Discrete-timesignalsarerepresentedmathematicallyassequencesofnumbers. Asequenceofnumbers x,inwhichthe nth numberinthesequenceisdenoted x[n], 1 is formallywrittenas
x ={x[n]}, −∞ <n< ∞, (1) where n isaninteger.Inapracticalsetting,suchsequencescanoftenarisefromperiodic samplingofananalog(i.e.,continuous-time)signal xa (t).Inthatcase,thenumericvalue ofthe nth numberinthesequenceisequaltothevalueoftheanalogsignal, xa (t), at time nT :i.e.,
x[n]= xa (nT), −∞ <n< ∞ (2)
Thequantity T isthe samplingperiod, anditsreciprocalisthe samplingfrequency. Althoughsequencesdonotalwaysarisefromsamplinganalogwaveforms,itisconvenienttoreferto x[n] asthe“nth sample”ofthesequence.Also,although,strictly
speaking, x[n] denotesthe nth numberinthesequence,thenotationofEq.(1)isoftenunnecessarilycumbersome,anditisconvenientandunambiguoustoreferto“the sequence x[n]”whenwemeantheentiresequence,justaswereferredto“theanalog signal xa (t).”Wedepictdiscrete-timesignals(i.e.,sequences)graphically,asshownin Figure1.Althoughtheabscissaisdrawnasacontinuousline,itisimportanttorecognize that x[n] isdefinedonlyforintegervaluesof n.Itisnotcorrecttothinkof x[n] asbeing zerowhen n isnotaninteger; x[n] issimplyundefinedfornonintegervaluesof n.
n Figure1 Graphicrepresentationofa discrete-timesignal.
Asanexampleofasequenceobtainedbysampling,Figure2(a)showsasegment ofaspeechsignalcorrespondingtoacousticpressurevariationasafunctionoftime,and Figure2(b)presentsasequenceofsamplesofthespeechsignal.Althoughtheoriginal speechsignalisdefinedatallvaluesoftime t ,thesequencecontainsinformationabout thesignalonlyatdiscreteinstants.Thesamplingtheoremguaranteesthattheoriginal
32ms (a)
256samples (b)
Figure2 (a)Segmentofacontinuous-timespeechsignal xa (t ).(b)Sequenceofsamples x[n] = xa (nT )obtainedfromthesignalinpart(a)with T = 125 μs.
thatis,thevalueoftheunitstepsequenceat(time)index n isequaltotheaccumulated sumofthevalueatindex n andallpreviousvaluesoftheimpulsesequence.Analternativerepresentationoftheunitstepintermsoftheimpulseisobtainedbyinterpreting theunitstepinFigure3(b)intermsofasumofdelayedimpulses,asinEq.(5).Inthis case,thenonzerovaluesareallunity,so
If A and α arerealnumbers,thenthesequenceisreal.If 0 <α< 1 and A ispositive, thenthesequencevaluesarepositiveanddecreasewithincreasing n,asinFigure3(c).
Discrete-TimeSignalsandSystems
For 1 <α< 0,thesequencevaluesalternateinsignbutagaindecreaseinmagnitude withincreasing n.If |α| > 1,thenthesequencegrowsinmagnitudeas n increases.
Theexponentialsequence Aαn with α complexhasrealandimaginarypartsthat areexponentiallyweightedsinusoids.Specifically,if α =|α|ejω 0 and A =|A |ejφ ,the sequence Aαn canbeexpressedinanyofthefollowingways:
Theintegerrestrictionon n resultsinsomesinusoidalsignalsnotbeingperiodic atall.Forexample,thereisnointeger N suchthatthesignal x3 [n]= cos(n) satisfies thecondition x3 [n + N ]= x3 [n] forall n.Theseandotherpropertiesofdiscrete-time sinusoidsthatruncountertotheircontinuous-timecounterpartsarecausedbythe limitationofthetimeindex n tointegersfordiscrete-timesignalsandsystems.
WhenwecombinetheconditionofEq.(17)withourpreviousobservationthat ω 0 and (ω 0 + 2πr) areindistinguishablefrequencies,itbecomesclearthatthereare N distinguishablefrequenciesforwhichthecorrespondingsequencesareperiodicwith period N .Onesetoffrequenciesis ωk = 2πk/N , k = 0, 1,...,N 1.Theseproperties ofcomplexexponentialandsinusoidalsequencesarebasictoboththetheoryandthe designofcomputationalalgorithmsfordiscrete-timeFourieranalysis.
Relatedtotheprecedingdiscussionisthefactthattheinterpretationofhigh andlowfrequenciesissomewhatdifferentforcontinuous-timeanddiscrete-timesinusoidalandcomplexexponentialsignals.Foracontinuous-timesinusoidalsignal x(t) = A cos( 0 t + φ),as 0 increases, x(t) oscillatesprogressivelymorerapidly.Forthe discrete-timesinusoidalsignal x[n]= A cos(ω 0 n + φ),as ω 0 increasesfrom ω 0 = 0 toward ω 0 = π , x[n] oscillatesprogressivelymorerapidly.However,as ω 0 increasesfrom ω 0 = π to ω 0 = 2π ,theoscillationsbecomeslower.ThisisillustratedinFigure5.In
Adiscrete-timesystemisdefinedmathematicallyasatransformationoroperatorthat mapsaninputsequencewithvalues x[n] intoanoutputsequencewithvalues y [n].This canbedenotedas y [n]= T {x[n]} (19)
andisindicatedpictoriallyinFigure6.Equation(19)representsaruleorformulafor computingtheoutputsequencevaluesfromtheinputsequencevalues.Itshouldbe emphasizedthatthevalueoftheoutputsequenceateachvalueoftheindex n may dependoninputsamples x[n] forallvaluesof n,i.e., y attime n candependonallor partoftheentiresequence x.Thefollowingexamplesillustratesomesimpleanduseful systems.
Figure6 Representationofa discrete-timesystem,i.e.,a transformationthatmapsaninput sequence x [n ]intoauniqueoutput sequence y [n ].
Thissystemcomputesthe nth sampleoftheoutputsequenceastheaverageof (M 1 + M 2 + 1) samplesoftheinputsequencearoundthe nth sample.Figure7showsaninput sequenceplottedasafunctionofadummyindex k andthesamples(soliddots)involved inthecomputationoftheoutputsample y [n] for n = 7, M 1 = 0,and M 2 = 5.Theoutputsample y [7] isequaltoone-sixthofthesumofallthesamplesbetweenthevertical dottedlines.Tocompute y [8],bothdottedlineswouldmoveonesampletotheright.
Figure7 Sequencevaluesinvolvedincomputingamovingaveragewith M1 = 0 and M2 = 5.
Classesofsystemsaredefinedbyplacingconstraintsonthepropertiesofthe transformation T {·}.Doingsooftenleadstoverygeneralmathematicalrepresentations,aswewillsee.Ofparticularimportancearethesystemconstraintsandproperties, discussedinSections2.1–2.5.
2.1MemorylessSystems
Asystemisreferredtoasmemorylessiftheoutput y [n] ateveryvalueof n depends onlyontheinput x[n] atthesamevalueof n
Example4AMemorylessSystem
Anexampleofamemorylesssystemisasystemforwhich x[n] and y [n] arerelatedby y [n]= (x[n])2 , foreachvalueof n. (22)
Discrete-TimeSignalsandSystems
ThesysteminExample2isnotmemorylessunless nd = 0; inparticular,that systemisreferredtoashaving“memory”whether nd ispositive(atimedelay)or negative(atimeadvance).ThemovingaveragesysteminExample3isnotmemoryless unless M 1 = M 2 = 0.
2.2LinearSystems
Theclassoflinearsystemsisdefinedbytheprincipleofsuperposition.If y
where a isanarbitraryconstant.Thefirstpropertyisthe additivityproperty,andthe secondthe homogeneity or scalingproperty.Thesetwopropertiestogethercomprise theprincipleofsuperposition,statedas
forarbitraryconstants a and b.Thisequationcanbegeneralizedtothesuperposition ofmanyinputs.Specifically,if
ConsidertheaccumulatorfromExample5.Wedefine x1 [n]= x[n n0 ].Toshowtime invariance,wesolveforboth y [n n0 ] and y 1 [n] andcomparethemtoseewhether theyareequal.First,
Next,wefind
Substitutingthechangeofvariables k 1 = k n0 intothesummationgives
Sincetheindex k inEq.(34)andtheindex k 1 inEq.(37)aredummyindicesof summation,andcanhaveanylabel,Eqs.(34)and(37)areequalandtherefore y 1 [n]= y [n n0 ].Theaccumulatorisatime-invariantsystem.
Thesystemdefinedbytherelation y [n]= x[Mn], −∞ <n< ∞, (38) with M apositiveinteger,iscalledacompressor.Specifically,itdiscards (M 1) samplesoutof M ;i.e.,itcreatestheoutputsequencebyselectingevery M th sample. Thissystemisnottimeinvariant.Wecanshowthatitisnotbyconsideringtheresponse y 1 [n] totheinput x1 [n]= x[n n0 ].Forthesystemtobetimeinvariant,theoutputof thesystemwhentheinputis x1 [n] mustbeequalto y [n n0 ].Theoutput y 1 [n] that resultsfromtheinput x1 [n] canbedirectlycomputedfromEq.(38)tobe
y 1 [n]= x1 [Mn]= x[Mn n0 ] (39)
Delayingtheoutput y [n] by n0 samplesyields
y [n n0 ]= x[M(n n0 )] (40)
Comparingthesetwooutputs,weseethat y [n n0 ] isnotequalto y 1 [n] forall M and n0 ,andtherefore,thesystemisnottimeinvariant. Itisalsopossibletoprovethatasystemisnottimeinvariantbyfindingasingle counterexamplethatviolatesthetime-invarianceproperty.Forinstance,acounterexampleforthecompressoristhecasewhen M = 2, x[n]= δ[n],and x1 [n]= δ[n 1].
Forthischoiceofinputsand M , y [n]= δ[n],but y 1 [n]= 0;thus,itisclearthat y 1 [n] = y [n 1] forthissystem.
2.4Causality
Asystemiscausalif,foreverychoiceof n0 , theoutputsequencevalueattheindex n = n0 dependsonlyontheinputsequencevaluesfor n ≤ n0 .Thisimpliesthatif x1 [n]= x2 [n] for n ≤ n0 ,then y 1 [n]= y 2 [n] for n ≤ n0 .Thatis,thesystemis nonanticipative.The systemofExample2iscausalfor nd ≥ 0 andisnoncausalfor nd < 0.Thesystemof Example3iscausalif M 1 ≥ 0 and M 2 ≥ 0;otherwiseitisnoncausal.Thesystemof Example4iscausal,asistheaccumulatorofExample5andthenonlinearsystemin Example6.However,thesystemofExample8isnoncausalif M> 1,since y [1]= x[M ]. Anothernoncausalsystemisgiveninthefollowingexample.
Example9TheForwardandBackwardDifferenceSystems
Thesystemdefinedbytherelationship
y [n]= x[n + 1]− x[n] (41) isreferredtoasthe forwarddifferencesystem.Thissystemisnotcausal,sincethe currentvalueoftheoutputdependsonafuturevalueoftheinput.Theviolationof causalitycanbedemonstratedbyconsideringthetwoinputs x 1 [n]= δ[n 1] and x2 [n]= 0 andtheircorrespondingoutputs y 1 [n]= δ[n]− δ[n 1] and y 2 [n]= 0 forall n.Notethat x1 [n]= x2 [n] for n ≤ 0,sothedefinitionofcausalityrequires that y 1 [n]= y 2 [n] for n ≤ 0,whichisclearlynotthecasefor n = 0.Thus,bythis counterexample,wehaveshownthatthesystemisnotcausal. The backwarddifferencesystem, definedas
y [n]= x[n]− x[n 1], (42) hasanoutputthatdependsonlyonthepresentandpastvaluesoftheinput.Because y [n0 ] dependsonlyon x[n0 ] and x[n0 1],thesystemiscausalbydefinition.
Asystemisstableinthebounded-input,bounded-output(BIBO)senseifand onlyifeveryboundedinputsequenceproducesaboundedoutputsequence.Theinput x[n] isboundedifthereexistsafixedpositivefinitevalue B x suchthat
(43)
Stabilityrequiresthat,foreveryboundedinput,thereexistsafixedpositivefinitevalue B y suchthat
Itisimportanttoemphasizethatthepropertieswehavedefinedinthissectionare propertiesof systems, notoftheinputstoasystem.Thatis,wemaybeabletofindinputs forwhichthepropertieshold,buttheexistenceofthepropertyforsomeinputsdoes notmeanthatthesystemhastheproperty.Forthesystemtohavetheproperty,itmust holdfor all inputs.Forexample,anunstablesystemmayhavesomeboundedinputs forwhichtheoutputisbounded,butforthesystemtohavethepropertyofstability,it
Discrete-TimeSignalsandSystems
mustbetruethatfor all boundedinputs,theoutputisbounded.Ifwecanfindjustone inputforwhichthesystempropertydoesnothold,thenwehaveshownthatthesystem does not havethatproperty.Thefollowingexampleillustratesthetestingofstabilityfor severalofthesystemsthatwehavedefined.
Example10TestingforStabilityorInstability
ThesystemofExample4isstable.Toseethis,assumethattheinput x[n] isbounded suchthat |x[n]|≤ B x forall n.Then |y [n]|=|x[n]|2 ≤ B 2 x .Thus,wecanchoose B y = B 2 x andprovethat y [n] isbounded.
Likewise,wecanseethatthesystemdefinedinExample6isunstable,since y [n]= log10 (|x[n]|) =−∞ foranyvaluesofthetimeindex n atwhich x[n]= 0, even thoughtheoutputwillbeboundedforanyinputsamplesthatarenotequaltozero. Theaccumulator,asdefinedinExample5byEq.(26),isalsonotstable.For example,considerthecasewhen x[n]= u[n],whichisclearlyboundedby B x = 1.For thisinput,theoutputoftheaccumulatoris
Thereisnofinitechoicefor B y suchthat (n + 1) ≤ B y < ∞ forall n;thus,thesystem isunstable.
Asincontinuoustime,aparticularlyimportantclassofdiscrete-timesystemsconsistsof thosethatarebothlinearandtimeinvariant.Thesetwopropertiesincombinationlead toespeciallyconvenientrepresentationsforsuchsystems.Mostimportant,thisclass ofsystemshassignificantsignal-processingapplications.Theclassoflinearsystemsis definedbytheprincipleofsuperpositioninEq.(24).Ifthelinearitypropertyiscombinedwiththerepresentationofageneralsequenceasalinearcombinationofdelayed impulsesasinEq.(5),itfollowsthatalinearsystemcanbecompletelycharacterized byitsimpulseresponse.Specifically,let hk [n] betheresponseofthesystemtotheinput δ[n k ],animpulseoccurringat n = k .Then,usingEq.(5)torepresenttheinput,it followsthat
AccordingtoEq.(48),thesystemresponsetoanyinputcanbeexpressedintermsofthe responsesofthesystemtothesequences δ[n k ].Ifonlylinearityisimposed,then hk [n] willdependonboth n and k ,inwhichcasethecomputationalusefulnessofEq.(48)is somewhatlimited.Weobtainamoreusefulresultifweimposetheadditionalconstraint oftimeinvariance.
Thepropertyoftimeinvarianceimpliesthatif h[n] istheresponseto δ[n],then theresponseto δ[n k ] is h[n k ].Withthisadditionalconstraint,Eq.(48)becomes
AsaconsequenceofEq.(49),anLTIsystemiscompletelycharacterizedbyitsimpulse response h[n] inthesensethat,giventhesequences x[n] and h[n] forall n,itispossible touseEq.(49)tocomputeeachsampleoftheoutputsequence y [n] Equation(49)isreferredtoasthe convolutionsum,andwerepresentthisbythe operatornotation
Theoperationofdiscrete-timeconvolutiontakestwosequences x[n] and h[n] andproducesathirdsequence y [n].Equation(49)expresseseachsampleoftheoutputsequence intermsofallofthesamplesoftheinputandimpulseresponsesequences.
ThenotationofEq.(50)fortheoperationofconvolutionasshorthandforEq.(49) isconvenientandcompactbutneedstobeusedwithcaution.Thebasicdefinitionof theconvolutionoftwosequencesisembodiedinEq.(49)andanyuseoftheshorthand forminEq.(50)shouldalwaysbereferredbacktoEq.(49).Forexample,consider y [n n0 ].FromEq.(49)weseethat
orinshorthandnotation
Substituting (n n0 ) for n inEq.(49)leadstothecorrectresultandconclusion,but blindlytryingthesamesubstitutioninEq.(50)doesnot.Infact, x[n n0 ]∗ h[n n0 ] resultsin y [n 2n0 ]
ThederivationofEq.(49)suggeststheinterpretationthattheinputsampleat n = k ,representedas x[k ]δ[n k ],istransformedbythesystemintoanoutputsequence x[k ]h[n k ],for −∞ <n< ∞, andthat,foreach k ,thesesequencesaresuperimposed (summed)toformtheoveralloutputsequence.ThisinterpretationisillustratedinFigure8,whichshowsanimpulseresponse,asimpleinputsequencehavingthreenonzero samples,theindividualoutputsduetoeachsample,andthecompositeoutputduetoall
