Homework 8 EE235, Spring 2012 Solutions 1. Find the (unilateral ...

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(b) What are the poles and zeros of this H(s)? Plot the corresponding pole-zero plot. There are no zeros for this function and a pole at s = −5. Imaginary Axis 6 4 2 0 −2 −4 −6 Pole−Zero Map −6 −4 −2 0 Real Axis 2 4 6 (c) With the given initial conditions, find the system response y(t) to an input x(t) = 3 5 e−2t u(t). Use the unilateral transform since we are given the initial condition RC = .2 sY (s) − y(0 − ) + 5Y (s) = 5X(s) Y (s)(s + 5) = 5X(s) + y(0 − ) Y (s) = 1 s + 5 (5X(s) + y(0− )) X(s) = 3 1 5 s + 2 Y (s) = 3 2 − (s + 5)(s + 2) s + 5 Y (s) = 1 1 2 − − s + 2 s + 5 s + 5 Y (s) = 1 3 − s + 2 s + 5 y(t) = (e −2t − 3e −5t )u(t) (d) Use the initial value theorem to find the initial value of y(t), i.e., y(0 + ). Confirm your answer with your result in part (c). 2
The initial value theorem states lims→∞sY (s) = y(0 + ) Y (s) = 1 3 − s + 2 s + 5 lims→∞sY (s) = lims→∞s( 1 = lims→∞ = 1 − 3 = −2 1 + 2 s − 3 s + 5 ) s + 2 1 − 3 1 + 5 s This is what we expect since the initial condition is -2 and the voltage across the capacitor cannot change instantaneously to match the forcing function. (e) Use the final value theorem to find the final value of y(t), i.e., the value of y(t) when t → +∞. Confirm your answer with your result in part (c). The initial value theorem states lims→0sY (s) = limt→∞y(t) Y (s) = 1 3 − s + 2 s + 5 lims→0sY (s) = lims→0s( 1 3 − s + 2 s + 5 ) = 0 1 2 = 0 − 03 5 This is what we expect because at t → ∞ we expect the voltage across the capacitor to match the voltage of the forcing function. At time t → ∞, x(t) = 0. 3. For an LTI system with transfer function H(s) = 3s(s−1) (s−2)(s+3) , give its natural modes and state if the system is stable/unstable/marginally stable. Justify your answer. The natural modes are e2t and e−3t . The poles to the system are at s = 2, −3. The system is unstable due to the pole at s = 2. 4. Consider a causal LTI system is described by d2y(t) − y(t) = 4x(t) dt2 with all initial conditions zero. An input x(t) = cos(5t)u(t) is applied. 3
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Page 5 and 6: (b) Does a stable and causal invers

Homework 8 EE235, Spring 2012 Solutions 1. Find the (unilateral ...

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