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Dottorato di Ri er a in Te nologie dell'Informazione

XXICi lo

ADVANCED MODULATION/DEMODULATION

SCHEMES FOR WIRELESS COMMUNICATIONS

Coordinatore:

Chiar.moProf. Carlo Morandi

Tutor:

Chiar.moProf. GiulioColavolpe

Dottorando: Aldo Cero

Gennaio2009

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List of gures viii

List of a ronyms ix

Foreword 1

1 Introdu tion 3

1.1 Ba kground and Obje tives . . . 3

1.2 Continuous PhaseModulation Signals . . . 7

1.3 GeneralFrameworks . . . 10

1.3.1 MAP SymbolDete tion Strategyand BCJRAlgorithm 10 1.3.2 Fa tor Graphsand SumProdu t Algorithm . . . 13

1.3.3 Iterative Joint Dete tion/De odingS hemes . . . 14

2 Capa ity Evaluation for CPM signals 17 2.1 Introdu tion . . . 17

2.2 InformationRate forChannelwith Memory . . . 19

2.2.1 InformationRate Denition . . . 19

2.2.2 Arnold andLoeligerMethod. . . 20

2.3 IRof CPMsoverAWGN . . . 22

2.3.1 OptimalMAP SymbolDete tion . . . 22

2.4 IRof CPMsoverChannelAe tedbyPhaseNoise . . . 26

2.4.1 WienerPhase NoiseModel . . . 26

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2.4.2 SATMODEPhaseNoiseModel . . . 28

2.4.3 Dete tors for IR Computation over Channels Ae ted by PhaseNoise . . . 38

2.4.4 Numeri alresults . . . 45

2.5 Capa ity Improvement: Shaper-Pre oderOptimization . . . 53

2.5.1 ProblemFormulation . . . 59

2.5.2 TheCarson's Bandwidthfor Correlated Input . . . 62

2.5.3 Numeri alResults . . . 67

3 CPM Redu ed-Complexity Soft-Output Dete tion 93 3.1 Introdu tion . . . 94

3.2 Dete tor Modeland ComplexityEvaluation . . . 96

3.3 AlgorithmsBased on MMDe omposition . . . 98

3.3.1 MMDe omposition . . . 98

3.3.2 MAPSymbolDete tionBased ontheMMDe omposition100 3.4 Complexity-Redu tion Based onthe GDe omposition . . . 106

3.4.1 TheG De omposition . . . 106

3.4.2 MAP SymbolDete tion Based ontheGDe omposition 108 3.5 Complexity-Redu tion Based onthe MADe omposition . . . . 109

3.5.1 TheMA De omposition . . . 109

3.5.2 MAP SymbolDete tion Based ontheMA De omposition111 3.6 Redu ed-Sear h Algorithms . . . 113

3.6.1 Rationale . . . 113

3.6.2 Optimization . . . 114

3.6.3 Appli ation to theTrellis of MMDete tor . . . 117

3.7 Numeri alResults . . . 118

3.7.1 Un oded CPMTransmissions . . . 119

3.7.2 Iterative De odingof SCCPM S hemes . . . 123

4 CPM Dete tion in the Presen e of Phase Noise 135 4.1 Introdu tion . . . 135

4.2 ChannelModelandIdeal Coherent MAPSymbol Dete tion . . 137

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4.3 Non-Bayesian Algorithms . . . 139

4.3.1 Rationale . . . 139

4.3.2 ProposedAlgorithm . . . 140

4.4 Bayesian Algorithms . . . 145

4.4.1 Derivation oftheAlgorithms . . . 145

4.4.2 ProposedAlgorithm . . . 150

4.4.3 Extension oftheAlgorithms . . . 154

4.4.4 Bayesian Algorithms basedon double-AR1Phase Noise Model . . . 155

4.5 Numeri alresults . . . 162

4.5.1 Simulationenvironment . . . 162

4.5.2 Channels ae tedbyWiener phasenoise . . . 164

4.5.3 Channels ae tedbytheSATMODEphase noise . . . . 167

4.5.4 SATMODEphasenoise ee tsat large spe trale ien y 172 5 Multi arrier s hemes over doubly-sele tive hannels 175 5.1 Introdu tion . . . 176

5.2 System Model . . . 178

5.2.1 UniformFilter-Bank . . . 183

5.2.2 Symbol-by-SymbolDete tion . . . 184

5.3 DFT-OFDM . . . 186

5.3.1 DFT-DMT . . . 188

5.3.2 DFT-PS . . . 189

5.4 DCT-OFDM . . . 190

5.4.1 DCT-MAN . . . 192

5.4.2 DCT-PS . . . 193

5.5 DTT-OFDM . . . 193

5.6 ODTT-OFDM . . . 195

5.6.1 Wilson BaseDerivation . . . 196

5.6.2 Modulator andDemodulator Derivation . . . 197

5.6.3 ODTT-PS . . . 200

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5.7 Signal Spe trumand Bandwidth Computation. . . 201

Con lusions 211

Bibliography 215

A knowledgements 227

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1.1 Blo kdiagram ofCPMmodulator . . . 10

2.1 Fa tor graph of the optimal MAP symbol dete torin the

n

-th

timeinterval. . . 24

2.2 SATMODEPSD of the ontinuous-time phasepro ess

θ(t)

. . . 29

2.3 SATMODEPSDmaskandPSDsobtainedfromvariousdouble-

AR1 approximations. . . 31

2.4 PSDsof slow andfastAR1 obtained withIA parameters. . . . 32

2.5 Timesnapshot ofthetwo phasenoise AR1 omponents, gener-

ateda ordingto the IA parameters. . . 33

2.6 Auto orrelationfun tionofthe ontinuous-timePNpro ess

θ(t)

and ofthe symbol-time PNpro ess

ψ n

at

256

kBaud.. . . . . . 35

2.7 Fa tor graphofDP-BCJR inthe

n

-thtime-interval. . . 40 2.8 FGof Double-DP-BCJR dete torinthe

n

-thtime-interval. . . 42 2.9 FGof I-DP-BCJRdete torinthe

n

-thtime-interval. . . 44 2.10 Information rate for a binary modulation with

2

-RC

h = 1/3

and D-DPdete tor. . . 46

2.11 Information rate for a binary modulation with

2

-RC

h = 1/3

and DPand I-DPdete tors. . . 48

2.12 Information rate for a binary modulation with

2

-RC

h = 1/3

:

omparison between D-DPand DP dete tors. . . 50

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2.13 Information rate for a quaternary modulation with

2

-RC

h = 1/5

and D-DPdete tor. . . . . . . . . . . . . . . . . . . . . . . 51

2.14 Information rate for a quaternary modulation with

2

-RC

h = 1/5

and DPand I-DPdete tors. . . . . . . . . . . . . . . . . . 52

2.15 Information rate for a quaternary modulation with

2

-RC

h = 1/5

: omparison between D-DP and DP dete tors. . . . . . . . 54

2.16 Information rate for a quaternary modulation with

2

-RC

h = 1/5

and DPand I-DPdete tors at

256

kBaud. . . . . . . . . . 55

2.17 Information rate for a quaternary modulation with

2

-RC

h = 1/5

and DPand I-DPdete tors at

2048

kBaud. . . . . . . . . 56

2.18 Interferen edue toadja ent hannels inFDMsystemswithdif-

ferent

bandwidthassumptions. . . 69

2.19 Interferen e whenpowerand information rateof interferers are

doubledrespe tto theCPMsignalof interest. . . 70

2.20 Spe tral e ien y for abinary2-RC with

h = 1/3

. . . . . . . . 72

2.21 PSDoftheCPMsignalwithi.u.d.inputsandwiththeoptimized

1st- and2nd-order Markov inputsdes ribedinTable2.4.. . . . 74

2.22 Spe tral PowerCon entrationfor a binary2-RCwith

h = 1/3

. 75

2.23 Auto orrelation fun tion of the CPM transmitted signal for a

binary2-RCwith

h = 1/3

.. . . . . . . . . . . . . . . . . . . . . 77

2.24 Spe tral e ien y for aquaternary3-RC with

h = 2/7

. . . . . 78

2.25 PSD and SPC of a CPM signal withi.u.d. input and with the

optimized 1st-order Markovinputdes ribed inTable2.6.. . . . 80

2.26 Auto orrelation fun tion of the CPM transmitted signal for a

quaternary3-RCwith

h = 2/7

. . . . . . . . . . . . . . . . . . . 81

2.27 Spe tral e ien y for ano tal2-RC with

h = 1/7

. . . . . . . . 82

2.28 PSD and SPC of a CPM signal withi.u.d. input and with the

optimized 1st-order Markovinputdes ribed inTable2.8.. . . . 85

2.29 Auto orrelation fun tionof the CPM transmitted signalfor an

o tal 2-RCwith

h = 1/7

. . . . . . . . . . . . . . . . . . . . . . 86

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2.30 Spe tral e ien y for a binary2-RCwith

h = 1/3

. Brute-for e

optimization w.r. t.the99.5% bandwidth. . . 88

2.31 Spe trale ien yforaquaternary2-RECwith

h = 1/4

.Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 89

2.32 Spe tral e ien y for a quaternary 3-RCwith

h = 2/7

.Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 90

2.33 Information rate for a binary 2-RC with

h = 1/3

. Brute-for e optimization w.r. t.the99.5% bandwidth. . . 90

2.34 Information ratefor a quaternary 2-REC with

h = 1/4

.Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 91

2.35 Information rate for a quaternary 3-RC with

h = 2/7

. Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 91

3.1 CPMdete torblo kdiagram. . . 97

3.2 Fa tor graph orrespondingto eqn. (3.15), in the

n

-thtimein- terval. . . 102

3.3 Signal pulsesfor a3RCmodulation with

h = 2/7

and

M = 4

. . 104

3.4 Fa tor graph orresponding to MMbased algorithm withsome sele tedse ondary pulses, inthe

n

-thtimeinterval. . . . . . . . 105

3.5 2RC modulationwith

h = 1/4

and

M = 4

.. . . . . . . . . . . . 121

3.6 2RC modulationwith

h = 1/7

and

M = 8

.. . . . . . . . . . . . 121

3.7 3RC modulationwith

h = 2/7

and

M = 4

.. . . . . . . . . . . . 122

3.8 3RC modulationwith

h = 2/7

and

M = 4

.. . . . . . . . . . . . 122

3.9 Transmitter and re eiver stru ture for the onsidered SCCPM s hemes. . . 124

3.10 2RCmodulationwith

h = 1/4

and

M = 4

.Comparisonbetween symbol andbit interleaver.. . . 126

3.11 2RC modulation with

h = 1/4

and

M = 4

: FCalgorithm with G-basedFE andDT-NZalgorithm. . . 127

3.12 2RC modulation with

h = 1/4

and

M = 4

: FCalgorithm with MA-based algorithm. . . 128

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3.13 3RCmodulationwith

h = 2/7

and

M = 4

:performan eof MM and DT-NZalgorithms. . . 131

3.14 3RC modulation with

h = 2/7

and

M = 4

:performan e ofFC algorithm withG-based andMA-based FE. . . 132

4.1 Fa tor graph orrespondingto eqn. (4.22). . . 148

4.2 MSKmodulation, odeword of

2000

bits.. . . . . . . . . . . . . 165

4.3 2RCmodulationwith

h = 2/7

and

M = 4

, odewordof

2000

bits.166

4.4 2RCmodulationwith

h = 1/7

and

M = 8

, odewordof

1760

bits.169

4.5 3RCmodulationwith

h = 2/7

and

M = 4

, odewordof

1760

bits.170

4.6 3RCmodulationwith

h = 2/7

and

M = 4

, odewordof

3520

bits.170

4.7 3RCmodulationwith

h = 2/7

and

M = 4

, odewordof

1760

bits.171

4.8 2RCmodulationwith

h = 1/5

and

M = 4

and oderate

r = 0.88

.172

5.1 Ideal ontinuous-time lter-bank system. . . 179

5.2 Pra ti al oversampled dis rete-timelter-bank system. . . 181

5.3 Systemmodelde omposedintothreeblo ks:themodulator,the

hannel and thedemodulator. . . 182

5.4 PSD ofthetransmitted signal

x(t)

for DFT-OFDM. . . . . . . 204

5.5 PSD ofthe transmitted signal

x(t)

for ODTT-OFDM. . . . . . 208

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CPM ontinuous phasemodulation

OFDM orthogonal frequen y-divisionmultiplexing

AWGN additive whiteGaussian noise

PN phasenoise

SCCPM serially- on atenated ontinuous phasemodulation

MAP maximumaposterioriprobability

APP aposterioriprobability

SISO soft-input soft-output

BER biterror rate

ICI inter- arrier interferen e

CIR hannel impulse response

DFT dis rete Fourier transform

DCT dis rete osine transform

DST dis rete sinetransform

DTT dis retetrigonometri transform

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ASK amplitude shift keying

CPE ontinuousphase en oder

pmf probabilitymassfun tion

SISO soft-input,soft-output

pdf probabilitydensity fun tion

BCJR Bahl,Co ke,Jelinek, Raviv

FSM nite-state ma hine

LLR log-likelihood ratio

FG fa tor graphs

SPA sumprodu talgorithm

QAM quadrature-amplitude modulations

PSK phase-shiftkeying

DVB-RCS digital video broad asting-return hannelsatellite

IR information rate

PSD powerspe traldensity

AR1 1storderauto-regressive

SPC spe tralpower on entration

CT ontinuous-time

ST symbol-time

ISI inter-symbol interferen e

ACF auto orrelationfun tion

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ML maximum-likelihood

RC raised- osine

i.u.d. independent and uniformlydistributed

SIR symmetri information rate

i.a.o.d. independent and asymptoti allyoptimally distributed

FM frequen ymodulation

FDM frequen y divisionmultiplexing

CC onvolutional ode

REC re tangular

MM Mengali andMorelli

G Green

PC prin ipal omponents

MA Moqvistand Aulin

FC full- omplexity

FE front end stage

DA dete tion algorithm

DT double-trellis

FT forward-trellis

NZ non-zero

MDT modieddouble-trellis

SER symbolerrorrate

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PLL phase-lo ked loops

GA genie-aided

LTI lineartime-invariant

DMT dis rete multitone

PS pulseshape

IBI inter-blo kinterferen e

MSE meansquareerror

CP y li prex

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Thisthesispresentstheresultsoftheresear ha tivityIhave arriedoutduring

thethree-yearperiod spent asa PhD.student at theDepartment ofInforma-

tionEngineeringoftheUniversityofParma.Mywork on ernsthestudyoftwo

dierentmodulations hemes,parti ularlysuitableforwireless ommuni ations

systems,namely ontinuousphase modulations(CPM) andmulti arrier mod-

ulations hemes.Indetail,Ifa ethemajorproblems inthedesignofpra ti al

systems employing CPM signals (i.e., the large re eiver omplexity and the

sensitiveness to arriersyn hronization) and I ope withtheproblem ofspe -

trale ien y evaluationandmaximization. Spe ial fo usisdevotedtotypi al

satellite hannels. Se ondly,Iaddresstheissuestemmingfrom themost riti-

aldrawba kofstandardorthogonalfrequen y-divisionmultiplexing(OFDM),

i.e., the in reased sensitivityto the hannel impulse response timevariations.

Hen e, inorder to perform reliable digitaltransmissions overdoubly-sele tive

hannels, I resort to the derivation of some alternative multi arrier modula-

tion formats, based on low- omplexity dis rete-time implementation s hemes

andinspiredbythetight onne tionbetweenmulti arrier modulation andthe

Gabor ommuni ation theory.

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Introdu tion

1.1 Ba kground and Obje tives

Wireless ommuni ations is the fastest growing se tor of the ommuni ation

industry.Inparti ular,su hagrowth on ernsbothwirelessterrestrial ommu-

ni ations (representedfor example by ellular phones,wireless lo alnetworks,

and wirelesssensornetworks) and satellite ommuni ations (whi h areproba-

blythemajor omponentofthewireless ommuni ationsinfrastru ture).How-

ever, many te hni al hallenges remainin designinglow ost, spe tral/energy

e ient wirelesssystems,inorderto be ompetitivewithrespe ttothe orre-

spondingwireline ounterparts.

Continuous phasemodulations (CPMs)and multi arrier s hemes, aretwo

modulation s hemes whi h seem to be very suitable for wireless ommuni-

ations. In parti ular, CPM is a wide lass of modulations hara terized by

ontinuous phase and onstant envelope. Thanks to the onstant envelope,

CPMs do not require power ampliers working in the linear region, but low

ost ampliers working in the saturation region an be employed. Moreover,

sin e the phase is ontinuous and onstrained to follow some well stru tured

variations, this lass of modulations shows very good bandwidth o upan y.

Hen e,CPMsignalshavere eivedanenthusiasti interestbytheinternational

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resear h ommunity in the past, but have so far been employed in a very

limited number of appli ations only.Among these, a CPMsignalling s heme,

the GMSK (Gaussian Minimum Shift Keying) modulation as been su ess-

fullyusedbytheGSMstandard.Morespe tral/energy e ient CPMs hemes

have found no ommer ial appli ations inthepast. Thereason is that CPMs

a hievegoodspe tral/energye ien y atthepri e ofan in reased omplexity

with respe t to linear modulations. In fa t, the main drawba k of the phase

ontinuity istheintrodu tion ofamemory inthemodulator anddemodulator

stages.Inparti ular,moree ient CPMs hemes,implyalarge omplexityin

the dete tionstage. The other major drawba k ofCPM signals,is thestrong

sensitivity to arrier syn hronization.

In the following work, rst of all we analyze the CPM signal from an in-

formationtheoreti point ofview.WeemploythemethodproposedbyArnold

and Loeliger to ompute the information rate of CPMs over Additive White

Gaussian Noise (AWGN) hannel and over hannels ae ted by phase noise

(PN). In parti ular, we onsider a Wiener PN pro ess and also a more pra -

ti al phase noise pro ess, typi al of some satellite real hannels (SATMODE

PN).Moreover,sin edespitethegoodspe tralpropertiesofCPMs,linearmod-

ulation oera mu h better e ien y,espe ially at medium and highspe tral

e ien ies, we propose a spe tral e ien y maximization method, where we

modify the input probability distribution. We restri t our sear h to Markov

inputs ofa givenorder andwe denote themaximumfound spe trale ien y,

asMarkov apa ity.

Se ondly,inordertoover omeoneofthemainCPMdisadvantages,wefa e

theproblem ofthe design of redu ed- omplexitys hemes for CPM signals.In

parti ular, we onsider serially- on atenated CPM(SCCPM) s hemes, whi h

admit a low- omplexity re eiver based on theserial on atenation of a CPM

modulatorwithanoutererror orre ting ode.Thesere eiversareparti ularly

interesting sin ethey ana hievepra ti allythesameperforman eofanopti-

mal re eiver, inalow omplexity way. Theoverallre eiver omplexitymainly

depends on that of theCPM dete tor and hen e, we fo us on the derivation

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of redu ed- omplexity CPM dete tors. In parti ular, we resort to dete tion

algorithms derived from the maximum a posteriori probability (MAP) sym-

bol dete tion strategy, sin e it allows us to obtain soft-de ision algorithms,

ne essaryinaSCCPMs heme.We onsidertwoalternativeapproa hesfor al-

gorithmderivation:onebasedonsomealternativerepresentationsoftheCPM

signal,andthe otherbasedonte hniquesfortrellis omplexityredu tion. The

ombination of the two approa hes is also investigated. We will show that,

for allthe onsidered CPMformats, at leastonelow- omplexity re eiverwith

optimal performan e, exists.

Then, we address the other CPM main drawba k (i.e., the sensitiveness

to ina urate arrier syn hronization) by deriving redu ed- omplexity soft-

input soft-output (SISO) dete tion algorithms suitable for iterative dete -

tion/de odinginthepresen eofPN.Inparti ular,PNestimationis arriedout

jointly to the dete tion stage. In this ase, we onsider two approa hes while

deriving thealgorithms: anon-Bayesian approa h,whi hdoesnotrequireany

assumptiononthestatisti alpropertiesofthephasenoise,sin ethePNissim-

ply onsideredasasequen eofunknownparameters tobeproperlyestimated,

and the Bayesian approa h,whi h onsistsofassuming a properprobabilisti

modelforthePN,andtoexploititfordete tionalgorithmderivation.Inparti -

ular, we proposeBayesian algorithms assumingboth aWiener PNmodel and

a statisti al model we have derived to des ribe SATMODE PN.We ompare

all the derived algorithms on hannels ae ted by the two PN and we relate

biterrorrate(BER)results withinformationrateresultspreviously obtained.

The other onsidered s enariois represented by multi arrier s hemes, em-

ployed in digital transmissions over doubly-sele tive hannels. We start by

onsidering orthogonal frequen y-division multiplexing (OFDM), whi h is an

e ientmodulations hemebelongingtothewide lassofmulti arriermodula-

tions. OFDMis today well understood and largely implemented interrestrial

networks as a mean to provide good spe tral and energy e ien y over fre-

quen y sele tive hannels. Example are both wireline appli ations (as in the

digital subs riber line (DSL) standards) as well as a wide range of wireless

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appli ations, ranging from the digital audio and video broad asting (DAB-

T, DVB-T, DVB-SH, DVB-H) standards, to the lo al and metropolitan area

networks (WLANs and WMANs). The hannel hara terizing these servi es

is frequen y sele tive, for whi h OFDMis parti ularly suitable, sin e OFDM

de omposeslineartime-invariant hannelsintoasetoforthogonal,interferen e-

freesub- hannels. However,themainOFDMdisadvantage isthestrong sensi-

tivity to the hannel impulse response (CIR) time-variations: inthe presen e

of a rapidly time-varying CIR, the orthogonality between the sub- arriers is

destroyedandinter- arrier interferen e (ICI)appears.Insu ha ase,wehave

two viablesolutions:

ˆ to derive re eivers with memory, able to ope with the interferen e or

omplex equalization te hniques;

ˆ the design of modulation formats su h as to redu e the interferen e

(rather than to opewith it).

Weresorttothese ondapproa h,byderivingmulti arriermodulations hemes

alternativetothe OFDM,toredu ethesensitivityto time-variations,inorder

to employthese modulations ondoubly-sele tive hannels

In parti ular, starting from a general lter-bank system, we propose an

oversampled dis rete-timesystemmodelinorderto getapra ti al implemen-

tation of various multi arrier modulation formats in realisti ommuni ation

systems.Weshowthatallmulti arriermodulationformats anbederivedfrom

su h a dis rete-time model, with a general prototype lter (rather than with

the lassi al re tangular lter) and a general timeand frequen y spa ing be-

tweenthe odedsymbols.Inotherwords,weapplypulseshapete hniquestoall

s hemes, extending the te hniquesalready proposedin theliterature. Finally,

inspired by the tight onne tion between multi arrier modulations and the

Gabor ommuni ation theory, we onsider the Wilson base, whi h is a lever

way to design well-lo alized and orthogonal frames in the windowed Fourier

framework, and we derive a novel pra ti al multi arrier modulation s heme,

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verypromisingondoublysele tive hannels.Itisimportanttoremarkthatfor

all modulations hemes we will derive,a fastimplementation exists.

1.2 Continuous Phase Modulation Signals

The omplexenvelope of aCPMsignal an be written as[1℄

s(t; x) = r 2E S

T exp (

j2πh

N −1 X

n=0

x n q(t − nT ) )

(1.1)

where

E S

istheenergy perinformation symbol,

T

isthesymbolinterval,

h

is

themodulationindex,

N

isthenumberofinformationsymbols,

x = {x n } N −1 n=0

is

theinformationsequen e,and

q(t)

isthephaseresponse, onstrainedtobesu h that

q(t) =

( 0 t ≤ 0

1

2 t ≥ LT ,

(1.2)

L

being the orrelation length. Several examples of ommonly used phase re- sponses arereportedin[1℄.

Weassume thatthemodulation index an bewritten as

h = r/p

(where

r

and

p

are relatively prime integers), and thattheinformation symbolsbelong to the

M

-ary alphabet

{±1, ±3, . . . , ±(M − 1)}

,

M

being a power of two.

In this ase, it an be shown [2℄ that the CPM signal in the generi time

interval

[nT, (n + 1)T ]

isgiven by

s(t; x) =

r 2E S T exp

( j2πh

L−1 X

l=0

x n−l q(t − (n − l)T ) )

exp (

jπh

n−L X

l=0

x l )

(1.3)

and itis ompletely dened bythe a tual symbol

x n

,the orrelative state

ω n , (x n−1 , x n−2 , . . . , x n−L+1 ) = x n−1 n−L+1

(1.4)

(whereas

x n−1 n−L+1

isave tor olle tingsymbolsfromthetimeinterval

n −L+1

to

n − 1

) andthephase state

π n

π n ,

"

πh

n−L X

l=0

x l

#

(1.5)

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(where

[·]

denotesthemodulo

 operator)whi h anbere ursivelydened as

π n = [π n−1 + π h x n−L ] .

(1.6)

For the initializationof there ursion (1.6),we willalwaysadopt thefollowing

onventions

π 0 = 0

(1.7)

x n = 0 ∀n < 0 .

(1.8)

Atanygiventimeepo h

n

,the orrelativestate

ω n

anassume

M L−1

dierent

values,whilethephasestate

π n

anassume

p

dierentvalues[2℄.Inparti ular, ifthenumerator

r

ofthemodulationindex

h

iseven,thephasestate anassume

thefollowing

p

values

π n

 0, π r

p , π2 r

p , . . . , π(p − 1) r p



(1.9)

while it an assumethefollowing

2p

possible values when

r

isodd:

π n

 0, π r

p , 2π r

p , . . . , π 2(p − 1) r p



.

(1.10)

However, inthelatter ase, whenthe time interval

n

isodd

π n

belongsto the

alphabet

A o = {2πhm, m = 0, 1, . . . , p − 1}

,while, when

n

is even, it belongs

to the alphabet

A e = {j2πh(m + 1/2), m = 0, 1, . . . , p − 1}

.

It isalsopossibleto provide a timeinvariant CPMstate representation by

onsidering the following symbolmapping

x n = 2¯ x n − (M − 1)

(1.11)

where

x ¯ n

isanintegerrepresentationoftheASKsymbols

x n

,

x ¯ n ∈ {0, 1, . . . , M−

1}

. We an also adopt the alternative integer representation

φ n

for thephase

state

π n

φ n ,

" n−L X

l=0

¯ x l

#

p

(1.12)

(25)

where

φ n ∈ {0, 1, . . . , p − 1}

independently of thetimeinterval andthere ur- sion (1.6) an berepla ed by

φ n = [φ n−1 + ¯ x n ] p .

(1.13)

The phase state dependen e on the time index

n

appears in the relation be-

tween the two phasestate denitions:

π n =

"

πh

n−L X

l=0

x l

#

=

"

2πh

n−L X

l=0

¯

x l − πh(M − 1)(n − L + 1)

#

= [2πhφ n − πh(M − 1)(n − L + 1)] 2π

(1.14)

where we have substituteddenitions(1.12) and (1.11) in(1.5).

Wenowre onsidertotheexpression(1.3)fortheCPMsignalinthegeneri

time interval

[nT, (n + 1)T ]

;wenote thatit an be expressedas

s(t; x) = es(t − nT ; x n , σ n )

(1.15)

where

e

s(t; x n , σ n ) , ¯ s(t; x n , ω n ) e n ∀t ∈ [0, T )

(1.16)

and

¯

s(t; x n , ω n ) ,

r 2E S T exp

( j2πh

X L l=0

x n−l q(t + lT ) )

(1.17)

and

σ n , (ω n , π n ) .

(1.18)

Hen e, we an onsider the CPM waveform (1.16) as the output of a nite-

statema hinewithinput

x n

andstate

σ n

dened in(1.18),whi h an assumes

pM L−1

values. In other words, as stated in [2℄, the CPM modulator an be

de omposed as a ontinuous-phase en oder (CPE) followed by a memoryless

modulator, so thaten oding operation an be studied independently of the

(26)

CPE Modulator Memoryless

x n (x n , σ n ) s(t; x e n , σ n )

Figure 1.1: Blo kdiagram of CPMmodulator de omposedinto a ontinuous-

phaseen oderand a memorylessmodulator

modulation.Moreover,thankstotheassumedstatedenition,theCPEresults

time-invariant andlinearmodulo-

p

and hen eit an be onsideredasa sortof

onvolutional en oder. Su h a s heme is represented in Fig. 1.1: the symbols

x n

are passed through a nite-state ma hine (the linear CPE), and then a time variant memoryless modulator generates the CPM waveforms (1.16) as

a fun tion of

(x n , σ n )

. Finally, su h a de omposition is very useful to show howaCPMmodulatorisespe iallysuitableto be on atenatedwithan outer

onvolutional en oder.For thisreason,thes ienti literaturehasreserved an

in reasing interest in SCCPM s hemes des ribed in [3 5℄, and based on low

omplexityiterative joint dete tion/de oding(see Se tion 1.3.3).

1.3 General Frameworks

In the following Se tion we des ribe some general frameworkswe will employ

inthe remainder ofthis work.

1.3.1 MAP Symbol Dete tion Strategy and BCJR Algorithm

Given a sequen e of transmitted symbols

x n

olle ted into ve tor

x

, where

x = (x 0 , x 1 , . . . , x N −1 )

anda hannel withmemory,we denoted bytheve tor

y

the su ient statisti sof there eived signal

r(t)

,extra tedbythere eiver.

Inparti ular,the

n

-thelement oftheve tor

y

anbeave tor,denotedinthe

followingby

y n

,sin e ingeneral,atea htimeepo h

n

thenumberofsu ient

statisti s an be greater than one. Thus, theMAP symboldete tion strategy

(27)

minimizing theaverage symbolerror probabilityis

ˆ

x n = argmax

x n

P (x n |y)

(1.19)

while MAPsequen e dete tionstrategy is

ˆ

x = argmax

x P (x|y)

(1.20)

where we denote bythe apitolletter

P (.)

a probabilitymassfun tion (pmf).

Inthefollowing work,wealways employMAP symboldete tion, sin e itpro-

videssoft-outputde isions,whileMAPsequen edoesn't.Indetail,MAPsym-

boldete tionstrategy omputes,asaby-produ t,theaposterioriprobabilities

P (x n |y)

whi h anbe onsideredasreliabilityestimatesonthe hosensymbols

ˆ

x n

; these estimates allow us to derive Soft-Input, Soft-Output (SISO) dete - tion (orde oding) algorithms, ne essaryinorder to implement iterative joint

dete tion/de oding s hemes.

In parti ular, by employing the Bayes rule, we an express MAP symbol

strategy in(1.19) as

ˆ

x n = argmax

x n p(y|x n )P (x n )

(1.21)

where

P (x n )

is the a priori probability of the symbol

x n

and we denote by

p(.)

aprobabilitydensityfun tion(pdf).Thus,inordertosatisfytheproposed maximizationalgorithm,weneed to omputethepdf

p(y|x n )

.Ifwe onsidera

hannelwithmemory,whi h anbedes ribedasanite-statema hine(FSM),

whose state is denoted by

σ n

,we an solve the MAP symbolproblem bythe

Bahl, Co ke, Jelinek, Raviv (BCJR) algorithm [6℄, based on a probabilisti

derivation.Inparti ular,

p(y|x n )

expressionis given by

p(y|x n ) = X

σ n

η f,nn ) η b,n+1n+1 ) F n (x n , σ n )

(1.22)

where

ˆ the fun tion

F n (x n , σ n )

is

F n (x n , σ n ) , p(y n |x n , σ n )

(1.23)

(28)

where

y n

is the ve tor olle ting all the su ient statisti s at the time

epo h

n

.

ˆ

η f,nn )

is alledforward metri and isdened as

η f,nn ) , p(y n−1 0 |σ n )P (x n )

(1.24)

wherewedenoteby

y n n 2 1

theve tor olle tingallstatisti s

y n

from

n = n 1

to

n = n 2

.

ˆ

η b,n+1 (σ n+1 )

is alledba kward metri and hasexpression

η b,n+1n+1 ) , p(y N −1 n+1 |σ n+1 ) .

(1.25)

Forward and ba kward metri s an be re ursively omputed through the fol-

lowing forward andba kward re ursions

η f,n+1n+1 ) = X

x n

X

σ n

η f,nn ) F n (x n , σ n ) P (x n )

(1.26)

η b,nn ) = X

x n

X

σ n+1

η b,n+1n+1 ) F n (x n , σ n ) P (x n ) .

(1.27)

Hen e theBCJR algorithm worksinthe following way:

ˆ rst,forward andba kward metri s

η f,nn )

and

η b,nn )

are omputed

by means of (1.24) and (1.25),for ea h timeepo h

n

and for ea hstate

value

σ n

;

ˆ then, the pdf

p(y|x n )

is derived by (1.22) exploiting

η f,nn )

,

η b,nn )

and

F n (x n , σ n )

values;

ˆ nally, the MAP strategy (1.21) an be implementedby omputing the

aposterioriprobabilities

P (x n |y)

.

We refer to[6℄ for more detailonthe BCJRalgorithm derivation.

Typi ally, when symbols

{x n }

are generated from an

M

-ary alphabet, we

hoosetheset

{ℓ a,n } a

of

M −1

logarithmi ratiosoftheaposterioriprobabilities

(29)

P (x n |y)

, asthe reliability estimates of the de ision on the symbol

x n

.

ℓ a,n

is

dened as

a,n = log P (x n = a|y)

P (x n = 0|y)

(1.28)

where

a ∈ {1, 2, . . . , M − 1}

.

a,n

isdenoted aslog-likelihood ratio (LLR).

1.3.2 Fa tor Graphs and Sum Produ t Algorithm

TheBCJRalgorithmdes ribedinSe tion1.3.1,solvestheMAPsymboldete -

tion problem. It an be alternatively derived by means of the Fa tor Graphs

(FG) and the Sum Produ t Algorithm (SPA), presented in [7℄. These tools

areparti ularly suited to nd the way a ompli ated global fun tion of many

variablesfa torsasa produ tof lo al fun tions, ea hof whi hdependson a

subsetofthevariables.Thisfa torization anbevisualizedwithaFG,whi his

abipartitegraphthatindi ateswhi hvariableisargumentofwhi hlo alfun -

tion. TheSPAworkson theFGand omputes themarginalfun tionsderived

fromthe globalfun tion.

Let

x = (x 1 , x 2 , . . . , x n )

be a olle tion of variables, where

x i

takes on

valuesonsome(usuallynite)domain

A i

,andlet

f (x)

amultivariatefun tion.

Suppose that

f (x)

fa tors into a produ t of several lo al fun tions

f j

, ea h

havinga subset

x j

of

x

,asargument:

f (x) = Y

j∈J

f j (x j )

(1.29)

where

J

is a dis rete index set. A fa tor graph isa bipartite graph whi h has

a variable node for ea h variable

x i

, a fa tor node for ea h fun tion

f j

, and

an edge onne ting variable node

x i

to fun tion node

f j

if and only if

x i

is

an argument of

f j

. The SPA is dened by the omputation rules at variable and at fa tor nodes and by a suitable node a tivation s hedule. Denoting by

µ x i →f j (x i )

a messagesent from the variablenode

x i

to thefa tor node

f j

,by

µ f j →x i (x i )

a messageintheoppositedire tion,and by

B i

theset offun tions

f j

having

x i

as argument, the message omputations performed at variable

(30)

and fa tornodesare, respe tively [7℄

µ x i →f j (x i ) = Y

h∈ B i \f j

µ h→x i (x i )

(1.30)

µ f j →x i (x i ) = X

∼{x i }

f j ({y ∈ B j }) Y

y∈B j \x i

µ y→f j (y)

(1.31)

where we indi ate by

P

∼{x i }

thesummary operator, i.e., asum overall vari-

ables ex luding

x i

.

Thus, we an be fa torize the pmf

P (x|y)

in order to nd, through the

SPA, the marginal a posteriori probabilities

P (x i |y)

, required by the MAP

symbol strategy (1.19). If the FG has y les, the SPA is inherently iterative

and onvergen etotheexa tmarginalpmfisnotguaranteed.Nevertheless,for

many relevant problems hara terized by FGs with y le, theSPA was found

toprovideverygoodresultsandthereforeitrepresentsaviablesolutiontothe

approximated marginalization of multivariate pmfs when exa t al ulation is

not feasible be ause of omplexity.

Finally, we dene the message-passing s hedule inthe SPA as thespe i-

ation of the order in whi h messages are updated. In general, espe ially for

graphs with y les,the so- alled ooding s hedule isadopted [8℄:inea h iter-

ation,allvariablenodesandsubsequently all fa tornodes,passnewmessages

to their neighbors.

1.3.3 Iterative Joint Dete tion/De oding S hemes

Whenwe onsider a ommuni ationsystem hara terized byan error orre t-

ing ode together with a hannel with memory, the ardinality of the over-

all system state an be very large, and thus, optimal MAP symbol dete tion

strategy at the re eiverbe omesunfeasible.In these ases,we an resort to a

suboptimaliterativejointdete tion/de odings heme,whi hexhibits omputa-

tional omplexityabsolutelylowerthanthe omplexityoftheoptimals heme,

but a performan e whi h tends to the optimal one (as veried by numeri al

(31)

results)[9℄.Inparti ular,herewedes ribetheoperationsofaserially on ate-

nateds heme, whi h is thes heme adoptedinChapters 3 and4 for dete tion

of CPMsignalinthe presen eof anouter error orre ting ode.

In an iterative on atenated joint dete tion/de oding s heme, ea h om-

ponent blo k (i.e.,the dete tor and the de oder) works separately, by imple-

mentingthe MAPsymbolstrategyoptimal forthesingleblo k,assumingthat

no other memory sour esarepresent inthe system.They employ a dete tion

(de oding)algorithmbasedontheMAPsymbolrule,whi hprovidesreliability

estimatesonthealgorithmde isions (Se tion1.3.1), sin eweneed soft-output

de isions in orderto on atenatethe two blo ks.In general, an iterative on-

atenated s heme is based on the following basi on ept: ea h omponent

blo k exploits the suggestion provided by the other omponent blo k, in or-

der to derive de isions whi h be omes more reliable with the iteration. In

detail, a serially on atenated s heme worksas follow. First of all, thedete -

torperformsan instan eof thedete tionalgorithm, operatingon the hannel

su ient statisti s

y

.Then, thesoft-de isionsitprodu es onea h symbol

x n

,

are forwarded to the de oder, whi h employs the dete tor a posteriori prob-

abilities, as a priori probabilities on

{x n }

while performing de oding. Thus,

it also produ es soft-de isions on the

x n

symbols, whi h are in turn passed

to the dete tor. The dete tor exploits su h reliability estimations as a priori

probabilities on

{x n }

and itstartsanewiterationof theserially on atenated s heme. Thejointdete tion/de odingstrategy ontinuesforaxednumberof

y les;then, hard nalde isions onthe symbols

{x n }

arederived.

In order to a elerate the onvergen e of theiterative dete tion/de oding

pro ess, ea h omponent blo k must re eive as input an information whi h

is not self-produ ed. With this aim,in [10,11℄ the on ept of extrinsi infor-

mation is introdu ed, whi h identies the reliabilityinformation produ ed by

a omponent blo k whi h does not depend on the information it re eived as

input. Indetail, ifwe denote by

a,n

out theLLRsdened in(1.28), produ ed by

a blo k and representing the reliability measure of a MAP symbol algorithm

onthede isiononthesymbol

x n

,theextrinsi information

a,n

e,out generated by

(32)

su h blo k,is given by

a,n

e,out

= ℓ a,n

out

− ℓ a,n

e,in

.

(1.32)

The FG/SPA toolintrinsi ally propagates extrinsi information, asdes ribed

by(1.30)-(1.31).

(33)

Capa ity Evaluation for CPM

signals

2.1 Introdu tion

Continuousphase modulations (CPMs)form a lass of onstant envelope sig-

naling formatsthataree ient inpowerand bandwidth[1℄.Inorder to om-

pare CPMs against linear modulations, e.g., phase-shift keying (PSK) and

quadrature-amplitude modulations(QAMs),aswellastooptimizethevarious

parameters whi h des ribe a CPM signal, namely phase response, alphabet

size,andmodulationindex,information theoreti arguments ouldbeused.In

parti ular, the availabilityof expressions for the hannel apa ity,interms of

bits per hannel use, and bandwidth o upan y of thesignal, would allow to

evaluate theamountof b/s arriedperbandunit (Hz),whi h isanimportant

omparison riterion for system designer. Unfortunately, neither losed-form

results for the apa ity of a CPM signal nor simple expressions for its band-

widthexist.

In[12℄and[13℄ananalyti alstudyoftheinformationtransferoveradditive

white Gaussian noise (AWGN) hannels for envelope onstrained waveforms

under a xed bandwidth denition is des ribed; however the results do not

(34)

seem beimmediately appli ableto theCPMs.Firstresultson CPMs apa ity

arereportedin[14℄,wheretheyarepresentedtojustifythe hooseofa ertain

outer odeinaserially- on atenated CPM(SCCPM)s heme. Theauthors do

not des ribe the method by whi h CPM apa ity is omputed and moreover,

the results they arry out do not mat h with results we will obtain in the

present work.In[15℄ multi- hannel apa ity resultsareapplied to thede om-

position [16℄ of CPM signal; unfortunately that method provides bounds too

looseon CPM apa ityto be onsidered of pra ti alinterest.

Finallyin[17,18℄and[19℄CPM apa ityis omputed byusingthemethod

by Arnold and Loeliger in 2001 [20℄ and in2006 [21℄. By this te hnique, the

information rate (IR) over a hannel with memory an be omputed by the

forward metri of a BCJR algorithm (explained in Se tion 1.3.1) applied at

theoutputofsu ha hannel. Thisworkwaspresentedinthe ontext oflinear

modulationsover hannels withmemory,but anbetransposedtoall hannels

withmemory.Hen e,representingCPMsignalsasasuitablenite-state,nite-

dimensional Markov pro ess (as done in Se tion 1.2), it is possible to derive

theinformation rateof the hannel omposedbythe on atenationof aCPM

modulator andan AWGN hannel. Moreover,in[19℄ theinformation rateof a

CPMsignalis omputedinthepresen eofaphasenoise(PN)pro ess,modeled

as a dis rete time Wiener random pro ess, whi h is a ommonly used model

intele ommuni ation s hemes.

The remainder of this hapteris organized as follows. First ofall, we pro-

vide a more exhaustive des ription of thealgorithm proposed byArnold and

Loeliger and we exploit it to derive the IR apa ity of a CPM signalover an

AWGN hannelandoveranAWGN hannelae tedbyphasenoise,modeledas

adis retetimeWienerpro ess.Then,weresorttoaphasenoise(PN)modelof

pra ti al interest, i.e.,the SATMODEphasenoisemodel,whi himpairs satel-

lite ommuni ations hara terized bylow- ost transmitter and re eiver. SAT-

MODEPN isamodelusually employedto testtheperforman eofDVB-RCS

(Digital Video Broad asting-Return Channel Satellite) systems.In parti ular,

inspe ting the phasenoise powerspe traldensity (PSD) [22℄,we see thatthe

(35)

dis rete-time phase pro ess an be well approximated by the sum of two 1st

orderauto-regressive (AR1)pro esses,andwe derive CPM'sIRinsu ha s e-

nario.Finally,we proposea novel iterative method,similarto thegeneralized

Blahut-Arimoto algorithm re ently proposed by Kav£i [23℄, to evaluate the

Markov apa ityofa CPMsignalover anAWGN hannel. Oneof thenovelty

of our approa h is that we maximize, with respe t to the input distribution,

thespe trale ien yinbps/Hzratherthanthemutualinformationinbitsper

hannel use. We address this problem by taking into a ount the bandwidth

o upan y of the CPM signal by means of the (properly modied) Carson's

rule bandwidth denition, and solving a linearly onstrained non-linear opti-

mization problem. The results show that Markov apa ity obtained with the

proposed input optimization algorithm strongly outperforms the apa ity for

independent and uniformly distributed inputs. In the numeri al results, we

also onsidera properbandwidth denitionrelatedtothespe tralpower on-

entration(SPC),although inthis asetheoptimization an beperformedby

resorting to abrute-for e approa h only.

2.2 Information Rate for Channel with Memory

2.2.1 Information Rate Denition

Mutualinformationrate

I(X; Y )

quantiestheamountofinformationthat an betransmittedovera hannelwithinputpro ess

X

andoutputpro ess

Y

and

it isexpressed inbits per hannel use. In the following work we will fo us on

the asewhereboth

X

and

Y

aredis rete-timestationarysequen es(ingeneral notofthe samelength),denotedwith

x

and

y

,respe tively.Frominformation theory[24,25℄,weknow thatfor ea h hannel

I(x; y)

an be expressedas

I(x; y) = h(x) − h(x|y)

 b/s Hz



(2.1)

(36)

where

h(x)

is thedierential entropy rate of a sour e generating the random dis rete-time stationarypro ess

x

h(x) , −E[log 2 p(x)] = Z +∞

−∞

p(x) log 2 1

p(x) dx

(2.2)

and

h(x|y)

is the onditional dierential entropy rate of the hannel input pro ess

x

given the hannel output

y

h(x|y) , −E[log 2 p(x|y)] =

Z Z +∞

−∞

p(x, y) log 2 1

p(x|y) dx dy

(2.3)

whi h depends only on the hannel hara teristi s (i.e.,the transition proba-

bilities

p(x|y)

). It an be shown that(2.1) isequivalent to

I(x; y) = h(y) − h(y|x)

 b/s Hz



.

(2.4)

2.2.2 Arnold and Loeliger Method

In[20,21,26℄, itisdes ribedamethodto omputethemutualinformation ofa

nite-statehiddenMarkovmodelemployingthe forward re ursionofthewell-

knownBCJRalgorithm.Su hamethod anbeextendedtoall hannelmodels

withan innite numberof states(for example an AWGN hannelae ted by

PN)ndinganauxiliary nite-state hannelwhi h wellapproximates thereal

hannel; hen e the algorithm allows to ompute a lower limit of the a tual

information rate, whi h tends to su h valuewhen thenumberof statesof the

auxiliary hannelgrows towards innity.

We now present the algorithm in [20℄. Given a ertain hannel input se-

quen e

x N 0 , (x 0 , x 1 , . . . , x N )

and the orresponding output sequen e

y N 0 , (y 0 , y 1 , . . . , y N )

,the omputation ofthe dierential entropyrate

h(y)

andthe

onditional dierential entropyrate

h(y|x)

ina simulation an be arriedout

thankstotheShannon-M Millian-Breimanntheorem[25,27℄whi hensuresthe

onvergen e withprobabilityone of

h(y) = − lim

N →+∞

1 N E 

log 2 p(y N 0 ) 

(2.5)

h(y|x) = − lim

N →+∞

1 N E 

log 2 p(y N 0 |x N 0 ) 

(2.6)

(37)

if

x N 0

and

y 0 N

arestationary ergodi nite-state hiddenMarkov pro esses.By repla ing (2.5) and(2.6) in(2.4),weget

I(x; y) = lim

N →+∞

1 N E



log 2 p(y N 0 |x N 0 ) p(y N 0 )



.

(2.7)

Hen e,from(2.7) itis learthatinorderto omputetheinformationrateitis

su ient to obtain thevalues of theprobability density fun tions

p(y N 0 )

and

p(y N 0 |x N 0 )

. Su h values an be ee tively omputed by the forward re ursion

of the BCJR algorithm, employed to implement a maximum a posteriori a

probability(MAP)symboldete tionstrategy.Inparti ular,bydening

σ n

the

hannelstate, theforward message

η f,nn )

isobtained as

η f,n+1n+1 ) = p(y n , y n−1 , ..., y 1 , σ n+1 )

= X

σ n

p(y n 0 , σ n+1 , σ n )

= X

σ n

p(y n |y 0 n−1 , σ n+1 , σ n )p(y n−1 0 , σ n+1 , σ n )

= X

σ n

p(y n |y 0 n−1 , σ n+1 , σ n )p(σ n+1 |σ n )η f,nn )

(2.8)

asshown in [28℄.At the

N

-thtime symbol interval,

p(y N 0 )

is obtained asthe

sumof the state metri s

p(y 0 N ) = X

σ N+1

η f,N +1 (σ N +1 ).

(2.9)

Finally, for evaluating the mean value of

log 2 p(y N 0 )

, we need to repeat the

simulation

K

times and,bydenoting with

ρ n

the

p(y N 0 )

measureat the

n

-th

simulation, we nd

E [log 2 p(y N 0 )] = lim

K→+∞

1 K

X K k=1

log 2k ).

(2.10)

(38)

2.3 IR of CPMs over AWGN

The method des ribed in Se tion 2.2.2 an be applied to ompute the CPM

informationrateforindependentanduniformlydistributedsymbolinput, over

AWGN hannel. In parti ular, in su h a ase, the omplex envelope of the

re eivedsignal an be written as:

r(t) = s(t; x) + w(t)

(2.11)

where

s(t; x)

istheCPMsignal(1.1)and

w(t)

isa omplex-valuedAWGNpro- ess with independent omponents, ea h with two-sided power spe tral den-

sity

N 0

.The value of

N 0

isassumed perfe tly known at the re eiver1 and we

alsoassumeindependent anduniformlydistributedsymbols

x

.Sin ethe han-

nel memory is on entrated in the CPM modulator whi h is des ribed as a

FSM model in Se tion 1.2, the expression for

σ n

is given by (1.18) and the

forward re ursion we need to ompute inthe Arnold and Loeliger method, is

theforward re ursion

η f,nn )

(2.17) oftheMAP symbol dete tionalgorithm

we will des ribe inSe tion 2.3.1.

2.3.1 Optimal MAP Symbol Dete tion

Anoverviewoftheoptimalalgorithm forMAP symboldete tion, whi h turns

out to be an instan e of the well-known BCJR algorithm [6℄, is given in the

following.

At ea h time epo h

n

and for ea h trial symbol

x n

and trial state

σ n

, let

us dene

y n (x n , σ n ) ,

Z (n+1)T

nT r(t)es (t − nT ; x n , σ n )dt

= e −jπ n

Z (n+1)T nT

r(t)¯ s (t − nT ; x n , ω n )dt

(2.12)

1

The onstant amplitude of a CPM signal makesthe dete tion algorithm veryrobust

toapossiblyina urateestimateofthe value of

N 0

[1℄. Hen e,the hypothesisofaperfe t

knowledgeof

N 0

isnot riti al.

(39)

where

e s(t; x n , σ n )

is the CPMwaveform in the interval

[nT, (n + 1)T ]

dened

in(1.16) and orrespondingto thetrialvaluesof

x n

and

σ n

,while

s ¯ (t; x n , ω n )

(dened in (1.17)) is the CPM waveform depending on just a tual symbol

x n

and orrelative state

ω n

. In pra ti e, samples

{y n (x n , σ n )}

are obtained

by sampling, at symbol rate, the output of a bank of lters mat hed to all

waveformswhi h an o ur inasymbolinterval. However, notethat mat hed

lters are

M L

insteadof

pM L

be ause the CPM phase state is onstant over

a symbol interval

[nT, (n + 1)T ]

; hen e we need a mat hed lter for ea h of

M L

ombinations of a tual symbol

x n

and orrelative state

ω n

.The ve tor

y

olle ting these samples gives a set of su ient statisti s for MAP symbol

dete tion [1℄.

For ea h symbol interval

n

,the MAP symbol dete tion strategy provides

thesymbol

x n

whi h satisfythefollowing ondition (Se tion1.3.1)

x n = argmax

ˆ x n

P (ˆ x n |y) .

(2.13)

Theprobabilities

P (ˆ x n |y)

anbeobtainedthroughthewell-knownBCJRalgo-

rithm(seeSe tion 1.3.1).Thisalgorithm anbederivedinaprobabilisti way,

asdone in[6℄,but also thanksto FGand SPA, whi h allow us to marginalize

the probability mass fun tion (pmf)

P (x, σ|y)

withrespe tto ea h variables

x n

,where

x = (x 0 , x 1 , . . . , x N −1 ) σ = (σ 0 , σ 1 , . . . , σ N )

and

y

is the ve tor olle ting all the su ient statisti s

y n (x n , σ n )

in (2.12),

with

n

from

0

to

N

.Inparti ular, we fa torize

P (x, σ|y)

as2

P (x, σ|y) ∝ p(y|x, σ) P (σ|x) P (x)

= P (σ 0 )

N −1 Y

n=0

F n (x n , σ n )T (x n , σ n , σ n+1 )P (x n )

(2.14)

2

Theproportionality symbol

is usedwhen two quantities dierfor apositive multi-

pli ativefa tor,irrelevantforthedete tionpro ess.

(40)

x n

η f,n+1 η b,n

σ n F n T σ n+1

P (x n ) P (x n ) P e (x n )

Figure2.1:Fa torgraphoftheoptimalMAPsymboldete torinthe

n

-thtime

interval.

where

T (x n , σ n , σ n+1 )

is the trellis indi ator fun tion, equal to one if

x n

,

σ n

,

and

σ n+1

satisfy the trellis onstraints and to zero otherwise, and

F n (x n , σ n )

isthe bran hmetri fun tion,dened as

F n (x n , σ n ) = exp

 1

N 0 Re{y n (x n , σ n )}



(2.15)

and where we have assumedindependent information symbols

P (x) =

N −1 Y

n=0

P (x n ).

(2.16)

In(2.14)wehaveintrodu edthefollowingnotation:

P (·)

indi atesapmf,while

p(·)

indi ates aprobability densityfun tion(pdf).

Hen e, we an drawna fa tor graph whose se tionat the

n

-thtimeinter-

val is represented in Fig 2.1, and by applying the SPA we derive the BCJR

algorithm, hara terized bythefollowing forward andba kward re ursions

η f,n+1n+1 ) = X

α n

X

σ n

P (x n )T (x n , σ n , σ n+1 )F n (x n , σ nf,nn )

(2.17)

η b,nn ) = X

α n

X

σ n+1

P (x n )T (x n , σ n , σ n+1 )F n (x n , σ nb,n+1n+1 )

(2.18)

Riferimenti

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