10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 1 Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg. PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY sote_logo.jpg dk_fejlec.gif INFOBLOKK 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 2 Peter Pazmany Catholic University Faculty of Information Technology Description digital signals and systems in transform (Z,DFT) domain Digitális jelek és rendszerek leírása és analízise transzformált tartományokban János Levendovszky, AndrásOláh, DávidTisza, Kálmán Tornai, GergelyTreplán Digitális-neurális-, éskiloprocesszorosarchitektúrákonalapulójelfeldolgozás Digital-and Neural Based Signal Processing & KiloprocessorArrays Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 3 Contents • Introduction• Review of basic definitions and operations on discrete time signals • Review of LTI system definition by difference equation • Motivation of using the z-transform for analysis of LTI systems. • Introduction of z-transform• Definition • RoC • Properties of z-transform • Poles-zeros (review from complex calculus) • Inverse z-transform • z-transform of elementary discrete time signals Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 4 Contents • z-transform of elementary operations on discrete time signals • z-transform of convolution • LTI systems in z-domain• Transfer function • Poles, zeroes of the transfer function • Z-transform and Fourier transform • Polynomial manipulation review• Polynomial long division • Partial fraction expansion • BIBO stability Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 5 Contents • Convolution properties in the z-domain• Serial combination of LTI systems • Parallel combination of LTI systems • Outline of special filter types• Linear phase filters • Minimum phase filters • All pass filters • Examples of z-transform in system analysis Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 6 LTI system as difference equation (review) EveryLTIsystemcanbedescribedinthetimedomainbyitssystemequationwhichisadiscretetimedifferenceequation. Thisequationcanbeanalyzedbywellknownmathematicalanalyticalmethodsinthetimedomain. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 7 Motivation of using transformed domains • Certaintransformationsenableustodealwiththerecurrencerelation(differenceequation)inthetransformeddomainasasimplealgebraicequation. • Aftersolvingthealgebraicequationswithsimplemathematicaltoolswecantransformbackthemintothetogetthetimedomainresponseofthesystem Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 8 Motivation of using transformed domains • LTIsystemscanbefullycharacterizedinthetransformeddomainbynumbers(polesandzeroes). • Fromthesenumberswecansimplytellimportantfeaturesofthesystem,e.g.stability,structuralbehavior,degreeofthesystem,frequencycharacteristics. • Incontinuoustimeweuseintegraltransformsindiscretetimeweusepowerseries,becausethesetransformshavenicepropertiesforderivationandconvolution. • LTIsystemsinthetransformeddomainwillberepresentedbydivisionofpolynomials. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 9 Geometric series (review) • Notethatatthetimedomainanalysiswemetwithgeometricsequences(e.g.homogenoussolution)andwhenweapplytheLaurentseriesthebasicreviewofthegeometricserieswillcomeinhandy.Wehavethefollowingclosedformsfortheseries: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 10 The z-transform • Thez-transform(bilateral)ofadiscretetimesignalisdefinedasaninfinitecomplexpowerseries(Laurentseries): • For compactness we use the notationfor the z-transform and for the inverse z-transform. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 11 The z-transform –example (entrant) • Let’stransformthefollowingseriesintothez-domain:Notethattheseriesisconvergentonlyforaregionofzvalues.Thisexampleisconvergentfor Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 12 The z-transform –example (leaving) • Let’s transform the following series into the z-domain: This example is convergent for • We have to define Region of Convergence. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 13 The z-transform –RoC • The region of convergence is defined as follows:all those numbers for which the series are convergent. • Example 3: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 14 The z-transform –RoC • Wecandecomposeasignaltoanentrantandtoaleavingpart: Thez-transformofasignalcanbealsodecomposedtotwoparts: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 15 The z-transform –RoC • The two parts have two different region of convergence Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 16 The z-transform –RoC C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\z_transform_diag.png Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 17 Properties of z-transform Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 18 Review of poles-zeroes (complex analysis) Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 19 Laurent series and inversez-transform Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 20 Inverse z-transform in practice • InanLTIsystemweexpectsystemexponentialtyperesponses(seetimedomainanalysishomogenoussolution)andexcitationlikeanswers(seetimedomainanalysisparticularpart). • Sousuallyitisenoughifwehaveatableofthemostcommonfunctionsz-transforms. • Andthenweneedtomanipulatethemintocorrectformandreversethemintothetimedomainfromatable.Thuswedon’tneedtoevaluatethecomplexclosedpathintegral. Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • We will present the z-transform, the visualization and the pole-zero plot of some elementary functions: • Pole-zero plot of an LTI system 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 21 Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Kronecker delta or unit sample 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 22 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\kronecker_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Unit step functionpole-zero plot: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 23 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\pole_zero_u.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 24 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitstep_z.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function (note that outside the RoC the function is not determined) 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 25 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitstep_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 26 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitstep_leaving_trafsame_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 27 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitstep_leaving_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Unit ramp function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 28 Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Pole-zero plot of the unit ramp function in the z-domain 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 29 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\pole_zero_ur.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 30 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitramp_z.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 31 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitramp_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 32 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\unitramp_leaving_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Exponential function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 33 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\exponential.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • z-transform of the exponential function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 34 Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Pole-zero plot of the exponential function in the z-domain 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 35 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\pole_zero_exp.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 36 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\exp_z.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 37 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\exp_z_valid.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • z-transform of sineand cosine functions 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 38 Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Poles and zeroes sine and cosine functions in the z-domain 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 39 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\pole_zero_cos.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 40 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\cos_z.png Digital Signal Processing: Transformed domain description Z-transform of elementary DT signals • Plot of function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 41 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\cos_z_valid.png Digital Signal Processing: Transformed domain description Operations of DT systems –basic operations 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 42 Digital Signal Processing: Transformed domain description Operations of DT systems –basic operations 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 43 Digital Signal Processing: Transformed domain description Operations of DT systems –basic operations 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 44 Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 45 LTI systems in the z-domain ALTIsysteminthetimedomainperformsadiscretetimeconvolutiononitsinputwithitsimpulseresponsefunction AnLTIsystemisfullycharacterizedbyitsimpulseresponsefunction(h(n)). Ifweexaminethesamesysteminthetransformeddomain,wecanusetheconvolutiontomultiplicationpropertyofthez-transform: LTI system Input,stimulus Output,system response Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 46 LTI systems in the z-domain ALTIsysteminthez-domainperformsamultiplicationonitsinput’sz-transformandthesystem’stransferfunction An LTI system is fully characterized by its transfer function H(z). LTI system Input,stimulus Output,system response Digital Signal Processing: Transformed domain description Transfer function of a system • Remember the linear constant-coefficient difference equation which described an arbitrary LTI system:Z-transforming it (using the time shift property and linearity): 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 47 Digital Signal Processing: Transformed domain description Transfer function of a system • We can define the transfer function of an LTI system: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 48 Digital Signal Processing: Transformed domain description Transfer function of a system –poles and zeroes • WedefinethepolesandzeroesofanLTIsystemasthepolesandzeroesofit’stransferfunction: • Forunderstandingwhatdoesapoleorazero“do”weneedtounderstandtheconnectionbetweenthez-transformandthediscreteFouriertransform. 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 49 Digital Signal Processing: Transformed domain description Z-transform and the Fourier transform • Remember the definition of the bilateral z-transform: • Ifwesubstitute wegettheFourierseriesofXwheretheFouriercoefficientsarethetimesamples. • Notethatusuallyweareinterestedinaperiodic,continuoustimefunction’sdecompositionintoFourierseries,wherethecoefficientsrepresenttheweightofthefrequencycomponent,butfromthedualitypropertywecandothisintheoppositedirection. 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 50 Digital Signal Processing: Transformed domain description Z-transform and the Fourier transform • Herethefrequencyfunctioniscontinuousandperiodic,andtheFouriercoefficientsarethetimesamples. • Xisafunctionoftheangularfrequency.Weviewtheoriginalwholecomplexzplaneonlyontheunitcircle. • Thefollowingillustrationshowstheconnection. • Ontheupperfiguretheredlineshowsthefunctionvaluesontheunitcircle.Onthelowerfigurethexaxis,theangularfrequencyisthe“unitcircle”,foldedouttoaline. 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 51 Digital Signal Processing: Transformed domain description Z-transform and the Fourier transform 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 52 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\exp_z_freq3D.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\exp_z_freq.png Digital Signal Processing: Transformed domain description Transfer function of a system –poles and zeroes • Ifweviewthesysteminthefrequencydomain(properlyintroducedlater)wecansaythatazerointhetransferfunctionatangularfrequencyomegaisattenuatingthatfrequencyandit’ssurroundings. • The attenuation strength is dependent how close ris to 1 (how close is the zero to the unit circle) • The next figure illustrates a simple zero and it’s frequency characteristics with zero at: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 53 Digital Signal Processing: Transformed domain description Transfer function of a system –poles and zeroes 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 54 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\simplezero3D.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\simplezero.png Digital Signal Processing: Transformed domain description Transfer function of a system –poles and zeroes • Ifweviewthesysteminthefrequencydomainapoleinthetransferfunction atanangularfrequencyomegaisamplifyingthatfrequencyandit’ssurroundings. • Theamplifyingstrengthisdependenthowcloseristo1(howcloseisthepoletotheunitcircle) • The next figure illustrates a simple two pole and it’s frequency characteristics with poles: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 55 Digital Signal Processing: Transformed domain description Transfer function of a system –poles and zeroes 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 56 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\simplepole.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\simplepole3D.png Digital Signal Processing: Transformed domain description Transfer function of a system • It can be useful to have the transfer function in the following form:It will be useful to perform the inverse z-transform because 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 57 Digital Signal Processing: Transformed domain description Transfer function of a system • Andfortherootfactoredterm Sowecancomputetheinversez-transformeasily.Therootsofthecharacteristicpolynomialarethepolesofthetransferfunction. 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 58 Digital Signal Processing: Transformed domain description Review of polynomial manipulation • We can manipulate polynomials like: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 59 Digital Signal Processing: Transformed domain description Review of polynomial long division • Polynomialsoflowerdegreecandivideapolynomialwithahigherdegree.WewillusethistoanalyzeLTIsystems. • Notethatyoulearnedthismethodinelementaryschooltodividenumbers. • The method will be illustrated on an example. 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 60 Digital Signal Processing: Transformed domain description Review of polynomial long division 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 61 Digital Signal Processing: Transformed domain description Review of polynomial long division 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 62 Digital Signal Processing: Transformed domain description Review of partial fraction expansion of polynomials 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 63 Digital Signal Processing: Transformed domain description Review of partial fraction expansion of polynomials 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 64 Digital Signal Processing: Transformed domain description Review of partial fraction expansion of polynomials 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 65 Digital Signal Processing: Transformed domain description Review of partial fraction expansion of polynomials 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 66 Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 67 LTI systems –BIBO stability review An LTI system is BIBO stable iffthe system’s impulse response is absolute summable. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 68 LTI systems –BIBO stability analysis in z-domain • IfweknowthepolesofthetransferfunctionwecanimmediatelyseeifthesystemisBIBOstableornot. • IftheallthepolesarewithintheunitcirclethesystemisBIBOstable. • IfapoleisoutsideorontheunitcirclethesystemisnotBIBOstable. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 69 LTI systems –BIBO stability analysis in z-domain • Example: C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\stabile_sys_ztraf_impulse_response.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\stabile_sys_ztraf_pole_zero.png Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 70 LTI systems –BIBO stability analysis in z-domain • Example: C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\instabile_sys_ztraf_impulse_response.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\instabile_sys_ztraf_pole_zero.png Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 71 Impulse response characteristics based on poles C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\abra.png Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 72 Convolution properties in the z-domain • Associativity –serial combination of LTI systems LTI system LTI system 1 LTI system 2 Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 73 Serial combination of simple IIR systems • If we write the system as • It can be realized that simple one poles one zeroes can be cascaded to get the original transfer function. LTI system Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 74 Convolution properties in the z-domain • Commutativity –switch ability of LTI systems LTI system LTI system 1 LTI system 2 LTI system LTI system 2 LTI system 1 Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 75 Convolution properties in the z-domain • Commutativity–switchabilityofimpulseresponseandexcitation LTI system 1 LTI system 2 Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 76 Convolution properties in the z-domain • Distributivity –parallel combination of LTI systems LTI system LTI system 1 LTI system 2 Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 77 Parallel combinations of simple IIR systems • Ifwewritethesystemas whichcanbeobtainedbypolynomiallongdivisionandpartialfractiondecomposition,itcanberealizedthataparallelimplementationofthesystemis: LTI system Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 78 Linear phase filters • Alinearphasefilterisafilterwherethephasefrequencyresponseofthesystemisalinearfunction. • Suchsystemshastheirimpulseresponsesymmetric: • E.g. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\linphase.png Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 79 Minimum phase filters • Wecallafilteraminimumphasefilter,ifthefilterandit’sinversearebothcausalandstable. • H(z)systemisstableandcausalifthepoles(rootsofA(z))arewithintheunitcircle,butwearefreetochoosethezeroes(rootsofB(z))ofsuchsystem. • TheinverseofH(z)willalsobestableandcausalifit’spolesarewithintheunitcircle(rootsofB(z)) • A filter is minimum phase if all it’s poles and zeroes are inside the unit circle. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 80 All pass filters • Afilterissaidtobeallpass(itpassesallthefrequencieswiththesameamplitudebutmaypasswithdifferentphase)if • Theallpassfiltershavetheirtransferfunctionintheform(notethatrealcoefficientscanberealizedbyhavingcomplexconjugatepolepairs) • Example C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\03_fejezet\allpass.png Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 81 Example 1 An LTI system is given by its system equation: a) Givetheimpulseresponseofthissystem,bysolvingitinthez-domain b) Givetheresponseofthesystemforthegivenstimulusbysolvingitinthez-domain c) Is it a BIBO stablesystem? Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 82 Example 1 –solution a Togivetheimpulseresponseofthissystem,bysolvingitinthez-domainweneedthetransferfunctionofthesystem: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 83 Example 1 –solution a Wehavethetransferfunction,weneedtoinversez-transformitbyeitherevaluatingthecontourintegralormanipulatingittoaformwhereweknowtheindividualtransformsofeachterm.Wechoosetodothelatterbyusingthepolynomiallongdivisionandthepartialfractionexpansion. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 84 Example 1 –solution a We apply the polynomial long division: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 85 Example 1 –solution a Do the partial fraction expansion to the residual part Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 86 Example 1 –solution a Wehavemanipulatedthetransferfunctiontohavetheform: we can do the inverse z-transform easily: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 87 Example 1 –solution a Let’scomparetheresultswiththesolutiongotfromthetimedomainanalysis: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 88 Example 1 –solution b Weknowthetransferfunction,weneedtocomputethez-transformofthestimulusx(n) Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 89 Example 1 –solution b We have the z-transform of the response, we need to inverse z-transform it Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 90 Example 1 –solution b We have manipulated the response to have the form: we can do the inverse z-transform easily: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 91 Example 1 –solution b Let’s compare the results with the solution got from the time domain analysis: Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 92 Example 1 –solution c c) ForthesystemtobeBIBOstablethenecessaryconditionistohavethesystemtransferfunction’spoleswithintheunitcircle. Thepolesareexactlyattheunitcircle,sothesystemisnotBIBOstable. Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 93 Summary of system specific functions and properties Time domain excitation response Impulse response System: Transformed, z-domain Transfer function System: Pole-zeroplot: Frequency(Fourier) domain Spectrum of the excitation: Spectrum of the response: Spectrum: Amplitude,phase char: Bode plot: freq.vs. amplorfreq. vs. phase plot Nyquistplot:parametric plot of in the complex plane or equivalently in polar coords: phase vs. amp. LTI system Digital Signal Processing: Transformed domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 94 Summary • Introduction and illustration of the z-transformation. • Properties of the z-transform. • Elementary signals transforms. • Use of the z-transform in the analysis of the LTI systems. • Filter realization forms based on the z-transform properties. • DFT and z-transform connection. • Examples.