数字信号处理与控制课件：Chapter 3 Discrete-time signals in the frequency domain(revised)

1Chapter3Discrete-TimeSignalsintheFrequencyDomain2HowtoRepresenttheDiscrete-TimeSignal?Time-domain(aweightedlinearcombinationofdelayedunitsamplesequences.)Transform-domain(1)

Frequencydomain;

(2)Zdomain(asequenceintermsofcomplexexponentialsequencesoftheform{}and{}.3SeveralFormsofFourierTransform(FT)FT--continuousintime,continuousinfrequencyx(t)t4FS--continuousintime,discreteinfrequencyx(t)T0where:53.1TheContinuous-TimeFourierTransformIt’sanusefultooltorepresentacontinuous-timesignalinfrequency-domain.ItbrieflycalledCTFT.Therelationshipbetweenit’stimeandfrequencyis:continuousandnon-periodicintime-domain,continuousandnon-periodicinfrequency-domain.63.1.1TheDefinitionThedefinitionofCTFTis:TheCTFToftenisreferredtoastheFourierspectrum.TheI-CTFT(inverseCTFT)is:7WedenotetheCTFTpairofabovetwoequationsas:TheCTFTisacomplexfunctionofintherange.Itcanbeexpressedinpolarformas83.1.2EnergyDensitySpectrumThetotalenergyofafinite-energycontinuous-timecomplexsignalisgivenby:93.1.2EnergyDensitySpectrum103.1.2EnergyDensitySpectrum113.1.2EnergyDensitySpectrumSowegetenergydensityspectrum

(能量密度谱)ofis:123.1.3Band-limitedContinuous-TimeSignalsAnideal

band-limitedsignalhasaspectrumas:Inpractice,foraband-limitedsignaloutsidethespecifiedfrequencyrange,thesignalenergyisarbitrarilysmall.Thebandwidthofaband-limitedsignal:lowpass,highpass,bandpass.133.1.3Band-limitedContinuous-TimeSignals143.2TheDiscrete-TimeFourierTransform离散时间傅里叶变换It’sanusefultooltorepresentadiscrete-timesignalinfrequency-domain.ItbrieflycalledDTFT.Therelationshipbetweenit’stimeandfrequencyis:discreteandnon-periodicintime-domain,continuousandperiodicinfrequency-domain.153.2.1DefinitionIt’sinversetransformprocess:Thediscrete-timeFouriertransformofasequenceisdefinedby:163.2.1Definition173.2.1Definition183.2.1Definition1920213.2.2BasicPropertiesIngeneral,X(ej)isacomplexfunctionoftherealvariablewandcanbewrittenas:X(ej)=Xre(ej)+jXim(ej)where,Xre(ej)andXim(ej)are,respectively,therealandimaginarypartsofX(ej),andarerealfunctionsofw.X(ej)canalternatelybeexpressedas:X(ej)=|X(ej)|ej() where()=arg{X(ej)}22|X(ej)|iscalledthemagnitudefunction.()iscalledthephasefunction.Bothquantitiesarerealfunctionsofw.Inmanyapplications,theDTFTiscalledtheFourierspectrum.Likewise,|X(ej)|and()arecalledthemagnitudeandphasespectra.Forarealsequencex[n],|X(ej)|and

Xre(ej)areevenfunctionsofw,whereas,()andXim(ej)areoddfunctionsofw.(P89)3.2.2BasicProperties23Note:X(ej)=|X(ej)|ej[()+2k]=|X(ej)|ej() foranyintegerk.Thephasefunctionq(w)cannotbeuniquelyspecifiedforanyDTFT.Unlessotherwisestated,weshallassumethatthephasefunctionq(w)isrestrictedtothefollowingrangeofvalues:-q(w) calledtheprincipalvalue.2425TheDTFTX(ej)ofasequencex[n]isacontinuousfunctionofwItisalsoaperiodicfunctionofwwithaperiod2p:263.2.3SymmetryRelations:x[n]isacomplexsequence27x[n]isarealsequence:283.2.4ConvergenceCondition293.2.4ConvergenceConditionIfwedenote:30ThenforallvaluesofwThus,theabsolutesummabilityofx[n]isasufficientconditionfortheexistenceoftheDTFTX(ej).3.2.4ConvergenceCondition313.2.4ConvergenceCondition323.2.4ConvergenceCondition333435363738394041424344454647CommonlyusedDTFTpairs483.2.5NormofaDTFTAmeasureofaFouriertransformX(ej)isgivenbyitsnorm.TheLp-normofX(ej)isdefinedbyBasedonthedefinition,thereare:L1-norm,L2-norm,…M-filefilternormcanbeusedtodetermineLp-norm.493.3DTFTTheorems503.3DTFTTheorems1.Linearity线性2.Time-Reversal时间反转3.Frequency-Shifting频移4.DifferentiationinFrequency5.Convolution卷积6.Modulation调变原理7.Parseval’sRelation

5152535455563.4EnergyDensitySpectrumofaDTSequence575859603.5Band-limitedDTSignals613.5Band-limitedDTSignals623.5Band-limitedDTSignals633.5Band-limitedDTSignals643.5Band-limitedDTSignals653.5Band-limitedDTSignals663.6DTFTComputationUsingMATLAB673.6DTFTComputationUsingMATLAB683.6DTFTComputationUsingMATLAB693.6DTFTComputationUsingMATLAB703.6DTFTComputationUsingMATLAB7172733.8DigitalProcessingofContinuous-TimeSignals(a)(b)Idealsamplingmodelst0t0t0Tt0t0t0T747576EffectofSamplingintheFrequencyDomainInfrequencydomain:Ifgiven:Wecanget:Hence77SamplingTheorem-(NyquistTheorem)Assumega(t)isaband-limitedsignalwithaCTFTGa(j)asshownbelow:ThespectrumP(j)ofp(t)havingasamplingperiodT=2/Tisindicatedbelow:78SamplingTheorem-(NyquistTheorem)TwopossiblespectraofGp(j):79SamplingTheorem-(NyquistTheorem)80SamplingTheorem-(NyquistTheorem)Fromabovediscussion,wecansee:(1)Ifga(t)isbandlimitedwith(2)IfwecallthefrequencyΩs/2satisfyingcondition(2)isNyquistfrequencyorfoldingfrequency,and(2)iscalledNyquistcondition.81SamplingTheorem-(NyquistTheorem)82SamplingTheorem-(NyquistTheorem)83SamplingTheorem-(NyquistTheorem)84over-sampling,under-sampling,criticalsamplingOver-sampling-ThesamplingfrequencyishigherthantheNyquistrate.Under-sampling-ThesamplingfrequencyislowerthantheNyquistrate.Criticalsampling-ThesamplingfrequencyisequaltotheNyquistrate.Note:Apuresinusoidmaynotberecoverablefromitscriticallysampledversion.85SamplingTheorem-(NyquistTheorem)Indigitaltelephony,a3.4kHzsignalbandwidthisacceptablefortelephoneconversation.Here,asamplingrateof8kHz,whichisgreaterthantwicethesignalbandwidth,isused.86SamplingTheorem-(NyquistTheorem)Inhigh-qualityanalogmusicsignalprocessing,abandwidthof20kHzhasbeendeterminedtopreservethefidelity.Hence,incompactdisc(CD)musicsystems,asamplingrateof44.1kHz,whichisslightlyhigherthantwicethesignalbandwidth,isused.87Effectofsamplingofanalogsinusoidalsignalsofdifferentfrequencies88Effectofsamplingofanalogsinusoidalsignalsofdifferentfrequencies8990919293%ProgramP3_4%IllustrationoftheSamplingProcessintheTimeDomainclear;closeall;t=0:0.0005:1;f=13;%frequencyofacontinous-timesignalxa=cos(2*pi*f*t);%continous-timesignalsubplot(2,1,1)plot(t,xa);gridxlabel('Time,msec');ylabel('Amplitude');title('Continuous-timesignalx_{a}(t)');axis([01-1.21.2])subplot(2,1,2);T=0.1;%samplingperiodn=0:T:1;xs=cos(2*pi*f*n);%sampleddiscrete-timesignalk=0:length(n)-1;stem(k,xs);gridxlabel('Timeindexn');ylabel('Amplitude');9495%===========================================%Illustrationoftheperfectreconstructedsignalfromthesamples%===========================================T=0.025;%samplingperiod,samplingfrequencyf=40Hzn=(0:T:1)';xs=cos(2*pi*f*n);t=t';ya=sinc((1/T)*t(:,ones(size(n)))-(1/T)*n(:,ones(size(t)))')*xs;figure;plot(t,xa,'r');grid;holdon;plot(n,xs,'o',t,ya);grid;xlabel('Time,msec');ylabel('Amplitude');title('Reconstructedcontinuous-timesignaly_{a}(t)');legend('Samples','OriginalCTsignal','ReconCTsignal')axis([01-1.21.2]);9697%=======================================%IllustrationofAliasingEffectintheTimeDomain=========================================T=0.1;%samplingperiod,samplingfrequencyf=10Hzn=(0:T:1)';xs=cos(2*pi*f*n);ya=sinc((1/T)*t(:,ones(size(n)))-(1/T)*n(:,ones(size(t)))')*xs;figure;plot(t,xa,'r');grid;holdon;plot(n,xs,'o',t,ya);grid;xlabel('Time,msec');ylabel('Amplitude');title('Reconstructedcontinuous-timesignaly_{a}(t)');legend('Samples','OriginalCTsignal','ReconCTsignal')axis([01-1.21.2]);9899%Program3_5%IllustrationofrelationbetweenCTFTofthesequencexa(t)andDTFT%ofthesequencexa[n]clear;t=0:0.005:10;xa=2*t.*exp(-t);subplot(3,2,1)plot(t,xa);gridxlabel('Time,msec');ylabel('Amplitude');title('Continuous-timesignalx_{a}(t)');subplot(3,2,2)wa=0:10/511:10;ha=freqs(2,[121],wa);%CTFTofxa.plot(wa/(2*pi),abs(ha));grid;xlabel('Frequency,kHz');ylabel('Amplitude');title('|X_{a}(j\Omega)|');axis([05/pi02]);100%==========================================================%ThesamplingfrequencysatisfiestheNyquistsamplingtheorem%==========================================================