ps9 - EE 350 PROBLEM SET 9 DUE: 3 December 2007 Reading...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
EE 350 PROBLEM SET 9 DUE: 3 December 2007 Reading assignment: Lathi Sections 6.1 through 6.5 and B.5 (pp. 24-33) Recitation sections meet the week of November 26. In problem 44 you will Fnd the Laplace transform of several elementary signals. Problem 45 shows how these basic transform pairs, along with several Laplace transform properties, can be used to quickly Fnd the Laplace transform of more complicated functions without evaluating the Laplace transform integral. The basic transform pairs obtained in Problem 44 will also allow you to Fnd the inverse Laplace transform using the partial fraction expansion method in Problem 46. Problem 44: (15 points) By direct integration, Fnd the Laplace transform F ( s ) and the region of convergence of F ( s ) for the following signals where a and b are positive real numbers: 1. (2 points) δ ( t ) 2. (2 points) u ( t ) 3. (3 points) e - at u ( t ) 4. (4 points) cos( bt ) u ( t ) 5. (4 points) sin( bt ) u ( t ) Problem 45: (20 points) 1. Let F ( s )= L{ f ( t ) } denote the unilateral Laplace transform of f ( t ). Prove the following properties of the Laplace transform, where t o 0 is a real constant and s o is a complex constant. (a) (2 points) Right shift in time: L{ f ( t - t o ) u ( t - t o ) } = F ( s ) e - st o ,t o > 0 (b) (3 points) Multiplication by t: L{ tf ( t ) } = - d ds F ( s ) (c) (3 points) ±requency shift: L ± e s o t f ( t ) ² = F ( s - s o ) 2. Using the elementary transform pairs derived in Problem 51 and the properties derived in part 1, Fnd the Laplace transform of the following signals where t o , a , and b are positive real parameters. (a) (2 points) u ( t - t o ) (b) (2 points) tu ( t ) (c) (2 points) te - at u ( t ) (d) (3 points) e - at cos( bt ) u ( t ) (e) (3 points) e - at sin( bt ) u ( t ) Note that this approach, particular in the case of the signals considered in parts (c) and (d), is much easier than Fnding the Laplace transform by direct integration.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Problem 46: (15 points) The inverse Laplace transform of F ( s ) can be calculated by expanding F ( s ) into partial fractions whose inverse transform is known. Carefully read sections 6.1-3 and B.5 in the text. Using the basic transform pairs in Table 6.1 on page 372 of the text and the techniques of partial fraction expansion, Fnd the inverse Laplace transforms of 1. (5 points) F ( s )= 2 s 3 +2 s 2 +17 s +15 s ( s + 3)( s +5) , 2. (5 points) F ( s 2 s 3 s 2 +40 s +24 ( s )( s + 3)( s 2 +4 s +8) , and 3. (5 points) F ( s 4 s 2 +7 s +1 s ( s +1) 2 . Problem 47: (12 points) ±or a causal and real-valued function f ( t ), a comparison of the unilateral Laplace transform F ( s ± 0 - f ( t ) e - st dt to the ±ourier transform F ( ω ± -∞ f ( t ) e - ωt dt = ± 0 f ( t ) e - ωt dt suggests that F ( ω F ( s ) | s = ω .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/23/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Pennsylvania State University, University Park.

Page1 / 6

ps9 - EE 350 PROBLEM SET 9 DUE: 3 December 2007 Reading...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online