1、Chapter 2 Linear Time-Invariant Systemsanalyze systems in time domain Chapter 4 The Continuous-Time Fourier Transformanalyze continuous-time systems in frequency domain Chapter 5 The Discrete-Time Fourier Transformanalyze discrete-time systems in frequency domain,Chapter 9 The Laplace Transformanaly
2、ze continuous-time systems in complex frequency domain Chapter 10 The Z-Transformanalyze discrete-time systems in complex frequency domain,2 Linear Time-Invariant Systems (LTI),Emphases in this chapter: The calculation of the convolution sum.2. The calculation of the convolution integral.3. The prop
3、erties of system represented by the unit impulse response h(t) or hn. 4. The calculation properties of,LTI,xn,yn=?,Question:,2 Linear Time-Invariant Systems (LTI),2.1 Discrete-time LTI system: The convolution sum,2 Linear Time-Invariant Systems (LTI),2.1.1 The Representation of Discrete-time Signals
4、 in Terms of Impulses,If xn=un, then,Figure 2.1,(The sifting property of n.),2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,2.1.2 The Discrete-time Unit Impulse Response and the Convolution Sum Representation of LTI Systems,(1) Definition: Unit Impulse(Sample) Response,For a discret
5、e-time LTI system, when the input signal is n , then the output signal is called as Unit Impulse Response hn .,2 Linear Time-Invariant Systems,(2) Convolution Sum of LTI System,Solution:,Question:,n hnn-k xkn-k ,hn-k=hkn,xk hn-k=xkhkn,2 Linear Time-Invariant Systems,( Convolution Sum ),So, defining:
6、,or,2 Linear Time-Invariant Systems,(3) Calculation of Convolution Sum,e. Summing:,a. Chang variable: xnxk, hnhk,b. Time Inversal: hk h-k,c. Time Shift: In order to obtain yn, thenh-k hn-k,d. Multiplication: xkhn-k,method 1: diagram(图解法),2 Linear Time-Invariant Systems,f. Change the value of n, and
7、repeat the processes of c,d and e. to obtain yn for all n.,2 Linear Time-Invariant Systems,Example 2.1 hn,xn are illustrated in the figure.,Determine and sketch yn = xn * hn for the following xn and hn.,n,0 1 2,hn,n,0 1,xn,1,0.5,2,2 Linear Time-Invariant Systems,Solution:,k,0 1,xk,0.5,2,k,-2 -1 0,h-
8、k,1,k,(n-2) (n-1) n -1 0,hn-k,1,n0,n0,1,k,(n-2) (n-1) n -1 0,xk hn-k,2 Linear Time-Invariant Systems,0 1,xk,0.5,2,-2 -1 0,h0-k,1,k,k,n=0,-2 -1 0 1,xkh0-k,1,k,n=0,0.5,2 Linear Time-Invariant Systems,k,0 1,xk,0.5,2,k,-1 0 1,h1-k,1,For n=1,n=1,k,0 1,xkh1-k,0.5,2,2 Linear Time-Invariant Systems,k,0 1,xk
9、,0.5,2,k,0 1 2,h2-k,1,For n=2,n=2,k,0 1,xkh2-k,0.5,2,k,0 1,xk,0.5,2,k,1 2 3,h3-k,1,For n=3,n=3,k,0 1 2,xkh2-k,2,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,k,0 1,xk,0.5,2,k,-2 -1 0 1 n-2 n-1 n,hn-k,1,For n-21 ( n3 ),k,(n-2) (n-1) n -1 0,xk hn-k,For n-21 ( n3 ),2 Linear Time-Invar
10、iant Systems,So, yn is as the following figure:,2 Linear Time-Invariant Systems,Example 2.3,k,0 1 2,xk,1,k,-2 -1 0 1 2 ,hk,1,2 Linear Time-Invariant Systems,k,0 1 2,xk,1,k,-2 -1 0 1 ,h-k,1,Solution:,n0,n0,2 Linear Time-Invariant Systems,So,2 Linear Time-Invariant Systems,Example 2.4,k,0 1 2 3 4,xk,1
11、,k,-1 0 1 2 3 4 5 6 7,hk,2 Linear Time-Invariant Systems,k,0 1 2 3 4,xk,1,k,-6 -5 -4 -3 -2 -1 0 1,h-k,1,Solution:,n0,2 Linear Time-Invariant Systems,k,0 1 2 3 4,xk,1,k,hn-k,1,0n4,0n4,k,-6 -5 -4 -3 -2 -1 0 1 n,xkhn-k,0n4,n-6 -2 -1 0 1 n4,2 Linear Time-Invariant Systems,k,0 1 2 3 4,xk,1,k,n-6 0 1 2 3
12、4 n,hn-k,5n6,n-60,n6,k,-6 -5 -4 -3 -2 -1 0 1 2 3 4,xkhn-k,2 Linear Time-Invariant Systems,k,0 1 2 3 4,xk,1,k,n-6 4 5 6 n,hn-k,6n10,0n-64,k,-6 -5 -4 -3 -2 -1 n-6 n-5 4 5,xkhn-k,6n10,2 Linear Time-Invariant Systems,k,0 1 2 3 4,xk,1,k,hn-k,n-64,4 n-6 n-5 n,n10,n10,2 Linear Time-Invariant Systems,So,2 L
13、inear Time-Invariant Systems,Example 2.5,2 Linear Time-Invariant Systems,k,xk,1,k,h-k,n0,-5 -4 -3 -2 -1 0 1 2,-5 -4 -3 -2 -1 0 1 2,1,hn-k,n-2 n-1 n 0 1 2,k,xk hn-k,n-2 n-1 n 0 1 2,2 Linear Time-Invariant Systems,n0,2 Linear Time-Invariant Systems,k,xk,1,k,h-k,n0,-5 -4 -3 -2 -1 0 1 2,-5 -4 -3 -2 -1 0
14、 1 2,1,hn-k, -1 0 1 n,k,xk hn-k,n-2 n-1 n 0 1 2,k,2 Linear Time-Invariant Systems,n0,So,2 Linear Time-Invariant Systems,Method 2: formula(公式法),2 Linear Time-Invariant Systems,Example:,Solution:,2 Linear Time-Invariant Systems,Method 3: multiplication (竖乘法),Example 2.1,Figure 2.3,2 Linear Time-Invari
15、ant Systems,Solution:,So,2 Linear Time-Invariant Systems,说明：两个有限长序列卷积和，可以用“竖乘法”求解。如果序列xn的变量的取值为（n1,n2),序列hn的变量的取值为（m1,m2),则卷积和的结果的序列yn的变量的取值为（n1+m1,n2+m2).,2 Linear Time-Invariant Systems,Example:,Solution:,We can know:,2 Linear Time-Invariant Systems,7 2 0 -52 0 2,x,14 4 0 -10,0 0 0 0,14 4 14 -6 0
16、-10,14 4 0 -10,So,2 Linear Time-Invariant Systems,addition,Example :,Solution:,n=0,1,2,3,4,5,即，利用普通乘法即可实现两个有限长序列的卷积和，只是乘法时不要进位。,2 Linear Time-Invariant Systems,addition,Example :,Solution:,n=-3,-2,-1,0,1,2,2 Linear Time-Invariant Systems,addition,等比数列求和公式：,有限项求和：（初值-终值公比）/（1-公比） Example:,2 Linear Ti
17、me-Invariant Systems,无限项求和：1/（1-公比） 条件：公比的绝对值小于1 Example:,addition,2 Linear Time-Invariant Systems,2.2 Continuous-time LTI system: The convolution integral,2.2.1 The Representation of Continuous-time Signals in Terms of Impulses,2 Linear Time-Invariant Systems,Considering a “staircase” signal :,To a
18、pproximate a continuous-time signal x(t):,2 Linear Time-Invariant Systems,Define,We have the expression:,Therefore:,2 Linear Time-Invariant Systems,or,2 Linear Time-Invariant Systems,The sifting property of (t):,For any continuous-time signal x(t):,Especially,2 Linear Time-Invariant Systems,2.2.2 Th
19、e Continuous-time Unit impulse Response and the convolution Integral Representation of LTI Systems,(1) Definition: Unit Impulse Response,LTI,x(t)=(t),y(t)=h(t),(2) The Convolution of LTI System,LTI,x(t),y(t)=?,2 Linear Time-Invariant Systems,LTI,(t),h(t),(t）,h(t),2 Linear Time-Invariant Systems,or y
20、(t) = x(t) * h(t),( Convolution Integral ),2 Linear Time-Invariant Systems,(3) Computation of Convolution Integral,a. Chang variable: x(t)x(), h(t)h(),b. Time Inversal: h() h(- ),c. Time Shift: h(-) h(t- ),d. Multiplication: x()h(t- ),e. Integrating:,Method 1: diagram(图解法）,2 Linear Time-Invariant Sy
21、stems,e. Integrating:,积分上下限的确定说明： 积分下限：为两相乘函数的左限的最大值。 积分上限：为两相乘函数的右限的最小值。,2 Linear Time-Invariant Systems,Example 2.6,Determine and sketch y(t)= x(t) * h(t) for the following x(t) and y(t).,h(t) ,x(t) are illustrated in the figure.,0,t,h(t),0,t,x(t),1,1,2 Linear Time-Invariant Systems,0,h(-),0,x(),1
22、,1,Solution:,0,h(t-),1,t,t0,t0,2 Linear Time-Invariant Systems,0,x(),1,Solution:,0,h(t-),1,t,t0,t0,0,x() h(t-),1,t,2 Linear Time-Invariant Systems,Example 2.7,x(),0 T,0 2T,-2T 0,h(),h(-),2 Linear Time-Invariant Systems,Solution:,x(),0 T,-2T+t t 0,h(t-),t0,t0,2 Linear Time-Invariant Systems,x(),0 T,-
23、2T+t 0 t T,h(t-),0tT,-2T+t 0 t T,x() h(t-),0tT,2 Linear Time-Invariant Systems,x(),0 T 2T,-2T+t 0 T t 2T,h(t-),Tt2T,-2T+t 0 T t 2T,x() h(t-),Tt2T,x(),0 T 2T 3T,0 -2T+t T 2T t 3T,h(t-),2Tt3T,x() h(t-),2 Linear Time-Invariant Systems,0 -2T+t T 2T t 3T,2Tt3T,x(),0 T 2T 3T,0 T -2T+t 2T 3T t,h(t-),t3T,t3
24、T,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,Example 2.8,x(),h(),3,0,0,h(-),-3,0,1,1,1,2 Linear Time-Invariant Systems,x(),0,Solution:,h(t-),-3,0,t0,t-3,t0,x() h(t-),0,t-3,2 Linear Time-Invariant Systems,x(),0,h(t-),-3,0,t-3,0t3,x() h(t-),-3 t-3 0,0t3,2 Linear Time-Invariant Sys
25、tems,x(),0,h(t-),-3,0,t-3,t3,x() h(t-),t3,0,2 Linear Time-Invariant Systems,Method 2: formula(公式法）,2 Linear Time-Invariant Systems,Example 2.8,Solution:,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,Example:,Solution:,2 Linear Time-Invariant Systems,常用公式：,2 Linear Time-Invariant Sy
26、stems,Problems : 2.1 2.3 2.4 2.5 2.10 2.11,2 Linear Time-Invariant Systems,2.3 Properties of Linear Time-Invariant System,Convolution formula:,2 Linear Time-Invariant Systems,The characteristics of an LTI system arecompletely determined by its impulse response.,2 Linear Time-Invariant Systems,2.3.1
27、The Commutative Property (交换性）,Discrete time: xn*hn=hn*xnContinuous time: x(t)*h(t)=h(t)*x(t),h(t),x(t),y(t)=x(t)*h(t),x(t),h(t),y(t)=h(t)*x(t),2 Linear Time-Invariant Systems,Prove :,2 Linear Time-Invariant Systems,Prove :,2 Linear Time-Invariant Systems,2.3.2 The Distributive Property（分配性）,Discret
28、e time: xn*h1n+h2n=xn*h1n+xn*h2nContinuous time: x(t)*h1(t)+h2(t)=x(t)*h1(t)+x(t)*h2(t),2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,Example 2.10,2 Linear Time-Invariant Systems,Solution:,2 Linear Time-Invariant Systems,Figure 2.24,2 Linear Time-Invariant Systems,2.3.3 The Associa
29、tive Property（结合性）,Discrete time: xn*h1n*h2n=xn*h1n*h2nContinuous time: x(t)*h1(t)*h2(t)=x(t)*h1(t)*h2(t),2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,补充,卷积的其它性质：,首先定义：,2 Linear Time-Invariant Systems,补充,微分性质：,积分性质：,时延性质：,2 Linear Time-Invariant Systems,补充,时延性质：,2 Linear Time-Inva
30、riant Systems,补充,Example:,Solution:,2 Linear Time-Invariant Systems,补充,2 Linear Time-Invariant Systems,1) the properties of (t) and n:,2.3.4 LTI system with and without Memory,2 Linear Time-Invariant Systems,Prove:,2 Linear Time-Invariant Systems,2) Memoryless system:Discrete time: yn=kxn, Continuou
31、s time: y(t)=kx(t),k (t),x(t),y(t)=kx(t)=x(t)*k(t),k n,xn,yn=kxn=xn*kn,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,Example:,2 Linear Time-Invariant Systems,0 1 2 3 t,Solution:,2 Linear Time-Invariant Systems,Example:,x(t) is shown as following figure:,2 Linear Time-Invariant Syst
32、ems,Solution:,If T2，then,2 Linear Time-Invariant Systems,If T2，then,2 Linear Time-Invariant Systems,补充,Examples: Compute,(1),2 Linear Time-Invariant Systems,补充,Solution: (1),2 Linear Time-Invariant Systems,补充,Examples: Compute,(2),2 Linear Time-Invariant Systems,补充,Solution: (2),Why?,because,对f1(t)积
33、分时，不能恢复原信号f1(t).,成立的条件是：,2 Linear Time-Invariant Systems,补充,We can do so:,2 Linear Time-Invariant Systems,2.3.5 Invertibility of LTI system,Original system: h(t) Reverse system: h1(t),So, for the invertible system:,h(t)*h1(t)=(t),hn*h1n=n,2 Linear Time-Invariant Systems,Example 2.11 2.12,Determine t
34、he inverse system of the following system.,2 Linear Time-Invariant Systems,(a) Solution:,(b) Solution:,2 Linear Time-Invariant Systems,2 Linear Time-Invariant Systems,2.3.6 Causality for LTI system,Causal discrete-time system satisfy the condition:hn=0 for n0 Causal continuous-time system satisfy th
35、e condition:h(t)=0 for t0,2 Linear Time-Invariant Systems,For a causal discrete-time LTI system,For a causal continuous-time LTI system,2 Linear Time-Invariant Systems,Only for linear systems, causality for systems is equivalent to the condition of initial rest;,Definition: the condition of initial
36、rest: Causal continuous-time system: if the input to a causal system x(t)=0 for tt0,then the output y(t)=0 for tt0.Causal discrete-time system: if the input to a causal system xn=0 for nn0,then the output yn=0 for nn0.,2 Linear Time-Invariant Systems,Definition: Causal signal: if xn is zero for n0 ,
37、or x(t) is zero for t0, then xn or x(t) is causal signal.,So, causality of an LTI system is equivalent to its impulse response( hn or h(t) ) being a causal signal.,2 Linear Time-Invariant Systems,2.3.7 Stability for LTI system,Definition of stability:Every bounded input produces a bounded output.,Th
38、e stability of a discrete-time system is equivalent to:,The stability of a continuous-time system is equivalent to:,2 Linear Time-Invariant Systems,So,the condition for |yn|A is,To prove :,If |xn|B,2 Linear Time-Invariant Systems,Continuous-time system:,If |x(t)|B,So,the condition for |y(t)|A is,2 L
39、inear Time-Invariant Systems,Example 2.13 Determine whether or not the following systems are stable.,(a) time-shift discrete-time system,(b) time-shift continuous-time system,(c) The accumulator,(d) The integrator,stable,stable,unstable,unstable,2 Linear Time-Invariant Systems,2.3.8 The Unit Step Re
40、sponse of LTI system,Discrete-time system:,hn,n,hn,un,sn=un*hn,The relationship between hn and sn:,The unit step response: sn,2 Linear Time-Invariant Systems,Continuous-time system:,h(t),(t),h(t),u(t),s(t)=u(t)*h(t),The relationship between h(t) and s(t):,The unit step response: s(t),2 Linear Time-I
41、nvariant Systems,2.4 Causal LTI Systems Described by Differential and Difference Equation,Discrete-time system: Difference Equation Continuous-time system: Differential Equation,2 Linear Time-Invariant Systems,2.4.1 Linear Constant-Coefficient Differential Equation,A .general Nth-order linear consta
42、nt-coefficient differential equation:,or,and initial condition:y(t0), y(t0), , y(N-1)(t0) ( N values ),2 Linear Time-Invariant Systems,the causal LTI systems satisfy the conditionof initial rest: if x(t)=0 for tt0, theny(t)=0 for tt0,y(t0) = y(t0) = = y(N-1)(t0) =0,2 Linear Time-Invariant Systems,Th
43、e complete solution of differential equation:,(1),A homogeneous solution of the homogeneous differential equation: ( natural response),2 Linear Time-Invariant Systems,The complete solution of differential equation:,(2),Zero-state response(零状态响应）:Initial states are zero,and the response is produced b
44、y input signal x(t).,Zero-input response(零输入响应）: The input signal x(t) is zero, and the response is produced by systems initial states y(t0),y(t0),2 Linear Time-Invariant Systems,2.4.2 Linear Constant-Coefficient Difference Equation,A. general Nth-order linear constant-coefficient difference equation:,or,and initial condition:y0, y-1, , y-(N-1) ( N values ),2 Linear Time-Invariant Systems,The complete solution of difference equation:,(1),Particular solution for a certain input.,( forced response),A homogeneous solution of the homogeneous difference equation: ( natural response),
