Signals on ? ?(T p ) 2011 Gianfranco Cariolaro.pdf

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Chapter10
SignalsonR/Z(Tp)

Acontinuous-timesignalthatverifiestheperiodicitycondition
s(t−t0)=s(t),t∈R,t0∈Z(Tp),
canbeformulatedbothonRandonthequotientgroupR/Z(Tp).Inthefirstcase
thesignalappearstobea“singular”signal,namelywithinfiniteenergy,Fourier
transformcomposedofdeltafunctions,
moreappropriately:theenergy(givenbytheenergyinoneperiod)becomesfinite,
theFouriertransformbecomesanordinary(discrete)function,etc.

TheHaarintegralonR/Z(Tp)()istheordinaryintegralextendedover
oneperiod
t0+Tp
dts(t)=s(t)dt,t0∈R.
R/Z(Tp)t0
Therefore,theconvolutionbecomes
t0+Tpt0+Tp
x∗y(t)=x(t−u)y(u)du=y(t−u)x(u)du.()
t0t0
Theimpulse,definedastheunitelementoftheconvolutionalgebra(see
),consistsofasequenceofdeltafunctionsappliedatthepointsofZ(Tp)
+∞
δR/Z(Tp)(t)=δ(t−nTp).
n=−∞
,UnifiedSignalTheory,509
-0-85729-464-7_10,©Springer-VerlagLondonLimited2011:.
51010SignalsonR/Z(Tp)

DifferentiatingasignaldefinedonR/Z(Tp),
ds(t)
s(t)=,
dt
givesasignalonR/Z(Tp).Theintegration
t
y(t)=s(u)du,t0∈R,()
t0
yieldsaperiodicsignalonlyifs(t)hasazeromeanvalueinaperiod;otherwise
y(t)becomesaperiodic.

Therelation
y(t)=x(at),a=0,()
convertsasignalx(t)definedonR/Z(Tp)intoasignaldefinedonthenewquo-
tientgroupR/Z(T/|a|).Forexample,ifa=1,wehaveatimeexpansion,andthe
p3
periodbecomes3Tp.

ThedualofR/Z(Tp)is
R/Z(Tp)=Z(F),F=1/Tp,

discretefrequencydomainZ(F),whereasthenonnegativebandandthepositive
bandarerespectively()
B0={0,F,2F,3F,...},B+={F,2F,3F,...}.

TheFouriertransformanditsinversearegivenby
t0+Tp
S(kF)=s(t)e−i2πkFtdt,kF∈Z(F),()
t0:.

(Tp)
+∞
s(t)=FS(kF)ei2πkFt,t∈R/Z(T).()
p
k=−∞
Hence,startingfromacontinuous-timeperiodicsignals(t),t∈R/Z(Tp),the
Fouriertransformgivesadiscrete-frequencyfunctionS(f),f∈Z(F).Theinverse
transformexpressesthesignalasasumofexponentials
FS(kF)ei2πkFt
withfiniteamplitudeandfrequencykF∈Z(F),thatis,withbothpositiveandneg-
ativefrequencies.
Forrealsignals,theHermitiansymmetryS(−kF)=S∗(kF)allowsarepresen-
,letting
S(kF)=A(kF)eiβS(kF),()
S
thegeneralrelationship()withIz={0}andI+={F,2F,...}gives
∞
s(t)=S0+2FAS(kF)cos2πkFt+βS(kF),()
k=1
whereS0=FS(0)isthemeanvalueinaperiod.

Letting
Δ
Sk=FS(kF),()
()givestheusualFourierseriesexpansionofaperiodicsignal
+∞
s(t)=Skei2πkFt,()
k=−∞:.
51210SignalsonR/Z(Tp)
and()givestheFouriercoefficients
t0+Tp
11−i2πkFt
Sk=S(kF)=s(t)edt()
TpTpt0
(),()and(),()aresubstantially
equivalent,althoughtheformerallowsthedirectapplicationofthegeneralFTrules.
Forexample,theParsevaltheorem(
t0+Tp+∞
22
s(t)dt=FS(kF),()
t0k=−∞
whereas,intermsofFouriercoefficients(),()itbecomes
t0+Tp+∞
122
s(t)dt=|Sk|.()
Tpt0
k=−∞

Symmetrypairs(evenandodd,realandimaginarysignals,Hermitianandanti-
Hermitian),,holdfortheFouriertransformonR/Z(Tp)
andthereforefortheFouriercoefficients.
:
realsignalHermitianFouriertransform
evenrealsignalevenrealFouriertransform
oddrealsignaloddimaginaryFouriertransform
(Tp)
,differentiationandintegrationcanbeconsid-
eredalsoonR/Z(Tp),butonlyinthetimedomain,sincethefrequencydomainis
discrete.
TimeDifferentiationandIntegrationTherulesaresubstantiallythesameseen
,namely
ds(t)F
−→i2πfS(f),
dt
t
F1
y(t)=s(u)du−→Y(f)=S(f),f=0.
t0i2πf:.


Theintegrationrulerequiresthatthemeanvalueofs(t)iszero,thatis,S0=
FS(0)=0;otherwisetheintegraly(t),thein-
determinacyintheoriginisremovedbycalculatingthemeanvalueY0,andthen
Y(0)=FY0.
RelationshipwiththeFourierTransformonRApplyingtheDualityTheorem
()withI=RandU=R/Z(Tp),weobtain:
(t),t∈R/Z(Tp),istheR→R/Z(Tp)periodicrepetitionof
apulsep(t),t∈R,thenthetransformS(f),f∈Z(F),istheR→Z(F)down-
samplingofP(f),f∈R:
F
s(t)=repTpp(t)−→S(kF)=P(kF).:.
51410SignalsonR/Z(Tp)
DurationandBandwidth
Theextensione(s)ofasignalonR/Z(Tp)isalwaysaperiodicset,whichcan
beexpressedintheforme(s)=J+Z(Tp),whereJisasubsetof[0,Tp).The
durationD(s)=mease(s)isevaluatedwithinaperiod(),andtherefore
D(s)≤,wehavethataperiodicsignalisstrictlytime-limitedina
periodifD(s)<Tp.
ThespectralextensionE(s)=e(S)isasubsetofZ(F).Forareallow-passsig-
nal,e(S)hasthesymmetricforme(S)={−N0F,...,−F,0,F,...,N0F},where
N0Fisthegreatestharmonicfrequencyofthesignal.
AswehaveseenonR,itispossibletoprovetheincompatibilitybetween
thestrictdurationandstrictbandlimitation,thatis,theincompatibilityofhaving
D(s)<TpandB(s)<∞atthesametime.
FourierTransformDamping
TheconsiderationsseenforcontinuousaperiodicsignalsaboutsignalandFTdamp-
ing(),wefindthat,if
thesignalhasregularitydegreen,thentheFouriertransformdecayswiththelow
O(1/fn)andthenthattheFouriercoefficientsSkwiththelowO(1/kn).Wesuggest
thereadertocheckthisstatementinthefollowingexamples.

(Tp).WenotethattheFT
evaluationisbasedonanintegralbetweenfinitelimits,whiletheinversetransform

.
TheSymmetryRule()takestheform
F
s(t)−→S(f)
F
S(t)−→s(−f)
R/Z(Tp)Z(F)
Z(T)R/Z(Fp)
Hence,fromaFourierpaironR/Z(Tp)oneobtainsapaironZ(T),andconversely
().
(1)–(6)(
thattheimpulseonR/Z(Tp)isgivenbyaperiodicrepetitionofdeltafunctions,
whiletheimpulseonZ(F)is1/F=Tpattheoriginandzerootherwise.
(7)–(10)Intheseexamplesthesignalsaregivenbyaperiodicrepetition,and
,thesquarewaveisthe:.

(Tp):.
51610SignalsonR/Z(Tp)
(Continued):.


repetitionoftherectangularpulsep(t)=rect(t/2a),whosetransformisP(f)=
2asinc(f2a).Therefore,itsFouriertransformis
S(kF)=2asinc(kF2a),kF∈Z(F).
(11).
(12)Seepair(19).
:APeriodicModulatedSignal
Considertheexponentialsignal()
s(t)=exp(iAsin2πFt),t∈R/Z(Tp),()
inwhichtheexponentisasinusoid(A0isarealamplitude).ItsFouriercoefficients
are
Tp
1iA0sin2πFt−i2πkFt
Sk=eedt
Tp0
2π
1i(A0sinu−ku)
=edu=Jk(A0),()
2π0:.
51810SignalsonR/Z(Tp)

whereJk(·)aretheBesselfunctionsofthefirstkind[1].Therefore,theFourierseries
expansionofsignal()is
+∞
eiA0sin2πFt=J(A)ei2πnFt.()
n0
n=−∞
TheBesselfunctionsofthefirstkind(ofintegerindex)maybedefinedbythe
integral
π
1i(nu−xsinu)
Jn(x)=edu
π0
.
Theyhavethefollowingproperties:
(1)Jk(A0)=J∗(A0),
k
(2)J−k(A0)=(−1)kJk(A0),
(3)Jk(−A0)=J−k(A0),
(4)+∞J2(A)=1.
k=−∞k0
ThesepropertiesareremarkableresultsofBesselfunctiontheorybutcanbeeasily
,signal(),(),()has
theHermitiansymmetry,andthereforeitsFouriercoefficientsSk=Jk(A0)arereal.
Next,considerthatashiftoft=1Tonthesignalgivestheconjugatesignal;then,
02p
forthetime-shiftrule,

1∗F−iπkFTp∗
st−Tp=s(t)−→S(kF)e=S(−kF),
2
whichforthe(real)FouriercoefficientsgivesSk(−1)k=S−k,thusobtaining(2).
Moreover,ifwereplaceA0with−A0,weobtaintheconjugatesignal,andsowe
prove(3).Finally,(4)canbededucedfromParseval’stheorem,writtenintheform
(),:.

-modulatedsignal
Exponentialsignal(),(),()isencounteredinModulation
Theory[2,3],specificallyinphaseorfrequency-modulatedsignalsoftheform
()
v(t)=V0cos(2πf0t+A0sin2πFt).()

Considertheinput–outputrelationshipofafilteronR
+∞
y(t)=g(t−u)x(u)du,t∈R.()
−∞
Then,itiseasytoseethatiftheinputisperiodicwithperiodTp,theoutputispe-
riodicwiththesameperiod,whiletheimpulseresponseg(t),t∈R,isingeneral
,therearetwopossibili-
ties.
ThefirstistorepresenttheinputandoutputonR,eveniftheyarebothperi-
odic,and()
thethreesignals(includingtheimpulseresponse)onR/Z(Tp).Infact,usingthe
integrationrule(see(),())
+∞t0+Tp+∞
f(u)du=f(u−kTp)du
−∞t0k=−∞
in()andtakingintoaccountthatx(u)isperiodicandthenx(u−kTp)=x(u),
weget
t0+Tp
y(t)=gp(t−u)x(u)du,t∈R/Z(Tp),()
t0
where
+∞
gp(t)=g(t−kTp),t∈R/Z(Tp).()
k=−∞:.
52010SignalsonR/Z(Tp)
Now,in()allthethreesignalsarerepresentedonR/Z(Tp).
Inthefrequencydomain,from()weobtain
Y(f)=Gp(f)X(f),f∈Z(F),()
whereGp(f)=G(f),f∈Z(F),istheR→Z(F)down-sampledversionofthe
,inthefrequencydomainwecanusethe
originalfrequencyresponseG(f),f∈R,whosevaluesareconsideredonlyfor
f∈Z(F).

[]Inthepreviouschapterwehaveseenthatadiscontinuoussignal
onRcanbedecomposedintoacontinuoussignalandapiecewiseconstantsignal.
FindthedecompositionforasignaldefinedonR/Z(Tp).
[]Findconditionsonthesignal


tt1
s(t)=A1repTprect+A2repTprect−,d=20%,
dTpdTp2
whichassurethatitsintegral,definedby(),,evaluatey(t)
anditsFouriertransform.
[]ComputetheFouriercoefficientsofthe“two-wave”rectified
sinusoid
s(t)=|cos2πF0t|,t∈R/Z(Tp).
[]AsignalwithminimumperiodT0canberepresentedon
R/Z(T0),butalsoonR/Z(3T0).Lets1(t)ands3(t)bethetworepresentations.
FindtherelationshipbetweenS1(f)andS3(f).
[]UsingtheFourierseriesexpansionofthesignal(),
provethatthemodulatedsignal()canbewrittenintheform
+∞
v(t)=V0Jk(A0)cos2π(f0+kF)t.
k=−∞
References
,ComplémentsdeMathématiques(EditionsdelaRevued’Optiques,Paris,1957)
,TheFourierIntegralandItsApplications(McGraw–Hill,NewYork,1962)
,DigitalCommunications,3rdedn.(McGraw–Hill,NewYork,1995)