测试信号英文版课件：Chapter2Discrete-Time Signals and Systems in the Time-Domain-L2

1、AgendaDiscrete-Time SignalsDiscrete-Time SystemsTime-Domain Characterization of LTI Discrete-Time Systems1Discrete-Time SystemsA discrete-time system processes a given input sequence xn to generates an output sequence yn with more desirable propertiesIn most applications, the discrete-time system is

2、 a single-input, single-output system:xnynInput sequenceOutput sequence2Discrete-Time Systems ClassificationLinear SystemShift (Time)-Invariant SystemLinear Time-Invariant SystemCausal SystemStable SystemPassive and Lossless Systems34Linear Discrete-Time SystemsDT systemDT systemDT systemDefinitionL

3、inear Discrete-Time SystemsAccumulator -For an inputthe output is Hence, the above system is linear5Nonlinear Discrete-Time SystemConsiderOutputs and for inputs and are given by6Nonlinear Discrete-Time SystemOutput yn due to an input is given by7Nonlinear Discrete-Time SystemOn the other handHence,

4、the system is nonlinear89Shift-Invariant SystemDT systemDefinitionDT systemShift-Invariant SystemTime-invariance property ensures that for a specified input, the output is independent of the time the input is being appliedIn other words, the input-output relationship of the system does not change ov

5、er time.10Shift-Invariant SystemExample - Consider the Modulator with an input-output relation given by11xnynwnwnShift-Invariant SystemFor an input the output is given by12Shift-Invariant SystemHowever from the definition of the ModulatorHence, the Modulator is a time-varying system13Linear Time-Inv

6、ariant SystemLinear Time-Invariant (LTI) System - A system satisfying both the linearity and the time-invariance propertyLTI systems are mathematically easy to analyze and characterize, Therefore, easy to design14Causal SystemIn a causal system, the -th output sample depends only on input samples xn

7、 for and does not depend on input samples forLet and be the responses of a causal discrete-time system to the inputs and , respectively 15Causal SystemThen for n Nimplies also that for 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

8、 the n-th sample of yn of the convolution sum 38Two methods to calculate the convolution sum in time-domainAnalytical method & Graphical methodExample 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 Analytical method (not recommended):y0=x0h0-0=-2;y1=x0h1-0+x1h1-1=-4y2=x0h2-0+

9、x1h2-1+x2h2-2=1y7=x4h7-4=-3390+04+3Graphical method (strongly recommended) :Determine the range for ynExample 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,.7Calculate the convolution sum 400+04+341y0=(-2)x1=-2;42y1=(-2)x2 + 0 x1=-443y2=(-2)x0 + 0 x1 + 1x1=144y2=3x(-1)=-3Stabil

10、ity Condition of a Discrete-Time LTI SystemBIBO Stability Condition - A discrete-time LTI system is BIBO stable if the output sequence yn remains bounded for any bounded input sequencexnA discrete-time LTI system is BIBO stable if and only if its impulse response sequence hn is absolutely summable,

11、i.e.S45Stability Condition of a Discrete-Time LTI SystemProof: Assume hn is a real sequenceSince the input sequence xn is bounded we haveThereforeSStability Condition of a Discrete-Time LTI SystemExample - Consider a causal discrete-time LTI system with an impulse responseFor this systemTherefore if

12、 for which the system is BIBO stableIf , the system is not BIBO stableSif47How to determine the stability of a discrete-time system? Definition If , we have . Time-domain (使用前提：系统是LTI的) Z-domain S(pre-condition: The system must be LTI)Causality Condition of a Discrete-Time LTI SystemA discrete-time

13、LTI system is causal if and only if its impulse response hn is a causal sequence, i.e., hk=0, for k0Example - The discrete-time system defined byis a causal system as it has a causal impulse response49Causality Condition of a Discrete-Time LTI SystemExample - The discrete-time accumulator defined by

14、is a causal system as it has a causal impulse response given by50Causality Condition of a Discrete-Time LTI SystemExample - The factor-of-2 interpolator defined byis noncausal as it has a noncausal impulse response given by51How to determine the causality of a discrete-time system? Definition Time-d

15、omain If its impulse response hn is a causal sequence (使用前提：系统是LTI的) Z-domain 52(pre-condition: The system must be LTI)Classification of Discrete-Time LTI SystemsBased on the Impulse Response Length:If the impulse response hn is of finite length, i.e.,then it is known as a finite impulse response (F

16、IR) discrete-time systemThe convolution sum description here is53Classification of Discrete-Time LTI SystemsThe output yn of an FIR LTI discrete-time system can be computed directly from the convolution sum as it is a finite sum of productsExamples of FIR LTI discrete-time systems will be given in Chapter 454