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# MAE107 Homework #5 Prof. M'Closkey Due Date The homework is due at 5PM on Tuesday, May 25, 2010, to David Shatto (38-138 foyer, Engineering 4)....

problem 1. i just don't know how to do it.
MAE107 Homework #5 Prof. M’Closkey Due Date The homework is due at 5PM on Tuesday, May 25, 2010, to David Shatto (38-138 foyer, Engineer- ing 4). Reading Please read the introduction to Chapter 17 and sections 17.1 thru 17.7. This material covers the bilateral and unilateral Laplace transform. Problem 1 This problem discusses correlation functions and their relation to input-output signals of linear systems. Consider the low-pass ﬁlter described by the diﬀerential equation ˙ V out + 1 RC V out = 1 RC V in . For the sake of brevity, deﬁne α = 1 / ( RC ). Thus, the impulse response of the circuit is h ( t ) = αe - αt μ ( t ) . Let the input to the circuit be V in ( t ) = ce - ct μ ( t ) , c > 0 . (1) In this problem, we don’t consider the limit as c → ∞ –the input is chosen to be this form because it makes the computations easier to carry out. Answer the following: 1. Compute V out given the input (1). Note that time is deﬁned on ( -∞ , ). 2. Compute the auto-correlation of (1). In other words, compute a new function of time given by R V in V in ( t ) = Z -∞ V in ( t + τ ) V in ( τ ) dτ, t ( -∞ , ) . You will need to consider the two cases when t < 0 and t 0. 3. Compute the cross-correlation , R V out V in , where V in is given by (1) and V out is the output calculated in part 1: R V out V in ( t ) = Z -∞ V out ( t + τ ) V in ( τ ) dτ. Again, you will need to consider the cases t < 0 and t 0. 1
4. Compute the response of the circuit when the input is R V in V in from part 2. In other words, compute, y ( t ) = Z -∞ h ( t - τ ) R V in V in ( τ ) dτ, t ( -∞ , ) , and show that y ( t ) = R V out V in ( t ), where R V out V in was computed in part 3. 5. Let α = 0 . 4 and c = 1. Plot, on a single ﬁgure, the input (1), and the response V out computed in part 1. On another ﬁgure, plot R V in V in and R V out V in . For both ﬁgures, use a time axis from - 10 to 10 seconds. Problem 2 For which of the signals shown below does the bilateral Laplace transform exist? As simple “yes” or “no” will suﬃce. For those signals that are Laplace transformable, please state the region of convergence. 1. u ( t ) = 1, for all t ( -∞ , ) 2. u ( t ) = ( e - t t 0 0 t > 0 3. u ( t ) = e - t for all t ( -∞ , ) 4. u ( t ) = ( e t t 0 0 t < 0 5. u ( t ) = e t for all t ( -∞ , ) Problem 3 1. Consider a signal u for which the bilateral Laplace transform exists and, furthermore, the ROC is { s C | Re( s ) > - a } where a is real and greater than zero (in other words, the ROC is a half-plane whose border is parallel to the -axis and passes through the point - a + j 0 on the real axis). An example of such a function is e - t μ ( t ). Show that the bilateral Laplace transform of the auto-correlation function R uu is ˆ R uu = L ( R uu ) = ˆ u ( s u ( - s ) , where ˆ u ( s ) = L ( u ) = Z -∞ u ( t ) e - st dt. What is the ROC for ˆ R uu ? 2
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