1. (Chapter I). A continuous-time system, with time t in seconds (s), input f(t), and output y(), is specified by the equation y(t) 1.5cos(250t) +0.8f(t) a. Is this system instantaneous (memoryle...
Do each of the following eight (8) problems. The problems have equal weight. For each problem, in order to receive maximum possible credit, show the steps of the solution clearly,and provide appropriate explanation. Return this exam with your answer sheets . Chapter continunous-time system, with time t in seconds () input fO, and output yo. is specified by the equation y(t) = 1.5cos(2x500 + 0.8ft). a. Is this system instantaneous (memoryless) or dynamic (with memory)? Justify your answer Show that...
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input f(t) and output y(t) is specified by the differential equation D2(D +1)y(t) (D - 3)f(t) Find the characteristic polynomial, characteristic equation, characteristic root(s), and characteristic mode(s) of this system. a. b. Is this system asymptotically stable, marginally stable, or unstable? Justify your answer.
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input f(t) and output y(t) is specified by the differential equation D2(D +1)y(t) (D - 3)f(t)...
2. (Chapter 2). A linear, time-invariant, continuous-time (LTIC) system with input 1(1) and output y(t) is specified by the differential equation D(D? + 1)y(t) = Df(t). a. Find the characteristic polynomial, characteristic equation, characteristic root(s), and characteristic mode(s) of this system. b. Is this system asymptotically stable, marginally stable, or unstable? Justify your answer.
Q4. An LTI continuous-time system is specified by dy(t).dyết + 4y(t) = f(t) dt2 *4 dt 4y(t) = f(t) a) Find its unit impulse response with the initial conditions yn (0) = 0, yn (0) = 1 where yn(t) is the zero input response and yn (0) = 1 is the 1st order derivative of yn(t). b) Please state the definition for stability, and then verify that whether this system is stable or not?
For a continuous time linear time-invariant system, the
input-output relation is the following (x(t) the input, y(t)
the
output):
, where h(t) is the impulse response function of the
system.
Please explain why a signal like e/“* is always an eigenvector
of
this linear map for any w. Also, if ¥(w),X(w),and H(w) are
the
Fourier transforms of y(t),x(t),and h(t), respectively.
Please
derive in detail the relation between Y(w),X(w),and H(w),
which means to reproduce the proof of the basic convolution
property...
1. Consider a continuous system whose input x(t) and output y(t) are related by dy(t) + ay(t) = x(t) dt where a is a constant. (a) Find y(t) with the condition y(0) = yo and x(t) = Ke-bu(t) (b) Express y(t) in terms of the zero-input and zero-state responses. 2. Consider the system in Problem 1. (a) Show that the system is not linear if y(0) = yo 70. (b) Show that the system is linear if y(0) = 0....
Consider the continuous-time system given by the equation y(t) = (v *v)(t). Is this system time- invariant? If yes, give a proof; if no, show why not by giving a counterexample.
In a continuous-time system, the laplace transform of the input X(s) and the output Y(s) are related by Y(s) = 2 (s+2)2 +10 a) If x(t) = u(t), find the zero-state response of the system, yzs(1). yzs() = b) Find the zero-input response of the system, yzi(t). Yzi(t) = c) Find the steady-state solution of the system, yss(t). Yss(t) =
In this chapter, we introduced a number of general properties of
systems. In particular,
a system may or may not be
(1) Memoryless
(2) Time invariant
(3) Linear
(4) Causal
(S) Stable
Determine which of these properties hold and which do not hold for
each of the
following continuous-time systems. Justify your answers. In each
example, y(t) denotes
the system output and x(t) is the system input.
(b) y(t) [cos(31)]x(1) (c) y() = 13, x(T)dT x(t) + x(t - 2...
1. For a system described in Figure 1. x(t) - input voltage, y(t) - output voltage. (a) Determine Continuous Time (C.T.) "Math Model" when R = 1/3 121, L = 1/2 [F], and C = 1 [F]. (b) Fine "Zero Input Response". y zit. for the C.T.system. when y(0) = 1 [V], y'(0) = 2 IV (c) Draw "Zero Input Response". y_zi(t) with respect to time 1 (2-D graph) (d) Find impulse response, h(!). of the Continuous Time (C.T.) system....