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Lecture 12

# Lecture 12 - Lecture 12 ECE 3101 Copyright P B Luh 1 ECE...

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Unformatted text preview: Lecture 12, 3/3/11 ECE 3101, Copyright P. B. Luh 1 ECE 3101 Signals and Systems • Reading Assignment: Sections 4.1, 4.3, B.5, and 10.3.1 • Problem Set 5: 4.1-1 (b, d, f, and h), 4.1-3 (b, d, f and h), 4.2-1 (b, d, f, and h), 4.2-2, 4.2-3 (b, d, f, and h), 4.2-9. Due Tuesday, 3/15/11 • Next class: Mon. 3/14/11 during discussion session • Exam. 2, Tue. 3/22/11, end of Ch. 3. Old sample Exam 1 • Last Time: Properties of the Laplace Transform ; dt e ) t ( f ) s ( F st – a 1 f 1 (t) + a 2 f 2 (t) a 1 F 1 (s) + a 2 F 2 (s) – f (1) (t) s F(s) - f(0- ); f (-1) (t) F(s)/s; t f(t) (-1)F (1) (s) – h(t) f(t) H(s) F(s) j c j c st ds e ) s ( F j 2 1 ) t ( f Lecture 12, 3/3/11 ECE 3101, Copyright P. B. Luh 2 – f(t-t 0 )U(t-t 0 ) e-st F(s); e s t f(t) F(s - s 0 ) – f(at) (1/a)F(s/a) for a > 0 – Initial Value Theorem: f(0 + ) = lim s sF(s) – Final Value Theorem: f( ) = lim s 0 sF(s) if all the poles of sF(s) have strictly negative real parts • Inverse Laplace Transform – F(s) = N(s)/D(s) ~ Rational function with real coefficients – Table Lookup • 0 0, (t) 1, (n) (t) s n • U(t) 1/s, t 1/s 2 , … t n-1 /(n-1)! 1/s n • e- t 1/(s+ ), t n-1 e- t /(n-1)! 1/(s+ ) n • sin t /(s 2 + 2 ), cos t s/(s 2 + 2 ) Lecture 12, 3/3/11 ECE 3101, Copyright P. B. Luh 3 • Today: – Inverse Laplace Transform (continued) – Solution of Differential and Integro-Differential Equations – State and Output Description • Next Time: Sections 4.4 and 4.5 – The Transformed Network Method – Block Diagrams 4 Inverse Laplace Transform by Partial Fraction Expansion • Decompose F(s) into basic terms so that tables can be used – Polynomial and rational functions of first and second orders – The method is relatively easy , and covers most of F(s) of interest. We shall consider 5 cases – Case 1. F(s) = N(s)/D(s) is not a strictly proper rational function , i.e., Order N(s) Order D(s) 1 s 2 s 1 s s 2 2 2 3 – Case 2. Strictly proper rational function (Order N(s) < Order D(s), and all the poles are distinct 2 s 3 s 3 s 2 – Case 3. Strictly proper with distinct complex poles 2 s 2 s s 1 2 – Case 4. Strictly proper with multiple poles 2 s 1 s 1 s s 2 2 – Case 5. With exponential terms , e.g., e-st F(s) 3 s 2 s e s 4 Lecture 12, 3/3/11 ECE 3101, Copyright P. B. Luh 5 Case 1. F(s) = N(s)/D(s) is not a strictly proper rational function , i.e., Order N(s) Order D(s) Example....
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Lecture 12 - Lecture 12 ECE 3101 Copyright P B Luh 1 ECE...

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