信号教学课件(华中科技大学)cha

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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 difference

2、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 arbitr

3、ary 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 opera

4、tion symbol:,LTI system is completely characterized by its response to the unit sample -hn .,Example 2.1,1,1,0.5,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

5、 = 1,1,10,hn = 0.5, 1, 0.5, 1, 0.5-2,xn*hn = 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,Example 2.2,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,Graph of yn in Exampl

6、e 2.2,2.2 CONTINUOUS-TIME LTI SYSTEMS: THE CONVOLUTION INTEGRAL,The Representation of Continuous-Time Signals in Terms of Impulses:,Mathematical representation for the rectangular pulses,as , the summation approaches an integral and is the unit impulse function,Compared with the Sampling property of

7、 the 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) : th

8、e response to the input . （单位冲激响应）,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):,From the definition of the convolution integral o

9、f two continuous-time signals,2T,h(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 t 2T,Thus, for T t 2T,Interval 3. For t T but t-2T 0, i.e. T t 2T,2T,h(t),t-2T t,1,x(),Interval 4. For t-2T 0, but t-2T T, i.e. 2T t 3T,Thus, for 2T t 3T,.,For 2T t

10、 3T,Interval 4. For t-2T T, or equivalently, t 3T, there is no overlap between the nonzero portions of and,hence,Summarizing,2.3 PROPERTIES OF CONVOLUTION OPERATION,2.3.1 The Commutative Property(交换律),2.3.2 The Distributive Property (分配律),Two equivalent systems: having same impulse responses,2.3.3 T

11、he Associative Property (结合律),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 and without Memory,2.4.2 Invertibi

12、lity of LTI Systems,Since,2.4 PROPERTIES OF LTI SYSTEMS,2.4.3 Causality for LTI Systems,2.4.4 Stability for LTI Systems,Suppose,Proof:,Then,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 summa

13、ble is also a necessary condition for the stability of a discrete-time LTI system,Let,where, is conjugate .,Then, xn is bounded by 1, that is,However,If,Then,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

14、)=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 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

15、 unit impulse response:,The unit impulse response of a continuous-time LTI system is the first derivative of the unit step response :,2.6 CAUSAL LTI SYSTEMS DESCRIBED BY DIFFERENTIAL AND DIFFERENCE EQUATIONS,Linear constant-coefficient differential equation,Linear constant-coefficient difference equ

16、ation is the mathematical representation of a discrete-time LTI system.,Linear 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(

17、初始静止): for a causal LTI system, if x(t)=0 for tt0, then y(t) must also equal 0 for tt0.,General Nth-order linear constant-coefficient differential equation:,N auxiliary conditions:,If R=4, C=1/2, and the input signal is,Then we obtain,From the eigenvalue of the homogeneous equation, we can write,Sin

18、ce 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 beginnin

19、g 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. (befor

20、e 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 Firs

21、t forward difference(一阶前向差分) is defined as xn = xn+1 xn,its First backward difference (一阶后向差分) is defined as xn = xn xn-1,General Nth-order linear constant-coefficient difference equation:,First resolution:,N auxiliary conditions:,Second resolution: (recursive method(迭代法),Example 2.6,Solve the diffe

22、rence 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 equation 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 equati

23、on,Determine the output recursively with the condition of initial rest and,Rewrite the given difference equation as,Starting from initial condition, we can solve for successive values of yn for n1:,Considering yn =0 for n0, the solution is,Question:,Example 2.8,Consider a continuous-time LTI system

24、described by the following differential equation with initial conditions of and input,From 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

25、t0, it becomes,h(t) is the solution for the homogeneous equation. Thus,And because the 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 corresp

26、onding terms on each side,Compute the integral in the interval of 0-, 0+ on both sides 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 sati

27、sfies the difference equation,And there should be,Substituting hn for yn and n for xn 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 recursi

28、ve method to obtain the hn for this system!,2.7 Block Diagram Representations of First-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

29、器) (延时器),Basic elements for the block diagram representation of causal discrete-time systems. (a) an adder (b) a multiplier (c) a delayer.,Block diagram representation for the causal discrete-time system described by the first-order difference equation (yn+ayn-1=bxn).,First-order differential equati

30、on :,differentiation,Three basic elements in block diagram: adder, multiplier and integrator(积分器) .,2.6 SUMMARY,A representation of an arbitrary discrete-time signal as weighted sums of shifted unit samples;,Convolution sum representation for the response of a discrete-time LTI systems;,A representa

31、tion of an arbitrary continuous-time signal a weighted integrals of shifted unit impulses;,Convolution integral representation for continuous-time LTI systems;,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,