# hw5 - 6.003 Homework 5 Due at the beginning of recitation...

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6.003 Homework 5 Due at the beginning of recitation on Wednesday, October 14, 2009 . Problems 1. Laplace Transforms Determine the Laplace transforms (including the regions of convergence) of each of the following signals: a. x 1 ( t ) = e 2 t u ( t 3) b. x 2 ( t ) = (1 (1 t ) e 3 t ) u ( t ) c. x 3 ( t ) = te −| t | d. 2 1 0 1 2 3 4 1 1 x 4 ( t ) t e. 2 1 0 1 2 3 4 1 x 5 ( t ) t 2. Inverse Laplace transforms Determine and sketch all possible signals with Laplace transforms of the following forms. For each signal, indicate the associated region of convergence. a. X 1 ( s ) = s + 2 ( s + 1) 2 b. X 2 ( s ) = 1 s 2 ( s 1) c. X 3 ( s ) = s + 1 s 2 + 2 s + 2 d. X 4 ( s ) = ± 1 e s s ² 2

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6.003 Homework 5 / Fall 2009 2 3. Symmetry Determine which of the following pole-zero 1 diagrams could represent Laplace transforms of even functions of time. Determine expressions for the time functions of those that can represent even functions of time. For those that cannot, explain why they cannot. 1 1 X 1 ( s ) 1 X 2 ( s ) 1 1 X 3 ( s ) 1 1 X 4 ( s ) How can you determine if a signal is even or not by looking at its Laplace transform and
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hw5 - 6.003 Homework 5 Due at the beginning of recitation...

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