2008.HW5.solutions - ECE 2704 Homework Number 5 Due at the...

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Unformatted text preview: ECE 2704 Homework Number 5 Due at the start of class on Wednesday, 12 March 2008 1. For what values of the complex number is the following true, Solution For any complex number , recall that Thus we require that and that , or equivalently that if . 2. Exercise 4.11 (page 478), parts (a), (b), (c), and (d). In each case, compute the Laplace transform and region of convergence. Do not us a table of Laplace transform identifies. Solutions , (a) For (b) For , Recall integration by parts, Note that if , thus for For (c) Recall the identity , then Note that this is almost exactly the problem addressed in part b. Using the exactly the same approach, integration by parts, we arrive at where we require that region of convergence is (d) For , . and . Since , the where for the leftmost term, we require that , which implies that . , and . Since for the rightmost term, we required that which implies that we require that both conditions are satisfied, the region of convergence is 3. Exercise 4.21 parts (a), (b), and (c) Solutions denote the Laplace transform of the signal (a) Let shifting property yields . Since , the time . Using linearity of the Laplace transform, (b) Since (c) Since , , the Laplace transform 4. Exercise 4.22 (a) Hint: express the signal compactly using unit step functions, but be careful. The function is not a timedelayed version of . Solution Note that However, this is not directly helpful since Instead, we can express as and then 5. Exercise 4.25. Do only the pairs Solutions (a) Since (b) Note that 6. Consider the differential equation with initial conditions Solutions and . Find the Laplace transform of . , we use the integration property, , . Recall that . is not a timedelayed version of . Computing the Laplace transform of both sides of the of differential equation yields which is expressed Thus, Multiplying top and bottom by yields 7. Consider the differential equation where and compute the Laplace transform of . Solution and Implies that Thus ...
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This note was uploaded on 03/29/2008 for the course ECE 2074 taught by Professor Stilwell during the Spring '08 term at Virginia Tech.

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