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 valuesof
E S /N 0
.Inparti ular,the DT-NZalgorithm outperformsallotherdete tion s hemes whenE S /N 0
is largerthan13
dB.Ontheotherhand,Fig. 3.8showsthatwe 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 4h
1/4 1/4 2/7Mapping 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
=2In Fig. 3.10 we onsider the on atenation of the above mentioned
onvolu-tional ode with a quaternary 2RC modulation having
h = 1/4
. Two odedbits 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 symbolin-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
isidenti altothatoftheoptimal BCJRalgorithmwith optimal FE. However, even when
Q = 1
, thus redu ing the number oflength-
T
mat hed lters from16
to12
, there is no loss with respe t to theoptimal 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
.Inthis ase,thenumberoftrellisstatesishalvedfrom16
to8
. In Fig. 3.11 the urve denoted as G-R with16
states represent the11
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
MM−P (4 states)
interleaver bit
interleaver symbol FC (16 states)
Figure 3.10: 2RC modulationwith
h = 1/4
andM = 4
. Comparisonbetweensymbolandbit 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
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
andM = 4
: FC algorithm withG-based FEandDT-NZalgorithm.
performan e when, working on thefull omplexity trellis, among all mat hed
ltersofthe set
Q = 1
,we onsiderthetwo lterswhi hassuretheautomatistate 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 notwork.
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
FC (16 states) MA, K=1 (16 states) MA, K=2 (16 states)
Figure 3.12: 2RC modulation with
h = 1/4
andM = 4
: FC algorithm withMA-based algorithm.
forman e of FC algorithm with FE based on MA de omposition is analysed.
Whilewhen
K = 1
(i.e. thefrontend is omposedbyjustone length-T
lter)MAdete torshowsanheavyloss,for
K
valuesgreaterorequalto2,ita hievestheperforman e ofFC algorithm withoptimal FE. In otherwords, bymeans
ofMA-based FEthe numberoflength-
T
lters an beredu ed fromM L = 16
to
2
: this solutions leads to a minimum in front end omplexity although aredu 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 statesFC 16 16
MM-P 7 4
G,
Q = 2
16 16G,
Q = 1
12 16MA,
K = 2
2 16MA,
K = 1
1 16Table 3.2:Front end omplexityand numberoftrellisstatesfor a2RC
modu-lationwith
h = 1/4
andM = 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 aresu 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
=3In Fig. 3.13, we onsider the on atenation of the above mentioned
onvolu-tional ode with a quaternary3RC modulation having
h = 2/7
. We adopt aDRP 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 isuseless 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 oftheFC algorithm for
Q ≥ 2
( urveswithQ > 2
are notreported intheguresin 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]
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
andM = 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 from64
to48
with noredu 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
ltersaresu ientinthefrontend.Also withthisCPMformat, 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]
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
andM = 4
: performan e of FC algorithm withG-based andMA-based FE.Algorithm Total numberof Trellis
length-
T
lters statesFC 64 112
MM-P 10 7
MM-PSS 26 28
MM-PSS-RS 26 7
G,
Q = 4
64 112G,
Q = 3
56 112G,
Q = 2
48 112G,
Q = 1
32 112MA,
K = 2
2 112MA,
K = 1
1 112Table 3.3:Front end omplexityand numberoftrellisstatesfor a3RC
modu-lationwith
h = 2/7
andM = 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.