10(1) 信号与系统 周建华 光电学院教学课件

10 1Thez Transform 10 Thez Transform 10 1Thez Transform LTI 1 Definition 10 1Thez Transform 10 1Thez Transform 10 1Thez Transform a Especially whenz ej aboveequationbecomesTheFouriertransformofsignalx n So therelationshipbetweentheFouriertransformandthez transformis 2 TherelationshipbetweenZ transformandtheFouriertransformofx n b Ontheotherhand 10 1Thez Transform 2 RegionofConvergence ROC ROC RangeofzforX z toconvergeRepresentation A InequalityB Regioninz plane 10 1Thez Transform Example10 1Determinethez Transformofx n anditsROC 10 1Thez Transform Solution 10 1Thez Transform Example10 2Determinethez Transformofx n anditsROC 10 1Thez Transform Solution 10 1Thez Transform Figure10 3 10 1Thez Transform and havesameZ transformrepresentation buttheirROCisdifferent Z Z Note forasignalx n wemustgiveoutthez transformwithitsROC 10 1Thez Transform 3 Thepole zeroplotofX z X z canberepresentedtheratiooftwopolynomials thenumeratorpolynomial thedenominatorpolynomial 10 1Thez Transform Definition ThezerosofX z therootsofthenumeratorpolynomialN z iscalledthezerosofX z ThepolesofX z therootsofthedenominatorpolynomialD z iscalledthepolesofX z 10 1Thez Transform TherepresentationofX z throughitspolesandzerosinthez planeisreferredtothepole zeroplotofX z Definition Inthez plane use X toindicatethepolesofX z anduse O toindicatethezerosofX z Ontheotherhand IfM N z X z X z have M N polesatinfinity IfM N z X z 0 X z have N M zerosatinfinity 10 1Thez Transform Example10 1DetermineX z itsROCanditspole zeroplot 10 1Thez Transform Figure10 2 Example 10 1Thez Transform Example10 3Determinethez Transformofx n itsROCanditspole zeroplot 10 1Thez Transform Figure10 4 10 1Thez Transform Example10 4Determinethez Transformofx n itsROCanditspole zeroplot Fugure10 5 10 2TheRegionofConvergenceforthez Transform Property1 TheROCofX z consistsofaringinthez planecenteredtheorigin 10 2TheROCofthez Transform 10 2TheROCofthez Transform Property2 theROCdoesnotcontainanypoles Property3 Ifx n isoffiniteduration thentheROCistheentirez plane exceptpossiblyz 0andz Example Solution 10 2TheROCofthez Transform Example10 5 10 2TheROCofthez Transform Property4 Ifx n isright sidesequence andifthecircle z r0isintheROC thenallvaluesofzforwhich z r0willalsointheROC 10 2TheROCofthez Transform Figure10 7right sidedsequencex n 10 2TheROCofthez Transform a ROCofaright sidedsequence Property5 Ifx n isleft sidedsequence andifthecircle z r0isintheROC thenallvaluesofzforwhich0 z r0willalsobeintheROC 10 2TheROCofthez Transform 10 2TheROCofthez Transform ROCofleft sidedsequence Property6 Ifx n istwosided andifthecircle z r0isintheROC thentheROCwillconsistofaringinthez planethatincludesthecircle z r0 10 2TheROCofthez Transform 10 2TheROCofthez Transform ROCoftwo sidedsequence Example10 6Determinethez transformofthefollowingsignals 10 2TheROCofthez Transform 10 2TheROCofthez Transform Solution ZerosofX z N 1polesofX z poleofX z 10 2TheROCofthez Transform Figure10 9 Example10 7Determinethez transformofthefollowingsignals 10 2TheROCofthez Transform 10 2TheROCofthez Transform Property7 Ifthez transformX z ofx n isrational thenitsROCisboundedbypolesorextendstoinfinity 10 2TheROCofthez Transform Property8 Ifthez transformX z ofx n isrational andifx n isrightsided thentheROCistheregioninthez planeoutsidetheoutmostpole i e outsidethecircleofradiusequaltothelargestmagnitudeofthepolesofX z Furthermore ifx n iscausal i e ifitisrightsidedandequalto0forn 0 thentheROCalsoincludesz 10 2TheROCofthez Transform Property9 Ifthez transformX z ofx n isrational andifx n isleftsided thentheROCistheregioninthez planeinsidetheinnermostpole i e insidethecircleofradiusequaltothesmallestmagnitudeofthepolesofX z otherthananyatz 0andextendinginwardtoanpossiblyincludingz 0 inparticular ifx n isanticausal i e ifitisrightsidedandequalto0forn 0 thentheROCalsoincludesz 0 10 2TheROCofthez Transform Example10 8 ConsiderallofthepossibleROCSofX z Figure10 12 10 2TheROCofthez Transform 10 3TheInversez Transform 10 3Theinversez Transform Show 10 3Theinversez Transform Thecalculationforinversez transformX z 1 Integrationofcomplexfunctionbyequation 2 usingfractionexpansion 10 3Theinversez Transform 3 Longdivision Taylor sseries 长除法 泰勒级数展开法 AppendixPartialFractionExpansion Considerafractionpolynomial 10 3Theinversez Transform 即 X z 是z的有理分式 把X z 表示成z 1的两个多项式之比形式 10 3Theinversez Transform DiscusstwocasesofD z 1 0 fordistinctroots andsameroots 我们这里对X z 以z 1进行部分分式展开 10 3Theinversez Transform Case1 Distinctroots thus 10 3Theinversez Transform CalculateA1 Generally 10 3Theinversez Transform Usingthefollowingrelationshipstoobtainx n 10 3Theinversez Transform 10 3Theinversez Transform Example Computetheinversez transformofX z Solution 10 3Theinversez Transform 10 3Theinversez Transform Case2 Sameroot So 10 3Theinversez Transform Forfirstorderpoles 10 3Theinversez Transform Multiplytwosidesby 1 p1z 1 r Forr orderpoles 10 3Theinversez Transform So 10 3Theinversez Transform 10 3Theinversez Transform using Wecanobtainx n 10 3Theinversez Transform Orusing Wecanobtainx n 10 3Theinversez Transform 10 3Theinversez Transform Example Determinetheinversez transform Solution 10 3Theinversez Transform 10 3Theinversez Transform 10 3Theinversez Transform Example10 910 1010 11Determinetheinversez transformofX z 10 3Theinversez Transform 3 10 3Theinversez Transform IfX z isnotrational computex n bythefollowingrelationships Longdivision Taylor sseries 长除法 泰勒级数展开法 a b Example10 1210 14Determinetheinversez transformofX z 10 3Theinversez Transform a b Example10 13Determinetheinversez transformofX z bylongdivision 10 3Theinversez Transform a 10 3Theinversez Transform Solution b 10 3Theinversez Transform Solution 10 5Propertiesofthez Transform 1 Linearity 10 5propertiesofthez Transform 10 5propertiesofthez Transform 线性性质 线性组合后的收敛域R是线性组合前两个信号的收敛域R1与R2的公共区域 如果在线性组合过程中出现零点与极点相抵消的情况 则收敛域可能会扩大 10 5propertiesofthez Transform Example 2 Timeshifting 10 5propertiesofthez Transform 10 5propertiesofthez Transform 3 Scalinginthez domain 可见 z平面上的尺度展缩 等效于x n 乘以指数序列 当z0为复指数时 z平面上的尺度展缩对应于Z平面上的点沿角度方向进行旋转 沿径向方向伸张或压缩 4 TimeReversal 10 5propertiesofthez Transform 5 Timeexpansion 10 5propertiesofthez Transform 6 Conjugation 7 Convolutionproperty 10 5propertiesofthez Transform Example10 1510 16 10 5propertiesofthez Transform 8 Differentiationinthez domain 10 5propertiesofthez Transform Example1