z变换与系统分析.ppt

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1、Discrete Time systems;Z-transformChapter 1 SummarylSignalslContinuous-time signallImpulse samplinglDiscrete-time signallQuantizationlSystemslFrequency responselImpulse responselTransfer functionChapter 1 SummarylSignal samplinglAliasing formulalShannon-Nyquist sampling theoremlSignal reconstructionl

2、Shannon reconstruction theoremlZero-order holdlPrefilter and postfilterlAnti-aliasing filterlAnti-imaging filterlAnalog Butterworth filterChapter 1 SummarylADClConverts analog input xa to N bits binary output blBinary counterlSARlFlashlDAClProduce analog output ya proportional to decimal value of N

3、bits binary number blWeighted resistorlR-2R ladderHomework Assignment#1Discrete Time SystemlDiscrete-time systemlInput x(k),output y(k)lCausal signalslContinuous Laplace transform and Fourier transform not availableExamplelHome MortgagelMonthly payment x(k)for a mortgage balance y(k)with an annual r

4、ate r compounded every monthlInitial condition y(-1)is the initial size of mortgagelWhat would be the monthly payment?lHow long does it take before paying the principle?ExamplelRunning average filterlm+1 years running average of evaluation scores x(k)from the kth yearlSimplify required floating-poin

5、t arithmetic operations(FLOPs)Z-tranformlZ-transform for a causal discrete-time signal x(k)lX(z)can be expressed as division of two polynomials for most signals lRegion of convergenceCommon SignalslUnit impulselZ-transform of unit impulse:lROC is entire complex planelUnit steplGeometric series:lZ-tr

6、ansform of unit step:lROC is|z|1Common SignalslCausal exponentialslDefinition:lDamping exponential:a1lUnit step:a=1lZ-transform:lROC:|z|aCommon SignalslExponentially damped sinelDefinition:lZ-transform:lTrigonometric form:Common SignalslExponentially damped cosinelDefinition:lZ-transform:lTrigonomet

7、ric form:Transfer FunctionlLinearity:Zx(k)+y(k)=X(z)+Y(z)lHomogeneity:Zax(k)=aX(z)lDelay property:lUnit delay:z-1X(z)=Zx(k-1)Transfer FunctionlZ-scale property:lZ-transform of sine:lZ-transform of cosine:lTime multiplication:ExamplelA pulse signal x(k)that has height of a and duration of MlDiscrete

8、signal:lZ-transform:lROC:|z|1ExamplelAn unit ramp signal x(k)lDiscrete signal:lZ-transform:lZ-transfer for x(k)=k2u(k):lZ-transfer for x(k)=k3u(k):Initial and Final ValuelInitial value:x(0)l lFinal value:y()l l(z-1)Y(z)has no poles on or outside of unit circle lSteady-stateCommon Z-transformZ-transf

9、orm propertiesInverse Z-TransformlZ-transform:lPolynomial expression:lPoles at 0 and lCommon pairs:lInverse z-transform:lContour in ROC including all poles of X(z)lTable for common functionsPartial Fraction MethodlZ-transform of x(k)=b(k-s):lZ-transform of x(k)=raku(k):lPartial fraction decompositio

10、n:lCoefficients:bj can be acquired by long division of z-1 polynomialsPartial Fraction MethodlInverse z-transform:lExample:ExampleResidue MethodlResidue Theorem:lResidue:lInverse z_transform:lPolynomial expression of X(z):lSimple residue:mi=1lMultiple residue:mi1Simple Pole ExamplelInverse z-transfo

11、rm:lExample:Multiple Pole ExampleSynthetic DivisionlPolynomials of z-1:l lm+r=nlLong division of bz-1 by az-1:l lIf r=0,x(k)=q(k)lIf r0,time delay of q(k)lx(k)=0,0krlx(k)=q(k-r),rkExampleSynthetic Division AlgorithmlExpress X(z)as division of two normalized polynomials in z-1lSet q(0)=b0lFrom 1kmlFr

12、om mnlZeros at z=0,mnlSystem Modesly(k)=H(z)X(z)=natural modes+forced modeslNatural modes are generated by poles of H(z)lForced modes are generated by poles of X(z)l lPole-zero cancellation can suppress modesExamplePole-zero Cancellation suppressed one natural mode and one forced mode,leaving only o

13、ne natural modeDC GainlStable systeml lAll natural modes decay to zeros when all poles of H(z)are within unit circlel lSteady-state response Y1(z)=H(z)U(z)l DC GainlDC gain=y1(k)/u(k)=aH(1)/a=H(1)lH(1)is the DC gain for a steady system with a transfer function H(z)lAC gain at other frequency can be

14、calculated with different z valueExamplelComb filterMatlab CommandslFilter(b,a,x)lB is vector of coefficients for numerator of transfer functionlReminder:in Matlab,b(1)=b0lA is vector of coefficients for denominator of transfer functionlReminder:in Matlab,a(1)=1lStem(y)Matlab CommandslRootslUsed to

15、find roots of a polynomiallRoots(a)find poleslRoots(b)find zeroslResiduelR,pole=residue(b,a)Signal Flow GraphslDifference equation:lTransfer function:lSignal flow:ScaleSumOutputSignal Flow GraphsSignal Flow GraphslDifference equations:lTransfer function:ARMA modelslAutoregressive modell l lMoving av

16、erage modell l ci=1 running average filterl lARMA model:HARMA(z)=HAR(z)HMA(z)ExampleXout/xin=abc/(1-cd-bce)ConvolutionlContinuous-time convolutionl lDiscrete-time convolutionl lZh(k)*x(k)=H(z)X(z)lAlternativeExamplelCausal sine wave x(k)to a system h(k),what is the output y(k)if initial condition is

17、 zeroImpulse ResponselImpulse response:system output when input is unit impulse function,(k),and initial conditions are zerolZx(k)=Z(k)=1l lY(z)=H(z)X(z)=H(z)ly(k)=h(k)=Z-1H(z)ExamplelImpulse response for following discrete-time systemFIR and IIR SystemslFinite-impulse response(FIR)systemslHave fini

18、te number of nonzero samples in impulse response h(k)lMA model:lTransfer function:lInfinite-impulse response(IIR)systemslHave infinite number of nonzero samples in impulse response h(k)lAR model:lNatural modes:ExamplelRunning average filter:BIBO StablelBounded-input bounded-output stable:lInput|x(k)

19、|BxlOutput|y(k)|ByBIBO StablelImpulse responselTime-domain stability constrainlBIBO stable if and only if impulse response h(k)is summableBIBO StablelImpulse response criterionlFIR system:finite number of sampleslTransfer function:lAll poles at 0 lImpulse response:lBIBO StablelIIR system:infinite nu

20、mber of sampleslNatural modes:lModes are convergent when|p|1lBIBO stable if and only if all poles within unit circle:|pi|1 Unstable|pi|=1 marginal unstableExamplelConsider a discrete-time system with following transfer function.lImpulse responselConsider following input.lZero-state responseJury Test

21、lDenominator polynomial coefficients of H(z)l lRoots(a)finds all poles of H(z)lDesigning a stable systemlAll poles within unit circlelStability criterion for coefficients alJury testlCriterion 1:a(1)0lCriterion 2:(-1)na(-1)0Jury TableJury TestlStability Conditionl lExamplelFind out the range for par

22、ameters a1 and a2 in following discrete-time system ExampleExamplelReal poles and complex poleslPoleslComplex pole:lReal pole:lImpulse responseExampleDiscrete-Time SystemFrequency ResponselDC gain:zero-state response to unit step input u(k)lDC gain=H(1)lFrequency response to signal that has frequenc

23、y components within 0 fs/2l lPolar formlMagnitude responselPhase responseFrequency ResponselSymmetry propertylFrequency in-fs/2 fs/2lSymmetry of conjugateslMagnitude response:even functionlPhase response:odd functionAll-Pass FilterlHas constant magnitude response for all frequency componentslApplica

24、tions in phase distortionslSimple all-pass filterImpulse ResponselDiscrete-time Fourier transformlImpulse responselConjugate of impulse responseZero-State ResponselConsider a sinusoidal inputl lZero-state responseZero-State ResponseFrequency ResponselSinusoidal steady-state responsel lGain:A(fa)lPha

25、se shift:(fa)lAll-pass filterlGain:1lPhase shift:lFor stable filter,|r|1,always negative phase shiftlPositive group delay in time ExamplelSecond-order digital filterExamplePeriodic InputslTruncated Fourier serieslDiscrete-time signallPeriodic steady-state responseExamplelSteady-state response with f

26、ollowing stable first-order filterl lFrequency responseExamplelTruncated Fourier series of a pulse trainlFor=0.01 sec,a=/4ExampleMatlab CommandslH f=f_freq(b,a,N,fs)lb=0.2,a=1-0.8,fs=2000 HzChapter 2 SummarylZ-transforml l l lInverse Z-transformlPartial Fraction MethodlResidue MethodlLong divisionCh

27、apter 2 SummarylTransfer functionl lImpulse responsel lZero-state responselNatural modes,forced modeslPole-zero cancellationlHarmonic forcingl Chapter 2 SummaryDifference equation:Transfer function:Signal flow:ScaleSumOutputChapter 2 SummarylARMA modelslAutoregression modelslMoving average modelslFI

28、R and IIR systemslFIR systemslFinite number of nonzero samples in impulse responselStable systemlIIR systemslinfinite number of nonzero samples in impulse responseChapter 2 SummarylBIBO Stability CriterionslBIBO stable if and only if all poles within unit circle:|pi|1 Unstablel|pi|=1 marginal unstablelJury testlStability criterion for coefficients alCriterion 1:a(1)0lCriterion 2:(-1)na(-1)0lJury tableChapter 2 SummarylFrequency responsel lDC gain:H(1)lMagnitude response(AC gain):A(f)lPhase response(Phase shift):(f)lSteady-state response for periodic input