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1、CHAPTER 2 LINEAR TIME- INVARIANT SYSTEMS 2.0 INTRODUCTION Representation of signals as linear combination of delayed impulses. Convolution sum(卷积和) or convolution integral(卷积 积分) representation of LTI systems. Impulse response and systems properties Solutions to linear constant-coefficient differenc
2、e and differential equations (线性常系数差分或微分方程). 2.1 DISCRETE-TIME SYSTEMS: THE CONVOLUTION SUM Derivation steps: Step 1: Representing discrete-time signals in terms of unit samples: Step 2: Defining Unit sample response hn : response of the LTI system to the unit sample n. n hn Step 3: Writing any arbi
3、trary input xn as: Step 4: By taking use of linearity and time-invariance, we can get the response yn to xn which is the weighted linear combination of delayed unit sample responses as following: The Convolution Sum Representation of LTI Systems convolution sum or superposition sum : Convolution ope
4、ration symbol: LTI system is completely characterized by its response to the unit sample -hn . Example 2.1 n xn 0 1 2 n hn -2 0 2 (a) Consider a LTI system with unit sample response hn and input xn, as illustrated in Figure (a). Calculate the convolution sum of these two sequences graphically. k xk
5、0 1 2 k h-k -2 0 2 (b) h-2h2 1 1 0.5 k xk 0 1 2 k hn-k n-2 n+2 If n4, Graph of yn in Example 2.1 From Example 2.1, we can draw the following table: Thus, we obtain a method for the computation of convolution sum, that is suitable for two short sequences. xn = 1,1,10hn = 0.5, 1, 0.5, 1, 0.5-2 xn*hn =
6、 0.5,1.5,2,2.5,2,1.5,0.5-2 0.5 1.5 2 2.5 2 1.5 0.5 0.5 1 0.5 1 0.5 xn h-2 h-1 h0 h1 h2 hn x0 x1 x2 ： x0h-2 x0h-1 x0h0 x0h1 x0h2 0 0 x1h-2 x1h-1 x1h0 x1h1 x1h2 0 0 x2h-2 x2h-1 x2h0 x2h1 x2h2 0 0 0 0 0 0 0 0 0 y- 2 y- 1 y0 y1 y3y2y4 0.5 1 0.5 1 0.5 1 1 1 0.5 1 0.5 1 0.5 + + 0.5 1 0.5 1 0.5 Example 2.2
7、 Consider an input xn and a unit sample response hn given by Determine and plot the output Using the geometric sum formula to evaluate the equation, we have n 2 1 yn Graph of yn in Example 2.2 2.2 CONTINUOUS-TIME LTI SYSTEMS: THE CONVOLUTION INTEGRAL The Representation of Continuous-Time Signals in
8、Terms of Impulses: t -02 k x(t) Staircase approximation to a continuous-time signal x(t) x(-2 ) t -2- x(0) 0 t 2 x() t -0 x(-) t Mathematical representation for the rectangular pulses as , the summation approaches an integral and is the unit impulse function Compared with the Sampling property of th
9、e unit impulse: Give the as the response of a continuous-time LTI system to the input , then the response of the system to pulse is Thus, the response to is As , in addition, the summing becomes an integral. Therefore, convolution integral or superposition integral : unit impulse response h(t) : the
10、 response to the input . （单单位冲激响应应） Convolution integral symbol: A continuous-time LTI system is completely characterized by its unit impulse response h(t) . Example 2.3 Consider the convolution of the following two signals, which are depicted in (a): 0 T 1 x(t) t 0 2T 2T h(t) t (a) 2T h(t) t-2T t 0
11、 T 1 x() For t 0 Interval 1. For t 0, there is no overlap between the nonzero portions of and , and consequently, From the definition of the convolution integral of two continuous- time signals, 2Th(t) t-2T 0 t T 1 x() For 0 t T . Thus, for 0 t T . Interval 2. For 0 t T, 2T h(t) t-2T T t 1 x() For T
12、 t 2T Thus, for T t T but t-2T 0, i.e. T t 0, but t-2T T, i.e. 2T t 3T Thus, for 2T t 3T . For 2T t T, or equivalently, t 3T, there is no overlap between the nonzero portions of and hence, x() 2T h(t) 0 T t-2T t 1 For t 3T Summarizing, 2.3 PROPERTIES OF CONVOLUTION OPERATION 2.3.1 The Commutative Pr
13、operty(交换律) in discrete time : in continuous time: 2.3.2 The Distributive Property (分配律) in discrete time : in continuous time: y(t) y2(t) y1(t) h2(t) x(t) h1(t) x(t) h1(t)+h2(t) y(t) Two equivalent systems: having same impulse responses 2.3.3 The Associative Property (结合律) in discrete time : in con
14、tinuous time: xnh1nynh2n xnhn=h1n*h2nyn xnhn=h2n*h1nynxnh2nynh1n Four equivalent systems 2.3.4 Convolving with Impulse 2.3.5 Differentiation and Integration of Convolution Integral Combining the two properties, we have 2.3.6 First Difference and Accumulation of Convolution Sum 2.4.1 LTI Systems with
15、 and without Memory for a discrete-time LTI system without memory: for a continuous-time LTI system without memory: 2.4.2 Invertibility of LTI Systems The impulse responses of a system and its inverse system satisfy the following condition: in discrete-time : in continuous-time: Since 2.4 PROPERTIES
16、 OF LTI SYSTEMS 2.4.3 Causality for LTI Systems for a causal discrete-time LTI system: for a causal continuous-time LTI system: 2.4.4 Stability for LTI Systems for a stable discrete-time LTI system: for a stable continuous-time LTI system: absolutely summable absolutely integrable Suppose Proof: The
17、n If Then Therefore, the absolutely summable is a sufficient condition to guarantee the stability of a discrete-time LTI system. To show that the absolutely summable is also a necessary condition for the stability of a discrete-time LTI system, Let where, is conjugate . Then, xn is bounded by 1, tha
18、t is However, IfThen 2.5 The Unit Step Response(单位阶跃响应) of an LTI System The unit step response, sn or s(t), is the output of an LTI system when input xn=un or x(t)=u(t). The unit step response of a discrete-time LTI system is the running sum of its unit sample response: The unit sample response of
19、a discrete-time LTI system is the first difference of its unit step response: The unit step response of a continuous-time LTI system is the running integral of its unit impulse response: The unit impulse response of a continuous-time LTI system is the first derivative of the unit step response : 2.6
20、 CAUSAL LTI SYSTEMS DESCRIBED BY DIFFERENTIAL AND DIFFERENCE EQUATIONS Linear constant-coefficient differential equation : input signal; : output signal. Ci (t)Vs R + Example 2.4 Linear constant-coefficient difference equation is the mathematical representation of a discrete-time LTI system. Linear
21、constant-coefficient differential equation is the mathematical representation of a continuous-time LTI system. We must specify one or more auxiliary conditions to solve a differential (difference) equation . Initial rest(初始静止): for a causal LTI system, if x(t)=0 for tt0, then y(t) must also equal 0
22、for t0, so let Taking x(t) and for t 0 into the original equation yields Thus So the solution of the differential equation for t0 is In Example 2.4, Taking use of the condition initial rest, we obtain Consequently, or for t0 Example 2.5 Jack saves money every month. It is known that at the beginning
23、 of the nth month the amount he saves into the bank is RMB xn yuan, and the rate of interest is per month. Suppose Jack wouldnt withdraw his bank deposits in whatever situation, try to give the difference equation relating xn and yn, which is the deposits of Jack at the end of the nth month. (before
24、 the bank calculates the interest) Solution:yn is consists of the sum of the following three parts: xn saved at the beginning of the nth month yn-1 interest at the end of the (n-1)th month yn-1 deposit of the (n-1)th month So the difference equation is also Difference: For sequence xn, its First for
25、ward difference(一阶前向差分) is defined as xn = xn+1 xn Second backward difference as 2x n = xn xn-1 = xn-2xn-1+xn-2 Analogously, Second forward difference can be constructed as 2xn = xn = xn+1 xn = xn+2-2xn+1+xn its First backward difference (一阶后向差分) is defined as xn = xn xn-1 General Nth-order linear c
26、onstant-coefficient difference equation: First resolution: N auxiliary conditions: Second resolution: (recursive method(迭代法) Example 2.6 Solve the difference equation and the initial condition is y0=1. The eigen equation is So the eigenvalue is a = 2 We can write Let Taking into the original equatio
27、n yields Thus The solution for the given equation is From the initial condition of y0=1, we have Consequently, Example 2.7 Consider the difference equation Determine the output recursively with the condition of initial rest and Rewrite the given difference equation as Starting from initial condition
28、, we can solve for successive values of yn for n1: Considering yn =0 for n0, the solution is What are the relationships between yp, yh, yzi and yzs ? Question: Example 2.8 Consider a continuous-time LTI system described by the following differential equation with initial conditions of and input From
29、 the definition of the zero-input response, we have In the case of zero input, Thus we can write Equivalently, Consequently, Next, solve for h(t). From the definition of the unit impulse response, we have And for t0, it becomes h(t) is the solution for the homogeneous equation. Thus, And because the
30、 system is a causal one, there should be The initial conditions used to determine A1 and A2 are But Let (2) Then (3) Taking equations (2) and (3) into equation (1), we have Comparing the coefficients of the corresponding terms on each side Compute the integral in the interval of 0-, 0+ on both sides
31、 of equation (2) to obtain Analogously to equation (3) to obtain Consequently, Then from We obtain So Then Example 2.9 Consider a causal LTI system described by Determine the unit sample response hn. For n0, hn satisfies the difference equation And there should be Substituting hn for yn and n for xn
32、 in the original difference equation, and let n=0, we obtain Its obvious that Taking use of h0, we make out the coefficient in hn: So In fact, for n=0, h0 also satisfy Thus, we can write You may also try the recursive method to obtain the hn for this system! 2.7 Block Diagram Representations of Firs
33、t-Order Systems Described by Differential and Difference Equations First-order difference equation : addition delay multiplication Three basic elements in block diagram: adder, multiplier and delayer. (方框图) (加法器)(乘法器) (延时器) Basic elements for the block diagram representation of causal discrete-time
34、systems. (a) an adder (b) a multiplier (c) a delayer. x1n+x2n + x2n x1n (a) axnxn a (b) D xn xn-1 (c) + D xnyn b -a yn-1 Block diagram representation for the causal discrete-time system described by the first-order difference equation (yn+ayn- 1=bxn). First-order differential equation : differentiat
35、ion x(t) Block diagram representation of an integrator + x(t) y(t) b -a Block diagram representation for the system described by the first-order differential equation Three basic elements in block diagram: adder, multiplier and integrator(积分器) . 2.6 SUMMARY A representation of an arbitrary discrete-
36、time signal as weighted sums of shifted unit samples; Convolution sum representation for the response of a discrete-time LTI systems; A representation of an arbitrary continuous-time signal a weighted integrals of shifted unit impulses; Convolution integral representation for continuous-time LTI sys
37、tems; Relating LTI system properties, including causality, stability, to corresponding properties of the unit impulse (sample) response; Some properties of systems described by linear constant- coefficient differential (difference) equations; Understanding of the condition of initial rest. Homework 2.21 (a) (c) 2.22 (a) (d) 2.28 (b) (e) (g) 2.29 (b) (e) (f) 2.32