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
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
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
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
5.7 Signal Spe trumand Bandwidth Computation. . . 201
Con lusions 211
Bibliography 215
A knowledgements 227
1.1 Blo kdiagram ofCPMmodulator . . . 10
2.1 Fa tor graph of the optimal MAP symbol dete torin the
n
-thtimeinterval. . . 24
2.2 SATMODEPSD of the ontinuous-time phasepro ess
θ(t)
. . . 292.3 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
at256
kBaud.. . . . . . 352.7 Fa tor graphofDP-BCJR inthe
n
-thtime-interval. . . 40 2.8 FGof Double-DP-BCJR dete torinthen
-thtime-interval. . . 42 2.9 FGof I-DP-BCJRdete torinthen
-thtime-interval. . . 44 2.10 Information rate for a binary modulation with2
-RCh = 1/3
and D-DPdete tor. . . 46
2.11 Information rate for a binary modulation with
2
-RCh = 1/3
and DPand I-DPdete tors. . . 48
2.12 Information rate for a binary modulation with
2
-RCh = 1/3
:omparison between D-DPand DP dete tors. . . 50
2.13 Information rate for a quaternary modulation with
2
-RCh = 1/5
and D-DPdete tor. . . . . . . . . . . . . . . . . . . . . . . 512.14 Information rate for a quaternary modulation with
2
-RCh = 1/5
and DPand I-DPdete tors. . . . . . . . . . . . . . . . . . 522.15 Information rate for a quaternary modulation with
2
-RCh = 1/5
: omparison between D-DP and DP dete tors. . . . . . . . 542.16 Information rate for a quaternary modulation with
2
-RCh = 1/5
and DPand I-DPdete tors at256
kBaud. . . . . . . . . . 552.17 Information rate for a quaternary modulation with
2
-RCh = 1/5
and DPand I-DPdete tors at2048
kBaud. . . . . . . . . 562.18 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
. . . . . . . . 722.21 PSDoftheCPMsignalwithi.u.d.inputsandwiththeoptimized
1st- and2nd-order Markov inputsdes ribedinTable2.4.. . . . 74
2.22 Spe tral PowerCon entrationfor a binary2-RCwith
h = 1/3
. 752.23 Auto orrelation fun tion of the CPM transmitted signal for a
binary2-RCwith
h = 1/3
.. . . . . . . . . . . . . . . . . . . . . 772.24 Spe tral e ien y for aquaternary3-RC with
h = 2/7
. . . . . 782.25 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
. . . . . . . . . . . . . . . . . . . 812.27 Spe tral e ien y for ano tal2-RC with
h = 1/7
. . . . . . . . 822.28 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
. . . . . . . . . . . . . . . . . . . . . . 862.30 Spe tral e ien y for a binary2-RCwith
h = 1/3
. Brute-for eoptimization 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. . . 892.32 Spe tral e ien y for a quaternary 3-RCwith
h = 2/7
.Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 902.33 Information rate for a binary 2-RC with
h = 1/3
. Brute-for e optimization w.r. t.the99.5% bandwidth. . . 902.34 Information ratefor a quaternary 2-REC with
h = 1/4
.Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 912.35 Information rate for a quaternary 3-RC with
h = 2/7
. Brute- for eoptimization w. r.t.the99.5% bandwidth. . . 913.1 CPMdete torblo kdiagram. . . 97
3.2 Fa tor graph orrespondingto eqn. (3.15), in the
n
-thtimein- terval. . . 1023.3 Signal pulsesfor a3RCmodulation with
h = 2/7
andM = 4
. . 1043.4 Fa tor graph orresponding to MMbased algorithm withsome sele tedse ondary pulses, inthe
n
-thtimeinterval. . . . . . . . 1053.5 2RC modulationwith
h = 1/4
andM = 4
.. . . . . . . . . . . . 1213.6 2RC modulationwith
h = 1/7
andM = 8
.. . . . . . . . . . . . 1213.7 3RC modulationwith
h = 2/7
andM = 4
.. . . . . . . . . . . . 1223.8 3RC modulationwith
h = 2/7
andM = 4
.. . . . . . . . . . . . 1223.9 Transmitter and re eiver stru ture for the onsidered SCCPM s hemes. . . 124
3.10 2RCmodulationwith
h = 1/4
andM = 4
.Comparisonbetween symbol andbit interleaver.. . . 1263.11 2RC modulation with
h = 1/4
andM = 4
: FCalgorithm with G-basedFE andDT-NZalgorithm. . . 1273.12 2RC modulation with
h = 1/4
andM = 4
: FCalgorithm with MA-based algorithm. . . 1283.13 3RCmodulationwith
h = 2/7
andM = 4
:performan eof MM and DT-NZalgorithms. . . 1313.14 3RC modulation with
h = 2/7
andM = 4
:performan e ofFC algorithm withG-based andMA-based FE. . . 1324.1 Fa tor graph orrespondingto eqn. (4.22). . . 148
4.2 MSKmodulation, odeword of
2000
bits.. . . . . . . . . . . . . 1654.3 2RCmodulationwith
h = 2/7
andM = 4
, odewordof2000
bits.1664.4 2RCmodulationwith
h = 1/7
andM = 8
, odewordof1760
bits.1694.5 3RCmodulationwith
h = 2/7
andM = 4
, odewordof1760
bits.1704.6 3RCmodulationwith
h = 2/7
andM = 4
, odewordof3520
bits.1704.7 3RCmodulationwith
h = 2/7
andM = 4
, odewordof1760
bits.1714.8 2RCmodulationwith
h = 1/5
andM = 4
and oderater = 0.88
.1725.1 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. . . . . . . 2045.5 PSD ofthe transmitted signal
x(t)
for ODTT-OFDM. . . . . . 208CPM 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
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
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
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
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.
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
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
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
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:
to derive re eivers with memory, able to ope with the interferen e or
omplex equalization te hniques;
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,
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
isthemodulationindex,
N
isthenumberofinformationsymbols,x = {x n } N −1 n=0
istheinformationsequen e,and
q(t)
isthephaseresponse, onstrainedtobesu h thatq(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
(wherer
and
p
are relatively prime integers), and thattheinformation symbolsbelong to theM
-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 bys(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 stateω 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 tingsymbolsfromthetimeintervaln −L+1
to
n − 1
) andthephase stateπ n
π n ,
"
πh
n−L X
l=0
x l
#
2π
(1.5)
(where
[·] 2π
denotesthemodulo2π
operator)whi h anbere ursivelydened asπ n = [π n−1 + π h x n−L ] 2π .
(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 orrelativestateω n
anassumeM L−1
dierentvalues,whilethephasestate
π n
anassumep
dierentvalues[2℄.Inparti ular, ifthenumeratorr
ofthemodulationindexh
iseven,thephasestate anassumethefollowing
p
valuesπ n ∈
0, π r
p , π2 r
p , . . . , π(p − 1) r p
(1.9)
while it an assumethefollowing
2p
possible values whenr
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
isoddπ n
belongsto thealphabet
A o = {2πhm, m = 0, 1, . . . , p − 1}
,while, whenn
is even, it belongsto 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
isanintegerrepresentationoftheASKsymbolsx n
,x ¯ n ∈ {0, 1, . . . , M−
1}
. We an also adopt the alternative integer representationφ n
for thephasestate
π n
φ n ,
" n−L X
l=0
¯ x l
#
p
(1.12)
where
φ n ∈ {0, 1, . . . , p − 1}
independently of thetimeinterval andthere ur- sion (1.6) an berepla ed byφ 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
#
2π
=
"
2πh
n−L X
l=0
¯
x l − πh(M − 1)(n − L + 1)
#
2π
= [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 expressedass(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 jπ 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
andstateσ n
dened in(1.18),whi h an assumespM L−1
values. In other words, as stated in [2℄, the CPM modulator an bede omposed as a ontinuous-phase en oder (CPE) followed by a memoryless
modulator, so thaten oding operation an be studied independently of the
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 sortofonvolutional 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) asa fun tion of
(x n , σ n )
. Finally, su h a de omposition is very useful to show howaCPMmodulatorisespe iallysuitableto be on atenatedwithan outeronvolutional 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 torx
, wherex = (x 0 , x 1 , . . . , x N −1 )
anda hannel withmemory,we denoted bytheve tory
the su ient statisti sof there eived signalr(t)
,extra tedbythere eiver.Inparti ular,the
n
-thelement oftheve tory
anbeave tor,denotedinthefollowingby
y n
,sin e ingeneral,atea htimeepo hn
thenumberofsu ientstatisti s an be greater than one. Thus, theMAP symboldete tion strategy
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 hosensymbolsˆ
x n
; these estimates allow us to derive Soft-Input, Soft-Output (SISO) dete - tion (orde oding) algorithms, ne essaryinorder to implement iterative jointdete 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 symbolx n
and we denote byp(.)
aprobabilitydensityfun tion(pdf).Thus,inordertosatisfytheproposed maximizationalgorithm,weneed to omputethepdfp(y|x n )
.Ifwe onsiderahannelwithmemory,whi h anbedes ribedasanite-statema hine(FSM),
whose state is denoted by
σ n
,we an solve the MAP symbolproblem bytheBahl, Co ke, Jelinek, Raviv (BCJR) algorithm [6℄, based on a probabilisti
derivation.Inparti ular,
p(y|x n )
expressionis given byp(y|x n ) = X
σ n
η f,n (σ n ) η b,n+1 (σ n+1 ) F n (x n , σ n )
(1.22)where
the fun tion
F n (x n , σ n )
isF n (x n , σ n ) , p(y n |x n , σ n )
(1.23)where
y n
is the ve tor olle ting all the su ient statisti s at the timeepo h
n
.
η f,n (σ n )
is alledforward metri and isdened asη f,n (σ n ) , p(y n−1 0 |σ n )P (x n )
(1.24)wherewedenoteby
y n n 2 1
theve tor olle tingallstatisti sy n
fromn = n 1
to
n = n 2
.
η b,n+1 (σ n+1 )
is alledba kward metri and hasexpressionη b,n+1 (σ n+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+1 (σ n+1 ) = X
x n
X
σ n
η f,n (σ n ) F n (x n , σ n ) P (x n )
(1.26)η b,n (σ n ) = X
x n
X
σ n+1
η b,n+1 (σ n+1 ) F n (x n , σ n ) P (x n ) .
(1.27)Hen e theBCJR algorithm worksinthe following way:
rst,forward andba kward metri s
η f,n (σ n )
andη b,n (σ n )
are omputedby means of (1.24) and (1.25),for ea h timeepo h
n
and for ea hstatevalue
σ n
; then, the pdf
p(y|x n )
is derived by (1.22) exploitingη f,n (σ n )
,η b,n (σ n )
and
F n (x n , σ n )
values; 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 anM
-ary alphabet, wehoosetheset
{ℓ a,n } a
ofM −1
logarithmi ratiosoftheaposterioriprobabilitiesP (x n |y)
, asthe reliability estimates of the de ision on the symbolx n
.ℓ a,n
isdened as
ℓ a,n = log P (x n = a|y)
P (x n = 0|y)
(1.28)where
a ∈ {1, 2, . . . , M − 1}
.ℓ 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, wherex i
takes onvaluesonsome(usuallynite)domain
A i
,andletf (x)
amultivariatefun tion.Suppose that
f (x)
fa tors into a produ t of several lo al fun tionsf j
, ea hhavinga subset
x j
ofx
,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 hasa variable node for ea h variable
x i
, a fa tor node for ea h fun tionf j
, andan edge onne ting variable node
x i
to fun tion nodef j
if and only ifx i
isan 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 byµ x i →f j (x i )
a messagesent from the variablenodex i
to thefa tor nodef j
,byµ f j →x i (x i )
a messageintheoppositedire tion,and byB i
theset offun tionsf j
havingx i
as argument, the message omputations performed at variableand 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 theSPA, the marginal a posteriori probabilities
P (x i |y)
, required by the MAPsymbol 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
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 symbolx 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 passedto 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 ontinuesforaxednumberofy 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
ℓ a,n
out theLLRsdened in(1.28), produ ed bya blo k and representing the reliability measure of a MAP symbol algorithm
onthede isiononthesymbol
x n
,theextrinsi informationℓ a,n
e,out generated bysu h blo k,is given by
ℓ 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).
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
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
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 essX
andoutputpro essY
andit isexpressed inbits per hannel use. In the following work we will fo us on
the asewhereboth
X
andY
aredis rete-timestationarysequen es(ingeneral notofthe samelength),denotedwithx
andy
,respe tively.Frominformation theory[24,25℄,weknow thatfor ea h hannelI(x; y)
an be expressedasI(x; y) = h(x) − h(x|y)
b/s Hz
(2.1)
where
h(x)
is thedierential entropy rate of a sour e generating the random dis rete-time stationarypro essx
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 essx
given the hannel outputy
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 toI(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 ey N 0 , (y 0 , y 1 , . . . , y N )
,the omputation ofthe dierential entropyrateh(y)
andtheonditional dierential entropyrate
h(y|x)
ina simulation an be arriedoutthankstotheShannon-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)
if
x N 0
andy 0 N
arestationary ergodi nite-state hiddenMarkov pro esses.By repla ing (2.5) and(2.6) in(2.4),wegetI(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 )
andp(y N 0 |x N 0 )
. Su h values an be ee tively omputed by the forward re ursionof the BCJR algorithm, employed to implement a maximum a posteriori a
probability(MAP)symboldete tionstrategy.Inparti ular,bydening
σ n
thehannelstate, theforward message
η f,n (σ n )
isobtained asη f,n+1 (σ n+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,n (σ n )
(2.8)asshown in [28℄.At the
N
-thtime symbol interval,p(y N 0 )
is obtained asthesumof 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 thesimulation
K
times and,bydenoting withρ n
thep(y N 0 )
measureat then
-thsimulation, we nd
E [log 2 p(y N 0 )] = lim
K→+∞
1 K
X K k=1
log 2 (ρ k ).
(2.10)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)andw(t)
isa omplex-valuedAWGNpro- ess with independent omponents, ea h with two-sided power spe tral den-sity
N 0
.The value ofN 0
isassumed perfe tly known at the re eiver1 and wealsoassumeindependent 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 theforward re ursion we need to ompute inthe Arnold and Loeliger method, is
theforward re ursion
η f,n (σ n )
(2.17) oftheMAP symbol dete tionalgorithmwe 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 symbolx n
and trial stateσ n
, letus 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 tknowledgeof
N 0
isnot riti al.where
e s(t; x n , σ n )
is the CPMwaveform in the interval[nT, (n + 1)T ]
denedin(1.16) and orrespondingto thetrialvaluesof
x n
andσ n
,whiles ¯ ∗ (t; x n , ω n )
(dened in (1.17)) is the CPM waveform depending on just a tual symbol
x n
and orrelative stateω n
. In pra ti e, samples{y n (x n , σ n )}
are obtainedby 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
insteadofpM L
be ause the CPM phase state is onstant overa symbol interval
[nT, (n + 1)T ]
; hen e we need a mat hed lter for ea h ofM L
ombinations of a tual symbolx n
and orrelative stateω n
.The ve tory
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 providesthesymbol
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 variablesx n
,wherex = (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 sy n (x n , σ n )
in (2.12),with
n
from0
toN
.Inparti ular, we fa torizeP (x, σ|y)
as2P (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.
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
-thtimeinterval.
where
T (x n , σ n , σ n+1 )
is the trellis indi ator fun tion, equal to one ifx n
,σ n
,and
σ n+1
satisfy the trellis onstraints and to zero otherwise, andF 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,whilep(·)
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