# Differential Equation

Assume that the resulting system is linear and time-invariant. x[n] O + r0n] D y[n] +1 3 -2 Figure P6. 5 (a) Find the direct form I realization of the difference equation. (b) Find the difference equation described by the direct form I realization. (c) Consider the intermediate signal r[n] in Figure P6. 5. (i) Find the relation between r[n] and y[n]. (ii) Find the relation between r[n] and x[n]. (iii) Using your answers to parts (i) and (ii), verify that the relation between y[n] and x[n] in the direct form II realization is the same as your answer to part (b). Systems Represented by Differential and Difference Equations / Problems P6-3

P6. 6 Consider the following differential 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 x[n] and y[n]. (b) For x[n] = [n], find r[n] for all n. (c) Find the system impulse response. P6. 8 Consider the system shown in Figure P6. 8. Find the differential 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: y[n] – ly[n – 1] = x[n] (P6. 9-1) (P6. 9-2) with x[n] = K(cos gon)u[n] Assume that the solution y[n] 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 Yh[flI – 12Yhn – 1] =0. (a) If we assume that Yh[n] = 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 x[n] = Re{Kej”onu[n]} and y[n] = Re{Ye”onu[n]}, where Y is a complex number 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 arbitrary LTI system H to the input r(t), then 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 arbitrary complex constant. (b) The polynomial p(s) in eq. (P6. 11-2) can 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 zero 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 solution 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 about citing these materials or our Terms of Use, visit: http://ocw. mit. edu/terms.