signal processing techniques for software radio-chp4

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1、Chapter4SamplingandDiscreteTimeSystemsInthischapter,weintroducethetheoryofsamplingandreviewsomerelevantre-sultsofdigitalsignalprocessing.Inaddition,anumberofmethodsofdesigningdigitalfiltersareintroduced.Moreover,theNyquistfilters/pulse-shapesthatwereintroducedinthepreviousch

2、apterarerevisitedanddiscussedfromthedigitalsignalprocessingpointofview.Designmethodsthatleadtooptimizedpulse-shapesarealsopresented.4.1SamplingSamplingtheoremisfundamentaltothetheoryofthesampledsignals.Accordingtothesamplingtheoremtheminimumratethataband-limitedsignalhasto

3、besampledinordernottoloseanyinformationcontentofitistwicethelargestfrequencycomponentinthesignal.Thatis,ifacontinuous-timesignalx(t)isband-limitedtoBHz,i.e.,

4、X(f)

5、=0,forf>B,thenthesamplesofx(t)takenatafrequencyfs≥2Baresufficientforreconstructionofx(t).Inotherwords,therewillb

6、enolossofinformationinusingthesampledsignalinplaceofitscontinuous-timeversion.Thefrequency2BiscalledtheNyquistrateofx(t).ThefrequencyBiscalledtheNyquistfrequencyofx(t).Thesamplingtheoremmaybeprovedasfollows.LetX∞δTs(t)=δ(t−nTs)(4.1)n=−∞anddefinethetrainofimpulsesofthesample

7、sofx(t)asX∞xs(t)=x(t)δTs(t)=x(nTs)δ(t−nTs).(4.2)n=−∞5354SamplingandDiscreteTimeSystemsChap.4Usingline17ofTable2.1andrecallingthatfs=1/Ts,theFouriertransformofδTs(t)isobtainedasX∞∆Ts(f)=fsδ(f−nfs).(4.3)n=−∞Sincetimedomainmultiplicationbecomeconvolutioninthefrequencydomain,f

8、rom(4.2)and(4.3),weobtainXs(f)=∆Ts(f)⋆X(f)X∞=fsX(f−nfs).(4.4)n=−∞Fig.4.1presentsasetofexamplegraphsofx(t),xs(t),X(f)andXs(f),foracasewherefs>2B.Fromthesefigures,onemayseethatx(t)canbereconstructedfromxs(t)bypassingxs(t)throughalowpassfilterthatselectstheportionofthespectruma

9、roundtheoriginandrejectstherestofthereplicasofX(f).AnexampleoftheresponseofsuchalowpassfilterisshownalongwiththegraphofXs(f).4.1.1Reconstructionofx(t)fromthesamplesx(nTs)Ifweassumethatx(t)issampledatarateabovetheNyquistandchoosetheideallowpassfilter(f1,

10、f

11、≤fs/2H(f)=Π=(4.5)

12、fs0,otherwiseaccordingtotheaboveresults,wehavex(t)=Tsxs(t)⋆h(t).(4.6)TakingtheinverseFour