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ece431notesCh1_2007_09_04

ece431notesCh1_2007_09_04 - CHAPTER1 DoesSampling Always...

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Unformatted text preview: CHAPTER1 DoesSampling Always LoseInformation? 1.1.Introduction Givenacontinuous-timewaveform x ( t ) ,supposeweextractsamples x ( t n ) atdistinct pointsintime t n .Isitpossibletoreconstruct x ( t ) forall t usingonlythedistinctvalues x ( t n ) ?Yourinitialreactionisprobably,“Noway!”However,ifweassumethat x ( t ) belongs toarestrictedclassofwaveforms,yourinitialanswermaychange. Example 1.1. Ifweknowthat x ( t ) isastraightline,then x ( t )= x ( t 1 )+ t- t 1 t 2- t 1 bracketleftbig x ( t 2 )- x ( t 1 ) bracketrightbig . Forthisclassofwaveforms,justtwosamples,takenatdistincttimes,allowreconstruction ofthecompletewaveformforalltime. Theforegoingexampleeasilygeneralizestotheclassofpolynomialsofdegreeatmost n .Inthiscase,just n + 1samples,takenatdistincttimes,determinethepolynomial. Asweshallsee,anywaveform x ( t ) intheclassofwaveformsbandlimitedto f c can bereconstructedfromitssamples { x ( n / f s ) } ∞ n =- ∞ if f s > 2 f c .Here f c iscalledthe cutoff frequency , f s iscalledthe samplingfrequency ,and2 f c iscalledthe Nyquistrate .The statementthatabandlimitedwaveformcanberecoveredfromitssamplesifthesampling frequencyisgreaterthantheNyquistrateiscalledthe SamplingTheorem . Nowconsideralineartime-invariantsystemwithimpulseresponse h ( t ) .Theresponse ofthissystemtoaninput x ( t ) isgivenbytheconvolutionintegral y ( t )= integraldisplay ∞- ∞ h ( t- τ ) x ( τ ) d τ . Iftheimpulseresponseisbandlimitedto f c ,thentheoutput y ( t ) isalsobandlimitedto f c . Hence, y ( t ) canberecoveredfromitssamples y ( n / f s ) if f s > 2 f c .Infact,wewillshowthat iftheinputisbandlimitedtoo,thentherequiredsamplescanbeobtainedfromthe discrete convolution, y ( n / f s )= ∞ ∑ m =- ∞ h ([ n- m ] / f s ) x ( m / f s ) . Looselyspeaking,thisexplainshowcomputersandcompactdisc(CD)playershandle audioinformation.Sincehumanscanonlyhearsoundsuptoabout f c = 20kHz,itsuffices tosamplemusicandspeechatabout f s = 2 f c = 40kHz.Asapracticalmatter,toallowfor non-idealelectronics,CDscontainsamplestakenat f s = 44 . 1kHz. 1 2 1DoesSampling Always LoseInformation? 1.2.ReviewofFourierAnalysis InSection1.2.1,werecallFourierseriesforcontinuous-time,periodicsignals.InSec- tion1.2.2,werecallthediscrete-timeFouriertransformfordiscrete-time,aperiodicsignals. Thedualitybetweenthesetwosituationsisthenreadilyapparent. InSection1.2.3,wemotivatethecontinuous-timeFouriertransformbyexaminingthe limitingformoftheFourier-seriesrepresentationoftruncationsofthetimesignal. 1.2.1.Continuous-TimePeriodicSignals Suppose x ( t ) isaperiodicsignalwithperiod T and Fourierseries expansion x ( t )= ∞ ∑ n =- ∞ x n e j 2 π nt / T . (1.1) Thenitiseasytoshowthatthe Fouriercoefficients aregivenby x n = 1 T integraldisplay T / 2- T / 2 x ( t ) e- j 2 π nt / T dt . Thiscanbedemonstratedbyconsideringtheintegral integraldisplay T / 2- T / 2 x ( t ) e- j 2 π mt / T dt = integraldisplay T / 2- T / 2 bracketleftbigg ∞ ∑ n =- ∞ x n e j 2 π nt / T bracketrightbigg e- j 2 π mt / T dt = ∞...
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ece431notesCh1_2007_09_04 - CHAPTER1 DoesSampling Always...

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