it wide spread

inthe frequency domain,asshown inFigure2.3(c). Onewayto reduce the

bandwidthisto round othe cornersof thepulse,but thenitisnot aNRZ

pulseanymore.

Assaid,NRZis aspecialcaseof PAM. Itserveswell asanintroductory

exampleofamodulation,butitisbettertodiscussthegeneralcaseindepth.

Thereforethe followingsectionwill discussPAMand insection 2.2.2,QAM

isdiscussed.

−6Ts −4Ts −2Ts 0 2Ts 4Ts 6Ts 0

0.5 1

Time (s)

× 1/ √ Ts

−3/Ts −2/Ts −1/Ts 0 1/Ts 2/Ts 3/Ts 0

0.5 1

Frequency (Hz)

×√ Ts (a)

−6Ts −4Ts −2Ts 0 2Ts 4Ts 6Ts 0

0.5 1

Time (s)

←α =30%

× 1/ √ Ts

α =100%

↓

× 1/ √ Ts

−3/Ts −2/Ts −1/Ts 0 1/Ts 2/Ts 3/Ts 0

0.5 1

Frequency (Hz)

←α =30%

×√ Ts (b)

←α =100%

×√ Ts (c)

−6Ts −4Ts −2Ts 0 2Ts 4Ts 6Ts 0

0.5 1

Time (s)

× 1/ √ Ts

−3/Ts −2/Ts −1/Ts 0 1/Ts 2/Ts 3/Ts 0

0.5 1

Frequency (Hz)

×√ Ts (c)

Figure2.3: Some pulseforms and their Fourier transforms: (a)sinc, (b)raised-cosine

and(c)NRZ.Thepulsesarescaledtomaintainunitenergyas

T s

changes.

−4Ts −2Ts 0 2Ts 4Ts 6Ts 8Ts 10Ts 12Ts

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

time (s)

Figure2.4: 2-PAMpulsetrainbuiltfromasincpulseusingthebits[0101101011] .

Thedottedlinesaretheindividual pulses. Theblackline isthesumofall

dottedlines.

of spectral ecient pulses are examined in detail. The sinc, raised-cosine

(rc)and root raised-cosine (rrc) pulses.

2.2.1 Pulse Amplitude Modulation

Pulse amplitude modulation (PAM) [16] is one of thesimplest modulation

techniques available, it can be used to modulate the signal by itself or as

apart of some more complex modulation like QAMand GMSK. Thepulse

train inFigure 2.2is infactaPAM signalmodulated withtheNRZ square

pulseshownto theleftofthetrain. AnotherexampleofaPAMsignalisthe

pulsetrain inFigure2.4.

Buthowdoyoubuildapulsetrain? Firstwehavetochooseapulse

v(t)

^,

andsome databits to modulate. Letschoosethesincpulse ofFigure2.3(a)

andthedatabits[01 011 01011]. Themathematicalexpressions forthe

sincpulse inthe timeand frequencydomain are:

sinc

(t) = sin(πt/T s ) πt/T s

(2.2)

and

sinc

(f ) =

 1, |f| ≤ 1/2T ^s

0, |f| > 1/2T ^s ,

^(2.3)

respectively, where

T s

^{isthe symbol time, i.e. thetimebetween two pulses}

A

S ₁

S ₂

Figure2.5: IllustratingA,whichishalfthedistancebetweentwoneighboringsymbols.

orthetimebetween thetopofthesinctotherstzerocrossingofthex-axis,

seeFigure 2.3(a).

Then we have to build the actual pulse train

s(t)

^{,by mapping thedata}

bits to symbols,e.g.

a n = ±1

^{,and summing the symbols multipliedwith a}

timeshiftedversionof the pulse

v(t)

^:

s(t) =

N

X

n=0

a _n v(t − nT s ).

^(2.4)

Theresultingmodulated signal,showninFigure2.4,isnowreadyfortrans-

missionthrougha basebandchannel.

Othersymbolsthan

±1

^{canbeused. Infactallrealnumberscanbeused}

astransmissionsymbols,but itiscustomto take one bitora clusterofbits

andusetheintegers

±1, ±3, ..., ±M − 1

^{multipliedwithAassymbols,where}

M

^{istheorderofPAM2and}

A

^{,ascalefactor,ishalfthedistancebetweentwo}

neighboring symbols inthe constellation 3

, see Figure 2.5. When specifying

anorder ofPAM,itis common to writeit intheform: M-PAM.

Thechoiceofsymbolconstellation aectsthe energy neededtotransmit

the signal, therefore the shape has to be chosen wisely. The mean symbol

energy

E av

^{for a random datasignalof order}

M

^{is calculatedbytakingthe}

meansquareof all pointsin thesymbolconstellation:

E _av = A ² M

M

X

m=1

(2m − 1 − M) ² = A ² M ² − 1

3 .

^(2.5)

The symbol transfer rate in a communication system is limited by the

symbol time

T s

^{, which is limited by how much} intersymbol interference 4

(ISI) thesystemcancope with. Ifno ISIisdesiredat theoptimalsampling

2

TheorderofPAMisthenumberofbitsinthecluster.

3

Agroupoftransmissionsymbolsiscalled asymbolconstellation.

4

Intersymbolinterferenceisthe nameofa problemcausedby symbolsthat areover-

points, thenthe maximum symboltransfer rate dependson the bandwidth

ofthe channelaccording tothe following expression:

T s = 1

2BW ,

^(2.6)

which can be derived from eq. (2.3) if the pulse maximizes the channel

bandwidth.

This makes it possible to send

2BW

symbols/second (Baud) through a channel witha bandwidth of

BW

^{Hz. However, it is not always you want}

to maximize the throughput this way since if there is unused bandwidth

available,itcanbeutilizedbytherrc-pulse tocreateapulsethatiseasierto

implementandtosynchronize. Therrc-pulseanditspropertiesarediscussed

insection2.3.1.

After the data is modulated and transmitted it will travel through a

channel to the receiver, but before it reaches the receiver, the channel will

distort the transmitted signal and noise will be added, see Figure 2.1. To

decode the signal to a data stream, the received signal must rst be sam-

pled and ltered by a matched lter, this is done to remove asmuch noise

aspossible 5

. Thereafter, a symbol synchronizer is used to synchronize the

samplerwiththesignalsothenextsamplewillalsobecorrectlysampled

anda detectoris usedto mapthesamples into symbols,which aredecoded

into binary digits, i.e. the output data. This can be an easy or hard task,

depending on how the channel aects the signal. If the channel is of the

multi-path type, i.e. the transmitted signalarrivesat thereceivervia mul-

tiple propagation paths, an equalizer must be used to compensate for the

channel distortions. If the channel ltering is changing, then the equalizer

must adapt to the changes, otherwise the performance of the detector de-

creases. Adaptiveequalizationusingtheleastmean square (LMS)algorithm

isdiscussed inchapter 3.

2.2.2 Quadrature Amplitude Modulation

Quadrature amplitude modulation (QAM) [16] is frequently used indier-

entcarriertransmissionsystemsfromtheoldestmodemtothenewestADSL

modems and wireless network cards. It is a more sophisticated amplitude

modulationtechnique that uses two PAM pulse trains to modulate two or-

thogonal carriers,

cos 2πf _c t

^and

sin 2πf _c t

^{, that are added together before}

transmission. The most elegant way to describe the modulation is with

complexnotation:

s(t) = Re



e ^{−2 iπf c t} ·

N

X

n=0

a n v(t − nT ^s )



,

^(2.7)

5

This descriptionis fora systemwithandigital matchedlter. Ifananalog lteris

(a) (b)

Figure2.6: Twoexamplesof16-QAMsymbolconstellations,onequadratic(a)andone

non-quadratic (b). Each dot is a symbol and represents a point on the

complexplane.

where

a n ∈ C

^{are the symbols sent and}

f c

^{is the carrier frequency. The}

modulated signal

s(t)

^{isa signalwithtwicethe bandwidthof its pulse}

v(t)

^,

shiftedinfrequency upto

f c

^Hz.

LikePAM,thesymbolconstellationinQAMcanbechanged bothinsize

andshape. Figure2.6showstwoexamplesof16-QAMsymbolconstellations.

The number (M) in M-QAM species only the size, not the shape of the

constellation, but typically the points areordered so thedistances between

their neighbors are equal. If energy consumption is of concern, the points

can also be ordered in levels where each level have the same distance to

thecenter of the constellation, see Figure2.6(b). The average energy for a

constellation is calculated bytaking themean squaresum of theEuclidean

distances ofthe pointsintheconstellation:

E _av = 1 M

M

X

m=1

kpoint m k ² ,

^(2.8)

whichis ageneralization of eq. (2.5).

Because QAMis built oftwo PAM signalsit inherits thecharacteristics

and restrictions of PAM and the choice of pulse follows the same criteria

as discussed later in this chapter. However, there are dierences between

the two modulation schemes. One dierence is that QAM can only trans-

mit symbols at

BW

^{Baud, compared to the}

2BW

^{Baud in PAM. This is}

compensated by the possibility to send complex symbols; the eective bit

throughput isthe same.

Another dierence from PAMis theeect ofnot sampling thesignalat

(a) (b)

Figure2.7: Two examplesof distorted 16-QAM constellations. The rstconstellation

(a)hasbeendistortedonlybyadditivewhiteGaussiannoise,thesecond(b)

isaAWGNdistortedreceivedconstellationwhensampling15%outofphase

withrespecttothecarriersignals.

willbearotationofthereceivedsymbolconstellation,whichwilldisturbthe

decoderand can make it decode thesymbols wrong. Figure 2.7 shows two

AWGNdistorted symbolconstellations;one correctlysampled andone that

is sampled too late. The phenomenon occurs because the two carriers get

mixedtogether whensampling at time

t

^when

2πf c t 6= nπ/2

^,where

n ∈ Z

^.

Tosolvethisproblem,theQAMdetectorneedsaphaselockloop(PLL)that

follows thecarrier signalsand synchronizesthesampler withthecarriers.

2.3 Pulses, Eye Diagrams and Matched Filters

Until now, onlyageneral descriptionofhowacommunicationsystemworks

andtwotypesofmodulationshasbeendiscussed. Noinformationabouthow

tochoosethe pulse,theperformance ofdierent pulsesor how thematched

lterworkshasbeen given. Thissectionwillhopefullygivesome insightson

thesetopics.

2.3.1 Pulse Properties

Thesinc pulse,

sinc

(t) = sin(πt/T s ) πt/T s

,

^(2.9)

isa socalled overlapping pulse and it isextremely bandwidthecient; but

seeFigure2.3(a). Overlappingpulseshave larger timespread than thenon-

overlappingones,which makesthem interferewithother pulses. Inthecase

ofthe sincpulse, itwill interferewithall other pulsesinthefuture andthe

past. This property makes a true sinc pulse impossible to implement, only

adequateapproximations can be made.

Animportant propertyofthesincpulseisthatitdoesnot interferewith

theother pulseswhenitcrossesthex-axis,whichisevery

T s

^{seconds;where}

T s

^{, assaid before, is the time between two pulses. The classof pulses that}

hasthis propertyfollowsa criterion calledthe Nyquist pulse criterion [9].

Denition 2.3.1 A pulse

v(t)

^{satises the Nyquist pulse criterion if}

v(t) = 0

^{, where}

t = nT s

^and

n = ±1, ±2, ...,

^{but not at}

t = 0

^.

Even though the sinc pulse obeys the Nyquist criterion, it can cause

problems when a sampler does not sample a signal at exactly

t = nT _s

^{, i.e.}

thetopof apulse. AsseeninFigure2.8(a),ifthesymbolat

4T s

^issampled

at

4.4T s

^{it willbe decodedto the digital digit zeroinsteadof one.}

The otherpulse mentioned, theraised-cosine pulse (rc-pulse)[16 ]:

rc-pulse

(t) = sin(πt/T _s ) πt/T _s

cos(απt/T _s )

1 − 4α ² t ² /T _s ² ,

^(2.10)

has a property to mitigate the symptoms of the intersymbol interference

problem. It hasan excess bandwidth factor

α

^{, which can be chosen inthe}

interval

[0, 1]

^{. The larger the excess bandwidth is, the shorter the time}

spreadof the pulsebecomes. Figure2.3(b) showshowtherc-pulse becomes

narrower when the excess bandwidthis increased. Figure2.8(a) shows how

apulse withno excessbandwidthinterferewiththe otherpulsesandaects

thewholepulsetrain,(b)showshowtheexcessbandwidthfactorcanreduce

thatproblembynarrowing the pulse.

One might think thattherc-pulseisa goodpulse tousebecauseit does

nothavethe sameISIproblemasthesincpulse itis, butonlyifusedwith

an ideal channel with no distortions. If the signal is transmitted through

a non-ideal channel we have to apply a matched lter, which will remove

the zero crossing property of the rc-pulse and introduce ISI. The reason is

that it can not keep the property when it is convolved with itself, and the

matched lter ofanysymmetrical pulse is the pulse itself. Thisis shownin

thesectionabout matched lters, section 2.3.3.

To solve this problem, we can use a variant of the rc-pulse called root

raised-cosine pulse (rrc-pulse) [16 ]. The rrc-pulse is an orthogonal 6

pulse

which,unliketherrc-pulse,doesnotfullltheNyquistcriterion. Thebenet

of thepulse is thatwhen itis convolved withitself, theresult isa rc-pulse.

This means that if a rrc-pulse train is ltered with its matched lter, the

resultingsignalwillbearc-pulsetrain. AccordingtoAnderson[9],itisthese

6

Anorthogonal pulseisuncorrelatedwithitselfshiftedbyanyintegermultipleof

T s

.

−4Ts −2Ts 0 2Ts 4Ts 6Ts 8Ts 10Ts 12Ts

−1 0 1 2

time (s) (a)

−4Ts −2Ts 0 2Ts 4Ts 6Ts 8Ts 10Ts 12Ts

−1 0 1 2

time (s) (b)

Figure2.8: Raised-cosinepulsetrainwith0%(a)and100%(b)excessbandwidth. The

signal (solid line)is the sumofthe pulses (dottedlines). Note howmuch

larger thepeaks arewhennoexcessbandwidthis usedcomparedto when

100%isused.Thepeaksarecreatedbytheinterferenceofotherpulses.

−0.5Ts 0 0.5Ts

−2 0 2

Time (s) (a)

−0.5Ts 0 0.5Ts

−2 0 2

Time (s) (b)

−0.5Ts 0 0.5Ts

−2 0 2

Time (s) (c)

Figure2.9: Eye diagramof rc-pulseswith excessbandwidthfactor

α

^{=0% (a),}

α

^=50%

(b)and

α

^=100%(c).

propertiesthatmakestherrc-pulsethe mostcommonlyusedpulseindigital

communicationsystems.

2.3.2 Eye Diagrams

The most common way to visualize the eect of a pulse shape in the time

domainisthroughan eyediagram. Theeyediagramisa plotof datapoints

repetitively sampled from a pseudo-random bit sequence and displayed by

anoscilloscope orinthis casesimulated inMATLAB, seeFigure2.9. What

is important in the eye diagramis the openinginthe center, also knownas

theeye. Theeye shows how much thepulses overlap eachother and where

itissafetosamplethepulse. Theheight oftheeyedeterminesifthesystem

will be sensitive to noise or not. The lower height, the more sensitive to

noise. Thewidthoftheeyedeterminesitssensitivitytotimingerrorsdueto

theprobabilityofdetectingthewrongsymbol. Figure2.9showstheeectof

theISI, withoutnoise,for theraisedcosine pulse withanexcess bandwidth

of0%, 50%and 100%.

2.3.3 Matched Filters

The matched lter is a lter that has been proven to maximize the output

signal-to-noise ratio (

SN R 0

^{). The lter itself is a time reversed copy of}

thepulse usedto modulate the signalbeforethe transmission.

SN R 0

^{is by}

denition

SN R 0 = y _s ² (t)

E[y _n ² (t)] ,

^(2.11)

where

y _s (t)

^{is the} transmitted signalltered by thematched lter

h(t)

^and

y _n (t)

^{is the lterednoise.}

y s (t) = Z t

0 s(t − τ)h(τ) dτ, y n (t) = Z t

0 n(t − τ)h(τ) dτ.

^(2.12)

Theorem 1 The matched lter maximizes the signal-noise ratio of the l-

tered signal (

SN R 0

^{)and has an impulse response that is a scaled andtime}

reversed version of the pulse used to modulate the signal.

Proof: We begin the proof by dening the transmitted signal to one

single pulse,

s(t) = v(t)

^{. Then we simplify the} denominator of eq. (2.11) andget

E[y ² _n (t)] = E

Z t

0 n(t − α)h(α) dα Z t

0 n(t − β)h(β) dβ



(2.13)

= Z t

0

Z t 0

E[n(t − α)n(t − β)]h(α)h(β) dα dβ

= N 0

2 Z t

0

Z t

0 δ(t − α − (t − β))h(α)h(β) dα dβ

= N 0

2 Z t

0

h ² (τ ) dτ,

where

N 0 /2 = σ _n ²

^{is thenoisevariance.}

To simplify the numerator of eq. (2.11),

y ² _s (t)

^{, we can use theCauchy-}

Schwartz inequalitywhich statesthat

Z ^∞

−∞

f 1 (t)f 2 (t) dt

 2

≤ Z ^∞

−∞

f ₁ ² (t) dt Z ^∞

−∞

f ₂ ² (t) dt,

^(2.14)

whereleftside ofthe inequalityequalstheright sidei

f 1 (t) = C · f ² (t)

^,for

anyarbitraryconstant

C ∈ R

^{. The}simplicationof

y _s ² (t)

^becomes

y _s ² (t) =

Z t 0

s(t − τ)h(τ) dτ

 2

≤ Z t

0

s ² (t − τ) dτ Z t

0

h ² (τ ) dτ.

^(2.15)

Inserting the simplied versions of

y _s ² (t)

^and

E[y _n ² (t)]

^{into eq. (2.11) and}

usingthe Cauchy-Schwartz inequalitygives

SN R 0 =

 R t

0 s(t − τ)h(τ) dτ

 2

N 0

2

R t

0 h ² (τ ) dτ ≤ 2 N 0

Z t 0

s ² (τ ) dτ,

^(2.16)

where the right part of the inequality is themaximum

SN R 0

^{possible and}

itis only obtained if thelter

h(t)

^{is a scaled and time reversed versionof}

thepulse

v(t)

^inthesignal

s(t)

^{. Thereforethematched ltermaximizesthe}

SN R 0

^{. The}

SN R 0

^{for a pulse}

v(t)

^{,can nowbe calculated as:}

SN R 0 = 2 N 0

Z t 0

v ² (τ ) dτ.

^(2.17)

Gaussian LPF FM modulator NRZ base-

band signal

GMSK signal

Figure2.10: GMSKmodulationscheme. TheNRZpulsetrainentersthesystem from

theleft,becomeslteredbyaGaussianlow-passlterandtransmittedby

theFMtransmitter. TheresultingsignalisGMSKmodulated.

2.4 Gaussian Minimum Shift Keying

Gaussian minimum shift keying (GMSK) [3] isanother type of modulation

than PAM and QAM, it isa frequency modulating technique not an am-

plitude modulating  that is a well tested and reliable; it hasbeen used in

the GSM cellphone networks for over 15 years. When the old analog cell-

phone networks were to be replaced by a digitalone, GMSK was chosento

be the modulation used. This was because it is possible to use the exist-

inginfrastructureofFMtransmittersto partlycodeanddecode thesignals.

Frequencymodulationisalsoanenergyecientmodulation that,becauseof

itsconstant envelope,can be ampliedwithout distortionbyhigheciency

classCampliers,whichmakesthemodulationwellsuitedforbatterydriven

applicationssuchascellular phones. Thepossibilityto usetheexistingFM

transmitterstocreate aGMSKsignalisalsoofgreatbenetforthis project

when theAdelareceiverhas abuilt-in FMreceiver circuit andthe newspa-

pers aredistributedvia anordinary FMradio channel.

AGMSK signalcan becreated byapplyingaGaussian low-passlterto

a NRZ pulse train and feed a FM transmitter with theltered pulse train.

Figure2.10showsanexampleofhowtoimplementaGMSKtransmitterand

themathematical expression of theGaussian low-pass lter usedinGMSK

is:

h(t) = 1

√ 2πσT _s exp

 −t ² 2σ ² T _s ²



,

^(2.18)

where

σ = pln(2)

2πBT _s

^(2.19)

and

BT s

^{is the} bandwidth-timeproduct.

If we take the NRZ pulse rect

(t/T _s )

^{and lter it through the lter}

h(t)

we getthenewpulse

v(t) = 1 2T s

 Q



2πBT s t − ^T 2 ^s

pln(2)



− Q

 2πBT s

t + ^T ₂ ^s pln(2)



,

^(2.20)

where

Q(t) = 1

√ 2π Z ^∞

t

e ^{− τ}

2

2 dτ .

^(2.21)

Thisnewpulse,showninFigure2.11,canbeusedwithPAMtogenerate

thesamelteredNRZpulsetrainasGMSKdoes. Thismeansthatwefound

−5 −4 −3 −2 −1 0 1 2 3 4 5 0

0.1 0.2 0.3 0.4 0.5

←−−−−−−−BTs=0.1

←−−−−−−−BTs=0.2

←−−−−−−−BTs=0.3

←−−−−−−−BTs=2.0

t/Ts

Ts ⋅ v(t)

Figure 2.11: FourGMSKpulseswithbandwidthparameter

BT s = 0 .1, 0.2, 0.3, 2

^.

The bandwidth-time product

BT s

^{aects the time duration and shape}

ofthe pulse. If

BT s

^{isa small value, the pulse will be widely spread inthe}

timedomain and therewill be alot ofISI.Tomitigate theeectsof ISI,an

adaptive lter can be used. The pulse used in the GSM cellphone system

hasavalueof

BT s = 0.3

^{and anadaptive lteris used.}

FormoreinformationaboutGMSKanditsimplementationdetails,Laster

[14 ],Chongburee[11 ] and [3]arerecommended.

2.5 Spectrum Analysis

ToseehowtheFMsystemaects thespectrumof thetransmitted signal,a

lab-test wasdone usinga FMmodulator. Thereceived signalinthis test is

notasdistortedasasignalofareal-worldtest,becausethedistortionsinthe

lab environment are not assevere as in reality. However, it will give some

insight ofhow the combination of aFMtransmitter andreceiveraectsthe

signal.

Before transmission, the FM modulator removes the frequency compo-

nents below 30Hz and amplies the high frequencies. To compensate for

the amplication, the receiver lowers the amplitude of the high frequencies

when demodulating the signal. This is done to get a higher SNR at the

high frequencies and it should, in theory, result in the transmitted signal

but often the resulting signal is either amplied or dampened at the high

frequencies,aswill be shownlater. Unfortunatelyit notpossible toretrieve

the frequenciesbelow30 Hz.

The test was done by constructing a white noise signal in MATLAB,

whichwastransmittedthroughtheFMmodulatorandrecordedbytheAdela

receiver. Thereafter it was taken into MATLAB where its power spectral

0 5 10 15 20

−115

−110

−105

−100

−95

−90

−85

−80

−75

−70

−65

Frequency (kHz)

Power/frequency (dB/Hz)

Figure2.12: Powerspectraldensityestimate,viaWelch,ofawhite noisesignaltrans-

mittedthroughaFMmodulatorandrecordedontheAdelareceiver. Notice

thenegativespikebetween0-30Hz,itgoesdownto-111dB.Thedipon19

kHzistheresultofasocalledpilottoneintheFMsystem.

of spectral estimation [8]. The spectrum of the received signal is shown in

Figure2.12. Noticetheeect ofthe30 Hz high-pass ltering.

Lookingatthepowerspectraldensityplot,itisclearthatitisnotpossible

tosendsignalswithfrequencycontentabove18.5kHz,becauseofthedipat

19 kHz and according to Wikipedia[6], the FMsystem is specied to only

transmit signalsupto 15kHz. Thedipisthe resultofa socalledpilot tone

intheFMsystem. For informationabout theFMsystemand itspilot tone,

seeLarsson[13].

One can also see a small dampening of the frequencies above 5 kHz.

Thiscanbecorrectedusinganequalizer,but thelowfrequenciescan notbe

restored, which can be a problem for baseband modulations such as PAM

andGMSK.

2.6 Discussion and Conclusion

Nowwehavediscussedthreemodulationtechniques,dierentpulsesandseen

theeectofthemodulation/demodulationintheFMsystem,butwhatmod-

ulationshouldbeusedinAdela? Inthis sectionthebenets anddrawbacks

of the investigated modulations will be discussed and nally a conclusion

2.6.1 Reasons for Using PAM

Pulseamplitude modulation wastestedbecauseof itssimplicityandeaseof

implementationonalimitedhardwareplatform. Itwasagoodstartingpoint

whenexperimentingwithtransmissions. Itsperformanceisgoodinbaseband

channels andthequalitiesmentionedbeforemakesitagoodcandidate. The

modulationhasalsobeenusedfordatatransmissionsoverFMbefore,inthe

ERMES[10 ]pagingsystem. TheERMESsystemusesa4-PAM modulation

to achieve a symbolrate of3125 Baudina FMchannel using 25kHz [4 ].

The drawbacks with using PAM lies inthe FMsystem. The FMtrans-

mittershasahigh-passlterthatremovesthelowerfrequenciesbelow30Hz,

aswas shown before. This can be avoided ifthe bandwidth

BW

^{is halved}

and the signalis modulatinga carrier above the halved

BW

^{or by using a}

PAM-like carrier modulation like Carrier-less Amplitude/Phase modulation

(CAP) [2]. Unfortunately, there was not enough time to investigate CAP

further.

2.6.2 Reasons for Using QAM

TheQAMmodulationwastestedbecauseithasaverybroadrangeofdevices

thathavebeenusingitforalongtime;itisareliablyandprovenmodulation.

Computermodemsusesittoremovetheeectsofapossiblebias-levelinthe

telephone system and because the FM system has a high-pass lter which

removes the frequencies below 30 Hz, QAM is a candidate modulation for

Adela.

ThedrawbacksofQAMarethecomplexityandtheneedofacarriersyn-

chronization loop. Thesynchronization can bedone using eithera software

PLLor a modiedsymbolsynchronizer.

2.6.3 Reasons for Using GMSK

Thestrongest reason for using GMSK is thesimplicity and the factthat it

hasbeen usedwithsuccess for over 15 years intheGSM cellphonesystem.

The dierencesbetween PAM via FM and GMSK are not signicant, both

modulationsareconstructed prettymuchthesame way. Itisonly thepulse

thatdiers,but PAM is easierto decode iftherrc-pulse isused, due to the

lesserISI. Therefore itisquestionable to useGMSK insteadofPAM.

2.6.4 Conclusion

The similarities between GMSK and PAM sent through a FM channel are

considerable,aGMSKsignalcanbebuiltusingaPAM signal. Itisonlythe

pulse thatdiers, but GMSK hasthe drawback of more ISIand it requires

a better equalizer and detector. Therefore the PAM modulation scheme is

−1.5 0 −1 −0.5 0 0.5 1 1.5 1

2 3 4 5 x 10 ⁵

Sample value (a)

NR of samples

−1.5 0 −1 −0.5 0 0.5 1 1.5

2 4 6 8 10 12 x 10 ⁴

Sample value (b)

NR of samples

Figure2.13: Histogramoftherawsamplesfroma23kBaud/s2-PAMsignalmodulated

byarrc-pulse with 30%excessbandwidth. Thesignal issampled at the

optimumsamplingpoint. (a)undistortedsignal. (b)lteredusinga30Hz

high-passlter.

The QAM modulation is a more sophisticated modulation than PAM,

it has the nice property that it will not be aected by the removal of the

lowfrequenciesintheFMsystem,but thebenetsofQAMdonot outweigh

itscomplexity comparedto PAM.ThehighfrequencydistortionsintheFM

systemarefargreaterthantheremovedlowfrequencies,thusitisabetteruse

of time to implement an equalizer than using a more complex modulation

that is not aected by the 30 Hz high-pass ltering. Therefore the PAM

modulationscheme waschosento beimplemented ontheAdelareceiver.

To see how much the low frequency ltering would aect a PAM sig-

nal, a test was performed in MATLAB, see Figure 2.13. The samples in

the histogram were sampled at the optimum sampling point. Notice how

muchwiderthe spikesbecomeswhentheltering isapplied. Itisclear that

the high-pass lter aects the performance of the whole system negative;

but it is still possible, in theory, to transmit a 8-PAM signal. That would

make it possible to send high quality audio, or multiple newspapers inone

69 kbit/23kBaud channel.

Demodulation

Todecode ananalogsignaltoadigitalbitstreamthesystemhastodemod-

ulatethesignal. To dothis,at leastthree operations have to be performed.

The rst, in a fully digital receiver system, is to sample and lter the re-

ceivedsignalwiththe matched lter,seeFigure3.1. Afterthat,thereceiver

hastolockonto thesymbolstream;nd therightphaseand pickasymbol-

sample 1

. Thethirdthingneededisasymboldetectorwhichdecodesasample

andmapsittoasymbol. Thesearetheminimumrequiredoperations,under

ideal circumstances. Under non-ideal circumstances an equalizer isneeded.

Itisanextremelyimportantpartinmostreal-worldcommunicationsystems

andits purposebeingto mitigatetheeects ofthechanneldistortions.

This chapter will discuss and explain one technique for each of these

three steps in the decoding process. First a symbol synchronization algo-

rithmcalleddecisiondirectedmaximumlikelihood (DDML)willbediscussed,

thereafter a symbol detector called maximum a posteriori (MAP) detector

andnally anequalizeralgorithm calledleast mean square (LMS)equalizer

arediscussed.

3.1 Symbol Synchronization

Symbolsynchronization isoneofthemostimportant partsofadigital com-

munication system. Without a synchronizer it is impossible to decode a

received signal. The taskof the symbolsynchronizer is to, with knowledge

1

Thesymbol-sampleisthesamplethatischosentobedecodedtoasymbol.

Matched filter Equalizer Synchronizer Detector

nT s

Sampler

of the symbol rate, actively calculate and correct the symbol phase. To

actively correctthephaseis important because theoscillatorson thetrans-

mittingandreceivingsystemsmightdriftintime,withrespecttoeachother;

i.e. it can never be guaranteed that they have exactly the same frequency

andphase.

The phase aects the error probability of the decoder, even if the re-

ceiversamplesthesymbolsatthesame symbolfrequencyasthetransmitter

is sending them. If a system samples, for example, Figure 2.8(b) at time

t = nT _s + φ

^{, where}

n ∈ Z

^{and the phase}

φ = 0.4T _s

^{; even very little noise}

can lead to detection errors. Thus the task of the symbol synchronizer is

to adjust the phase to make the system sample the signal at the optimal

samplingpoint thus improving the chances ofdecoding thecorrectsymbol.

In this section a symbol synchronizer calleddecision directed maximum

likelihood (DDML) [16 ] synchronizer is discussed. Decision directed means

thatthealgorithm takesthe detectedsymbolintoconsiderationwhentrack-

ingthesignal. Itwaschosenafteraliteraturestudyofthreesynchronization

algorithmsthe DDML, non-decisiondirected maximum likelihood [16]and

early-late gate [9] synchronization algorithms  because it is relatively easy

to implement and it givesa betterresult locking onto thesymbols than the

otherinvestigatedtechniques. Itisversatile,andsuitableforbasebandPAM

signalsbutitcan alsobemodiedto workwithcarriermodulations suchas

QAM.

Sampler

VCO d

dt

å d

dt y[n+ φ ]

N

a n

n+ φ y[n]

φ ^[n]

Figure3.2: Decisiondirectedmaximum-likelihoodsynchronizer.

Briey described one can say that theDDML synchronizer consist of a

dierentiator, a VCO 2

controlled sampler and a sliding window averaging

lter, seeFigure 3.2. It uses thereceived signal afterthe oversamplingand

the matched lter, which works as a correlator. The resulting correlation

signal,

y[n]

^{inFigure3.2,isusedforbothsymbol}synchronizationandsymbol detection. The symbol synchronizer uses the correlation signal for nding

the points on thesignal which give the highestcorrelation, i.e. theoptimal

samplingpoints. Tondthesepoints,thecorrelationsignalisdierentiated;

sampledagain, i.e. the optimal sampling points arechosen, and multiplied

by the detected symbol from the detector. See Figure 3.3 for the eect of

2

−4Ts −2Ts 0 2Ts 4Ts 6Ts 8Ts 10Ts 12Ts

−1.5

−1

−0.5 0 0.5 1 1.5

time (s)

Figure3.3: Pulsetrain(dashedline)afterthematchedlterandtheVCOinputsignal

(solidline)beforetheslidingwindowaveragelter. TheVCOinputsignal

hasbeenampliedtentimesinthisplot.

thisoperation. Ifthe correction signalisnegative,itmeansthatthesystem

is sampling the signaltoo late and the VCO will make the sampler sample

abit earlier. Ifthesignalispositive,theoppositewill happen. Observe the

zero crossings of the correction signal, the signal hasdiscontinuities in the

pointswhere

y[n]

^{crossesthe time axis.}

Aftermultiplication,thesignalislteredbyanaveragingslidingwindow

lter,whichresults inavery lowfrequencycontrolsignal, usedbytheVCO

to correct the next sampling point. The ltering is needed to remove the

noise which is amplied by the derivative. Mathematically the correction

signalfor the n:thsample can be expressedwiththefollowing expression:

φ[n] = C N

N

X

i=1

a _n−i d

dt y[n − i],

^(3.1)

wheretheconstant

C

^isthegain,

N

^{isthelengthoftheslidingwindowlter,}

y[n]

^{is then:th sample fromthe received signal after theltering and}

a n

^is

thecorrespondingdetected symbol.

This signal makes it possible for the system to cope with a certain fre-

quencydierence withrespectto the transmitting system, butif thedier-

ence becomes to large, the system will not be able to lock onto the signal

andthedecodingofthesymbolswillbreakdown. However, thisisgenerally

3.2 Symbol Detection

Thetaskof the symboldetectoristo decodea sample tothe symbolwhich

is the most probable. A simple way is to calculate which symbol in the

constellation that hasthe value nearest to the sample; i.e. nd thesymbol

s i

^{,which minimizes the Euclidean distance}

d =

^min

_i k r − s i k .

^(3.2)

This type of detector is called maximum likelihood (ML) [16 ] detector and

isaverypopulardetectorbecause ofits simplicity andgoodperformanceif

the symboldistribution of the transmitted data isunknown or independent

andidentically distributed (IID).

If the data is not IID, this type of detector will give inferior results

and therefore the probability density function of thetransmission symbols

shouldbetakeninto consideration but ifthedistribution isunknown, the

ML detector is the one which provides the bestperformance. Thisis what

moresophisticated detectors such asthemaximum a posteriori (MAP)[16 ]

detector does, but at the price of increased computational complexity. It

is done by nding the symbol

s _i

^{that maximizes the} probability of being detected given thatthe sample

r

^{hasbeen received:}

P (s i |r) = P (s i

^{sent and}

r

^received

)

P (r

^received

) .

^(3.3)

This entire expression is not needed to calculate what symbol has the

greatest probability, for example the denominator can be reduced because

itis independent of what symbol is tested inthe numerator. However, the

numerator canalso berewritten as

P (s _i

^{sent and}

r

^received

) = P (r

^received|

s _i

^sent

)P (s _i

^sent

),

^(3.4)

which in turn can be simplied because the probability that

r

^{is received,}

given that

s i

^is transmitted, is identical to the probabilitythat the noise

n

equalsthedierence between

r

^and

s _i

^:

P (n = r − s ⁱ ) = P (r

^received|

s i

^sent

).

^(3.5)

Thisisbecausethe channelnoiseisassumedto beadditiveandindependent

ofthe signal.

These simplications results in eq. (3.6) where the symbol

s i

^which

maximizestheexpressionisthesymbolthatwasmostprobablytransmitted.

P (n = r − s ⁱ )P (s i ).

^(3.6)

Thefunction

P (n = r−s i )

^{caneitherbemodeledorcalculatedusingreceived}

LMS algorithm Sync & detect

x[n] d[n] d[n]

e[n]

^

- +

Figure3.4: Blockschematicofasystemusingtheleastmeansquarealgorithm.

Thistypeofdetector, whichtakestheobservedinformation about

r

^and

thepriorknowledgeabout

s i

^into considerationwhen makingthedecision a posteriori  that is, after the observation  is called maximum a posteriori

detector. TheinterestedreaderisreferredtoAnderson[9 ]forfurtherreading.

3.3 Adaptive Equalization

An equalizerisa necessarypartof the decoderprocess. Itstaskisto elimi-

natechanneldistortionandminimizeISI. Withouttheequalizeritisalmost

impossible to decode data with satisfying results under non-ideal circum-

stances.

In this section some theory about how to construct a minimum mean

square error equalizer is discussed, a short explanation of why we can not

usethis inreal-timeand howto makea suitablesolution.

Thereareseveralwaystoconstructanequalizer;oneisto,duringthede-

signphaseofthecommunicationsystem,constructanequalizerfor aknown

channel. Thistype of predetermined equalizeronly workson theparticular

channelit was constructed for and ifthe channel varieswithtime, theper-

formance of the equalizer will degrade. This is not a good solution. For a

digital communication systeman equalizer thatdoes not know thechannel

inadvanceandone thatiscapableofadapting toa timeinvariant channel's

varyingchannelresponseisneeded. Tosolvethisproblemanadaptiveequal-

izercanbedesigned usingtheleastmean square (LMS)[12]algorithm. This

algorithm  which is probably the most commonlyused adaptive equalizer

algorithmcanbeusedtondtheltercoecientvector

w

,inaFIRlter, whichminimizes themean square

ξ n = Ee ² _n ,

^(3.7)

of the error signal

e n

^{of a} transmitted signal that is wide sense stationary (WSS) and distorted by additive white Gaussian noise. A block schematic

ofa systemusingthe LMSalgorithm isshowninFigure3.4.

Tond the meansquareerror, wehave to dene whattheerror is. Lets

dene the error as the dierence between the desired symbol

d _n

^{and the}

estimatedsymbol

d ˆ n

^,

e n = d n − ˆ d n ,

^(3.8)

wherethe desiredsymbol

d ˆ _n

^{isthedot productbetween theltercoecient}

vector

w

andthe inputsamplevector

x

,

d ˆ _n = w · x,

^(3.9)

w = [w 0 w 1 w 2 . . . w m ] ^T , x = [X _n X _n−1 X _n−2 . . . X _n−m ] ^T .

The elements of the sample vector

x

;

X _n , X _n−1 , . . . , X _n−m

^{; are dened as}

WSSstochasticvariables.

Tominimizethemeansquareerror

ξ n

^{,wemustndthevaluesof}

w

that resultinmakingthegradientof

ξ _n

^{,withrespectto}

w

,equalzero,i.e. nding

w

sothat

∇ξ ⁿ = 2E e n ∇e ⁿ  = −2E[e n x ] = 0,

^(3.10)

where

∇e ⁿ = ∇(d ⁿ − ˆ d n ) = −x.

^(3.11)

As seen in eq. (3.10), the gradient

∇ξ ⁿ

^{equals zero when}

E[e n x ] = 0

^.

Tosolvethis,wecan rewritetheequationintomatrix notationusingacross

correlationvector anda autocorrelationmatrix as

E[e n x ] = r _dx − R ^xx w = 0,

^(3.12)

where

r _dx

is the cross correlation vector between

d _n

^and

x

.

R _xx

is the autocorrelationmatrix of

x

.

r _dx = 

E[d n X n ] E[d n X n−1 ] · · · E[d ⁿ X n−m ]  T

(3.13)

R _xx =











E[X n X n ] E[X n X n−1 ] . . . E[X n X n−m ] E[X n−1 X n ] E[X n−1 X n−1 ] . . . E[X n−1 X n−m ]

.

.

.

.

.

.

.

.

.

.

.

.

E[X n−m X n ] E[X n−m X n−1 ] . . . E[X n−m X n−m ]











(3.14)

The lter coecient vector

w

can now be calculated by solving eq. (3.12), i.e. calculate the inverse of the autocorrelation matrix

R _xx

multiplied by

r _dx

:

w = R ⁻¹ _xx r _dx .

^(3.15)

Unfortunately this requires information about the symbol distribution

andthedistributionofthereceivedsamples,whichwillnotbevalidifsome-

thingchangesinsidethechannel. Thiscanbesolved byapproximating

R _xx

and

r _dx

inreal-time by calculating these using previous samples, inverting the matrix and doing the matrix-vector multiplication. This is a proces-

sor intensive process and is rarely, if ever, used inpractice. However, it is

possible to iteratively calculate thelter coecients

w

usingthemethodof

The basic ideaof the steepestdecent method isasfollows. Let

w _n

be a vector that minimizes the mean square error

ξ n

^{at time}

n

^{. At time}

n + 1

^,

w _n

will not be optimal any more and we have to calculate

w _n+1

bytaking astepofsize

µ

^{downthequadraticerror surfacetogetclosertotheoptimal}

solution. This is done by subtracting the gradient of the error, at time

n

^,

multipliedbyastep size

µ

^:

w _n+1 = w n − µ∇ξ,

^(3.16)

where

µ

^{is limitedby}

λ max

^{,themaximum eigenvalueof the}autocorrelation matrix

R _xx

:

µ < 1 λ max

.

^(3.17)

For information on how to calculate

λ max

^{,seeHayes[12 ].}

But to calculate

∇ξ

^{, we still need some} statistical information about

e _n

^and

x

, otherwise we can not calculate

E[e _n x ]

^{. F}ortunately, the LMS algorithm will converge even if we do not have that information. We can

approximate

E[e n ]

^with

e n

^and

E[x]

^with

x

, i.e.

E[e n x ] ≈ e ⁿ x

. The new updated equationfor

w

is now:

w _n+1 = w _n + 2µe _n x ,

^(3.18)

Ifthereceivedsignalandthetransmittedsymbolsarejointly wide sense

stationary processes,which is assumed, theLMS equalizer will converge to

an optimal solution. Figure3.5 and 3.6shows the eect of theequalizer on

a2-PAM signalthat wastransmitted througha FMchannelwhen theFM-

modulator wascongured to useanerroneous preemphasis 3

setting. Notice

howmuch bettertheconditions arefor symboldetection aftertheequaliza-

tion. Onecannotethattheamplitudeofthebasebandsignaldirectlyaects

theresult of the equalized signal. If thetransmitted baseband signalhas a

low amplitude, then the samples in Figure 3.5(a) will be close to zero and

theequalized outputwill bemore wide spread.

For further reading about adaptive ltering, the book Statistical digital

signalprocessing andmodeling byMonsonH.Hayes[12] is recommended.

3

Thepreemphasis settingis a parameterused onthe FM-modulatorto control how

muchthe higherfrequencies are amplied beforetransmission. If the receiver doesnot

−0.4 0 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 1000

2000 3000 4000 5000

Sample value (a)

NR of samples

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

0 2000 4000 6000 8000 10000 12000

Sample value (b)

NR of samples

Figure3.5: Histogramofa2-PAMsignal,modulatedusinga30%excessbandwidthrrc-

pulse. Thesignal was transmittedthroughaFMmodulatorand recorded

inthe Adela receiver. (a)shows the rawsamples. (b)shows the samples

afterequalization. TheequalizedhistogramhastwoGaussianshapedparts

partlybecauseofthe30Hzhigh-passlteringintheFMsystem. Thelength

oftheequalizerwas33symbolsand

µ = 0.25

^.

−0.5 0 0.5

−0.3

−0.2

−0.1 0 0.1 0.2 0.3

Time (s) (a)

−0.5 0 0.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

Time (s) (b)

Figure3.6: Eye diagramofthe signal inFigure 3.5. Thediagramhas beencalculated

usingthesampledsignal. Thereforeitisnoisierthaninrealitybecauseofthe

errorsthesymbolsynchronizermakes. Theleftgure(a)istheeyediagram

beforeequalizationandtherightdiagram(b)isafterequalization.