Consider the linear system described by x(t)= A(t)x(t), x(f,)= x0 Show that the following relation is...
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...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...
f) Use the definition of a linear system to show if the following system is linear: 3dy0)-2)-). t)-xt). dt g) Use the definition of a linear system to show if the following system is linear: 3t yle)-2x(). h) Use the definition of a linear system to show if the following system is linear: 24 3)1). dy(t) dt
Consider a causal, linear and time-invariant system of continuous time, with an input-output relation that obeys the following linear differential equation: y(t) + 2y(t) = x(t), where x(t) and y(t) stand for the input and output signals of the system, respectively, and the dot symbol over a signal denotes its first-order derivative with respect to time t. Use the Laplace transform to compute the output y(t) of the system, given the initial condition y(0-) = V2 and the input signal...
Q2. Consider the system described by the following differential equation x(t) Ax(t) where and x(0) -
(d) [5] The input-output relation for DT system is described by following system equation y[n] = 31[] State if the system possesses the following properties: Linear BIBO-Stable Casual Memoryless Time-Invariance
Let a linear system with input x(t) and output y(t) be described by the differential equation . (a) Compute the simplest math function form of the impulse response h(t) for this system. HINT: Remember that with zero initial conditions, the following Laplace transform pairs hold: Let the time-domain function p(t) be given by p(t) = g(3 − 0.5 t). (a) Compute the simplest piecewise math form for p(t). (b) Plot p(t) over the range 0 ≤ t ≤ 10 ....
3. Consider a linear time invariant system described by the differential equation dy(t) dt RCww + y(t)-x(t) where yt) is the system's output, x(t) ?s the system's input, and R and C are both positive real constants. a) Determine both the magnitude and phase of the system's frequency response. b) Determine the frequency spectrum of c) Determine the spectrum of the system's output, y(r), when d) Determine the system's steady state output response x()-1+cos(t) xu)+cost)
The input signal x(1) of the LTI system is given by the following relation x(t) = xo(1–17), where ,1)= 1 3/4 +2,05151 / and it is passed through a filter with frequency response given below: (-;,O< f 54 H(S)= ,-45 f<0 (0, otherwise Determine the power of the signal x. (t)
Consider the LTI system described by the following information X(s) = 2. S-2 where x(t) = 0 for t > 0, and y(t) = -e2'u(-t) + e-t u(t). 3 Determine H(s), and its region of convergence. (5 points). Determine h(t). (5 points). a. b. Using the system function found in part(a), determine the output y(t) if the - o <t < + o. ( 10 points). c. input is given by: est,