Analysis of the faster-than-Nyquist optimal linear multicarrier system

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Eprints ID: 17471

To cite this version:

Marquet, Alexandre and Siclet, Cyrille and Roque, Damien Analysis

of the faster-than-Nyquist optimal linear multicarrier system. (2017) Comptes Rendus

Physique, vol. 18 (n° 2). pp. 168-177. ISSN 1631-0705

Analysis

of

the

faster-than-Nyquist

optimal

linear

multicarrier

system

Analyse du système linéaire optimal pour les communications

multiporteuses au-delà de la cadence de Nyquist

Alexandre Marquet

a

,

∗

,

Cyrille Siclet

a

,

Damien Roque

b a_{UniversitéGrenobleAlpes,CNRS,GIPSA-Lab,38000Grenoble,France}

b_{Institutsupérieurdel’aéronautiqueetdel’espace(ISAE–SUPAERO),UniversitédeToulouse,31055Toulouse,France}

a

b

s

t

r

a

c

t

Keywords: Multicarriermodulations Faster-than-Nyquistsignaling Interferenceanalysis Performanceanalysis Mots-clés : Modulationsmultiporteuses Transmissionau-delàdelacadencede Nyquist

Analysedel’interférence Analysedesperformances

Faster-than-Nyquist signalization enables a better spectral efficiency at the expense of anincreasedcomputationalcomplexity.Regardingmulticarriercommunications,previous workmainly reliedonthe studyof non-linearsystemsexploitingcoding and/or equali-zationtechniques, with noparticular optimization ofthe linearpart ofthe system. In thisarticle,weanalyzetheperformance oftheoptimallinearmulticarriersystemwhen usedtogetherwithnon-linearreceivingstructures(iterativedecodinganddirectfeedback equalization),orinastandalonefashion.We alsoinvestigatethe limitsofthenormality assumptionoftheinterference,usedforimplementingsuchnon-linearsystems.Theuseof thisoptimal linearsystemleadstoaclosed-form expressionofthe bit-errorprobability that can be used to predict the performance and help the design of coded systems. Ourwork alsohighlightsthegreat performance/complexitytrade-offoffered bydecision feedbackequalizationinafaster-than-Nyquistcontext.

Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Les communications au-delà de la cadence de Nyquist permettent une augmentation de l’efficacité spectrale en contrepartie d’une complexité plus élevée. Concernant les communications multiporteuses, les travaux menés jusque-là se sont principalement focalisés sur l’étude des systèmes non linéaires exploitant des techniques de codage et/ou d’égalisation, sansconsidération ou optimisation particulière de la partie linéaire du système. Dans cet article, nous analysons le comportement du système linéaire multiporteuse optimal lorsqu’ilest utilisé seulou avecdes structuresde réception non linéaires (décodage itératif et égalisation à retour de décision). Nous nous intéressons égalementauxlimitesdel’hypothèse denormalitédel’interférence, laquelleest utilisée lorsde l’implémentationdecessystèmes non linéaires.L’utilisationdu systèmelinéaire optimal permet d’obtenir uneexpression analytique de la probabilité d’erreur, laquelle peutalorsêtreutiliséepourprédirelesperformancesetaideràlaconceptiondesystèmes codés.Cetravailmetaussienavantleboncompromisperformances/complexitéoffertpar

*

Correspondingauthor.

1. Introduction

Mostcurrentcommunicationsystemsenableperfectreconstruction ofthetransmittedsymbols:synthesis andanalysis families,usedrespectivelyfortransmissionandreception,arebiorthogonal(theyareRieszbases).Inthescopeof bandlim-ited transmission,thisisonlyguaranteed whenthe Nyquistcriterionis respected,whichimposes a symbolrate R lower

than the bilateral bandwidth B (R

≤

B) [1].On the contrary,overriding this criterion makes it possibleto transmit ata highersymbolrateR whilekeepingthesamebandwidth(R

>

B),whichimprovesspectralefficiency (definedastheratio betweenthebitrateofthetransmissionD andthebilateralbandwidthB ofthesignal).However,thistechnique,referredto asfaster-than-Nyquist (FTN),inducesaninterferencebetweenpulse-shapes(orinter-pulse-interference–IPI).

Inacontextofovercrowdedradiofrequencyresources, FTNcommunicationsallowforareductionofthespectral occu-pancyatagivenbitrateor,equivalently,fora higherbitrate atagivenspectraloccupancy. Unlikeamoreclassicalwayto improvespectralefficiencyconsistinginanaugmentationofthemodulationalphabet’ssize(everyotherparametersbeing fixed), FTNsystemsdonot risethesensitivitytothenoiseifIPIisproperlycompensatedatthereceiver’sside.Moreover, lotsof transmissionsystems aredesigned towork withmultipath,andpotentially mobile, channels. Through such chan-nels, multicarrier modulations are particularlyefficient asthey permit tochoose matched pulse-shapes accordingto the time–frequencyselectivityofthechannel,thusreducingequalizationcomplexityatthereceiver’ssize[2].FTNtransmission techniquecanbeextendedtothisfamilyofmodulations[3].Inthiscase,denotingT0themulticarriersymboldurationand

F0 theinter-carrierspacing,onecanshowthattransmissionandreceptionfamilies arenolongerbiorthogonalifF0T0

<

1

(theycan,however,formovercompleteframes),whichleadstoIPIinbothtimeandfrequency.

Thisarticleisbasedontheuseoftheoptimalmulticarrierlinearsystemwithrespecttothe signal-to-interference-plus-noise ratio (SINR)criterion, as developed in[4],and whichrelies on the useof tight Gabor framesin transmission and reception.Wewillinvestigatetheperformanceofthissystemincombinationwithdecisionfeedbackequalization(DFE)and low-densityparitycheckcoding(LDPC).

Thisarticleisconstructedasfollows.Section2presentstheinput–outputrelationsofthesystem,basedonframetheory. ThistheoreticalframeworkallowsforthedeterminationoftheSINR,aswell astheclosed-form expressionofthebit-error probability for a transmissionover an additive white Gaussian noise (AWGN) channel. Section 3 goesinto thedetails of two interferencemitigationtechniques.The firstone isbased onaLDPC code,while theother oneuses aDFE structure. Section4underlinesthelimitsoftheGaussianapproximationoftheinterferencebymeansofsimulationsandthenpresents the performance of the two interference mitigation structures mentioned in Section 3. Conclusion andperspectives are finallygiveninSection5.

2. OptimallinearmulticarriersystemwithAWGN

2.1. Input–outputrelationofthelinearmulticarriersystem

Wedenotec

= {

cm,n

}

(m,n)∈

∈ 

2

()

asequenceofzero-mean,independent,andidenticallydistributed(IID)coefficients,

withvariance

σ

2

c and



⊂

Z2.Theequivalentbasebandmulticarriersignalisdefinedby

s(t

)

=



(m,n)∈

cm,ngm,n

(t),

t

∈

R (1)

with g

= {

gm,n

}

(m,n)∈ a Gaborfamily withparameters F0

,

T0

>

0 and whoseelements are givenby its generator g(t)

∈

L

2

(

R

)

:

gm,n

(t)

=

g(t

−

nT0

)

ej2πmF0t (2)

As a consequence,the information carriedby c is regularly spreadin thetime–frequency plane (Fig. 1) witha minimal distanceF0 infrequencyandT0 intime.

Inpractice,thetransmissionislimitedtoM subcarriersandK symbolssuchthat



= {

0

,

. . . ,

M

−

1

}

× {

0

,

. . . ,

K

−

1

}

is afiniteset,inducingthatthesum(1)alwaysconverges.Itcanhoweverpossessalotofterms,soitisimportanttoensure itsstability.Denoting

H

g

=

Vect

(

g

)

theclosureofthelinearspanofthefamily g,1 thestabilityof(1)isguaranteedwhen

g isaBesselsequence,whichmeansthatthereexistsaboundBg

>

0 suchthat

1 _{TheclosureofanormedvectorspaceE containsalltheelementsofE,togetherwithitslimitelements.Forexample,theclosureofthesetofthe}

Fig. 1. Time–frequencyrepresentationofamulticarriersignal.Inthisexample,thegenerator g(t)andtheparametersT0andF0arechoseninorderto

showaclearseparationinfrequency,butnotintime.



(m,n)∈



gm,n

,x



2

≤

Bg



x



2

∀

x

∈

H

g (3)

with

·, ·

and

·

theusualinnerproductandnormof

L

2

(

R

)

,definedby

x,y

=

+∞



−∞

x∗

(t)

y(t)dt

,



x

 =



x,x

∀

x,y

∈

L

2

(

R

)

(4)

where

·

∗ denotes the complexconjugation. Toperfectlyreconstruct c fromtheknowledge of s(t),it isfurthermore nec-essary(and sufficient)for g to bea linearlyindependent family. g isthena Rieszbasisof

H

g,inotherwords alinearly

independentfamilyforwhichthereexisttwobounds0

<

Ag

≤

Bg suchthat:

Ag



x



2

≤



(m,n)∈



gm,n

,

x



2

≤

Bg



x



2

∀

x

∈

H

g (5)

The density

ρ

of g must then be lower than or equal to one:

ρ

=

1

/(F

0T0

)

≤

1. In this article, we take the opposite

casewhere

ρ

>

1 in orderto risethe spectralefficiencyofthe system(everyother parameters beingfixed).Fora linear receiver,thisinterferenceisconsideredasanadditivenoise,addedtotheoneinducedbythechannel,yielding toahigher errorprobability.However, when

ρ

>

1,thereexistlinearlydependentGabor families,whichconstituteredundantframes of

L

2

(

R

)

.Thesearefamiliesforwhich(5)isvalidforx

∈

L

2

(

R

)

.Asaconsequence,thestabilityof(1)isalwaysguaranteed

and

H

g

=

L

2

(

R

)

,butg cannotbeabasisof

L

2

(

R

)

.

Wesupposeaperfectchannelwithadditivenoiseandwespecifyalinearreceiversuchthatc isestimatedby

ˆ

cp,q

=



ˇ

gp,q

,r



∀(

p,q)

∈ 

(6)

withg

ˇ

= {ˇ

gm,n

}

(m,n)∈areceptionfamily,r(t)

=

s(t)

+

z(t)thereceivedsignalandz(t)azero-meanwhiteGaussiancomplex circularnoise,independentofthesymbolsandcharacterizedbyitspowerspectraldensity

γ

z

(

f

)

=

2N0for f

∈

R:E

(z(t))

=

0

andE

(z

∗

(t)z(t



))

=

2N0

δ(t

−

t

)

,withE

(

·)

theexpectationoperator.

2.2. Interferenceandnoiseanalysis

Itispossibletorewrite(6)inordertohighlightinterferenceandnoiseterms:

ˆ

cp,q

=

cp,q



ˇ

gp,q

,

gp,q







˜ cp,q:useful signal

+



(m,n)∈\{(p,q)} cm,n



ˇ

gp,q

,

gm,n







ip,q:interference

+



g

ˇ

p,q

,

z



 

zp,q:noise (7)

In[4]weshowedthattheSINRismaximizedwheng

ˇ

=

1

/

Agg isatightframe,whichmeansthat(5)istruewithAg

=

Bg.

Inthatcase,thefollowingrelationshold:



g



2

=

Ag

/

ρ

(8) Es

=

Ag 2

ρ

σ

2 c (9)

σ

_i2

=

E

(

|

ip,q

|

2

)

= (

ρ

−

1

)

σ

c2 (10)

σ

_z2

=

E

(

|

zp,q

|

2

)

=

ρ

Ag 2N0 (11)

with Es theper-symbolenergy,

σ

i2 theinterference’svarianceand

σ

z2 the noise’svarianceafterfiltering,sothattheSINR

canbewrittenasfollows:

SINR

=

1

ρ

−

1

+

N0 Es

(12)

Weobserve thattheinterferencetermip,q isa randomvariableindependentofthenoise, andthat itcorresponds tothe

sumofahighnumberofzero-mean,independentrandomvariablesc

˜

m,n followingthesamekindofdistribution,buthaving

differentvariances

σ

2 ˜ cm,n:

˜

cm,n

=

cm,n



ˇ

gp,q

,

gm,n



and

σ

_˜cm2 ,n

=

σ

2 c

ˇ

g,gm−p,n−q



2 (13)

Thenecessaryconditionstoapply thecentrallimittheoremarethusnot fulfilled.Thisisconfirmedbyoursimulationsin Section 4.1,whichshow thattheinterferencedoesnot exactlyfollowa normaldistribution.However,we noticethatitis agoodapproximationwhen

ρ

≤

8

/

5.Inthisscenario,thetransmissionbecomessimilartoacasewerethesymbolswould havebeentransmittedoveranon-dispersivechannel withAWGN, andasignal-to-noise ratioequalto(12).Regardingthe filterednoisetermzp,q,itisworthwhilenotingthatitiszero-meanGaussian,butnotnecessarilywhite.

2.3. Approximationofthebit-errorprobability

We now restrict ourselves tothe case wherethe symbols c arechosen from a quadraturephase-shift keying(QPSK) alphabet.Inthiscase,consideringboththeinterferenceandthenoisetobeGaussian,thetheoreticalerrorprobabilityfora transmissionoveraperfectchannelwithAWGNisgivenbythefollowingformula:

Pe

=

Q

√

SINR

=

Q



1

(

ρ

−

1

)

+

N0 2Eb



(14)

with Q

(

·)

thecomplementary cumulativedistributionfunction (CCDF)ofthestandardnormaldistributionand Eb

=

Es

/

2

theper-bitenergy[5,chapter4].

3. Interferencemitigationstructures

3.1. LDPCforwarderrorcorrection

Byconsideringtheinterferenceasanoise,a straightforwardreceptionstrategyreliesonthecompensationofboththe noiseandtheinterferencebychannelcoding.Inthiswork,wechooseaLDPCcodebecauseofitsgoodcorrectingcapabilities atagivencodingrate(seeSection4.3forsimulationresults).Decodersforthisfamilyofcodesrelyonalgorithmsusingsoft inputsintheformoflog-likelihoodratios(LLR),givenby

L



bl

(c

p,q

)

|ˆ

cp,q



=

ln



Pr



bl

(c

p,q

)

=

0

|ˆ

cp,q



Pr



bl

(c

p,q

)

=

1

|ˆ

cp,q





(15)

wherebl

(c

p,q

)

isthelthbitofsymbolcp,q.UsingtheGaussianapproximationoftheinterference,itispossibletowritethe

probabilitydensityfunction(PDF)ofthesumofthenoiseandtheinterferenceterm

ν

p,q

=

zp,q

+

ip,qas:

fν

(x)

=

1

π σ

2 ν exp



−|

x

|

2

σ

2 ν



(16)

where,given(10)and(11)

σ

_ν2

=

σ

z2

+

σ

i2

= (

ρ

−

1

)

σ

c2

+

ρ

Ag

2N0 (17)

Moreover,from(7)and(8),onecan write

ν

p,q

= ˆ

cp,q

−

cp,q

/

ρ

,which,giventhatthesymbolsareIID,followinga uniform

distribution,leadsto L



bl

(c

p,q

)

|ˆ

cp,q



=

ln

⎛

⎜

⎝



c s.t. bl(c)=0exp



_−|ˆ cp,q−c/ρ|2 σ2 ν



c s.t. bl(c)=1exp



_−|ˆc p,q−c/ρ|2 σ2 ν

⎞

⎟

⎠

(18)

SimulationsofSection4.3useLLRcomputedvia(18),meaningthatwesupposethatthePDFoftheinterferenceiswell approximatedbyaGaussianfunction,which,asshowninSection4.1,isrelevantfor

ρ

≤

8

/

5.

Fig. 2. Flowchartofareceiverexploitingourproposedper-blockiterativeDFE.c,ˆcˆ(i)andcˇ(i)arematricescontainingtheelementsˆcp,q,cˆp(i,)q,cˇ(pi,)qrespectively

(wherep indexesthelinesandq thecolumns).

3.2. Decisionfeedbackequalization

Another waytomitigate interferencerelies onequalization schemes.In thispart,we derive an equalizationstructure that usespreviouslyestimatedsymbolstocanceltheinterferenceterm.Indeed,asshownin(7),ifthistermiscompletely canceled, thentheperformanceoftheFTNsystemisidenticalto thatofanorthogonal system(i.e.a systemwithdensity

ρ

≤

1)intermsofbiterrorrate(BER).

Letussupposethatthegenerator g(t)usedintransmissionisknownbythereceiver,thentheonlymissingparameterto computetheinterferencetermip,qisthetransmittedsymbolssequencec.TheDFEpresentedhereusesaper-blockiterative

approach: symbolsobtainedafterthresholdingatthepreviousiteration areusedtocompute andremove theinterference terminthecurrentiteration(Fig. 2).Estimatedsymbolsafterinterferencecancellationaregivenby

ˆ

c(pi,)q

=

cp,q



ˇ

gp,q

,

gp,q



+



(m,n)∈\{(p,q)}



cm,n

− ˇ

c(mi−,n1)



ˇ

gp,q

,

gm,n



+



g

ˇ

p,q

,

z



,

i

∈ {

0

, . . . ,

NI

−

1

}

(19)

where NI isthetotalnumberofiterations,c

ˆ

(pi,)q andc

ˇ

(pi,)qarerespectivelythesymbolsestimatedandthesymbolsobtained

afterthresholdingatiterationi.Inthefirstiteration,wesetc

ˇ

p(−,q1)

=

0 sothatc

ˆ

(

0)

p,q

= ˆ

cp,q

,

(p,

q)

∈ 

.

Wecanseethatintheperfectcase,wherethesymbolsafterthethresholddetectoratiterationi

−

1 areidenticaltothe symbolssent



ˇ

cp(i−,q1)

=

cp,q

, (p,

q)

∈ 

,theinterferencetermiscompletelyremoved atiterationi.However,thisisnever thecaseinpracticesinceerrorpropagationoccurs(BERfloorathighSNR).

4. SimulationsonAWGNchannel

4.1. Empiricalanalysisoftheinterference

Asintroducedin2.2,itisnecessarytoempiricallystudythestatisticalpropertiesoftheinterference.Tothisextent,we measured 3

.

6

×

106 realizationsoftheinterferencetermip,qthrough thetransmissionof K

=

50000 multicarriersymbols

takingtheirvaluesinaQPSKalphabet,overM

=

128 subcarriers,usingtransmissionandreceptiongeneratorsyieldingtight frames, for differentvaluesofthe density

ρ

andover aperfect noise-free channel.These realizationsofthe interference termwerethen standardizedinordertofacilitatethecomparisonoftheir cumulativedistributionfunction(CDF)andPDF tothestandardGaussiandistribution’sone.Weobservedasimilarstatisticalbehaviorforboththerealandimaginaryparts ofip,q,aswellasvariousgeneratorfunctionsyieldingtightframes.

Consideringthe transmissionofIIDbits overaperfectnoise-free channel (SINR

=

1

/(

ρ

−

1

)

)anddenoting Fi,ρ

(x)

the

CDFoftheinterferenceforadensity

ρ

,onecanexpressthebit-errorprobabilityas

Pe

(

ρ

)

=

1

−

Fi,ρ

√

SINR

=

1

−

Fi,ρ



1

ρ

−

1



(20)

In order to evaluate the validity of the Gaussian approximation of the interference in the context of error probability estimation,wecompare Pe

(

ρ

)

and Q

√

1

/(

ρ

−

1

)



forvariousvaluesof

ρ

inFig. 3.Wecanseethat,althoughitdoesnot strictly followaGaussian distribution,such anapproximation appropriatelyfits thebit-errorprobabilitygiven

ρ

>

16

/

15. Moreover,when

ρ

≤

16

/

15 theQ-functionseemstobeanupperboundofthebit-errorprobability.

InordertoverifytherelevanceofthisapproximationforcomputingLLRusedby soft-inputreceptionalgorithms(such asLDPCdecoders),wecomparethePDFoftheinterference fi,ρ(x)totheoneofastandardGaussiandistribution fN (0,1)

(x)

for variousvaluesof thedensityinFig. 4.We seethat theGaussian approximation isrelevant fortheinterferencePDF’s estimationgiven

ρ

≤

8

/

5.However,forhighervaluesofthedensity,theapproximation errorcanbecomesufficientlyhigh tointroducesignificanterrorswhencomputingLLR,yieldingdegradedperformance.

Fig. 3. Comparison of the CCDF of the interference and its Gaussian approximation (Q-function) with respect to the densityρ.

Fig. 4. Comparison of the PDF of the interference and its Gaussian approximation with respect to the densityρ.

4.2. Linearsystemperformance

Simulations in this Section as well as in Sections 4.3 and 4.4 make use of transmissions of K

=

5000 multicarrier symbolsover M

=

128 subcarriers witha QPSKconstellation.Theywere runforvarious generators(yieldingtight frames ornot). Tightframesare obtainedusingthedualityprinciple oftheWexler–Raztheorem[6, Theorem9.3.4].Indeed,this theoremstatesthatg andg generate

ˇ

dualGaborframeswithtime–frequencyparametersT0,F0ifandonlyiftheygenerate

biorthogonalRiesz–Gaborsequenceswithtime–frequencyparameters1

/F

0,1

/T

0.Whatismore, g generatesatightGabor

frame withtime–frequency parameters T0, F0 ifandonlyifit generates an orthogonalGabor sequence withparameters

1

/F

0, 1

/T

0.Thus, orthogonal generatorsused inthe case

ρ

<

1 correspondto tight framegenerators when

ρ

>

1.Thus,

the two orthogonal generators obtained in [7] form tight frames, asshown in [4]. The first one, which maximizes the time–frequencylocalizationisdenotedbyTFL,andthesecondone,whichminimizestheout-of-bandenergyisdenotedas OBE.Forthesamereasons,thesquare-root-raised-cosine(SRRC)withtheroll-offfactor

α

=

ρ

−

1 aswell astheT0-width

rectangular(RECTT0) generatoryield tightframes.Whensucha generatorisusedinbothtransmissionandreception,itis

sufficienttosetitsnormto1

/

√

ρ

inordertoobtaintightframeswithAg

=

1.Bycontrast,althoughtheRECTρT0andRECTT0

generatesdualframes,theyarenotcanonicaldualandusingoneofthemfortransmissionandtheotherforreceptiondoes notleadto apairoftight frames.Finally,therectangulargeneratorofwidth

ρ

T0 (RECTρT0) doesnotformcanonicaldual

frameswhenusedbothintransmissionandreception.

Fig. 5 showsthatthe SINRis perfectlypredictedby (12)whentransmission andreceptiongenerators yield tightdual canonical frames. Performance getsworse whenthiscondition isnot respected, which isin linewiththe resultsshown in[4].Inthiscase,wecanseethatitisbettertousethesamefamiliesfortransmissionandreception,eveniftheyarenot dualframes,thanusingnon-tightdualframes.

Fig. 6confirms therelevance oftheclosed-formexpressionofthebit-errorprobability(14).ItmeansthattheGaussian approximation of the interference term is also accurate, even for

ρ

>

8

/

5, because in this case, the BER is sufficiently high compared to the approximation error so that the relative error is kept low. However, for strong values of Eb

/N

0

Fig. 5. SINR versus Es/N0, withρ=16/15.

Fig. 6. BER versusρ, with Eb/N0=20 dB.

Fig. 7. BER versus Eb/N0, withρ=16/15.

(

≥

14 dB)andfor

ρ

closetoone(

ρ

=

16

/

15),thelimitsofthisapproximationbecomenoticeable(Fig. 7).Inthecontextof a non-codedsystem, theBERrapidlyincreaseswiththedensity(Fig. 6), andalower boundoftheBERappears whenthe noise powerbecomesnegligiblecomparedtotheinterference’sone (Fig. 7).Theseresultsconfirmtheneedfornon-linear detectorsenablingabetterinterferencemitigation.

Fig. 8. OutputBERasafunctionoftheinputBERfortherate1/2 LDPCcodeoftheDVB-S2standardoveranAWGNchannel.Inthisconfiguration,the convergencethresholdisgivenatBERin=0.15.

Fig. 9. BERasafunctionofEb/N0forasystemusingtherate1/2 LDPCcodeoftheDVB-S2standard(10iterationsofthedecoder),adensityρ=4/3 and

aTFLgenerator.TheconvergencethresholdhappensatthevalueofEb/N0correspondingtotheexpectedBERin(determinedusingFig. 8). 4.3. PerformancewithLDPCcoding

BERcurvesofcodedsystemsusingiterativestructuressuchasturbo-codesorLDPCarecharacterizedbytheir so-called

convergencethreshold correspondingtothelowest Eb

/N

0valueallowingabetterBERattheoutputofthedecoder(denoted

asBERout)thanatitsinput(denotedasBERin)[8].FortransmissionshappeningoveraperfectchannelwithAWGN,itisalso

possibletocharacterizesuchacodedsystemwithacurverepresentingBERoutasafunctionofBERin.Onthiskindofcurve,

theconvergencethresholdisgivenwithaparticularvalue oftheinputBER.As aconsequence,thankstothe closed-form expressionoftheerrorprobabilitygivenin(14),onecanfindthehighestdensity

ρ

allowingthecodedsystemtoconverge foragivenvalueofEb

/N

0.

For instance, Fig. 8 shows that a system using the rate 1

/

2 LDPC code defined in the DVB-S2 standard [9] has its convergencethresholdforaninput BER ofapproximately 0

.

15.Referringtothe Fig. 9,wecan seethatwhenthiscodeis used together witha multicarrier FTNsystemusing tight frames, the convergence thresholdhappens for Eb

/N

0

=

2 dB,

whichactuallyisthesignal-to-noiseratiocorrespondingtoaninputBERof0

.

15.

4.4. Performancewithdirectfeedbackequalization

Ourproposedper-blockiterativeDFE,althoughitsverylowcomplexity,caneffectivelymitigateinterferenceforlowSNR values,yieldingcloseproximitybetweenFTNmulticarrierDFEandorthogonalsystemBERperformance,asshownbyFig. 10. WeobservethattheBERperformanceofthisDFEdoesnotdependonthegeneratorsusedintransmissionandreceptionas longasthey yieldtightcanonicaldualframes.Forstrongvaluesof Eb

/N

0,wecanseetheerrorpropagationphenomenon

Fig. 10. BER as a function of Eb/N0for a multicarrier FTN system using a DFE, a densityρ=8/7 and a TFL generator.

Fig. 11. BER as a function ofρfor a multicarrier FTN system using a DFE, Eb/N0=8 dB and a TFL generator.

When usedwitha linearreceiverandnothingelse, Fig. 11 showsthat thisequalizerisnot effectivewithhighvalues ofthedensity.However,itwouldbesimpletogetbetterperformance byaddinganerror-correctingcode,allowing foran improvementofthereliabilityoftheestimatedsymbols.Anotherapproachwouldconsistinmodifyingthisequalizertolet itworkwithsoftinputsandproducesoftoutputs,asawaytointegrateitinaturbo-equalizationstructure[10,8].

5. Conclusion

This articleis based on theuse ofthe optimalFTN linearmulticarrier system derived in [4]. Thissystem enablesan increase in thesignalingdensityin time and/or frequency.Consequently,spectral efficiency isimprovedaswell, butthis comesattheexpenseofunavoidableinter-pulseinterference.

WeshowthataGaussianapproximationoftheinterferenceisaccurateforthesakeofabit-errorprobabilityestimation, forwhichwegiveaclosed-formexpression.Twointerferencemitigationtechniquesarealsopresented,thefirstoneinvolves anerror–correctioncodewithiterativedecoding(LDPCcode),whilethesecondoneisbasedonaper-blockiterativedecision feedback equalizer.The latterisshownto be veryeffective forlowvalues ofthedensity(

ρ

≤

8

/

7),butsuffers fromthe errorpropagationphenomenonforhighvaluesofthesignal-to-noiseratio.ItisshownthroughsimulationsthattheGaussian approximationoftheinterferencetermisalsoaccurateforthesakeoflog-likelihoodratioscomputation,neededforexample by LDPC codesandturbo-codesdecoders, ensuringthat

ρ

≤

8

/

5.Inthiscontext,we introduceamethodallowing forthe predictionoftheperformanceofthisoptimallinearsystemwhenassociatedwithan error–correctioncodeusingiterative decoding,withoutrequiringsimulations,thusfacilitatingsystemdesign.

Future work mayinclude interference mitigation by means of turbo-equalization [10], a more in-depth study of the statistical propertiesoftheinterference term,or anevaluation ofthisoptimallinearsystemover time and/or frequency-selectivechannels.

References

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[3]F.Rusek,J.B.Anderson,ThetwodimensionalMazolimit,in:InternationalSymposiumonInformationTheoryProceedings,2005,p. 970.

[4]C.Siclet,et al.,Onthestudyoffaster-than-Nyquistmulticarriersignalingbased onframetheory,in:Proc. InternationalSymposiumonWireless CommunicationsSystems,ISWCS,2014,p. 251.

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