Z变换与系统分析.ppt

Discrete Time systems; Z-transform Chapter 1 Summary lSignals lContinuous-time signal lImpulse sampling lDiscrete-time signal lQuantization lSystems lFrequency response lImpulse response lTransfer function Chapter 1 Summary lSignal sampling lAliasing formula lShannon-Nyquist sampling theorem lSignal reconstruction lShannon reconstruction theorem lZero-order hold lPrefilter and postfilter lAnti-aliasing filter lAnti-imaging filter lAnalog Butterworth filter Chapter 1 Summary lADC lConverts analog input xa to N bits binary output b lBinary counter lSAR lFlash lDAC lProduce analog output ya proportional to decimal value of N bits binary number b lWeighted resistor lR-2R ladder Homework Assignment #1 Discrete Time System lDiscrete-time system lInput x(k), output y(k) lCausal signals lContinuous Laplace transform and Fourier transform not available Example lHome Mortgage lMonthly payment x(k) for a mortgage balance y(k) with an annual rate r compounded every month lInitial condition y(-1) is the initial size of mortgage lWhat would be the monthly payment? lHow long does it take before paying the principle? Example lRunning average filter lm+1 years running average of evaluation scores x(k) from the kth year lSimplify required floating-point arithmetic operations (FLOPs) Z-tranform lZ-transform for a causal discrete-time signal x(k) lX(z) can be expressed as division of two polynomials for most signals lRegion of convergence Common Signals lUnit impulse lZ-transform of unit impulse: lROC is entire complex plane lUnit step lGeometric series: lZ-transform of unit step: lROC is |z|1 Common Signals lCausal exponentials lDefinition: lDamping exponential: a1 lUnit step: a=1 lZ-transform: lROC: |z|a Common Signals lExponentially damped sine lDefinition: lZ-transform: lTrigonometric form: Common Signals lExponentially damped cosine lDefinition: lZ-transform: lTrigonometric form: Transfer Function lLinearity: Zx(k)+y(k)=X(z)+Y(z) lHomogeneity: Zax(k)=aX(z) lDelay property: lUnit delay: z-1X(z)=Zx(k-1) Transfer Function lZ-scale property: lZ-transform of sine: lZ-transform of cosine: lTime multiplication: Example lA pulse signal x(k) that has height of a and duration of M lDiscrete signal: lZ-transform: lROC: |z|1 Example lAn 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 Value lInitial value: x(0) l lFinal value: y() l l(z-1)Y(z)has no poles on or outside of unit circle lSteady-state Common Z-transform Z-transform properties Inverse Z-Transform lZ-transform: lPolynomial expression: lPoles at 0 and lCommon pairs: lInverse z-transform: lContour in ROC including all poles of X(z) lTable for common functions Partial Fraction Method lZ-transform of x(k)=b(k-s): lZ-transform of x(k)=raku(k): lPartial fraction decomposition: lCoefficients: bj can be acquired by long division of z-1 polynomials Partial Fraction Method lInverse z-transform: lExample: Example Residue Method lResidue Theorem: lResidue: lInverse z_transform: lPolynomial expression of X(z): lSimple residue: mi=1 lMultiple residue: mi1 Simple Pole Example lInverse z-transform: lExample: Multiple Pole Example Synthetic Division lPolynomials of z-1: l lm+r=n lLong 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, 0kn lZeros at z=0, m1 Unstable |pi|=1 marginal unstable Example lConsider a discrete-time system with following transfer function. lImpulse response lConsider following input. lZero-state response Jury Test lDenominator polynomial coefficients of H(z) l lRoots(a) finds all poles of H(z) lDesigning a stable system lAll poles within unit circle lStability criterion for coefficients a lJury test lCriterion 1: a(1)0 lCriterion 2: (-1)na(-1)0 Jury Table Jury Test lStability Condition l lExample lFind out the range for parameters a1 and a2 in following discrete-time system Example Example lReal poles and complex poles lPoles lComplex pole: lReal pole: lImpulse response Example Discrete-Time System Frequency Response lDC gain: zero-state response to unit step input u(k) lDC gain=H(1) lFrequency response to signal that has frequency components within 0 fs/2 l lPolar form lMagnitude response lPhase response Frequency Response lSymmetry property lFrequency in -fs/2 fs/2 lSymmetry of conjugates lMagnitude response: even function lPhase response: odd function All-Pass Filter lHas constant magnitude response for all frequency components lApplications in phase distortions lSimple all-pass filter Impulse Response lDiscrete-time Fourier transform lImpulse response lConjugate of impulse response Zero-State Response lConsider a sinusoidal input l lZero-state response Zero-State Response Frequency Response lSinusoidal steady-state response l lGain: A(fa) lPhase shift: (fa) lAll-pass filter lGain: 1 lPhase shift: lFor stable filter, |r|1 Unstable l|pi|=1 marginal unstable lJury test lStabil