Question

Term brua 2019 Instructor: Ahnet A 1. Determine the Laplace transform and the associated region of convergence and pole-zero plot for each of the following functions of time (b) r(t)te-24 elsewhere (d) a(t) (t)+u(t) 2. Determine the function of time, a(t), for each of the following Laplace transforms and their associated regions of convergenice: )부부, Rds) > 1 d)승부 R1(s) >-1 3. Consider an LTI systern with input r(t)-ε-lu(t) and impulse response h(t)-e-2u(t). (a) Determine the Laplace transforms of ar(t) and h(t) (b) description Using the convolution property, determine the Laplace transform Y(s) of the output g(t) (c) From the Laplace transform of y(t) as obtained in part (2), determine y(t). (d) Verify your result in part (3) by explicitly convolving z(t) and h(t) 4. Consider a continuous-time LTI system for which the input(t) and output (t) are related by the differential equation d dt Let X(s) and Y(s) denote Laplace transforms of a(t) and y(t), respectively, and let H(s) denote the Laplace transform of h(t), the system impulse response (a) Determine H(s) as a ratio of two polynomials in s. Sketch the pole-zero pattern of H(s) (b) Determine h(t) for each of the following cases: 1. The system is stable. 2. The system is causal. 3. The system is neither stable nor causal 5. The system function of a causal LTI system is 8 +1 Determine and sketch the response y(t) when the input is r(t) = e-H

Answer 1

From the given problem a) The expression for x(t) x(t) = e^2 t u (t) + e^3 t u (- t) Applying Laplace transform to the given x(t) function x(s) = integral^infinity_- infinity x(t) e^- st dt = integral^infinity_- infinity [e^2 t u (t) + e^3 t u (- t)]e^- st dt = integral^0_- infinity e^2 t . e^- st at + integral^0_- infinity e^- (s - 3) t dt = e^- (s - 2) t/- (s - 2) Vertical-bar ^0_- infinity + e^- (s - 3) t/- (s - 3) Vertical-bar ^0_- infinity = -1/(s - 2) (1 - 0) - 1/s - 3 (1 - 0) x(s) = -1/s - 2 - 1/s - 3 = 5 - 2s/(s - 2) (s - 3) The region of convergence ROC is Re {s} < 2 \r\n Thus the Laplace Transform of the function is 5 - 2s/(s - 2) (s - 3): Re {s} < 2, Pole. zero plot for the transfer function b) The expression of x(t) is given by x(t) = t e^- 2 (t) Apply Laplace transform to the function x(s) = integral^infinity- infinity x(t)e^- st dt = integral^infinity_- infinity