测试信号英文版课件：Final Review

1、Final ReviewWhy signals should be processed?Signals are carriers of informationUseful and unwantedExtracting, enhancing, storing and transmitting the useful informationHow signals are being processed?- Analog Signal Processing vs. Digital Signal Processing 2The framework of DSPPrF: antialiasing filt

2、ering PoF: smooth out the staircase waveform3Chapter 2 3(Digital) Signal, System, Transformation 4Representation of Digital Signalsbraces&arrowmathematical functiongraphics5Basic Operations of SequenceN 0 and N 0 * N61.一般用画图法求解 2.卷积和还可以用z变换来做Energyaverage powerabsolutely summable7Aperiodic signalPer

3、iodic signal bounded sequenceDiscrete-Time SystemsWhat is the impluse response of an LTI-system how to obtain it based on its definition?8LTIDiscrete-Time SystemsInput-Output relationship9LTIyn = xn hn*hnDiscrete-Time Systems ClassificationLinear SystemShift (Time)-Invariant SystemLinear Time-Invari

4、ant SystemCausal SystemStable System1011Linear Discrete-Time SystemsDT systemDT systemDT systemDefinitionLinear Discrete-Time SystemsAccumulator -For an inputthe output is Hence, the above system is linear12Nonlinear Discrete-Time SystemConsiderOutputs and for inputs and are given by13Nonlinear Disc

5、rete-Time SystemOutput yn due to an input is given by14Nonlinear Discrete-Time SystemOn the other handHence, the system is nonlinear1516Shift-Invariant SystemDT systemDefinitionDT systemShift-Invariant SystemIn the case of sequences and systems with indices n related to discrete instants of time, th

6、e above property is called time-invariance propertyTime-invariance property ensures that for a specified input, the output is independent of the time the input is being applied17Shift-Invariant SystemExample 1 - Consider the Modulator with an input-output relation given by18xnynwnwnShift-Invariant S

7、ystemFor an input the output is given by19Shift-Invariant SystemHowever from the definition of the ModulatorHence, the Modulator is a time-varying system20Linear Time-Invariant SystemLinear Time-Invariant (LTI) System - A system satisfying both the linearity and the time-invariance propertyLTI syste

8、ms are mathematically easy to analyze and characterize, Therefore, easy to design21Stability Condition in Terms of the DefinitionWe consider here the bounded-input, bounded-output (BIBO) stability definitionIf yn is the response to an input xn and if for all values of nthen for all values of n22Stab

9、ility Condition in Terms of the DefinitionExample - The M-point moving average filter is BIBO stable:For a bounded input we have23Stability Condition in Terms of the Impulse ResponseBIBO Stability Condition - A discrete-time LTI system is BIBO stable if the output sequence yn remains bounded for any

10、 bounded input sequencexnA discrete-time LTI system is BIBO stable if and only if its impulse response sequence hn is absolutely summable, i.e.S24Stability Condition in Terms of the Impulse ResponseExample - Consider a causal discrete-time LTI system with an impulse responseFor this systemTherefore

11、if for which the system is BIBO stableIf , the system is not BIBO stableSif25Stability Condition in Terms of the Pole Locations (Causal system ONLY)Conclusion: All poles of a causal stable transfer function H(z) must be strictly inside the unit circle The stability region (shown shaded) in the z-pla

12、ne is shown below1Re zj Im zunit circlestability region26Stability Condition in Terms of the Pole LocationsExample - The factored form ofiswhich has a real pole at z = 0.902 and a real pole at z = 0.943Since both poles are inside the unit circle, H(z) is BIBO stable27Stability Condition in Terms of

13、the Pole LocationsExample - The factored form ofiswhich has a real pole on the unit circle at z = 1 and the other pole inside the unit circleSince one pole is not inside but on the unit circle, H(z) is unstable28How to determine the stability of a discrete-time system? Definition If , we have . Time

14、-domain (使用前提：系统是LTI的) Z-domain The ROC of H(z) includes the unit circle (使用前提：系统是LTI的)29SAll poles of H(z) must be strictly inside the unit circle (使用前提：系统是LTI且因果的)Causality Condition in Terms of the DefinitionIn a causal system, the -th output sample depends only on input samples xn for and does n

15、ot depend on input samples forLet and be the responses of a causal discrete-time system to the inputs and , respectively 30Causality Condition in Terms of the DefinitionThen for n Nimplies also that for n NFor a causal system, changes in output samples do not precede changes in the input samples31Ca

16、usality Condition in Terms of the DefinitionExamples of causal systems:Examples of noncausal systems:32Causality Condition in Terms of the DefinitionA noncausal system can be implemented as a causal system by delaying the output by an appropriate number of samplesFor example a causal implementation

17、of the factor-of-2 interpolator (see below) is given by33Causality Condition in Terms of the Impulse Response A discrete-time LTI system is causal if and only if its impulse response hn is a causal sequence, i.e., hk=0, for k=M, and usually N should be the power of 2 (512, 1024, ).Digital spectrum a

18、nalysis of continuous-time signal Using the DFT时域的有限化和离散化一频域的有限化和离散化二误差产生原因及解决办法三谱分析时DFT参数的选择四谱分析时DFT参数的选择1）估计待分析信号中频率范围和频率上限 。2）选定抽样频率 。3）根据分析精度 F，确定连续信号有效长度 。4）确定点数N。5）选窗口。Definition61z-Transform62Table : Commonly Used z-Transform Pairs63Rational z-TransformsHow to find Zeros, poles?-remember to c

19、heck zeros and poles at 0 or 四种类型序列的收敛域a)有限长序列 b) 右边序列 c) 左边序列 d) 双边序列ROC of a Rational z-Transform65z-Transform Propertiesz-Transform PropertiesExample - Consider the two-sided sequenceLet and with X(z) and Y(z) denoting, respectively, their z-transformsNowand66z-Transform PropertiesUsing the linea

20、rity property we arrive atThe ROC of V(z) is given by the overlap regions of and If , then there is an overlap and the ROC is an annular regionIf , then there is no overlap and V(z) does not exist 67z-Transform PropertiesExample - Determine the z-transform and its ROC of the causal sequenceWe can ex

21、press xn = vn + v*n whereThe z-transform of vn is given by68z-Transform PropertiesUsing the conjugation property we obtain the z-transform of v*n asFinally, using the linearity property we get69z-Transform Propertiesor,Example - Determine the z-transform Y(z) and the ROC of the sequenceWe can write

22、where 70z-Transform PropertiesNow, the z-transform X(z) of is given byUsing the differentiation property, we arrive at the z-transform of as71z-Transform PropertiesUsing the linearity property we finally obtain72z-Transformz-Transform: analysis equationInverse z-Transform: synthesis equationtime dom

23、ainz-domainROC=73Inverse z TransformPartial-Fraction Expansion By the residual methodWrite time-domain sequence according to ROC74Inverse Transform by Partial-Fraction ExpansionSimple Poles: In most practical cases, the rational z-transform of interest G(z) is a proper fraction with simple polesLet

24、the poles of G(z) be at ,A partial-fraction expansion of G(z) is then of the form75Inverse Transform by Partial-Fraction ExpansionThe constants in the partial-fraction expansion are called the residues and are given byTherefore, the inverse transform gn of G(z) is given by76Inverse Transform by Part

25、ial-Fraction ExpansionExample - Let the z-transform H(z) of a causal sequence hn be given byA partial-fraction expansion of H(z) is then of the form77Inverse Transform by Partial-Fraction ExpansionNowand7879Inverse Transform by Partial-Fraction ExpansionHencehn is known to be a causal sequence, thus

26、 the ROC of H(z) should be80Inverse Transform by Partial-Fraction ExpansionHenceThe inverse transform of the above is therefore given byInverse Transform by Partial-Fraction ExpansionNow suppose hn is known to be a two-sided sequence, thus the ROC of H(z) should be8182Inverse Transform by Partial-Fr

27、action ExpansionHenceThe inverse transform of the above is therefore given byk-domainHk-domainHej Discrete Time-domainhnz-domainHzContinuous Time-domainhtS-domainHas-domainHaj Sampling & AtoDLaplacian -TransformInverse L -TransformS=j Bilinear -TransformInverse z -Transformz-Transformz=ej DTFTIDTFTS

28、amplingInterpolationInterpolationDFTIDFTInverse F -TransformF -Transform83Convolution SumThe summationis called the convolution sum of the sequences xn and hn and represented compactly asyn = xn hn*84Convolution SumInterpretation/Solution:1) Time-reverse hk to form2) Shift to the right by n sampling

29、 periods if n 0 or shift to the left by n sampling periods if n 0 to form3) Form the product4) Sum all samples of vk to develop the n-th sample of yn of the convolution sum 85Example xn=-2,0,1,-1,3,n=0,1,2,3,4 hn=1 ,2, 0, -1,n=0,1,2,3, solve yn,n=0,1,.7 1、使用图形法Determine the range for ynCalculate the convolution sum 2、Y(z)=X(z)H(z), yn=Z-1Y(z) 860+04+387z-Transform PropertiesC