DiscreteTimeSystem實(shí)用教案

1、1 Now consider single-input single-output discrete-time system defined by the input/output difference equationMiiNiiinxbinyany01where n is the integer-valued discrete-time index, xn is the input, and yn is the output. Here it is assumed that the coefficients a1, a2, , aN and b0, b1, b2, , bM are con

2、stants.(2.1)第1頁/共34頁第一頁，共35頁。2 Since Eq. (2.1) is a linear difference equation with constant coefficients, the system defined by the equation is linear, time invariant, and finite dimensional. The integer N in (2.1) is the order or dimension of the system. Also, any discrete-time system in the form

3、of Eq. (2.1) is causal since the output yn at time n depends only on previous values of the output and the current and previous values of the input xn. Unlike linear input/output differential equations, linear input/output difference equations can be solved by a direct numerical procedure. More prec

4、isely, the output yn for some finite range of integer values of n can be computed recursively as follows. First, rewrite (2.1) in the form 第2頁/共34頁第二頁，共35頁。3MiiNiiinxbinyany01(2.2) Then setting n=0 in (2.2) gives y0 a1y1 a2y2 aNyN + b0 x0 b1x1 bMxMThus the output y0 at time 0 is a linear combination

5、 of y1, y2, , yN and x0, x1, , xM.Setting n=1 in (2.2) gives y1 a1y0 a2y1 aNyN+1 + b0 x1 b1x0 bMxM+1So y1 is a linear combination of y0, y1, , yN+1 and x1, x0, , xM+1.第3頁/共34頁第三頁，共35頁。4 If this process is continued, it is clear that the next value of the output is a linear combination of the N past

6、values of the output and M+1 values of the input. At each step of the computation, it is necessary to store only N past values of the output (plus, of course, the input values). This process is called an Nth-order recursion. Here the term recursion refers to the property that the next value of the o

7、utput is computed from N previous values of the output (plus the input values). The discrete-time system defined by (2.1) or (2.2) is sometimes called a recursive discrete-time system or a recursive discrete-time filter since its output can be computed recursively. Here it is assumed that at least o

8、ne of the coefficients ai in (2.1) is nonzero. If all the ai are zero, the input/output difference equation (2.1) reduces to 第4頁/共34頁第四頁，共35頁。5In this case, the output at any fixed time point depends only on values of the input xn, and thus the output is not computed recursively. Such systems are sa

9、id to be nonrecursive. Finally, from (2.1) or (2.2) it is clear that the computation of the output response yn for n0 requires that the N initial conditions y1, y2, , yN must be satisfied. In addition, if the input xn is not zero for n0, the evaluation of (2.1) or (2.2) also requires the M initial i

10、nput values x1, x2, , xM. Miiinxbny0第5頁/共34頁第五頁，共35頁。6Consider the discrete-time system given by the second-order input/output difference equation yn 1.5yn1 +yn2 2xn2 (2.3)Write (2.3) in the form (2.2) results in the input/output equation yn 1.5yn1 yn2 +2xn2 (2.4)Now suppose that the input xn is the

11、 discrete-time unit-step function un and that the initial output values are y2=2 and y1=1. Thus setting n=0 in (2.4) gives y0 1.5y1 y2 +2x2 (1.5)(1) 2 +(2)(0)= 0.5第6頁/共34頁第六頁，共35頁。7Setting n=1 in (2.4) givesy1 1.5y0 y1 +2x1 (1.5)(0.5) 1 +(2)(0)= 1.75 Continuing the process yieldsy2 1.5y1 y0 +2x0 (1.

12、5)(1.75) 0.5 +(2)(1)= 0.125 y3 1.5y2 y1 +2x1 (1.5)(0.125) 1.75 +(2)(1)= 3.5625 and so on. In solving (2.1) and (2.2) recursively, the process of computing the output yn can begin at any time point desired. In the development above, the first value of the output that was computed was y0. If the first

13、 desired value is the output yq at time q, the recursive process should be started by setting n=q in (2.2). In this case, the initial values of the output that are required are yq 1, yq 2, , yq N 第7頁/共34頁第七頁，共35頁。8By solving (2.1) or (2.2) recursively, it is possible to generate an expression for th

14、e complete solution yn resulting from initial conditions and the application of the input xn. The process is illustrated by considering the first-order linear difference equation yn = ayn 1 +bxn, n =1, 2, (2.5)with the initial condition y0. First, setting n =1, n =2 and n =3 in (2.5) gives y1 = ay0

15、+bx1, (2.6) y2 = ay1 +bx2, (2.7) y3 = ay2 +bx3, (2.8)Inserting the expression (2.6) for y1 into (2.7) gives y2 = a( ay0 +bx1) +bx2, = a2y0 abx1 +bx2, (2.9)第8頁/共34頁第八頁，共35頁。9Inserting the expression (2.9) for y2 into (2.8) yields y3 = a(a2y0 abx1 +bx2) +bx3, = a3y0 +a2bx1 abx2+bx3, (2.10)From the pat

16、tern in (2.6), (2.9) and (2.10), it can be seen that for n1, niinnibxayany1)(0)(This equation gives the complete output response yn for n1 resulting from initial condition y0 and the input xn applied for n1. 第9頁/共34頁第九頁，共35頁。10As an application of the difference equation framework, in this section i

17、t is shown that a linear constant-coefficient input/output differential equation can be discretized in time, resulting in a difference equation that can be then solved by recursion. This discretization in time actually yields a discrete-time representation of the continuous-time system defined by th

18、e given input/output differential equation. The development begins with the first-order case. 第10頁/共34頁第十頁，共35頁。11Consider the linear time-invariant continuous-time system with the first-order input/output differential equation )()()(tbxtaydttdy (2.11) where a and b are constants. Eq. (2.11) can be

19、discretized in time by setting t=nT, where T is a fixed positive number and n takes on integer values only. This results in the equation )()()(nTbxnTaydttdynTt (2.12) 第11頁/共34頁第十一頁，共35頁。12Now the derivative in (2.12) can be approximated by TnTyTnTydttdynTt)()()(If T is suitable small and y(t) is con

20、tinuous, the approximation (2.13) to the derivative dy(t)/dt will be accurate. This approximation is called the Euler approximation of the derivative. Inserting the approximation (2.13) into (2.12) gives)()()()(nTbxnTayTnTyTnTy (2.13) (2.14) To be consistent with the notation that is being used for

21、discrete-time signals, the input signal x(nT) and the output signal y(nT) will be denoted by xn and yn, respectively; that is, xn= x(t)| t=nT and yn= y(t)| t=nT第12頁/共34頁第十二頁，共35頁。13In terms of this notation, (2.14) becomes 1nbxnayTnynyFinally, multiplying both sides of (2.15) by T and replacing n by

22、 n 1 results in a discrete-time approximation to (2.11) given by the first-order input/output difference equation yn yn 1 = aTyn 1+ bTxn 1, or yn = (1 aT)yn 1+ bTxn 1, (2.16)The difference equation is called the Euler approximation of the given input/output differential equation (2.11) since it is b

23、ased on the Euler approximation of the derivative. (2.15) 第13頁/共34頁第十三頁，共35頁。14The discrete values yn= y(nT) of the solution y(t) to (2.11) can be computed by solving the difference equation (2.16). The solution of (2.16) with initial condition y0 and with xn=0 for all n given by yn=(1 aT)n y0, n =0

24、, 1, 2, (2.17)The exact solution y(t) to (2.11) with initial condition y(0) and zero input is given by (2.18) 0 ),0(e)(tytyatTo analyze the approximation error between (2.17) and the exact solution (2.18) of y(t), set t=nT in (2.18) gives the following exact expression for yn yn=eanT y0= (eaT)n y0,

25、n =0, 1, 2, (2.19)Further, inserting the expansion 第14頁/共34頁第十四頁，共35頁。15221e3322TaTaaTaTfor the exponential into (2.19) results in the following exact expression for the values of y(t) at the times t=nT: (2.20) Comparing (2.17) and (2.20) shows that (2.17) is an accurate approximation if 1 aT is a g

26、ood approximation to the exponential eaT. This will be the case if the magnitude of aT is much less than 1, in which case the magnitude of aT will be much smaller than the quantity 1 aT. , 2 , 1 , 0 ,02213322nyTaTaaTnyn第15頁/共34頁第十五頁，共35頁。16Consider the RC circuit given in Fig. 2-1. The circuit has t

27、he input/output differential equation)(1)(1)(txCtyRCdttdywhere x(t) is the current applied to the circuit and y(t) is the voltage across the capacitor. (2.21) Fig. 2-1第16頁/共34頁第十六頁，共35頁。17 The difference equation (2.22) can be solved recursively to yield approximate values yn of the voltage on the c

28、apacitor resulting from initial voltage y0=0, input current xn= x(nT)= u(nT) and R=C=1. The recursion can be carried out using the MATLAB program in the course text.Writing (2.21) in the form (2.11) reveals that in this case, a=1/(RC) and b=1/C . Hence, the discrete-time representation (2.16) for th

29、e RC circuit is given by 1 11nxCTnyRCTny(2.22) 第17頁/共34頁第十七頁，共35頁。18 To compare with the exact solution of (2.21), the plots of the resulting output (the unit-step response) for the approximation are displayed in Fig. 2-2(a) for T=0.2 and Fig. 2-2(b) for T=0.1 along with the exact unit-step response

30、 y(t)=(1et)u(t). Obviously, the approximation error in Fig. 2-2(b) is smaller than that in Fig. 2-2(a) as the sampling interval T becomes smaller. Fig. 2-2(a)Fig. 2-2(b)第18頁/共34頁第十八頁，共35頁。19The discretization technique for first-order differential equations described above can be generalized to seco

31、nd- and high-order differential equations. In this second-order case the following approximations can be used:TnTyTnTydttdynTt)()()(2.23) TdttdydttdydttydnTtTnTtnTt|/ )(|/ )()(22(2.24) Combining (2.23) and (2.24) yields the following approximation to the second derivative:222)()(2)2()(TnTyTnTyTnTydt

32、tydnTt(2.25) 第19頁/共34頁第十九頁，共35頁。20The approximation (2.25) is the Euler approximation of the second derivative. Now consider a linear time-invariant continuous-time system with the second-order input/output differential equation)()()()()(010122txbdttdxbtyadttdyadttyd(2.26) Setting t=nT in (2.26) and

33、 using the approximations (2.23) and (2.25) results in the following time discretization of (2.26): 1 1 12201012nxbTnxnxbnyaTnynyaTnynyny(2.27) 第20頁/共34頁第二十頁，共35頁。21Replacing n by n 2 in (2.27) and multiplying both sides of (2.27) by T2 yields the difference equation yn +(a1T 2) yn 1+ (1 a1T + a0T2)

34、 yn 2 = b1T xn 1+( b0T2 b1T) xn 2 (2.28)Eq. (2.28) is the discrete-time approximation to the second-order input/output difference equation (2.26). The discrete values y(nT) of the solution y(t) to (2.26) can be computed by solving the difference equation (2.28). To solve (2.28), the recursion will b

35、e started at n=2 so that the initial values y0= y(0) and y1= y(T) are required. The initial value y(T) can be generated by using the approximationTyTyy)0()()0(2.29) where denotes the derivative of y(t). )0(y 第21頁/共34頁第二十一頁，共35頁。22Solving (2.29) for y(T) gives y1= y(T) = y(0) + (2.30)with the initial

36、 values y0 and y1, the second-order difference equation (2.28) can be solved using the MATLAB program in the course text.)0(yT 第22頁/共34頁第二十二頁，共35頁。23Consider the series RLC circuit shown in Fig. 2-3. As indicated, the input x(t) is the voltage applied to the circuit and the output y(t) is the voltag

37、e vC(t) across the capacitor. We have known that the differential equation for the circuit is given by)(1)(1)()(22txLCtvLCdttdvLRdttvdCCC(2.31) Fig. 2-3 Series RLC circuit第23頁/共34頁第二十三頁，共35頁。24Eq. (2.31) is a second-order differential equation that can be written in the form (2.26) with a1=R/L, a0=1

38、/(LC), b1=0, b0=1/(LC) (2.32)Inserting (2.32) into the discretized equation (2.28) yields221 1222nxLCTnvLCTLRTnvLRTnvCCC(2.33) Eq. (2.33) is the difference equation approximation of the RLC circuit. The voltage vC(t) across the capacitor will be computed using the discretization (2.33) in the case w

39、hen R=2, L=C=2, vC(0)=1, , and vC(t)=sin(t)u(t). 1)0(Cv 第24頁/共34頁第二十四頁，共35頁。25To solve the difference equation (2.33) for n 2, the initial conditions are x0=sin(0)=0, x1= sin(T), vC0=1, and from (2.30), we getTvTvTvvCCCC1)0(0) )( 1 Now the second-order difference equation (2.33) can be solved using

40、the MATLAB program. To compare with the exact solution vC(t)=0.5(3+t)etcos(t)u(t) to the differential equation (2.31), the plots of the resulting output for the approximation are displayed in Fig. 2-4(a) for T=0.2 and Fig. 2-4(b) for T=0.1 along with the exact solution.第25頁/共34頁第二十五頁，共35頁。26From the

41、 plots it is seen that there is a significant error in the approximation in Fig. 2-4(a) for T=0.2. To obtain a better approximation, the discretization interval T can be decreased to be 0.1, the result is shown in Fig. 2-4(b). In fact, as T0, the approximation should approach the true response value

42、s.Fig. 2-4(a)Fig. 2-4(b)第26頁/共34頁第二十六頁，共35頁。27Problems2.1 For the difference equation yn+ 1.5yn 1= xn, use the method of recursion to compute yn for n=0, 1, 2, 3, when xn=0 for all n and y1=2, and then find a complete solution for yn.2.2 Consider the following differential equations: (a) (b) Using E

43、ulers approximation of derivatives with T arbitrary and input x(t) arbitrary, derive a difference equation model.)(3)(2)(txtydttdy)()(2)(3)(22txtydttdydttyd第27頁/共34頁第二十七頁，共35頁。28 If this process is continued, it is clear that the next value of the output is a linear combination of the N past values

44、of the output and M+1 values of the input. At each step of the computation, it is necessary to store only N past values of the output (plus, of course, the input values). This process is called an Nth-order recursion. Here the term recursion refers to the property that the next value of the output i

45、s computed from N previous values of the output (plus the input values). The discrete-time system defined by (2.1) or (2.2) is sometimes called a recursive discrete-time system or a recursive discrete-time filter since its output can be computed recursively. Here it is assumed that at least one of t

46、he coefficients ai in (2.1) is nonzero. If all the ai are zero, the input/output difference equation (2.1) reduces to 第28頁/共34頁第二十八頁，共35頁。29In this case, the output at any fixed time point depends only on values of the input xn, and thus the output is not computed recursively. Such systems are said

47、to be nonrecursive. Finally, from (2.1) or (2.2) it is clear that the computation of the output response yn for n0 requires that the N initial conditions y1, y2, , yN must be satisfied. In addition, if the input xn is not zero for n0, the evaluation of (2.1) or (2.2) also requires the M initial inpu

48、t values x1, x2, , xM. Miiinxbny0第29頁/共34頁第二十九頁，共35頁。30Consider the discrete-time system given by the second-order input/output difference equation yn 1.5yn1 +yn2 2xn2 (2.3)Write (2.3) in the form (2.2) results in the input/output equation yn 1.5yn1 yn2 +2xn2 (2.4)Now suppose that the input xn is the discrete-time unit-step function un and that the initial output values are y2=2 and y1=1. Thus setting n=0 in (2.4) gives y0 1.5y1 y2 +2x2 (1.5)(1) 2 +(2)(0)= 0.5第30頁/共34頁第三十頁，共35頁。31In terms of this nota