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# HW4 - MAE107 Homework#4 Prof MCloskey Due Date The homework...

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MAE107 Homework #4 Prof. M’Closkey Due Date The homework is due by Friday, 5PM, May 14, 2010, to David Shatto in 38-137 E4. Problem 1 Consider the periodic signal u shown in Fig. 1. Answer the following questions: 1. Compute the Fourier series coeﬃcients, c k , using the “complex” representation, i.e., ﬁnd a formula for c k in u ( t ) = 1 T X k = -∞ c k e jkω 0 t , ω 0 = 2 π T , c k = Z T/ 2 T/ 2 u ( t ) e - jkω 0 t dt. 2. Use Matlab to plot an approximation of u using the following truncated Fourier series: u ( t ) 1 T 10 X k = - 10 c k e jkω 0 t . In other words you will sum only the terms in the Fourier series corresponding to k = - 10 , - 9 ,..., - 1 , 0 , 1 ,..., 9 , 10. Plot u and this approximation on the same ﬁgure. Your plot axes must extend from 0 to 1 seconds and -0.2 to 1.2 for the “ y ” axis. Hand in your code. Problem 2 Consider the ﬁrst order linear system ˙ x + 3 x = 3 u. (1) You will compute the periodic response of this system to the periodic input from Problem 1 in two ways. The ﬁrst method will use time-domain calculations, and the second method will use a Fourier series approach. 1. Time-domain approach. Imagine applying u from Problem 1 to this system since t = -∞ . No matter what the initial condition was when the input was ﬁrst applied, its eﬀect on x via a homogeneous solution of the ODE will have decayed to zero by the time we start taking our measurements. The system is in “steady-state” response at this point. Here is a general fact that we proved: an asymptotically stable linear system when driven by a periodic input, reaches a periodic (with same period as the input) steady-state once the eﬀect of initial conditions has decayed . Your objective is to compute the periodic steady 1

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ï 2 ï 1.5 ï 1 ï 0.5 0 0.5 1 1.5 2 ï 1 ï 0.5 0 0.5 1 1.5 2 Periodic function seconds Figure 1: A periodic signal –you will ﬁnd a closed form representation for the Fourier series coeﬃ- cients for this signal. state of this system when u is the signal from Problem 1. Here is a big hint: if the system is in periodic steady-state then x ( T ) = x (0), where T is the period of u . You can ﬁnd x (0) (or, what is the same, x ( T )) from x ( T ) = e - 3 T x (0) + Z T 0 h ( T - τ ) u ( τ ) dτ, where you set x ( T ) = x (0) = c (ﬁnd c ). Note that h is the impulse response of this system, and the term e - 3 T x (0) represents the homogeneous solution. Another way to think about this
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HW4 - MAE107 Homework#4 Prof MCloskey Due Date The homework...

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