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Q.4) [25 Marks] a) [15] Consider a CT LTI system described by the following differential equation...
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider...
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider a differential equation system for which the input x(t) and output y(t) are related by the differential equation d’y(t) dy(t) -6y(t) = 5x(t). dt dt Assume that the system is initially at rest. a) Determine the transfer function. b) Specify the ROC of H(s) and justify it. c) Determine the system impulse response h(t).
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t)...
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using the pole-zero plot technique a) b) What can be said about the stability of this stem?
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using...
Laplace Transform 5. Given a causal LTI system with pole-zero cancellation such as H(s)= S+1 what...
Laplace Transform
5. Given a causal LTI system with pole-zero cancellation such as H(s)= S+1 what is the region of convergence and why. (5+1)(3+2) i. ROC = undefined ii. ROC = Re(s) > 0 iii. ROC = Re(s) >-2 iv. ROC = Re(s) >-1
Problem 3. The input and the output of a stable and causal LTI system are related...
Problem 3. The input and the output of a stable and causal LTI system are related by the differential equation dy ) + 64x2 + 8y(t) = 2x(t) dt2 dt i) Find the frequency response of the system H(jw) [2 marks] ii) Using your result in (i) find the impulse response of the system h(t). [3 marks] iii) Find the transfer function of the system H(s), i.e. the Laplace transform of the impulse response [2 marks] iv) Sketch the pole-zero...
2.6.1 Consider a causal continuous-time LTI system described by the differential equation u"(t) +...
2.6.1-2.6.62.6.1 Consider a causal contimuous-time LTI system described by the differential equation$$ y^{\prime \prime}(t)+y(t)=x(t) $$(a) Find the transfer function \(H(s)\), its \(R O C\), and its poles.(b) Find the impulse response \(h(t)\).(c) Classify the system as stable/unstable.(d) Find the step response of the system.2.6.2 Given the impulse response of a continuous-time LTI system, find the transfer function \(H(s),\) the \(\mathrm{ROC}\) of \(H(s)\), and the poles of the system. Also find the differential equation describing each system.(a) \(h(t)=\sin (3 t) u(t)\)(b)...
4. Laplace Transform. (15 pts) Find the Laplace Transform of the following signals and sketch the...
4. Laplace Transform. (15 pts) Find the Laplace Transform of the following signals and sketch the corresponding pole-zero plot for each signal. In the plot, indicate the regions of convergence (ROC). Write X(s) as a single fraction in the form of DO (a) (5 pts)-(t-e*ta(t) + e-8tu(t). Show that X(s) =は,,늚. with ROC of Re(s) >-6. (b) (5 pts)-(t) = M(-t) +Au(-t). (c) (5 pts)-(t) 6(t)-a(-t). (s+6) (s+8)
A continuous-time LTI system has unit impulse response h(t). The Laplace transform of h(t), also called...
A continuous-time LTI system has unit impulse response h(t). The
Laplace transform of h(t), also called the “transfer function” of
the LTI system, is
.
For each of the following cases, determine the region of
convergence (ROC) for H(s) and the corresponding h(t), and
determine whether the Fourier transform of h(t) exists.
(a) The LTI system is causal but not stable.
(b) The LTI system is stable but not causal.
(c) The LTI system is neither stable nor causal
8...
Assume amplitude a = 4 The input to an LTI system is shown in the graph...
Assume amplitude a = 4
The input to an LTI system is shown in the graph below. Assume a = 4. X(t) 20 t @ by 0 Ingineering Given that the Laplace transform of the output is Y(s) = - (s + 3)(1 – e-45)2 s(s + 5)2 a) Find the transfer function of the system and the region of convergence for o = Re(s). H(s) = RoC: For regions of convergence, answer in interval notation e.g. (-INF, a),(a,b) or...
15. A dynamical system is modeled by the following differential equation under zero initial conditions: d’y(t)...
15. A dynamical system is modeled by the following differential equation under zero initial conditions: d’y(t) d’y(t) dy(t) du(t) + 5 + dt4 + 15 dt3 + 2y(t) = 8 + 10u(t) dt2 dt dt d4y(t) Write the system's state equation and the system's output equation.
2. The transfer function of a CT LTI system is given by H(s) (s2 +6s +10) (s2 -4s +8) a) Draw the...
2. The transfer function of a CT LTI system is given by H(s) (s2 +6s +10) (s2 -4s +8) a) Draw the pole-zero plot of the transfer function. b) Show all possible ROC's associated with this transfer function. c) Obtain the impulse response h(t) associated with each ROC of the transfer function. d) Which one (if any) of the impulse responses of part c) is stable?
2. The transfer function of a CT LTI system is given by H(s) (s2...
3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider a differential equation system for which the input x(t) and output y(t) are related by the differential equation d’y(t) dy(t) -6y(t) = 5x(t). dt dt Assume that the system is initially at rest. a) Determine the transfer function. b) Specify the ROC of H(s) and justify it. c) Determine the system impulse response h(t).
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using the pole-zero plot technique a) b) What can be said about the stability of this stem?
For the system described by the following differential equation d3y(t) d2y(t) d2x(t) dy(t) 3 dt dx(t) 9 dt y(t) 5x(t) 7 2 6 dt3 dt2 dt2 Express the system transfer function using...
Laplace Transform
5. Given a causal LTI system with pole-zero cancellation such as H(s)= S+1 what is the region of convergence and why. (5+1)(3+2) i. ROC = undefined ii. ROC = Re(s) > 0 iii. ROC = Re(s) >-2 iv. ROC = Re(s) >-1
Problem 3. The input and the output of a stable and causal LTI system are related by the differential equation dy ) + 64x2 + 8y(t) = 2x(t) dt2 dt i) Find the frequency response of the system H(jw) [2 marks] ii) Using your result in (i) find the impulse response of the system h(t). [3 marks] iii) Find the transfer function of the system H(s), i.e. the Laplace transform of the impulse response [2 marks] iv) Sketch the pole-zero...
2.6.1-2.6.62.6.1 Consider a causal contimuous-time LTI system described by the differential equation$$ y^{\prime \prime}(t)+y(t)=x(t) $$(a) Find the transfer function \(H(s)\), its \(R O C\), and its poles.(b) Find the impulse response \(h(t)\).(c) Classify the system as stable/unstable.(d) Find the step response of the system.2.6.2 Given the impulse response of a continuous-time LTI system, find the transfer function \(H(s),\) the \(\mathrm{ROC}\) of \(H(s)\), and the poles of the system. Also find the differential equation describing each system.(a) \(h(t)=\sin (3 t) u(t)\)(b)...
4. Laplace Transform. (15 pts) Find the Laplace Transform of the following signals and sketch the corresponding pole-zero plot for each signal. In the plot, indicate the regions of convergence (ROC). Write X(s) as a single fraction in the form of DO (a) (5 pts)-(t-e*ta(t) + e-8tu(t). Show that X(s) =は,,늚. with ROC of Re(s) >-6. (b) (5 pts)-(t) = M(-t) +Au(-t). (c) (5 pts)-(t) 6(t)-a(-t). (s+6) (s+8)
A continuous-time LTI system has unit impulse response h(t). The
Laplace transform of h(t), also called the “transfer function” of
the LTI system, is
.
For each of the following cases, determine the region of
convergence (ROC) for H(s) and the corresponding h(t), and
determine whether the Fourier transform of h(t) exists.
(a) The LTI system is causal but not stable.
(b) The LTI system is stable but not causal.
(c) The LTI system is neither stable nor causal
8...
Assume amplitude a = 4
The input to an LTI system is shown in the graph below. Assume a = 4. X(t) 20 t @ by 0 Ingineering Given that the Laplace transform of the output is Y(s) = - (s + 3)(1 – e-45)2 s(s + 5)2 a) Find the transfer function of the system and the region of convergence for o = Re(s). H(s) = RoC: For regions of convergence, answer in interval notation e.g. (-INF, a),(a,b) or...
15. A dynamical system is modeled by the following differential equation under zero initial conditions: d’y(t) d’y(t) dy(t) du(t) + 5 + dt4 + 15 dt3 + 2y(t) = 8 + 10u(t) dt2 dt dt d4y(t) Write the system's state equation and the system's output equation.
2. The transfer function of a CT LTI system is given by H(s) (s2 +6s +10) (s2 -4s +8) a) Draw the pole-zero plot of the transfer function. b) Show all possible ROC's associated with this transfer function. c) Obtain the impulse response h(t) associated with each ROC of the transfer function. d) Which one (if any) of the impulse responses of part c) is stable?
2. The transfer function of a CT LTI system is given by H(s) (s2...
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