lecture 1sampling of signals

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1、Lecture1SamplingofSignalsbyGrahamC.GoodwinUniversityofNewcastleAustraliaLecture1Presentedatthe“ZaborszkyDistinguishedLectureSeries”December3rd,4thand5th,20071RecallBasicIdeaofSamplingandQuantizationQuantizationSamplingt1t3t2t4t02Inthislecturewewillignorequantizatio

2、nissuesandfocusontheimpactofdifferentsamplingpatternsforscalarandmultidimensionalsignals3OutlineOneDimensionalSamplingMultidimensionalSamplingSamplingandReciprocalLatticesUndersampledSignalsFilterBanksGeneralizedSamplingExpansion(GSE)RecurrentSamplingApplication:Vide

3、oCompressionatSourceConclusions4Sampling:Assumeamplitudequantizationsufficientlyfinetobenegligible.Question:SaywearegivenUnderwhatconditionscanwerecoverfromthesamples?5AWellKnownResult(Shannon’sReconstructionTheoremforUniformSampling)Considerascalarsignalf(t)consisti

4、ngoffrequencycomponentsintherange.Ifthissignalissampledatperiod,thenthesignalcanbeperfectlyreconstructedfromthesamplesusing:6LowpassfilterrecoversoriginalspectrumHenceorProof:Samplingproducesfolding7ASimple(butsurprising)Extensionwhere[RecurrentSampling]isaperiodicse

5、quenceofintegers;i.e.,LetNotethattheaveragesamplingperiodise.g.average58Non-uniformUniform09-1101920xxxxxx05101520xxxxx9Claim:Providedthesignalisbandlimitedtowhere,thenthesignalcanbeperfectlyreconstructedfromtheperiodicsamplingpattern.where=averagesamplingperiodProof

6、:WewilldefertheprooftolaterwhenwewilluseitasanillustrationofGeneralizedSamplingExpansion(GSE)Theorem.10OutlineOneDimensionalSamplingMultidimensionalSamplingSamplingandReciprocalLatticesUndersampledSignalsFilterBanksGeneralizedSamplingExpansion(GSE)RecurrentSamplingAp

7、plication:VideoCompressionatSourceConclusions11MultidimensionalSignalsDigitalPhotographyDigitalVideox1x2x1x2x3(time)12OutlineOneDimensionalSamplingMultidimensionalSamplingSamplingandReciprocalLatticesUndersampledSignalsFilterBanksGeneralizedSamplingExpansion(GSE)Recu

8、rrentSamplingApplication:VideoCompressionatSourceConclusions13Howshouldwedefinesamplingformulti-dimensionalsignals?UtilizeideaofSam