Signal Processing Techniques for Software Radio-chp2

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1、Chapter2FourierAnalysisandLinearTime-InvariantSystemsInthischapter,webrieflyreviewthebasictoolsfromthetheoryofsignalsandlineartime-invariant(LTI)systemsthatareusedintheanalysisanddesignofcommuni-cationsystems.AnLTIsystemischaracterizedinthetimedomainbyitsimpulseresponseandinthefrequencydomainby

2、itstransferfunction.Moreover,theim-pulseresponseandtransferfunctionofanLTIsystemarerelatedthroughFouriertransform.Atthesametime,theFouriertransformisananalysistoolthatisusedtoexpandasignalasasumofspectralcomponents;afiniteorinfinitesetofsinu-soidalwaveforms.Thisleadstothenotionofsignalspectra.Si

3、ncesignalspectraandtransferfunctionsarefundamentaltothedesignandanalysisofcommunicationsystems,mostofthediscussionsinthischapterarerelatedtothesetopics.2.1FourierSeriesAccordingtothetheoryofFourier,aperiodicsignalxT0(t)withperiodofT0andfrequencyoff0=1/T0canbeexpressedasX∞x(t)=xej2πnf0t(2.1)T0n

4、n=−∞√wherej=−1andthexn’saretheFourierseriescoefficientsofthesignalx(t).TheFouriercoefficientsxn’scanbeevaluatedbyperformingtheintegral1Zα+T0x=x(t)e−j2πnf0tdt(2.2)nT0T0αwhereαisanarbitraryconstantthatdeterminesthestartoftheperiodandischosensuchthattosimplifytheevaluationoftheintegral.34FourierAnaly

5、sisandLinearTime-InvariantSystemsChap.2Accordingto(2.1),theperiodicsignalxT0(t)canbesynthesizedbyadding(com-plex)sine-wavesoffrequencies0,±f0,±2f0,···.Thecoefficientx0whichassociateswiththefrequencyof0iscalledDC(directlycoupled)componentofxT0(t).Thefrequencyf0iscalledthefundamentalfrequencyofxT0

6、(t),andaccordinglyx1andx−1arereferredtoasthefundamentalcomponentsofxT0(t).Theremainingtermsarereferredtoasharmonics.WhenxT0(t)isreal-valued,wehave1Zα+T0x=x(t)ej2πnf0tdt−nT0α"Z#∗1α+T0=x(t)e−j2πnf0tdtT0α=x∗(2.3)nwhichimplies

7、xn

8、=

9、x−n

10、and6xn=−6x−n.(2.4)Thatis,themagnitudeofxnisanevenfunctionofnan

11、ditsphaseisanoddfunctionofn.Inotherwords,theFourierseriescoefficientsofareal-valuedsignalxT0(t)haveHermitiansymmetry.TheFourierseriesexpansionof(2.1)isknownastheexponentialFourierseriesandisapplicabletobothreal-andcomplex-valuedperiodicsi