信号与系统英文版课件：Chap2 Linear Time-Invariant Systems

1、Linear Time-Invariant SystemsChapter 2 Lecture 1Signals and Systems Spring 2015Homework #22.1, 2.3, 2.5, 2.7, 2.8, 2.10, 2.11, 2.12, 2.19, 2. 20, 2.23, 2.27, 2.40, 2.46, 2.47Linear Time Invariant systemA system is linear time invariant (LTI) if it is both linear and time invariant. LTI SystemDescrip

2、tion of CT LTI SystemsA CT LTI system can be described by a (linear constant-coefficient) differential equation and a set of initial conditions.Differential Eq(initial cond)f(t)y(t)CT LTI Systemf(t)y(t)Differential eq:Initial conditions:N: order of the systemlinear equationconstant coefficients (ak,

3、 bl)lineartime-invariantDifferential Eq(initial cond)f(t)y(t)Differential eq:Initial conditions:y(t) = ?Differential Eq(initial cond)f(t)y(t)Differential eq:Initial conditions:General solution:Plugging in the input signals, yp(t) is any solution that meets the differential equation.Ak obtained from

4、initial condhomogeneous solutionparticular solutionZero-State ResponseWhen all initial conditions are zero, the system response is called zero-state response, denoted by yf(t).Zero-Input ResponseWhen the inputs are zero, the system response is called zero-input response, denoted by ys(t).Total Respo

5、nseTotal response is the sum of the zero-input response and the zero-state responseCT LTI systeminput signalinitial condtotal responseDescription of DT LTI SystemsA DT LTI system can be described by a (linear constant-coefficient) difference equation and a set of initial conditions.Difference Eq(ini

6、tial cond)fnynDT LTI SystemfnynDifference eq:Initial conditions:N: order of the systemDifference Eq(initial cond)fnynlinear equationconstant coefficients (ck, dl)lineartime-invariantDifference eq:Initial conditions:General solution:Plugging in the input signals, ypn is any solution that meets the di

7、fferential equation.Ak obtained from initial condhomogeneous solutionparticular solutionZero-State ResponseWhen all initial conditions are zero, the system response is called zero-state response, denoted by yfn.Zero-Input ResponseWhen the inputs are zero, the system response is called zero-input res

8、ponse, denoted by ysn.Total ResponseTotal response is the sum of the zero-input response and the zero-state responseDT LTI syteminput signalinitial condtotal responseResponse of DT LTI SystemsDifference Eq(zero initial cond)hn is response of DT LTI system for input nfn = nyn = hnImpulse response of

9、DT LTI systemsEquivalent Descriptions of DT LIT SysA DT LTI system can be described by either its difference equation or impulse response.A DT LTI System described by its impulse responseDifference EqImpulse response hnfnynImpulse response hnExample 1 Given a DT LTI system with difference eq Determi

10、ne its impulse response hn.Solution: fn = nExample 2Given a DT LTI system with difference eq Determine its impulse response hn.Solution: fn = nzero initial condFIR and IIR SystemsA system is called a finite impulse response (FIR) system, if its impulse response hn has finite time duration. A system

11、is called an infinite impulse response (IIR) system, if its impulse response hn has infinite time duration. E.g.FIRIIRConvolution SumSuppose the impulse response hn and the input fn of a DT LTI system are both known. We want to find the zero-state response yn. For any input fn, we havefffLTIn hn Con

12、volution Sumyn is a convolution sum of fn and hn, which is often denoted asAt a given time n = n0, we haveCalculating the Convolution SumFollowing these steps to computeTime reversalTime delayMultiplicationSummation Convolution Sum Example 1Given a DT LTI system with impulse response hn = n un and i

13、nput signal fn = n un. Determine the system response yn. = 2, = 0.5Solution: (1) For n 0, fkhn-k = 0, so (2) For n 0, Employing the unit step signal to getLinear Time-Invariant SystemsChapter 2 Lecture 2Signals and Systems Spring Convolution Sum Example 2Given the impulse response hn and the input s

14、ignal fn of a DT LTI system. Determine the corresponding system response.Solution (method 1): 1) For n 2, n-4n-3n-2n-1hn-knn-6n-52) For n = 3,n-4n-3n-2n-1hn-knn-6n-5 3) For n = 4, n-4n-3n-2n-1hn-knn-6n-54) For n = 5, n-4n-3n-2n-1hn-knn-6n-5 5) For n = 6, n-4n-3n-2n-1hn-knn-6n-5 6) For n = 7, n-4n-3n

15、-2n-1hn-knn-6n-5 7) For n 8, n-4n-3n-2n-1hn-knn-6n-5 Taking together 1)-7), we obtainSolution (method 2): Write down the expression of fn which leads to the same result as method 1.LTISolution (method 3): Construct a polynomial based on fn = 1, 2, 1 for n = 1, 2, 3 as Similarly, construct a polynomi

16、al based on hn = 1, 2, 3 for n = 2, 3, 4 asfn0n0 the same result as we got previously. If the length of fn and hn are nf and nh respectively, the length of yn = fn*hn isny = nf + nh - 1 Note that since and we can compute fn from yn and hn, or compute hn from yn and fn. e.g. In this example, if yn an

17、d hn are given, thenDescription of CT LTI SystemsA CT LTI system can be described by a differential equation and a set of initial conditions.Differential Eq(initial cond)f(t)y(t)Response of CT LTI SystemsImpulse response of CT LTI systemsDifferential Eq(zero initial cond)f(t) = (t)y(t) = h(t)Equival

18、ent Descriptions of CT LIT SysA CT LTI system can be described by either its differential equation or impulse response.f(t)y(t)h(t)Differential Eq(zero initial cond)f(t)y(t)ExampleAssume the response y(t) of a CT LTI system to the input f(t) is Determine the system impulse response h(t). Solution: S

19、ince the impulse response is the response to the input f(t) = (t), we haveAccording to the sifting property of QuestionSuppose the impulse response h(t) and the input f(t) of a CT LTI system are both known. How to find the system (zero-state) response y(t)?For any f(t), we sample it with interval L

20、to get f(kL), k = 0, 1, 2, Based on f(kL), construct a broken line signal to approximate f(t)The segment centered at t = kL is expressed asThus, the broken line signal can be written asLet LTIConvolution IntegralSystem response y(t) is a convolution integral of the input signal f(t) and the impulse

21、response h(t), often denoted asCalculating the Convolution IntegralFollowing these steps to computeTime reversalTime delayMultiplicationIntegration Convolution Integral Example 1Given the input signal and the impulse response of a CT LTI system. Determine the system response. Solution: tttttt(1) For

22、 , (2) For , Taking together (1) and (2), we obtainConvolution Integral Example 2Given the input f(t) and impulse response h(t) of a CT LTI system. Determine the system response y(t). (1) For , (2) For , (3) For , (4) For , (5) For , If the length of f(t) and h(t) are tf and th respectively, the len

23、gth of y(t) = f(t)*h(t) isty = tf + thLinear Time-Invariant SystemsChapter 2 Lecture 3Signals and Systems Spring Impulse Response of Specific LTI SystemsA LTI system is memoryless, iff where K is a constant. Proof: DT LTI systems memoryless implies N = M = 0for DT LTI systemsfor CT LTI systemsFor CT

24、 LTI systems memoryless implies N = M = 0Impulse Response of Specific LTI SystemA LTI system is causal, iffThe impulse response of a causal LTI system is a causal signal, and vice versa. That is, for a LTI system, the system causality consists with the impulse response causality. for DT LTI systemsf

25、or CT LTI systemsImpulse Response of Specific LTI SystemA LTI system is stable, ifffor DT LTI systemsfor CT LTI systemsabsolutely summableabsolutely integrableProof: For CT LTI sys If , then Thus, the system is stable (i.e. ) iffCauchy-Schwarz inequalityCauchy-Schwarz InequalityCauchy-Schwarz inequa

26、lity states that for vectors x and y and their inner product If and , then FurtherExampleGiven the impulse response of the LTI system Determine if it is memoryless, causal, stable?Solution: (1) not memoryless(1) , (2)causalunstable(2)not causalstablenot memorylessProperties and Operations of LTI Sys

27、temsCommutative propertyDistributive propertyAssociative propertyInvertibility of LTI systems Differentiation IntegrationUnite step response of LTI systemsCommutative PropertyProof: For DT LTI systems, Let , then ThereforeFor CT LTI system, Let , then ThereforeCommutative PropertyThe input signal an

28、d impulse response play the same role when computing the system response. f(t)y(t)h(t)h(t)y(t)f(t)Distributive Propertyf(t)h1(t)y(t)h2(t)+f(t)y(t)h(t) = h1(t) + h2(t)Associative PropertyProof: Let , then and Define Associative Propertyf(t)y1(t)h1(t)y(t)h2(t)f(t)y(t)h(t) = h1(t) * h1(t)f(t)y2(t)h2(t)

29、y(t)h1(t)Example 1 Determine the impulse response of the system below. Solution: From the block diagram Example 2Given a DT LTI system diagram where Determine the system response yn.Solution: From the diagram, where associative propdistributive propLinear Time-Invariant SystemsChapter 2 Lecture 4Signals and Systems Spring Invertibility of LTI SystemsFor an invertible system h(t) there exists an inverse system hinv(t), which, when connected in series with the original system, produces an output equa