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 tananyagánakkomplexfejlesztésekonzorciumikeretben ***A projektazEurópaiUniótámogatásával, azEurópaiSzociálisAlaptársfinanszírozásávalvalósulmeg. 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 time domain Digitális jelek és rendszerek idõbeli leírása és analízise András Olás, Gergely Treplán, Dávid Tisza Digitális-neurális-, éskiloprocesszorosarchitektúrákonalapulójelfeldolgozás Digital-and Neural Based Signal Processing & KiloprocessorArrays Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 3 Contents • Introduction• Review of sampling and quantizing • Most important categories of discrete time signals • Basic definitions and operations on discrete time signals• Elementary discrete time signals • Elementary operations on discrete time signals • Convolution • Even-odd decomposition • Discrete Time system• Linearity • Time invariance Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 4 Contents • Causality • Stability, BIBO stability • LTI systems• Definitions of an LTI system • Impulse response of an LTI system • Properties of the convolution relation to the LTI systems • FIR, IIR systems • Block diagrams, signal flow diagrams of LTI systems • Basic flow graph types of a system• Direct form 1, 2 • Transposed forms • Serial, parallel forms • Description of the LTI systems by difference equation Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 5 Contents • General method for solving LTI system type difference equations • Examples• Stability • Time invariance • Causality • LTI system analysis in time domain Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 6 Introduction –review of signals • A signal is a physical quantity in space, time and other dimensions in the physical reality (e.g. voltage, current) • In mathematical sense a signal is a model of the physical quantity, a function of one or more independent variables (usually time or frequency) e.g.: • A discrete time (DT) signal is a function of time where the domain of time consist of a discrete set. It is a sequence in a mathematical sense e.g.: C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\function.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\sequence.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 7 Introduction –review of signals • A DT signal has values only where the domain has elements, it is undefinedat other places. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\sequence.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 8 Introduction –review of sampling, quantization Signal Time Value Analog signal x(t) Continuous Continuous Sampled signal x(n) orx(nT) Discrete Continuous Quantized signal Discrete Discrete Coded signal cn Discrete Binary ADC x(t) x(nT) . x(n) Sampling Quantization T .T Optimal representation cn Coding Compressing Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 9 Introduction –review of sampling, quantization • Weassumethatwehaveasufficientlyprecisequantizersoweneglecttheeffectofquantizationwhenweareanalyzingdiscretetimesystemsandsignals(weareworkingonlywiththesampledsignal) • Theconsequenceofthesamplingtheoremisthatthesignalsaretobeanalyzedarereconstructablewithoutlossonlyfromthesampledversions. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 10 Categories of discrete time signals The most important properties of the DT signals are • Time behavior, amplitude behavior, periodicity:support (finite, infinite, entrant), energy, power, even-oddness C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\limited support, entrant.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 11 Categories of discrete time signals C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\even odd.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 12 Categories of discrete time signals C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\periodic energy.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 13 Categories of discrete time signals • If the energy of a signal is finite, the average power is null. • There exist several signal of which it’s energy is infinite but the average power is finite. E.g. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\powersignal.png Digital Signal Processing: Time domain description Elementary DT signals • Kronecker delta or unit sample • Unit step function • Unit ramp function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 14 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\kronecker_delta.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\unitstep.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\unitramp.png Digital Signal Processing: Time domain description Elementary DT signals • Exponential function 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 15 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\exponential.png Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Addition: • Multiplication with constant: • Multiplication: • Time shift: • Accumulation: • Discrete time convolution: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 16 Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Addition: • Multiplication with constant: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 17 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\addition.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\mul_with_const.png Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Multiplication: • Time shift: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 18 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\multiplication.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\timeshift.png Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Accumulation: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 19 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\accumulate.png Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Discrete Time convolution: • Upper figure:• Blue dots: • Purple dots: • Red bars: • Lower figure:• Blue dots: • Red dot (sum of red bars):value of convolution at time instant n 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 20 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol13.png Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Discrete Time convolution: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 21 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol5.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol9.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol13.png Digital Signal Processing: Time domain description Operations on DT signals –basic operations • Discrete Time convolution: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 22 C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol17.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol20.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\convolution\convol23.png Digital Signal Processing: Time domain description Operations on DT signals –algebraic properties of the discrete time convolution: • Linear operator • Commutativity • Associativity • Distributivity • Associativity with scalar multiplication • Multiplicative identity • Complex conjugation 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 23 Digital Signal Processing: Time domain description Operations on DT signals –algebraic properties of the discrete time convolution: • Integration: • Differentiation: • Time invariance: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 24 Digital Signal Processing: Time domain description Operations on DT signals –Signal representation by convolution (multiplicative identity): Similarly as the definition of the Dirac delta we can use the same structure to define an arbitrary DT signal with convolution. • Dirac delta (continuous time): • Kronecker delta alternate definition (the property comes from definition): • Signal representation by convolution:every signal can be represented as a series of weighted Kronecker deltas 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 25 Digital Signal Processing: Time domain description C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\evenodd_func.png Operations on DT signals –even-odd decomposition AnimportantpropertyofaDTsignalforit’sanalysisisthatitcanbedecomposedtoevenandoddparts: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 26 Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 27 Discrete Time system A DT system is an object which operates in a discrete time fashion. We can describe a DT system with the input-output model: • A DT system has an input (inputs) • A DT system has an output (outputs)• Single input single output –SISO • Single input multiple output –SIMO • Multiple input single output –MISO • Multiple input multiple output –MIMO • In this course we are dealing with SISO systems only Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 28 Discrete Time system A DT system’s input-output model is described by a mapping, a rule between the input and the output. A DT system is a function which operates on the input. We can describe the simplest DT systems with the basic signal operations mentioned earlier. Discrete Time system Input,stimulus Output,system response Digital Signal Processing: Time domain description Operations of DT systems –basic operations • Addition: • Multiplication with constant: • Multiplication: • Time shift:Shift register: • Accumulation: • Discrete time convolution: 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 29 T T T T T T Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 30 Discrete Time system -Linearity • A DT system is linear if: • A DT system is non linear if that does not hold. • Example of a linear DT system: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 31 Discrete Time system -Linearity • Example of a non linear DT system: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 32 Discrete Time system –Time invariance • ADTsystemistimeinvariantifthesystemoperatorisinvarianttotimeshifts(thesystemdoesthe“same”onMonday,Tuesday,andoneveryholidaysaswell) • ADTsystemistimevariantifthesystemisdependentofthetimewhenisitevaluated. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 33 Discrete Time system –Time invariance • Example of a time invariant DT system: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 34 Discrete Time system –Time invariance • Example of a time variant DT system: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 35 Discrete Time system -Causality • ADTsystemiscausalifthesystem’snextstatecanbefullydeterminedbythecombinationoftheinputatthattimeinstantandtheinput’spreviousvalues.Inotherwordsthesystemisfullydeterminedbythepastandthepresentinput. • A typical causal system could be a real time audio processing codec or physical phenomenon Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 36 Discrete Time system -Causality • ADTsystemisanticausalifitonlydependsonthefutureinputvalues • ADTsystemisacausalornon-causalifthesystem’sresponseneedsbothfutureandpastinputvalues • Atypicalacausalsystemcouldbeanofflinecompressionalgorithm(e.g.zip),wherewecanseekforfuturesamplestodeterminethe“best”valueatthepresent Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 37 Dynamical system –stability • Adynamicalsystemissaidtobestabileifafteratimeinanabstractwidesenseitstayssomewhereanddoesnotmoveelsewhere. • Therearedifferenttypesofstabilitydefinedforadynamicalsystem.E.g.:• Lyapunovstability–ifallsolutionsofaDSstartnearanequilibriumpointandstaysclosetoit,thesystemissaidtobeLyapunovstabile • Marginalstability,asymptoticstability • Orbitalstability,structuralstability, • BIBOstability,etc. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 38 Discrete Time system –BIBO stability • ADTsystemissaidtobeBoundedInput–BoundedOutput(BIBO)stabileifforeveryfiniteamplitudeexcitationitproducesafiniteamplituderesponse. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 39 LTI systems AdiscretetimesystemisanLTI(lineartimeinvariantsystem)ifthelinearityandthetimeinvariancepropertyholds. AnLTIsystemisfullycharacterizedbyitsh(n)impulseresponsefunction. LTI system Input,stimulus Output,system response Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 40 LTI systems BecauseanLTIsystemcanbeviewedasaconvolutionbyit’simpulseresponsefunction(h(n))everypropertywhichtheconvolutionholdsistrueforanLTIsystemaswell. LTI system Input,stimulus Output,system response Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 41 LTI systems –impulse response (constructive definition) Theimpulseresponseofasystemcanbedefinedwiththemultiplicativeidentitypropertyoftheconvolution: Theimpulseresponseofasystemisthesystem’sresponsetotheKroneckerdelta. LTI system Input,stimulus Output,system response Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 42 LTI systems –consequences of the convolution properties • Associativity–serial combination of LTI systems LTI system LTI system 1 LTI system 2 Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 43 LTI systems –consequences of the convolution properties • 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: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 44 LTI systems –consequences of the convolution properties • Commutativity –switch ability of impulse response and excitation LTI system 1 LTI system 2 Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 45 LTI systems –consequences of the convolution properties • Distributivity –parallel combination of LTI systems LTI system LTI system 1 LTI system 2 Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 46 LTI systems –causality An LTI system is causal iff it’s impulse response is an entrant DT signal. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 47 LTI systems –BIBO stability AnLTIsystemisBIBOstableiffthesystem’simpulseresponseisabsolutesummable. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 48 LTI systems –BIBO stability ConsequentlyifanLTIsystemhasanimpulseresponsewithfinitenumberelements(limitedsupport),thesystemisalwaysBIBOstabile. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 49 LTI systems –FIR, IIR systems AnLTIsystem(orequivalentlyanLTIfilter)issaidtobeofFIRtype(finiteimpulseresponse)ifit’simpulseresponsehaslimitedsupport(finitelength). Ansystem/filterissaidtobeofIIRtype(infiniteimpulseresponse)ifit’simpulseresponsehasinfinitesupport. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\firh.png C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\iirh.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 50 LTI systems –flow graph representation • EveryLTIsystemcanbederivedbythecombinationofthefollowingbasicoperations:addition,multiplication,timeshift • Every FIR system is an IIR system • There exist IIR systems where the impulse response function can be represented by a closed form formula so they are implementable by a finite number of basic operations. They are calledrecursive IIR systems. • We deal with FIR and Recursive IIR. FIR type systems Recursive IIR systems IIR type systems Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 51 LTI systems –flow graph representation AFIRtypeLTIsystemcanbeviewedasapurelyfeedforwardtypeflowgraph: C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\fir_block.png FIR LTI Output, system response Input, stimulus Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 52 LTI systems –flow graph representation MostpracticallyusefulIIRfiltercanberepresentedbyafiniteelementfeedbackandfeedforwardtypeflowgraph: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 53 LTI systems –flow graph representation C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\iir_block.png IIR LTI Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 54 LTI systems –system equation Implementablesystemscanberepresentedbyfiniteelementaddition,multiplicationandtimeshiftoperations.Wearedealingwithsuchsystems. TheseLTIsystemscanbedescribedbyalineardifferenceequation(systemequation): Theorderofthissystem/filteristhemaximumnumberoftimeshiftusedeitherinthefeedforwardorthefeedbackpathforthesystem: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 55 Basic flow graph types –Direct form I Direct form I implementation of a filter is the direct readout implementation of the system equation: C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\iir_block.png IIR LTI Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 56 Basic flow graph types –Direct form II Direct form II implementation of a filter is the direct readout of the modified system equation: DF-II is a canonical representation respect to time delays. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\iir_block_DFII.png Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 57 Basic flow graph types –Transposed forms TransposedSISOfilterscanbeconstructedbyexchangingthesignalflowdirections. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\IIR_block_DFI_tr.png IIR LTI –DF-I Transposed Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 58 Other flow graph types Thereexistseveralothertypeofblockdiagramswhichwedon’tdealwithwithinthiscourse.E.g. • Lattice–laddertypeconstruction • SOS(secondordersections)• Canbeconnectedinserialorparallelfashion • Specialtypeserialandparallelconstructedfilters EverystructuretypecanimplementthesameLTIsystem,butfortheactualimplementation(letitbebyhardwareorsoftware)allhaveconsequencesoperatingproperties.E.g.internalnumericalstability,scalability,numericaloutputprecision,numberofmultiplication,adder,timeshiftelementsused. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 59 Canonical representations Acanonicalrepresentationrespecttoanelementtype(e.g.timeshift)istheimplementationwhichhastheleastnumberofelementfromthattype. DirectformIItypeisthecanonicalimplementationrespecttothetimeshiftoperation. C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\iir_block.png DF-I C:\home\tisda\itk.ppke.hu\wsn\tamop\Munka_NEURODSP\02_fejezet\iir_block_DFII.png DF-II Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 60 System complexity –SISO, MIMO SISO–singleinputsingleoutput SIMO–singleinputmultipleoutput MISO–multipleinputsingleoutput MIMO–multipleinputmultipleoutput ItisoftenusefulornecessarytobreakmorecomplexsystemsaparttoSISOsystemsandjointhemtogetherviaknownoperationstogettheoriginalbehaviorandbeanalyzable. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 61 LTI system description by difference equation EveryLTIsystemcanbedescribedinthetimedomainbyit’ssystemequationwhichisadiscretetimedifferenceequation. Thisequationcanbeanalyzedbywellknownmathematicalanalyticalmethods. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 62 LTI system description by difference equation Adifferenceequationforaspecifictimestepcanbecomputedrecursivelyifweknowtheinitialconditions: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 63 Difference equation analysis in time domainreview Asolutiontoadifferenceequationcanbecomposedtoahomogenousandaparticularsolution: Thehomogeneoussolutioniswhenthesystemhasnoexcitation: Theparticularsolutionisthecomposition: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 64 Homogenous solution Forsolvingthehomogenouspartweneedthecharacteristicequationofthedifferenceequation: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 65 Homogenous solution Bysolvingthecharacteristicpolynomialwegetthehomogenoussolutionwithunknowncoefficients: Ifthecharacteristicpolynomialhascomplexconjugaterootpairsthenthecorrespondingcoefficientswillbealsocomplexconjugate: Ifthecharacteristicpolynomialhasmultiplicityinit’sroots,thenwespeakofinternalresonance,andthesolutionwillhavetheformof: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 66 Particular solution Theparticularpartisdeterminedbytheexcitationofthesystem: Thereexistacoupleofmethodsforsolvingtheparticularsolution,weusethemethodofundeterminedcoefficients. Withthismethod,weexpecttheresponsetobeofaspecifictypeandwewanttoguessthecoefficientsofthespecifiedfunctionfamily. Forthemostcommonexcitationswegiveatablewiththeguessedresponsetype. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 67 Particular solution The guessed function families: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 68 Particular solution Forfindingtheparticularsolutionwiththeguessedfunctions,weneedtosolvethelineareq.systemtofindthevaluesoftheundeterminedcoefficientsbybacksubstitutingtheguessedfunctionandtheexcitationtothesystemequationhavingallterms(n>max(N,M)). Thetermshavingthetimeinsideshouldcancelouteachother. Thispartoftheparticularsolutionisvalidfromtimen.M+1 Tofindthepartoftheparticularsolutionfortimesn=M+1soweneedtosolveforthehomogenouspart’scoefficients: But we need to correct the particular solution with the additional terms for the time n=M+1usingthemethodofundeterminedcoefficients: weneedtobacksubstitutetheexcitationandtheguessedfunctionbacktothesystemequationusingallterms. Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 82 Example 1 –solution b Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 83 Example 1 –solution b iv. We need to solve: butforthisweneedtherecursivesolutionsforthefirstMtimeinstantfortherelaxedsystem: Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 84 Example 1 –solution b Digital Signal Processing: Time domain description 10/5/2011. TÁMOP –4.1.2-08/2/A/KMR-2009-0006 85 Example 1 –solution b we need to solve for the homogenous part’s coefficients: Butweneedtocorrecttheparticularsolutionwiththeadditionaltermsforthetimen