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Unformatted text preview: 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 > . (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....
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This note was uploaded on 02/02/2011 for the course MAE 107 taught by Professor Tsao during the Spring '06 term at UCLA.
- Spring '06