Wavelet modulation: An alternative modulation with low energy consumption

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Wavelet modulation: An alternative modulation with

low energy consumption

Marwa Chafii, Jacques Palicot, Rémi Gribonval

To cite this version:

Marwa Chafii, Jacques Palicot, Rémi Gribonval. Wavelet modulation: An alternative modulation

with low energy consumption. Comptes Rendus Physique, Centre Mersenne, 2017, 18 (2), pp.156-167.

�10.1016/j.crhy.2016.11.010�. �hal-01445465�

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Contents lists available atScienceDirect

Comptes

Rendus

Physique

www.sciencedirect.com

Energy and radiosciences / Énergie et radiosciences

Wavelet

modulation:

An

alternative

modulation

with

low

energy

consumption

La

modulation

en

ondelettes :

une

modulation

alternative

à

faible

consommation

d’énergie

Marwa Chaﬁi

a

,

∗

,

Jacques Palicot

a

,

Rémi Gribonval

b a_{CentraleSupélec,}_IETR,_Campus_de_Rennes,₃₅₅₇₆_{Cesson-Sévigné}_cedex,_France b_IRISA,_Inria_–_Bretagne_Atlantique,₃₅₀₄₂_Rennes_cedex,_France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Keywords: Waveletmodulation Wavelettransform OFDM Peak-to-Average-PowerRatio Meyerwavelet Haarwavelet Mots-clés : Modulationenondelettes Transforméeenondelettes OFDM Facteurdecrête OndelettesdeMeyer OndelettedeHaar

Thispaper presentswavelet modulation,basedonthediscrete wavelettransform,as an alternative modulation with low energy consumption. The transmitted signal has low envelope variations, whichinduces a good efficiency for the power amplifier. Wavelet modulation is analyzed and compared for different wavelet families with orthogonal frequencydivisionmultiplexing(OFDM)intermsofpeak-to-averagepower ratio(PAPR), power spectral density (PSD) properties, and the impact of the power amplifier on the spectral regrowth. The performance in terms of bit error rate and complexity of implementationarealsoevaluated,andseveraltrade-offsarecharacterized.

r

é

s

u

m

é

Danscet article,nousprésentonslamodulationenondelettes, baséesurlatransformée discrèteenondelettes,commeunemodulationàfaibleconsommationd’énergie.Lesignal généréparcettemodulationa,eneffet,defaiblesvariationsdepuissanceparconstruction. Nousanalysonslamodulationenondelettespourplusieurs famillesd’ondelettesetnous comparonssesperformancesaveccellesdel’OFDM,entermesdefacteurdecrête(PAPR), depropriétésdeladensitéspectraledepuissanceainsiquedel’impactdel’ampliﬁcateur depuissancesurlaremontéedeslobessecondaires.Lesperformancesentermesdetaux d’erreurbinaireetdecomplexitéd’implémentationsont égalementévaluées,etplusieurs compromissontcaractérisés.

*

Correspondingauthor.

E-mailaddresses:marwa.chaﬁi@supelec.fr(M. Chaﬁi),jacques.palicot@centralesupelec.fr(J. Palicot),remi.gribonval@inria.fr(R. Gribonval).

http://dx.doi.org/10.1016/j.crhy.2016.11.010

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Thiswork is anextended version ofour studyonthe Meyerwaveletmodulation, published inthe conferencepapers

[1,2].Comparedto[1,2],wepresenthereamorecompletestudycoveringother waveletfamiliesandanalyzingthepower ampliﬁereffectonthespectralregrowth.

1. Introduction

Incommunications, an importanteffectof thelarge amplitude variations of thetransmitted multicarrier signal isthe weakpowereﬃciencyofthepowerampliﬁer.Thiscausesintensiveenergyconsumptionandgrowingneedforcooling.Our objectiveistoproposeanalternativetotheclassicalmulticarriermodulationsinordertoreduceenergyconsumption.

Orthogonalfrequency division multiplexing(OFDM)is amodulation schemeadoptedby various wirelineandwireless communicationstandards asa modulationtechnique for datatransmission. However, theOFDM signal suffersfromhigh powerfluctuations.Thevariationsofitsenvelopegeneratenon-lineardistortionswhenweintroducetheOFDMsignalinto apoweramplifier (PA).InordertoavoidthecompressionrangeofthePAandamplifythesignalinthe lineardomain of thePA characteristic,an inputback-off (IBO) isperformedattheinput ofthe PA.However, whentheinput back-off gets larger,powerefficiencygetspoorer,whichinduceshighenergyconsumption.

PAenergyconsumptionmayrepresentmorethan60% oftheenergyconsumptionoftheLTEmacrobase-station trans-mitter[3].Reducingtheenvelopevariations, andthen improvingtheampliﬁer’seﬃciency,contributestoreducing energy consumption atthe levelof mobile equipments (batteriesthat last longer), andatthe level ofthe basestations (energy savings andreducedelectricitybillfortelecom operators).Therefore,thiscouldhelp thereduction ofCO2 emissionsand preventenvironmentalpollution.

Thepeak-to-averagepower(PAPR)isarandomvariablethathasbeenintroducedtomeasurethepowervariationsofthe OFDMsignal.MuchresearchhasbeenconductedinordertoreducethePAPRandanalyzeitsprobabilitydistribution.Inour previouswork [4],wehavestudiedthecomplementarycumulativedistributionfunction(CCDF)ofthePAPR,andwehave shownthatitdependsonthewaveformusedinthemodulation.ThePAPRreductionproblemcanthereforebeformulated asanoptimizationproblem.In[5],wehaveshownthathavingatemporalsupportstrictly lessthan asymbolperiodisa necessaryconditiononthewaveformsforabetterPAPRthanOFDM.Sincethewaveletbasissatisﬁesthiscondition,weare interestedinexploringwaveletmodulationandenergyconsumptionthroughtheevaluationofitsPAPR.

Inthispaper,waveletmodulationisproposed asan alternativemodulationwithlow energyconsumption. Power con-sumptionisevaluated through thestudyofPAPR performance,andcompared withthat ofOFDM.The simulationresults showthattheproposedschemeforwaveletmodulationachievessigniﬁcantgainsintermsofPAPRcomparedwithOFDM, atthecostofaffordableincreasedcomplexity.Acomparisonintermsofbiterrorrateandpowerspectraldensitybetween theOFDMsystemandthewaveletmodulationsystemisalsoinvestigatedinthiswork.

2. Waveletmodulation

Waveletshavebeenappliedinseveralwirelesscommunicationapplicationssuchasdatacompression,sourceand chan-nel coding, signal denoising and channel modeling. Moreover, wavelets have been proposed as a modulation basis for multicarrier modulation systems. The resulting systembased on wavelet modulationis often named in theliterature as Wavelet-OFDM[6]oralsoknownasorthogonalwaveletdivisionmultiplexing(OWDM)[7].

Themostcommonschemeintheliteratureiswaveletpacketmodulation(WPM),whichisageneralizedformofwavelet modulation.WPMhasbeenintroducedintheliteraturebyLindsey[8],whostudiedtheapplicationofthewaveletpacket basis inorthogonalmultiplexingofdata.Inthispaper,we havechosento focusonwaveletmodulationforits signiﬁcant PAPRperformancegains,aswillbestudiedinSection3.

2.1. Waveletbasis

Let

ψ

and

φ

betwofunctions

∈

L2

₍

_R)

_,1_of_a_ﬁnite_support2

_[

₀

_,

_T

0

]

,suchthat:

ψ

2

_{= φ}

2

₌

₁ ₍₁₎ and +∞

−∞

ψ (

t

)

=

0 (2)

The norm

.

ofa function g

∈

L2

₍

_R)

_is_deﬁned _as:

_g

2

₌

+∞

−∞

|

g

(

t

)

|

2dt.

ψ

and

φ

can be named themother wavelet functionandthemotherscalingfunctionrespectively.

LetL betheeffectivenumberofscalesselected;wedeﬁnecontractedversions

ψ

jand

φ

jofthefunctions

ψ

and

φ

,for

j

≥

J

−

L,

1 _The_space_of_square_integrable_functions.

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ψ

j

(

t

)

=

2

j

/2

_{ψ (}

₂j_t

₎

₍₃₎

φ

j(t

)

=

2j/2

φ (

2jt

)

(4)

Thecontractedfunctions

ψ

j and

φ

j haveasupportof

[

0

,

T₂0j

]

.

Foreveryscale j,wedeﬁnetranslatedversions

ψ

j,kand

φ

j,kofthefunctions

ψ

jand

φ

jasfollows:

ψ

j,k(t

)

= ψ

j(t

−

2−jkT0

)

=

2 j /2

ψ (

2jt

−

kT0

)

(5)

φ

j,k

(

t

)

= φ

j(t

−

2−jkT0

)

=

2 j_/2

φ (

2jt

−

kT0

)

(6)

Thecontractedtranslatedfunctions

ψ

j,kand

φ

j,khaveasupportof

[

kT₂j0

,

(k+1)T0

2j

]

.For j

∈ [

[

J

−

L

,

J

−

1

]

andk

∈ [

[

0

,

2j

−

1

]

,

wedeﬁnethewaveletbasisas:

φ

J−L,k

k=2J−L−1 k=0 J−1

j=J−L

ψ

j,k

k=2j−1 k=0 (7)

where J isthenumberofscalesconsidered.

2.2. Expressionofthetransmittedsignal

Wavelet modulationis a multicarrier modulationsystembased on thewavelet basis insteadof theFourierbasis. The modulationsystem

{

gm

}

m∈[[0,M−1]]isrepresentedbythewaveletfunctions

{ψ

j,k

}

j∈[[J−L,J−1]],k∈[[0,2j₋₁_]] andthescaling

func-tions

{φ

J−L,k

}

k∈[[0,2J−L₋1]] oftheﬁrstscale.Thewaveforms

{

gm

}

m∈[[0,M−1]]canbeexpressedas:

{

gm

}

m∈[[0,2J0+1₋₁_]]

:= {ψ

J0,k

}

k∈[[0,2J0₋1]]

∪ {φ

J0,k

}

k∈[[0,2J0₋1]]

{

gm

}

m∈[[2J0+1_,₂J₋₁_]]

:= {ψ

j,k

}

j∈[[J0,J−1]],k∈[[0,2j−1]]

Thetransmittedsignalbasedonwaveletmodulationisthendeﬁnedasfollows:

x

(

t

)

=

n J−1

j=J−L 2j₋₁

k=0 wj,k

ψ

j,k

(

t

−

nT0

)

+

n 2J−L

−1 q=0 aJ−L,qφJ−L,q(t

−

nT0

)

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•

J

−

1:lastscaleconsidered,withM

=

2J_,

•

L:effectivenumberofscalesconsidered( J

−

L

≤

j

≤

J

−

1),

•

wj,k:waveletcoeﬃcientslocatedatk-thpositionfromthescale j,

•

aJ−L,k:approximationcoeﬃcientslocatedatthek-thpositionfromtheﬁrstscale J

−

L,

• ψ

j,k

=

2

j_/₂

ψ(

2j_t

₋

_kT₀₎_:_the_wavelet_orthogonal_functions,

• φ

J−L,k

=

2

J−L

2

_{φ (}

₂J−L_t

−

_kT₀₎_:_the_scaling_orthogonal_functions_at_the_scale _J

−

_L.

The motherwaveletfunctionandthemotherscaling functioncorrespond to j

=

0,k

=

0.Toeachscale j corresponds 2j

translatedwaveletfunctions.Fromonescaletothenext,thenumberofwaveletfunctionsisthenmultipliedbytwo.

2.3. Variants

Several variants of wavelet modulation can be considered, depending on the number of scales L selected. Since the scalingfunctionsareconsideredonlyfortheﬁrstscale, J

−

L thendeﬁnesthenumberofthescalingfunctions

φ

j,k inthe

modulationsystem.Fig. 1depictsthewaveletmodulationsystemfordifferentvaluesofL,forM

=

8 carriers.Byconvention, when L

=

0,thereare 2J scaling functions

φ

j,k andnowaveletfunction

ψ

j,kconsidered inthewaveletbasis(thecaseof

singlecarriermodulation).

Note that theposition ofthefunctionsin Fig. 1is not acoincidence, butithas an importance,since itgives an idea aboutthetime-frequencylocalizationofthewaveforms

(

t

,

f

)

.

2.4. Somewavelets

In thissection,we outlinesome importantwaveletfamilies:Meyerwavelets,whichallowcompact frequencysupport, andDaubechieswavelets,whicharecharacterizedbyacompacttemporalsupport.

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Fig. 1. Some variants of wavelet modulation for M=8.

Fig. 2. Haar mother wavelet and scaling function.

Fig. 3. Daubechies-3 mother wavelet and scaling function.

Haarwavelet TheHaarwaveletistheoldestandsimplestwavelet, andhasaclosedformexpressioninthetime and fre-quencydomains.ThemotherHaarwaveletandthemotherHaarscalingfunctionarepresentedinFig. 2,andareexpressed asfollows:

ψ (

t

)

=

⎧

⎪

⎨

⎪

⎩

1 √ T0 if 0

≤

t

≤

T0 2

−

_√1 T0 if T0 2

≤

t

≤

T0 0 else (9)

φ (

t

)

=

₁ √ T0 if 0

≤

t

≤

T0 0 else (10)

Daubechieswavelets The Daubechies-p wavelets are characterized by a number p of zero moments.They havecompact supportandthereforearecalculatedfromconjugatemirrorﬁlters fl _of_ﬁnite_impulse_response_[9]_._In_particular,_when _{p is}

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Fig. 4. Dmey mother wavelet and scaling function.

Fig. 5. Daubechies-20 mother wavelet and scaling function.

equal to1,weget theHaarwavelet. Fig. 3andFig. 5 displaythefunctions

φ

and

ψ

for p

=

3 and p

=

20: Daubechies-3 (db3)andDaubechies-20(db20).

Meyerwavelet TheMeyerwaveletisafrequency-band-limitedorthogonalwavelet,whichhasbeenproposedbyYvesMeyer in 1985[10,11]. Meyerwavelets are indefinitely differentiableorthonormal wavelets,which are well localized anddecay fromtheircentralpeakfasterthananyinversepolynomial.DmeyisadiscreteformatapproximationoftheMeyerwavelet, anditcanapproximatetheMeyerwaveletbasedonafiniteimpulseresponse(FIR)filter,asdepictedinFig. 4.3Asaresult, thefastwavelettransformcanapproximatetheMeyerwavelettransform,andthediscretewavelettransform(DWT)canbe applied.

3. PAPRperformance

AcomparisonintermsofPAPRperformanceispresentedinFig. 6andFig. 7forthewaveletmodulationvariantsL

=

1 and L

=

7 respectively.Toevaluate thePAPR performance,the complementarycumulative distributionfunction (CCDF)of thePAPRissimulated,whichistheprobabilitythatthePAPRexceedsadeﬁnedvalue

γ

.Thesimulationsettingsconsidera numberofcarriersM

=

128 ( J

=

7)andthe16-QAM(quadratureamplitudemodulation)constellation.

Differentwaveletsareselected:theHaarwavelet,theDaubechieswavelets3 and20,andtheDmeywavelet.TheCCDFfor differentwaveletsareshiftedtotheleft comparedwiththatofOFDM.Therefore,waveletmodulationachievesbetterPAPR performancethanOFDMassummarizedinTable 1.Inparticular,theHaarwaveletreachesthebestperformance.Intuitively, thePAPRperformancegaincanbeexplainedbythefactthatinwaveletmodulation,onlyL carriersareoverlappinginthe sameinstant,whichdoesnotpromotetheadditionofthelargepeakpower.

Followingthesamereason,the L

=

1 variantachievesbetterPAPRperformancethanthe L

=

7 variant.Whenthe num-ber of overlapping carriers L gets smaller, the PAPR performance gets then better. This can also be interpreted by the considerationthatineverydecompositionlevel,themulticarriersystemcanbeseenasasinglecarriersystem,becausethe

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Fig. 6. PAPR performance comparison for L=1 and J=7 (M=128).

Fig. 7. PAPR performance comparison for L=7 and J=7 (M=128).

Table 1

PAPRperformanceofwaveletmodulationcomparedwithOFDMforthe16-QAMconstellation.

Wavelet name Haar Daubechies-3 Dmey Daubechies-20

PAPR Gain compared with OFDM at CCDF=10−3_{for L}₌_{1 and J}₌_{7 (M}₌₁₂₈₎ _{4.7 dB} _{3.3 dB} _{2.8 dB} _{1.8 dB} PAPR Gain compared with OFDM at CCDF=10−3_{for L}₌_{7 and J}₌_{7 (M}₌₁₂₈₎ _{2.4 dB} _{1.9 dB} _{1.6 dB} _{1.1 dB}

waveformsofthesamescale j havethesamebandwidthandareonlyshiftedintime.Asthesinglecarriersystemdoesnot sufferfromincreasedPAPR,thesmallerL gets,thelesstheeffectofthePAPRisobserved.

Thus, we canconclude thatthe PAPR performance ofwaveletmodulation dependsonthe variantconsidered andthe wavelet familyselected. Moreover, all the waveletmodulation schemes achieve better PAPR performance than OFDM. In otherwords,itcanbestatedthatwaveletmodulationconsumeslessenergythantheFourierfunctionsusedintheOFDM system.

4. PSDperformance

PSDis arelevant criterion inthe evaluationofthe spectral efficiencyofa modulation scheme.It highlightsthe main properties ofthe used bandwidth andthe side lobe effects that characterize bandwidth efficiencyand adjacent channel interference. Inthis section, we compare the PSDperformance ofdifferent waveletmodulation with theOFDM scheme, beforeandafterthepoweramplifier.

4.1. PSDperformancecomparison

PSDsfordifferentwaveletsaredisplayed inFigs. 8,9,10,and11asfunctionsofanormalizedfrequency,andcompared withOFDM.PSDissimulatedusingMatlab,andestimatedviatheperiodogrammethodwitharectangularwindow.Before applying wavelet modulation based on the inverse discrete wavelettransform (IDWT), zero padding by a factor of 4 is performedontheinputsignalinthefrequencydomain.Thenumberofcarriersconsidered inthesimulationsisM

=

128 ( J

=

7)andthevariantusedisL

=

1.

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Fig. 8. PSD of Haar wavelet modulation.

Fig. 9. PSD of Daubechies-3 wavelet modulation.

Fig. 10. PSD of Dmey wavelet modulation.

Fig. 11. PSD of Daubechies-20 wavelet modulation.

The most striking observation is the properties of PSD Haar wavelet. The width of the main lobe of its PSD is the double of that ofOFDM. Moreover, it suffers from very large side lobes. Manyapplications can not tolerate these poor spectrumcharacteristics.ItistruethatHaarcanbeﬁlteredtoreducethesidelobeseffect,butthiswillchangethesystem’s performance. Similarly, the Daubechies-3 waveletshows signiﬁcantly degraded PSDperformance. In contrast, Dmey and Daubechies-20waveletsachievealmostcomparableperformancetothatofOFDM.

HaarandDaubechies-3reachthebestPAPRperformanceasdiscussedinSection3,andyet,presentthepoorestspectrum characteristics.Thechoiceofasuitablewaveletislimitedbythetrade-offbetweenPAPRperformanceandPSDproperties.

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Fig. 12. PSD Comparison between Dmey wavelet modulation and OFDM at the output of the power ampliﬁer of the Saleh model.

Fig. 13. PSD Comparison between Dmey wavelet modulation and OFDM at the output of the power ampliﬁer of the Rapp model.

4.2. Powerampliﬁereffect

Inthispart,weevaluatetheeffectofapoweramplifieronthePSDproperties.WeintroducetheOFDMsignal,theDmey waveletmodulationsignal,andtheDaubechies-20waveletmodulationsignal intomemorylesspoweramplifiers basedon Saleh’smodel [12] andRapp’smodel[13].The samesimulation parameters ofSection 4.1are considered inthissection. Theinputback-offconsideredinSaleh’smodelis5 dB,whileitisfixedat0dBforRapp’smodel.Therelationshipbetween the inputandoutput voltages isusually expressedby the AM/AMfunction. The AM/PMfunction characterizes thephase shiftbetweentheinputandoutputvoltagesasafunctionoftheinputvoltage.TheAM/AMandAM/PMfunctionsofSaleh’s amplifiermodelaredefinedasfollows:

FAM/AM

=

α

a

|

x

(

t

)

|

1

+ β

a

|

x

(

t

)

|

2

,

FAM/PM

(

|

x

(

t

)

|) =

α

p

|

x

(

t

)

|

2 1

+ β

p

|

x

(

t

)

|

2 (11)

where

α

a

=

2

.

1587 and

β

a

=

1

.

1517 (

αp

=

4

.

0033 and

β

p

=

9

.

1040,respectively)areusedtocomputetheamplitude gain

(thephasechangerespectively)foraninputsignalx

(

t

)

.

TheAM/AMfunctionfortheRapp’sampliﬁermodelisgivenby:

FAM/AM

=

|

x

(

t

)

|

1

+ (

|x(_Ot)|

)

2S

1/2S

(12)

where

|

x

(

t

)

|

isthemagnitudeoftheinputsignal,S

=

0

.

5 isthesmoothnessfactor,andO

=

1 istheoutputsaturationlevel. AsdepictedinFigs. 12–15,OFDMfacesaspectralregrowthcomparedwiththeotherwavelets,duetoitshighPAPR.The spectralregrowthiscausedbythenon-linearityoftheamplifier.Whenamodulatedsignalwithlargeenvelopefluctuations is introduced into the non-linear amplifier, its bandwidth is broadeneddue to the non-linearities that generate mixing products betweentheindividualfrequency componentsofthespectrum.The spectralregrowthleads to adjacentchannel interference.

5. BERperformance

The BER performance ofthewaveletmodulation andtheOFDM systems isevaluated usingthe parameters illustrated inTable 2.The achievedperformance forother valuesof L (L

∈ [

[

1

,

7

]

) issimilartothatofthestudiedcaseL

=

1,witha slightBERdegradationinafrequency-selectivechannel.

5.1. AWGNandﬂatfadingchannel

Inthispart,acomparisonbetweentheBER ofOFDM,theDmeywaveletandtheDaubechies-20waveletforAWGN and ﬂatfading channels, ispresentedinFig. 16.Sincetheseareall orthonormalwaveformsets, itisonlyto beexpectedthat

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Fig. 14. PSD Comparison between Daubechies-20 wavelet modulation and OFDM at the output of the power ampliﬁer of the Saleh model.

Fig. 15. PSD Comparison between Daubechies-20 wavelet modulation and OFDM at the output of the power ampliﬁer of the Rapp model.

Fig. 16. BER performance comparison for MMSE equalizer in AWGN and ﬂat fading channels.

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Table 2

Simulationparameters.

Parameters Deﬁnition Values

M Number of carriers 128

S Number of frames 100

SNR Signal to Noise Ratio in dB 0:5:30 and 0:5:35

nloop Number of iteration loops 100,000

F Intercarrier spacing 15 kHz

L Effective number of scales considered 1

CP Cyclic preﬁx 25%

Table 3

ChanneldelayandpowerproﬁleofETUchannelforLTEstandard.

Discrete delay [ns] 0 50 120 200 230 500 1600 2300 5000

Average path gains [dB] −1.0 −1.0 −1.0 0.0 0.0 0.0 −3.0 −5.0 −7.0

Fig. 18. Wavelet transmission chain in a frequency selective channel.

Fig. 19. IDWT(j)implementation.

theperformanceofallofthemwillbe thesame.ThesimulationsconﬁrmthatwaveletmodulationisasgoodasOFDMin termsofBERperformance.

5.2. Frequencyselectivechannel

Inafrequency-selectivefading channel,acyclicpreﬁxisaddedinthetimedomainto thetransmittedsignalbasedon waveletmodulation,anda frequencydomain equalizationis performedatthereceiver sideusing eitherthe zeroforcing (ZF) equalizeror theminimum mean-square error (MMSE) equalizeraspresented in Fig. 18. ForOFDM, ZFequalizer or MMSEequalizerisusedaftertheFFTdemodulation.

Theextendedtypicalurban(ETU)modelforLTEmultipathchannelstandard[14],deﬁnedbythechanneldelayandthe power proﬁlein Table 3,is used inthissection. As depictedinFig. 17,waveletmodulation basedon the Dmey wavelet andtheDaubechies-20waveletreachesa gainof6

.

5 dB intermsofSNRforaBER of10−3 andthe4-QAMconstellation, comparedwithOFDM.Forhigherconstellations(16-QAM)andforlowSNRvalues,theperformanceofOFDMiscomparable tothatofwaveletmodulation.StartingfromSNR

=

25 dB,waveletmodulationoutperformsOFDM.

Itisimportanttohighlightthatwhenusingchannelcodingtechniques,thegainintermsofSNRwillbelesssigniﬁcant andwilldependalsoontheeﬃciencyofthecodingtechniqueimplemented.Wehavechosennottousecodingtoevaluate onlytheeffectofthemodulationschemeontheBERperformance.

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Table 4

Numberofnon-zerocoeﬃcientsfordifferentwaveletﬁlters.

Wavelet name Haar Daubechies-3 Dmey Daubechies-20

Parameter K 2 6 47 30

6. Complexityofimplementation

6.1. Implementation

Inordertoimplementthewaveletmodulationsystemexpressedin(8),weapplytheMallatalgorithm[9].Forawavelet modulation signalbased onthewaveletsof L scales andthescalingfunctionsofthescale J

−

L, theIDWT

(

j

)

shouldbe performed L times. L can be also be interpreted asthe numberofdecomposition levels. LetCn be a vector of M input

complex symbols Cm,n.The 2J−L ﬁrst Cm,n symbols correspond to the 2J−L scaling coeﬃcients

(

aJ−L,q

)

q∈[[0,2J−L₋1]]. The

second 2J−L _complex_symbols_correspond_to_the_wavelet_coeﬃcients

₍

_w

J−L,k

)

k∈[[0,2J−L₋1]]oftheﬁrstscale J

−

L.First,one IDWT

(

j

)

is performed,which givesinitsoutput 2J−L+1 _scaling _{coeﬃcients.} _After_that, _the_next ₂J−L+1 _coeﬃcients_from

thevectorCnareextractedandconsideredaswaveletcoeﬃcients,andthesecondIDWT

(

j

)

isperformed.Thenextsymbols

areprocessedinthesamewayuntilthelastscale j

=

J

−

1 isreached.Thevector Cncanbeexpressedthenas: Cn

= (

aJ−L,0

,

aJ−L,1

, . . . ,

aJ−L,2J−L−1

)

.

(

wJ−L,0

,

wJ−L,1

, . . . ,

wJ−L,2J−L−1

)

.

(

wJ−L+1,0

,

wJ−L+1,1

, . . . ,

wJ−L+1,2J−L+1₋₁

)

.

. . .

.

(

wj,0

,

wj,1

, . . . ,

wj,2j₋₁

)

.

. . .

.

(

wJ−1,0

,

wJ−1,1

, . . . ,

wJ−1,2J−1₋₁

)

(13)

Thesymbol

.

in(13)standsfortheconcatenationoperator.Fig. 19deﬁnestheimplementationofonedecompositionlevel j. According totheMallatalgorithm,theIDWT

(

j

)

consistsinupsamplingby afactoroftwoandfilteringtheapproximation coefficients(scalingcoefficients)andthedetailcoefficients(waveletcoefficients),respectively,byalow-pass fl anda high-pass fh_filter,_whose_responses_are_derived_from_the_wavelet_considered.

6.2. Complexity

Sincewaveletmodulationneedstwo moreblocks(IDWT,DWT)comparedwithOFDM,aspresentedinFig. 18,its com-plexityisthereforehigherthanthatofOFDM.Letuscomputethecomplexityofthewaveletmodulation(IDWT)andwavelet demodulation(DWT)blocks.AccordingtotheMallatAlgorithm[9],IDWT

(

j

)

consistsinupsamplingbyafactoroftwoand filtering theapproximationcoefficientsaj,k (scalingcoefficients) andthe detailcoefficients wj,k (wavelet coefficients),

re-spectively, by a low-pass filter fl anda high-pass filter fh. Let K be the length of the filters fl and fh (K non-zero coefficients).Waveletmodulationiscalculatedwith

J

j=J−L+1 2jK

≤

J

j=1 2jK

=

2M K (14)

The complexity order in terms of the number of additions and multiplications is therefore

O(

M K

)

. Knowing that the complexity order of the FFT or the IFFT is

O(

M log2

(

M

))

, the complexity increase order is about

O(

log2K(M)

)

, which is

affordablesinceK isbounded,andthenumberofcarriersM isusuallylarge.

Thenumbersofnon-zerocoeﬃcients K foreachoneofthewaveletsstudiedinthisworkaredisplayedinTable 4.The Haar wavelethasthe smallestcomplexity,but,aswe haveseen inSection 3,ithas theworst PSD.The Dmey waveletis computationallymoredemandingthantheDaubechies-20wavelet, butitoutperformstheDaubechies-20waveletinterms of PAPR,while havingthesame BER performance. Thechoice betweentheDmey waveletand theDaubechies waveletis subjecttothePAPRperformancerequiredandthecomplexityofimplementationtolerated.

7. Conclusion

Waveletmodulationhasbeenproposedinthispaperasanalternativemodulationwithlowenergyconsumption.Based onsimulationresults,wehaveshownthatwaveletsachievebetterPAPRperformancethanOFDM.TheHaarwaveletreaches the best PAPR performance, butit suffers frompoor spectral characteristics. The Dmey andthe Daubechies-20 wavelets havebeenproposedaswaveletsthatprovidethebesttrade-off.Thepropertiesofwaveletmodulationhavebeencompared with those of uncodedOFDM. We haveshown that OFDM suffers froma spectral regrowth larger than that of wavelet modulation,whileintroducingthetransmittedsignalintoapowerampliﬁer.WhilemaintainingthesameBERperformance intheAWGNchannelandtheﬂatfadingchannel,thevariantL

=

1 ( J

=

7,M

=

128)ofthewaveletmodulationoutperforms uncodedOFDMinthefrequency-selectivechannelbyupto6

.

5 dB,atBERof10−3_,_for_the_MMSE_equalizer_and_the_4-QAM

(13)

dependsontheapplication,andissubjecttotrade-offPAPRandspectraleﬃciency(Haarvs.Dmey),andtotrade-offPAPR andcomplexityofimplementation(Dmeyvs.Daubechies-20).

Ourfutureworkwillbetostudyasuitablechannelcodingforwaveletstocompareitsperformancewiththatof Coded-OFDM under a frequency-selective channel. A less complex equalization as well aspilot insertion methods for wavelet modulationarealsoopensubjectstobeinvestigated.

Acknowledgements

This work has received ﬁnancial support from the French Government granted to the CominLabs excellence labora-tory and managed by the French National Research Agency in the “Investissements d’Avenir” program under reference No. ANR-10-LABX-07-01.Theauthorswouldalsoliketothankthe‘RégionBretagne’,France,forsupportingthiswork.

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