版權(quán)說明:本文檔由用戶提供并上傳,收益歸屬內(nèi)容提供方,若內(nèi)容存在侵權(quán),請(qǐng)進(jìn)行舉報(bào)或認(rèn)領(lǐng)
文檔簡介
信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheAnalysisofDiscrete-TimeSystemsinthez-DomainThez-TransformDefinitionofthez-TransformDefinitionofthez-TransformIntuitionontheRelationbetweenZTandLTLT:Let:Definitionofthez-TransformDefinitionBilateral(two-sided)z-Transform:Unilateral(one-sided)z-Transform:Thetransformpairnotation:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Thez-TransformCommonz-transformpairsCommonz-transformpairsUnitSampleSequenceCommonz-transformpairsOne-sideExponentialSequencewhereaisarealorcomplexnumber.UnitStepSequenceCommonz-transformpairswhere
aisarealorcomplexnumber.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheRegionofConvergenceforthez-TransformDefinitionTheRegionofConvergenceforthez-TransformThesetofallcomplexnumberszsuchthatthesummationontheright-handside
convergesiscalledtheregionofconvergence(ROC)ofthez-transformF(z).F(z)converges:f(k)z-kisabsolutelysummableFinite-durationsequenceTheRegionofConvergenceforthez-Transformf(k)=0,k
<k1,k>k2,k1<k2k1<0,k2>0:
k1<0,k2
0:k10,k2
>0:0<|z|<
|z|<
|z|>0Example:CausalsequenceTheRegionofConvergenceforthez-Transformf(k)=0,k<0Example:z-planeak
(k),aisarealorcomplexnumber.AnticausalsequenceTheRegionofConvergenceforthez-TransformExample:f(k)=0,k≥0f(k)=-ak
(-k-1),aisarealorcomplexnumber.Two-sidedsequenceTheRegionofConvergenceforthez-Transformk=-∞→+∞
0<R1<R2<:R1<|z|<R2
R1>R2
:
ROCdoesnotconvergeTheRegionofConvergenceforthez-TransformROCisboundedbypolesorextendstoinfinity.F(z)isrational:f(k)ROCrightsidedoutsidetheoutermostpole——outsidethecircleofradiusequaltothelargestmagnitudeofthepolesofF(z)leftsidedinsidetheinnermostnonzeropole——insidethecircleofradiusequaltothesmallestmagnitudeofthepolesofF(z)otherthananyatz=0andextendinginwardtoandpossiblyincludingz=0.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——LinearityIff1(k)
F1(z),
1<
z
<
1,f2(k)
F2(z),
2<
z
<
2,thenLinearityExample:Iff1(k)
F1(z),
1<
z
<
1,f2(k)
F2(z),
2<
z
<
2,thenLinearityExample:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——TimeShiftingTimeShiftingExample:Bilateralz-TransformIff(k)
F(z),
<
z
<
,thenwheremisapositiveinteger.TimeShiftingProof:Unilateralz-Transform——RightshiftIff(k)
F(z),
z
>
,thenwheremisapositiveinteger.TimeShiftingUnilateralz-Transform——RightshiftIff(k)=0,k<0,thenExample:Iff(k)
F(z),
z
>
,thenwheremisapositiveinteger.TimeShiftingUnilateralz-Transform——LeftshiftIff(k)
F(z),
z
>
,thenwheremisapositiveinteger.Proof:TimeShiftingUnilateralz-Transform——LeftshiftIff(k)
F(z),
z
>
,thenwheremisapositiveinteger.Example:
(k+1)信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——Scalinginthez-DomainScalinginthez-DomainProof:Iff(k)
F(z),R1<|z|<R2
,thenaisanonzerorealorcomplexnumber.ROCofF(z):ROCof
:Scalinginthez-DomainIff(k)
F(z),R1<|z|<R2
,thenaisanonzerorealorcomplexnumber.Example:
aksin(
k)
(k),0<a<1Scalinginthez-DomainIff(k)
F(z),R1<|z|<R2
,thenaisanonzerorealorcomplexnumber.Example:(-1)k
(k)信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——ConvolutionConvolutionProof:Iff1(k)
F1(z),
1<z<
1,f2(k)
F2(z),
2<z<
2,thenConvolutionIff1(k)
F1(z),
1<z<
1,f2(k)
F2(z),
2<z<
2,thenExample:(k+1)
(k)LTIsystems:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——DifferentiationandIntegralinthez-DomainDifferentiationinthez-DomainProof:Iff(k)
F(z),
<
z
<
,then
wherekisanypositiveinteger.Differentiationinthez-DomainIff(k)
F(z),
<
z
<
,then
wherekisanypositiveinteger.Example:Ifa=1,thenDifferentiationinthez-DomainIff(k)
F(z),
<
z
<
,then
wherekisanypositiveinteger.Integralinthez-DomainProof:Iff(k)
F(z),
<
z
<
,then
(misaninteger,andk+m>0)Integralinthez-DomainIff(k)
F(z),
<
z
<
,then
(misaninteger,andk+m>0)Example:Integralinthez-DomainIff(k)
F(z),
<
z
<
,then
(misaninteger,andk+m>0)m=0,k>0:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——Reflectioninthek-domainReflectioninthek-domainProof:Iff(k)
F(z),
<
z
<
,then
Example:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——SummationSummationProof:Iff(k)
F(z),
<
z
<
,then
Example:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)Propertiesofthez-Transform——Initial-ValueTheoremandFinal-ValueTheoremInitial-ValueTheoremProof:Iff(k)=0,k<0,andf(k)
F(z),then
Example:0Thez-transformofacausalsequencef(k)isfindf(0).Final-ValueTheoremProof:Iff(k)=0,k<0,f(k)
F(z),a<
z<,0≤a<1,then
Final-ValueTheoremIff(k)=0,k<0,f(k)
F(z),a<
z<,0≤a<1,then
Example:f(k)=0,k<0. aisarealnumber,findf(
).Final-ValueTheorem√√××Final-ValueTheoremIff(k)=0,k<0,f(k)
F(z),a<
z<,0≤a<1,then
Example:f(k)=0,k<0. aisarealnumber,findf(
).Final-ValueTheoremIfF(z)isrationalandthepolesof(z-1)F(z)havemagnitudes<1,then
Example:Thez-transformofacausalsequencef(k)is
Poles:信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheInversez-TransformTheInversez-Transform(IZT)Integral:DefinitionalongacounterclockwiseclosedcircularcontourthatiscontainedintheROCofF(z).AlternativeproceduresPower-seriesexpansionsPartialfractionexpansionsROCandtheInversez-TransformROCf(k)Causalsequence|z|>af1(k)e
(k)Anticausalsequence|z|<bf2(k)e
(-k-1)Two-sidedsequencea<|z|<b
f1(k)e(k)+
f2(k)e
(-k-1)信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)TheInversez-Transform——PartialfractionexpansionsPartialfractionexpansionsRationalpolynomial:Procedure:PartialfractionexpansionsF(z)f(k)×zIZTPartialfractionexpansions
DistinctPolesSupposethatthepolesz1,z1,…,zNofF(z)aredistinctandareallnonzero.(1)|z|>2;(2)|z|<1;(3)1<|z|<2(1)Example:Partialfractionexpansions
DistinctPolesSupposethatthepolesz1,z1,…,zNofF(z)aredistinctandareallnonzero.(1)|z|>2;(2)|z|<1;(3)1<|z|<2(2)Example:Partialfractionexpansions
DistinctPolesSupposethatthepolesz1,z1,…,zNofF(z)aredistinctandareallnonzero.(1)|z|>2;(2)|z|<1;(3)1<|z|<2(3)Example:Partialfractionexpansions
DistinctPolesz1,2=ae±jbROC:|z|>
Complex
Poles:Partialfractionexpansions
DistinctPolesz1,2=ae±jbComplex
Poles:Example:PartialfractionexpansionsRepeatePolesSupposethatthepolez1isrepeatedrtimes.Matchingcoefficients:Example:PartialfractionexpansionsExample:Step1DividethroughtoobtainwhereF1(z)isstrictlyproper.Step2CarryoutthepartialfractionexpansionofF1(z)and,knowingtheROC,obtaintheinversez-transform.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)z-DomainAnalysis—TransformoftheInput/outputDifferenceEquationTransformoftheInput/outputDifferenceEquationLTIsystem:Input:f(k)=0,k<0Initialstate:y(-1),y(-2),…,y(-n)z-Transform:Y(z)=Yzi(z)+Yzs(z)IZT:y(k)=yzi(k)+yzs(k)TransformoftheInput/outputDifferenceEquationExample:y(k)-y(k-1)-2y(k-2)=f(k)+2f(k-2),y(-1)=2,y(-2)=-0.5,f(k)=e(k).Findyzi(k),yzs(k),y(k),k≥0.TransformoftheInput/outputDifferenceEquationExample:y(k)-y(k-1)-2y(k-2)=f(k)+2f(k-2),y(-1)=2,y(-2)=-0.5,f(k)=e(k).Findyzi(k),yzs(k),y(k),k≥0.TransformoftheInput/outputDifferenceEquationExample:y(k)-y(k-1)-2y(k-2)=f(k)+2f(k-2),y(-1)=2,y(-2)=-0.5,f(k)=e(k).Findyzi(k),yzs(k),y(k),k≥0.信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)z-DomainAnalysis—TheSystemFunctionTheSystemFunction(TransferFunction)DefinitionDeterminationofthesystemfunction(1)
H(z)=Yzs(z)/F(z)(2)H(z)=Z[h(k)]SystemFunctionofInterconnectionsSeriesconnectionH(z)ParallelconnectionH(z)Parallelconnection
H(z)SystemFunctionforInterconnectionsofLTISystemsExample:Determinethezero-stateoftheLTIsystem.Pole-zeroPlotoftheSystemFunctionPole-zeroplotExample:Aplotofthelocationsinthecomplexplaneofthepolesandzeros.ZerosrootsofN(z)=0——○ZerosrootsofD(z)=0——×zeros:z=0poles:z=1信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)z-DomainAnalysis—BlockDiagramRepresentationofDiscrete-timeSystemsinthez-DomainBlockDiagramRepresentationofDiscrete-timeSystemsinthez-DomainMultiplicationbyacoefficientAdderUnitdelayelement(f(-1)=0)信號(hào)與系統(tǒng)SignalsandSystems吉林大學(xué)CausalityandStabilityofDiscrete-TimeSystemsCausalityandStabilityofDiscrete-TimeSystemsCausalityk-domain:LTIsystemcausality
h(k)=0,k<0Proof:Necessity:Letf(k)=d(k)
f(k)=0fork<0,theny(k)=h(k).Ifthesystemiscausal,thenh(k)=0fork<0.Sufficiency:
f(k)=0,k<0
k-i<0(i>k),f(k-i)=0,thenIfh(k)=0,k<0
h(i)=0,i<0,then
yzs(k)=0,k<0CausalityandStabilityofDiscrete-TimeSystemsCausalityk-domain:LTIsystemcausality
h(k)=0,k<0z-domain: ,|z|>R0
AdiscreteLTIsystemiscausalifandonlyifth
溫馨提示
- 1. 本站所有資源如無特殊說明,都需要本地電腦安裝OFFICE2007和PDF閱讀器。圖紙軟件為CAD,CAXA,PROE,UG,SolidWorks等.壓縮文件請(qǐng)下載最新的WinRAR軟件解壓。
- 2. 本站的文檔不包含任何第三方提供的附件圖紙等,如果需要附件,請(qǐng)聯(lián)系上傳者。文件的所有權(quán)益歸上傳用戶所有。
- 3. 本站RAR壓縮包中若帶圖紙,網(wǎng)頁內(nèi)容里面會(huì)有圖紙預(yù)覽,若沒有圖紙預(yù)覽就沒有圖紙。
- 4. 未經(jīng)權(quán)益所有人同意不得將文件中的內(nèi)容挪作商業(yè)或盈利用途。
- 5. 人人文庫網(wǎng)僅提供信息存儲(chǔ)空間,僅對(duì)用戶上傳內(nèi)容的表現(xiàn)方式做保護(hù)處理,對(duì)用戶上傳分享的文檔內(nèi)容本身不做任何修改或編輯,并不能對(duì)任何下載內(nèi)容負(fù)責(zé)。
- 6. 下載文件中如有侵權(quán)或不適當(dāng)內(nèi)容,請(qǐng)與我們聯(lián)系,我們立即糾正。
- 7. 本站不保證下載資源的準(zhǔn)確性、安全性和完整性, 同時(shí)也不承擔(dān)用戶因使用這些下載資源對(duì)自己和他人造成任何形式的傷害或損失。
最新文檔
- 暑期實(shí)踐報(bào)告2000字(31篇)
- 繁峙縣南循環(huán)工程砂礫墊層施工方案
- Benalaxyl-Standard-生命科學(xué)試劑-MCE
- Anti-Rat-CD28-Antibody-JJ316-生命科學(xué)試劑-MCE
- 預(yù)防溺水心得體會(huì)
- 小學(xué)生我心中的好老師演講稿(31篇)
- 有趣教學(xué)教育課件
- 紀(jì)念館裝修貸款合同
- 旅游景區(qū)開發(fā)調(diào)研居間合同
- 塑料回收廢料清運(yùn)協(xié)議書
- 混凝土結(jié)構(gòu)無損檢測(cè)課件
- 水痘護(hù)理課件
- 地質(zhì)公園調(diào)研分析報(bào)告
- 刺猬杰斐遜和一樁懸案
- 浙江省A9協(xié)作體2023-2024學(xué)年高二上學(xué)期期中聯(lián)考英語試題2
- 多邊形的內(nèi)角和-說課課件
- 《恐龍世界》課件
- 如何提高學(xué)生的學(xué)習(xí)解決問題能力
- 基于同伴互評(píng)反饋機(jī)制的高中英語寫作教學(xué)實(shí)證研究
- 小學(xué)閱讀教學(xué)中隨文練筆的思考與探索
- 跑分后掛式卡取錢判決書
評(píng)論
0/150
提交評(píng)論