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.