**3.7 Numeri
al Results**

**3.7.2 Iterative De
oding of SCCPM S
hemes**

rithm and that of the redu ed-sear h algorithms when they all work on a

7-state trellis, it is lear that theformer solution is more ee tive at low

val-ues of

### E _{S} /N _{0}

^{,}

^{whereas}

^{the}

^{latter}

^{solutions}

^{are}

^{more}

^{ee tive}

^{at}

^{large}

^{values}

of

### E _{S} /N _{0}

^{.}

^{In}parti ular,the DT-NZalgorithm outperformsallotherdete tion s hemes when

### E _{S} /N _{0}

^{is}

^{larger}

^{than}

### 13

^{dB.}

^{On}

^{the}

^{other}

^{hand,}

^{Fig.}

^{3.8}

^{shows}

thatwe ana hievetheoptimalperforman ebytakingintoa ount alsosome

sele ted se ondary omponents of the MM de omposition, and applying the

MM-PSS algorithm, whi h works on a 28-state trellis, or even the

MM-PSS-RSalgorithm,whi hworksona7-statetrellis.Hen e,asinallother onsidered

s enarios,the hoi eofresortingtodete tions hemesbasedonMM

de ompo-sitionismoreee tivethanapplyingredu ed-sear hte hniquestotheoptimal

BCJRalgorithm,whateveristhe onsidered FE(an approximatedversion

de-rived byG orMA de ompositions,aswell asthefull omplexityfront end).

### interl.

### outer

### encoder mapper CPM

### modulat.

### SISO interl.

### CPM

### deinterl.

### front end

### processor SISO

### decoder

### decisions Channel

### r(t)

### (FE) (DA)

Figure 3.9: Transmitter and re eiver stru ture for the onsidered SCCPM

s hemes.

Fig. 3.10 3.11and 3.12 3.13and 3.14

Pulse 2RC 2RC 3RC

### M

^{4}

^{4}

^{4}

### h

^{1/4}

^{1/4}

^{2/7}

Mapping Gray Natural Natural

Outer ode CC (7,5)

### r = 1/2

^{CC}

^{(7,5)}

### r = 1/2

^{CC}

^{(7,5)}

### r = 1/2

Codeword 2048 1760 2000

length

Interleaver Bit or Symbol Bit Bit

Iterations 20 10 10

Table3.1: Detailsof the onsideredSCCPM s hemes.

SCCPM S hemes with

### L

^{=2}

In Fig. 3.10 we onsider the on atenation of the above mentioned

onvolu-tional ode with a quaternary 2RC modulation having

### h = 1/4

^{.}

^{T}

^{wo}

^{ oded}

bits aremapped into one quaternary symbol byGraymapping. The

redu ed- omplexitysoft-outputdete torbasedonthe MM-Palgorithm hasonly

### p = 4

states, whereas the optimal dete tor has

### pM ^{L−1} = 16

^{states.}

^{A}

^{symbol}

^{}

in-terleaver of length 1024 symbols and a bit interleaver of length 2048 bits are

onsidered,and20iterationsareperformed.

11

In[38℄some onsiderationshave

been arried out about theadvantagesand disadvantages,intermsof

onver-gen ethresholdanderroroor,ofsymbolinterleaverswithrespe ttobit

inter-leaversinSCCPMs hemes.Inparti ular,ithasbeenshownthatsystemswith

symbol interleaving usually have a lower onvergen e threshold and a higher

erroroor.Inoursimulation,theerroroordoesnot appear,buttheresulton

the onvergen e threshold is onrmed. In any ase, it an be observed that,

irrespe tivelyoftheusedinterleaver,theMM-Palgorithm,with4trellisstates,

exhibitsa negligibleperforman elosswithrespe tto theFCalgorithm.

In Fig. 3.11 we analyze the same SCCPM s heme of Fig. 3.10, but now

with a bit interleaver of length 1760 bits and natural mapping. As expe ted,

theperforman eoftheoptimalFCalgorithmwithaFEbasedontheG

de om-positionand

### Q = 2 ^{L−1} = 2

^{is}

^{identi al}

^{to}

^{that}

^{of}

^{the}

^{optimal}

^{BCJR}

^{algorithm}

with optimal FE. However, even when

### Q = 1

^{,}

^{thus}

^{redu ing}

^{the}

^{num}

^{ber}

^{of}

length-

### T

^{mat hed}

^{lters}

^{from}

### 16

^{to}

### 12

^{,}

^{there}

^{is}

^{no}

^{loss}

^{with}

^{respe t}

^{to}

^{the}

optimal performan e. Note that, inboth ases, unless expli itredu ed-sear h

te hniquesareapplied,thereisnoredu tioninthenumberoftrellisstates.An

automati redu tioninthe numberoftrellisstatesmaybeobtained whenthe

solution des ribed in Se tion 3.4.2, and denotedas G-R,is adopted. As

men-tioned, this solution onsistsof takinginto a ount only a subset of mat hed

ltersofthe ase

### Q = 1

^{.}

^{In}

^{this}

^{ ase,}

^{the}

^{number}

^{of}

^{trellis}

^{states}

^{is}

^{halved}

^{from}

### 16

^{to}

### 8

^{.}

^{In}

^{Fig.}

^{3.11}

^{the}

^{ urve}

^{denoted}

^{as}

^{G-R}

^{with}

### 16

^{states}

^{represent}

^{the}

11

Aditheredrelativeprime(DRP)interleaver[61℄isusedinboth ases.

## 10 ^{−6} 10 ^{−5} 10 ^{−4} 10 ^{−3} 10 ^{−2} 10 ^{−1} 10 ^{0}

## 0 0.5 1 1.5 2 2.5 3

## BER

*E* _{b} */N* _{0}

_{b}

### MM−P (4 states)

### interleaver bit

### interleaver symbol FC (16 states)

Figure 3.10: 2RC modulationwith

### h = 1/4

^{and}

### M = 4

^{.}

^{Comparison}

^{between}

symbolandbit interleaver.

### 10 ^{-5} 10 ^{-4} 10 ^{-3} 10 ^{-2} 10 ^{-1} 10 ^{0}

### 2 3 4 5 6 7 8 9 10

### BER

*E* _{b} */N* _{0}

_{b}

_{0}

### G-R (8 states) G-R (16 states) G, Q=1 (16 states) G, Q=2 (16 states) DT-NZ (4 states) FC (16 states)

Figure 3.11: 2RC modulation with

### h = 1/4

^{and}

### M = 4

^{:}

^{FC}

^{algorithm}

^{with}

G-based FEandDT-NZalgorithm.

performan e when, working on thefull omplexity trellis, among all mat hed

ltersofthe set

### Q = 1

^{,}

^{we}

^{ onsider}

^{the}

^{two}

^{lters}

^{whi h}

^{assure}

^{the}

^{automati }

state redu tion. The performan e loss is signi ant and when we exploit the

state redu tion ( urve denoted as G-R with

### 8

^{states)}

^{the}

^{algorithm}

^{does}

^{not}

work.

In Fig. 3.11, the urve related to the MM algorithm and working on a

4-state trellis is not shown sin e, as demonstrated in Fig. 3.10, it exhibits a

negligibleperforman elosswithrespe tto theFCalgorithm.Onthe ontrary,

the performan e of the best redu ed-sear h algorithm, namely the DT-NZ,

applied to the trellis of optimal BCJR algorithm, is also reported. Although

theDT-NZalgorithm worksona 4-statetrellis, thedegradation islarge.

In Fig.3.12, CPMformat and s enarioare thesame of Fig.3.11 and

### per-10 ^{-5} 10 ^{-4} 10 ^{-3} 10 ^{-2} 10 ^{-1} 10 ^{0}

### 1.5 2 2.5 3 3.5 4 4.5 5 5.5

### BER

*E* _{b} */N* _{0}

_{b}

_{0}

### FC (16 states) MA, K=1 (16 states) MA, K=2 (16 states)

Figure 3.12: 2RC modulation with

### h = 1/4

^{and}

### M = 4

^{:}

^{FC}

^{algorithm}

^{with}

MA-based algorithm.

forman e of FC algorithm with FE based on MA de omposition is analysed.

Whilewhen

### K = 1

^{(i.e.}

^{the}

^{front}

^{end}

^{is}

^{ omposed}

^{by}

^{just}

^{one}

^{length-}

### T

^{lter)}

MAdete torshowsanheavyloss,for

### K

^{values}

^{greater}

^{or}

^{equal}

^{to}

^{2,}

^{it}

^{a hieves}

theperforman e ofFC algorithm withoptimal FE. In otherwords, bymeans

ofMA-based FEthe numberoflength-

### T

^{lters}

^{ an}

^{be}

^{redu ed}

^{from}

### M ^{L} = 16

to

### 2

^{:}

^{this}

^{solutions}

^{leads}

^{to}

^{a}

^{minimum}

^{in}

^{front}

^{end}

^{ omplexity}

^{although}

^{a}

redu tioninthe number oftrellisstates isnot a hieved.

It is interesting to re all that, for all onsidered CPM formats, the

DT-NZ algorithm and the MM-P algorithm exhibit a similar performan e when

un oded transmissions are onsidered (see Figs. 3.5-Fig. 3.8 and the relevant

dis ussion). Hen e, we an state that the DT-NZ algorithm is very ee tive

in produ ing hard de isions, but denitely inee tive in produ ing soft

de i-sions to be exploited initerative de oding. In Table 3.2the number of trellis

statesand the front end omplexityare ompared forall onsidered dete tion

algorithms. It is important to observe that sin e we are onsidering SCCPM

s hemes, the omputational omplexity is mainly determined by the number

of trellis states sin e the front end stage pro esses ea h odeword just one

timewhile thedete tionalgorithm operateson ea h odeword one timeevery

iteration.

Algorithm Totalnumber of Trellis

length-

### T

^{lters}

^{states}

FC 16 16

MM-P 7 4

G,

### Q = 2

^{16}

^{16}

G,

### Q = 1

^{12}

^{16}

MA,

### K = 2

^{2}

^{16}

MA,

### K = 1

^{1}

^{16}

Table 3.2:Front end omplexityand numberoftrellisstatesfor a2RC

modu-lationwith

### h = 1/4

^{and}

### M = 4

^{.}

In on lusion, while the FC algorithm with a FE based on the MA

de- omposition exhibits the lowest FE omplexity (just two length-

### T

^{lters}

^{are}

su ient)without omplexityredu tionintheDAstage,theMM-Palgorithm

allows us to a hieve the optimal performan e with the minimum trellis

om-plexityandaveryredu edfrontend omplexity.Hen e,iftheFCdete torwith

a MA-based FE and the MM-P algorithm an be onsidered as two equally

validsolutionsifun odedCPMtransmissionsare onsidered, MM-Palgorithm

isthe bestsolution for allSCCPM s hemes with

### L = 2

^{.}

SCCPM S hemes with

### L

^{=3}

In Fig. 3.13, we onsider the on atenation of the above mentioned

onvolu-tional ode with a quaternary3RC modulation having

### h = 2/7

^{.}

^{W}

^{e}

^{adopt}

^{a}

DRP bit interleaver of length 2000 bits, and the oded bits are mapped into

quaternarysymbolsbynatural mapping.Atthe re eiverside,iterative

de od-ing is employed, performing 10 iterations. In this ase, the MM-P algorithm

provides, intermsofnumber ofstates, aredu tion fa torequal to 16 with

re-spe ttotheFCalgorithm but,asexpe teda ordingtothedis ussion arried

out inSe tion 3.3.2,the orresponding performan edegradationissigni ant.

Hen e,to a hieve theoptimalperforman e,we have totake intoa ountsome

sele ted se ondary omponentsand applythe MM-PSS algorithm,whi h

pro-vides a redu tion fa tor equal to 4. Unlike the results in Fig. 3.8, related to

an un oded transmission of the same CPM format, Fig. 3.13 also shows that

theappli ation oftheMM-PSS-RSalgorithm ausesalimited,butstill

signi- ant,performan edegradation.Ontheotherhand,theMM-PSS-RSalgorithm

worksonatrelliswithonly7statespertimeepo h,providingaredu tion

fa -tor equal to 16 with respe t to the optimal FC algorithm, and an be thus

onsideredasa onvenienttradeo.Even inthis ase,thealgorithmsbasedon

redu ed-sear h te hniques and dire tly applied to the optimal FC algorithm

arenota onvenient solution: thebestofthem,namelytheDT-NZalgorithm,

annot a hieve the performan e of the MM-PSS-RS algorithm, even when it

workson alarger trelliswith28 states.

In Fig. 3.14 we onsider the same SCCPM s heme and thesame s enario

ofFig.3.13andweshowstheperforman eoftheFCdete tionalgorithmwhen

we onsiderG-basedandMA-based frontend stages.Regarding theFEbased

on the G de omposition, sin e the solution with

### Q = 1

^{does}

^{not}

^{work,}

^{it}

^{is}

useless to assess the performan e of the G-R algorithm (with redu ed DA)

and itisne essaryto in reasethevalueof

### Q

^{.}

^{Fig.}

^{3.14}demonstratesthatthe performan e of the approximated G-based FE is lose to the performan e of

theFC algorithm for

### Q ≥ 2

^{( urves}

^{with}

### Q > 2

^{are}

^{not}

^{reported}

^{in}

^{the}

^{gure}

sin ethey oin idewiththeFC urvewhenoptimalFEis onsidered).Inother

### 10 ^{-5} 10 ^{-4} 10 ^{-3} 10 ^{-2} 10 ^{-1} 10 ^{0}

### 2 2.5 3 3.5 4 4.5 5

### BER

*E* _{b} */N* _{0} [dB]

_{b}

_{0}

### FC (112 states) MM-P (7 states) MM-PSS (28 states) MM-PSS-RS (7 states) DT-NZ (28 states)

Figure 3.13: 3RC modulation with

### h = 2/7

^{and}

### M = 4

^{:}performan e of MM and DT-NZalgorithms.

words, as summarized in Table 3.3, the G de omposition in this ase allows

a redu tion in thenumber of length-

### T

^{mat hed}

^{lters}

^{from}

### 64

^{to}

### 48

^{with}

^{no}

redu tion in the number of trellis states. Same on lusions hold for the MA

de omposition:itisnot possibletoa hievearedu tioninthenumberoftrellis

statesbut

### K = 2

^{length-}

### T

^{lters}

^{are}

^{su ient}

^{in}

^{the}

^{front}

^{end.}

^{Also}

^{with}

^{this}

CPMformat, MAalgorithm isthe dete torwhi h assures theminimumfront

end omplexity.

In on lusion,forallthe onsideredSCCPMs hemes,theapproa h

provid-ingthe simplestfront end is thatbasedon theCPM de omposition proposed

byMoqvist and Aulin, while the most onvenient solution in terms of trellis

omplexity results that based on the CPM de omposition proposed by

Men-galiandMorelli,whi halsoredu estheFE omplexity,possibly ombinedwith

### 10 ^{-5} 10 ^{-4} 10 ^{-3} 10 ^{-2} 10 ^{-1} 10 ^{0}

### 2 2.5 3 3.5 4 4.5 5 5.5

### BER

*E* _{b} */N* _{0} [dB]

_{b}

_{0}

### FC (112 states) MA, K=1 (112 states) MA, K=2 (112 states) G, Q=1 (112 states) G, Q=2 (112 states)

Figure 3.14: 3RC modulation with

### h = 2/7

^{and}

### M = 4

^{:}performan e of FC algorithm withG-based andMA-based FE.

Algorithm Total numberof Trellis

length-

### T

^{lters}

^{states}

FC 64 112

MM-P 10 7

MM-PSS 26 28

MM-PSS-RS 26 7

G,

### Q = 4

^{64}

^{112}

G,

### Q = 3

^{56}

^{112}

G,

### Q = 2

^{48}

^{112}

G,

### Q = 1

^{32}

^{112}

MA,

### K = 2

^{2}

^{112}

MA,

### K = 1

^{1}

^{112}

Table 3.3:Front end omplexityand numberoftrellisstatesfor a3RC

modu-lationwith

### h = 2/7

^{and}

### M = 4

^{.}

properte hniques for redu ed trellis sear h (see theMM-PSS-RS algorithm).

For that reason, the hoi e of resorting to dete tion s hemes based on MM

de ompositionensuresthe bestperforman e/ omplexity tradeo.

CPM Dete tion Algorithms in

the Presen e of Phase Noise

We onsider ontinuousphase modulations (CPMs)initeratively de oded

se-rially on atenated s hemes. The problem of designing low- omplexity

sub-optimal dete tion algorithms for maximum a posteriori symbol dete tion for

CPMs, already des ribed in Chapter 3 for the ase of a oherent hannel, is

fa edhereforthe aseofatransmissionoveratypi alsatellite hannelae ted

byphasenoise.Anideal frequen y syn hronization isassumed. We des ribe a

ouple of dierent approa hes for dete tion, whi h basi ally dierin theway

they model the phase noise (PN) and perform the phase tra king. We also

exploit the Mengali and Morelli de omposition (Se tion 3.3.1), whi h allows

to approximate the CPM signal as the superposition of linearly modulated

omponents, redu ing thenumberof statesrequired to des ribe thesystem.

4.1 Introdu tion

Two major problems in the design of pra ti al systems employing CPM

sig-nals arethe large omplexity inthere eiver(in termsof front end lters and

numberofstatesinthetrellis)andthesensitivenessto ina urate arrier

syn- hronization [1℄. The former problem is addressed in Chapter 3 for the ase

of an ideal oherent re eiver, showing that the algorithms ensuring themost

onvenient performan e/ omplexity tradeo are those based on the Mengali

and Morelli (MM) de omposition [16,46℄, whi h allows to redu e thenumber

ofstatesrequiredtodes ribethesystem.InthisChapter,weaddressthelatter

problem, onsidering the transmissionofa CPMsignaloveratypi al satellite

hannelwhi h introdu es phasenoise.

Wefo usonalgorithms whi h an ope withstrongphase noise,assuming

an ideal frequen y syn hronization. Although several soft-input soft-output

(SISO) dete tion algorithms suitable for iterative dete tion/de oding have

been re ently designed for linear modulations transmitted over hannels

af-fe tedbyatime-varyingphase(seeforexample[62 64℄andreferen estherein),

less attention hasbeen devoted to CPM signals. An ex eption is represented

by[65,66℄ where,based onthe approa h in[63,67℄, joint dete tion and phase

syn hronizationisperformedbyworkingonthetrellisoftheCPMsignaloron

anexpandedtrellisandusingmultiple phaseestimatorsinaper-survivor

fash-ion 1

. We onsider here various algorithms for MAP symbol dete tion,

distin-guishing two main approa hes: theNon-Bayesian approa h and theBayesian

approa h.Thenon-Bayesian approa hdoesnotrequireanyassumptiononthe

statisti al properties of the phase noise, sin e the phase noise is simply

on-sidered as a sequen e of unknown parameters to be properly estimatedthe

above mentioned algorithms presented in [65℄ follow the Non-Bayesian

ap-proa h. On the other hand, the rationale of the Bayesian approa h onsists

of assuming aproperprobabilisti modelfor the phasenoise,for example the

Wienermodel[29,30℄,and toexploit itfor derivingalgorithms forMAP

sym-boldete tion, asalsodone in[63℄ forthe aseof linearmodulations. Forsome

algorithmsderived bysu hanapproa h,wealsoproposeatrivialextensionto

the aseof phase noise modeled bythe sum of two rst order auto-regressive

Gaussian pro esses(SATMODE PNmodelproposed inSe tion 2.4.2).In this

1

Asaparti ular aseofthisgeneralapproa h,theuseofasinglephaseestimatorisalso

onsideredin[66℄totradeperforman eagainst omplexity.

Chapter it is shown that Bayesian approa h ensures a better performan e,

even ifthe a tualphase noise does not mat h theassumed Wienermodel. As

in Chapter 3, we again resort to the MM de omposition to redu e the

om-plexity of the front end ltering stage and the size of the trellis required to

des ribethe modulatorstate.Moreover,forthederivationoftheMAPsymbol

dete tion algorithm, we adopt the framework based on fa tor graphs (FGs)

and the sum-produ talgorithm (SPA)[7℄ (see Se tion 1.3.2 for an overview),

sin e the lassi alprobabilisti arguments an not be exploited whentheMM

de ompositionisemployed(Se tion3.3). Inparti ular,toderive algorithmsof

pra ti al omplexity, we manage all ontinuous random variables in the FG

bymeans ofthe anoni al distribution approa h [7,68℄, whi hallowsto

imple-ment a Bayesian algorithm without dramati ally in reasing the omplexity of

there eiverwithrespe tto the ideal oherent ounterpart.