Differential Equation Essay Example for Free

Differential Equation EssayAs pluse that the resulting system is linear and time-invariant. xn O + r0n D yn +1 3 -2 Figure P6. 5 (a) Find the purport form I realization of the difference equation. (b) Find the difference equation described by the manage form I realization. (c) Consider the intermediate head rn in Figure P6. 5. (i) Find the relation between rn and yn. (ii) Find the relation between rn and xn. (iii) Using your answers to parts (i) and (ii), verify that the relation between yn and xn in the direct form II realization is the same as your answer to part (b). Systems Represented by Differential and Difference Equations / Problems P6-3P6. 6 Consider the undermentioned first derivative equation governing an LTI system. dx(t) dytt) dt + ay(t) = b di + cx(t) dt dt (P6. 6-1) (a) Draw the direct form I realization of eq. (P6. 6-1). (b) Draw the direct form II realization of eq. (P6. 6-1). Optional Problems P6. 7 Consider the block diagram in Figure P6. 7. The system is causal and is initially at rest. r n x n + D y n -4 Figure P6. 7 (a) Find the difference equation relating xn and yn. (b) For xn = n, find rn for all n. (c) Find the system impulse response. P6. 8 Consider the system shown in Figure P6. 8. Find the differential coefficient equation relating x(t) and y(t). x(t) + r(t) + y t a Figure P6. 8 b Signals and Systems P6-4 P6. 9 Consider the following difference equation yn lyn 1 = xn (P6. 9-1) (P6. 9-2) with xn = K(cos gon)un Assume that the solution yn consists of the sum of a particular solution y,n to eq. (P6. 9-1) for n 0 and a homogeneous solution yjn satisfying the equation YhflI 12Yhn 1 =0. (a) If we assume that Yhn = Az, what value must be chosen for zo? (b) If we assume that for n 0, y,n = B cos(Qon + 0), what are the values of B and 0? Hint It is convenient to view xn = ReKejonun and yn = ReYeonun, where Y is a complex numeral to be determined. P6. 10 Show that if r(t) satisfies the homogeneous differential equation m d=r( t) dt 0 and if s(t) is the response of an impulsive LTI system H to the input r(t), and thence s(t) satisfies the same homogeneous differential equation. P6. 11 (a) Consider the homogeneous differential equation N dky) k=0 dtk (P6. 11-1) k=ak Show that if so is a solution of the equation p(s) = E akss k=O N = 0, (P6. 11-2) then Aeso is a solution of eq. (P6. 11-1), where A is an commanding complex constant. (b) The polynomial p(s) in eq. (P6. 11-2) empennage be factored in terms of its roots S1, ,S,. p(s) = aN(S SI)1P(S tiplicities.Note that S2)2 . . . (S Sr)ar, where the si are the distinct solutions of eq. (P6. 11-2) and the a are their mul U+ 1 o2 + + Ur = N In general, if a, gt 1, then not only is Ae a solution of eq. (P6. 11-1) but so is Atiesi as long as j is an integer greater than or equal to vigor and less than or Systems Represented by Differential and Difference Equations / Problems P6-5 equal to oa 1. To illustrate this, show that if ao = 2, then Atesi is a so lution of eq. (P6. 11-1). Hint Show that if s is an arbitrary complex number, then N ak dtk = Ap(s)te t + A estI Thus, the most general solution of eq. P6. 11-1) is p ci-1 ( i=1 j=0 Aesi , where the Ai, are arbitrary complex constants. (c) Solve the following homogeneous differential equation with the specified aux iliary conditions. d 2 y(t) 2 dt2 + 2 dy(t) + y(t) = 0, dt y(0) = 1, y() = 1 MIT OpenCourseWare http//ocw. mit. edu Resource Signals and Systems Professor Alan V. Oppenheim The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information nearly citing these materials or our Terms of Use, visit http//ocw. mit. edu/terms.