Assume that the resulting system is linear and time-invariant. x[n] O & r0n] D sumado a[n] +1 a few -2 Number P6. 5 (a) Discover the direct form I realization in the difference equation. (b) Find the difference equation described by the direct type I understanding. (c) Consider the intermediate signal ur[n] in Number P6. 5. (i) Get the relationship between l[n] and y[n]. (ii) Find the relation among r[n] and x[n]. (iii) Utilizing your answers to parts (i) and (ii), verify that the relation among y[n] and x[n] inside the direct contact form II realization is the same as your answer to component (b).
Systems Represented by Differential and Difference Equations / Complications P6-3
P6. 6 Consider the following differential box equation governing an LTI system. dx(t) dytt) dt + ay(t) = b di & cx(t) dt dt (P6. 6-1) (a) Draw the direct type I realization of frequency.
(P6. 6-1). (b) Draw the direct type II realization of eq. (P6. 6-1). Optional Challenges P6. six Consider the block diagram in Figure P6. several. The system is causal and it is initially at rest. r [n] x [n] + D y [n] -4 Number P6. six (a) Get the difference equation relating back button[n] and sumado a[n]. (b) For x[n] = [n], find r[n] for any n. (c) Find the system impulse response. P6. eight Consider the device shown in Figure P6. 8. Discover the gear equation relating x(t) and y(t). x(t) + r(t) + sumado a t a Figure P6. 8 b Signals and Systems P6-4 P6. on the lookout for Consider this difference formula: y[n] ” ly[n ” 1] sama dengan x[n] (P6. 9-1) (P6. 9-2) with x[n] sama dengan K(cos gon)u[n] Assume that the solution y[n] involves the total of a particular solution y,[n] to frequency. (P6. 9-1) for and 0 and a homogeneous solution yjn] gratifying the formula Yh[flI ” 12Yhn ” 1] =0. (a) If we assume that Yh[n] = Az, what value must be picked for zo? (b) Whenever we assume that to get n zero, y,[n] sama dengan B cos(Qon + 0), what are the values of B and 0? [Hint: It truly is convenient to watch x[n] sama dengan Re Kejonu[n] and y[n] = Re Yeonu[n], exactly where Y is a complex quantity to be established. P6. 15 Show that if r(t) satisfies the homogeneous differential box equation meters d=r(t) dt 0 and if s(t) is a response of your arbitrary LTI system H to the input r(t), then s(t) fulfills the same homogeneous differential formula. P6. 10 (a) Consider the homogeneous differential equation N dky) k~=0 dtk (P6. 11-1) k=ak Display that if so is known as a solution from the equation p(s) = Elizabeth akss k=O N = 0, (P6. 11-2) after that Aeso’ can be described as solution of eq. (P6. 11-1), in which a is an arbitrary complicated constant. (b) The polynomial p(s) in eq. (P6. 11-2) could be factored in terms of it is roots S1, ¦, T,.: p(s) sama dengan aN(S ” SI)1P(S tiplicities.
Note that ” S2)2… (S ” Sr)ar, where the si are the specific solutions of eq. (P6. 11-2) and the a happen to be their mul U+ you o2 & + Your = And In general, if the,; gt; one particular, then not merely is Ae’ a solution of eq. (P6. 11-1) but so is usually Atiesi’ given that j can be an integer greater than or equal to absolutely no and less than or Devices Represented by Differential and Difference Equations / Problems P6-5 corresponding to oa ” 1 . To illustrate this, show that if ao = two, then Atesi is a remedy of frequency. (P6. 11-1). [Hint: Show that if t is an arbitrary sophisticated number, then N o ve dtk = Ap(s)te’ to + A estI Hence, the most general solution of eq. P6. 11-1) can be p ci-1 ( i=1 j=0 Aesi, where the Ai, are arbitrary complex constants. (c) Resolve the following homogeneous differential formula with the specified aux iliary conditions. d 2 y(t) 2 dt2 + a couple of dy(t) & y(t) = 0, dt y(0) sama dengan 1, y'() = you MIT OpenCourseWare http://ocw. über. edu Useful resource: Signals and Systems Professor Alan Sixth is v. Oppenheim This may not match a particular course on ÜBER OpenCourseWare, nevertheless has been given by the author as an individual learning resource. For information regarding citing these types of materials or our Terms of Use, check out: http://ocw. mit. edu/terms.
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