Answer Happy

Problem 1. (20 marks) Continuous-time Signal Analysis. (a) (9 marks) An LTIC system is specified by the equation Dº (D+1)y(t) = (D2 + 2)x(t) (i) (4 marks) Find the characteristic polynomial, characteristic equation, characteristic roots, and characteristic modes of this system. (Hints: For repeated roots r, the corresponding characteristic modes are e" and te".) (ii) (5 marks) Find yo(t), the zero-input component of the response y(t) for t > 0, if the initial conditions are yo(0) = 4, yo(0) = 3, and yo(0) = -1. (b) (11 marks) The unit impulse response of an LTIC system is h(t) = e-tu(t). Find the system response y(t) when all the initial conditions are zero, if the input r(t) is: (i) (2 marks) u(t) (ii) (2 marks) e-tu(t) (iii) (2 marks) e-2 u(t) (iv) (2 marks) sin 3tu(t) (v) (3 marks) u(t), but h(t) is changed to h(t) = (1 - 2t)e-2u(t)