3. (l’+2° +1²=4') Topic: Laplace transform, CT system described by differential equations, LTI system properties. Consider...
Q.4) [25 Marks] a) [15] Consider a CT LTI system described by the following differential equation (assume zero initial conditions): dºy(t) _6dy(t) + 3 dy(t) = 2x(6) dt3-6 dt2 +8 dt = 2x(t) [5] Using Laplace transform and its properties determine the transfer function H(s) [5] Draw the pole-zero diagram of H(s) (5) Write down all possible Region-of-Convergence (ROC) for the H(s) (iii) [5] white b) (10) Determine the signal x(t) ( assume it to be right-sided signal) when the...
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...
2. Using Laplace transform, solve the system of differential equations d.x: dy dt where x(0)1 2. Using Laplace transform, solve the system of differential equations d.x: dy dt where x(0)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)...
3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t)...
Use the Laplace transform to solve the given system of differential equations. Use the Laplace transform to solve the given system of differential equations. of + x - x + y = 0 dx + dy + 2y = 0 x(0) = 0, y(0) = 1 Hint: You will need to complete the square and use the 1st translation theorem when solving this problem. x(t) = y(t) =
dy(D), 5) Consider a causal LTI system S described by the following differential equation: 2 + 3y(t) = x(t). Draw a block digram representation for S. Then, convert this differential equation into an integral equation, and draw a corresponding block diagram representation. dt
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...
Question 2 A linear time-invariant (LTI) system has its response described by the following second-order differential equation: d'y) 3-10))-3*0)-6x0) dy_hi dx(t) where x() is the input function and y(t) is the output function. (a) Determine the transfer function H(a) of the system. (b) Determine the impulse response h(t) of the system.