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.
The Essay on Our Conceptualization Of The Solar System
The human conceptualization of the solar system dates back to the beginning of time. The early Egyptians worshipped the sun as a source of life and then the area called space was becoming a curiosity to humans. Throughout history, our knowledge of the solar system has increased and there is still much to learn. Through the research and studies of Brahmagupta, Ptolemy, Kepler, Brahe, Copernicus, ...
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, > 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.
The Essay on Dconcentrations Of Solutions Determine The Mass Of A Potato
Introduction: The way to get the full results of this lab was through the process of osmosis. Osmosis is the movement of water across a membrane into a more concentrated solution to reach an equilibrium. When regarding cells osmosis has three different terms that are used to describe their concentration. The first of these words is isotonic. Cells in an isotonic solution show that the water has no ...
