Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part7

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part7

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Chapter 2 The Laplace Transformation 2 8 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (2.26) and the proof of (2.23) follows from (2.25) and (2.26). 2.2.8 Integration in Complex Frequency Domain Property This property states that integration in complex frequency domain with respect to corresponds to division of a time function by the variable , provided that the limit exists. Thus, (2.27) Proof: Integrating both sides from to , we obtain Next, we interchange the order of integration, i.e., and performing the inner integration on the right side integral with respect to , we obtain 2.2.9 Time Periodicity Property The time periodicity property states that a periodic function of time with period corresponds to the integral divided by in the complex frequency domain. Thus, if we let be a periodic function with period , that is, , for we obtain the transform pair L f τ () 0 t d τ    Fs s ---------- = s ft t t -------- t0 lim t s d s 0 e st dt = s s d s 0 e s d s = s d s e s s df t t d 0 = s s d s 1 t -- e s t d 0 t --------e t d 0 L t == = T 0 T e 1 e sT Tf t ft nT + = n1 2 3 ,,, =
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 2 9 Copyright © Orchard Publications Properties and Theorems of the Laplace Transform (2.28) Proof: The Laplace transform of a periodic function can be expressed as In the first integral of the right side, we let , in the second , in the third , and so on. The areas under each period of are equal, and thus the upper and lower limits of integration are the same for each integral. Then, (2.29) Since the function is periodic, i.e., , we can write (2.29) as (2.30) By application of the binomial theorem, that is, (2.31) we find that expression (2.30) reduces to 2.2.10 Initial Value Theorem The initial value theorem states that the initial value of the time function can be found from its Laplace transform multiplied by and letting .That is, (2.32) ft nT + () ft 0 T e st dt 1e sT ----------------------------- L {} 0 e f t 0 T e T 2T e 3T e +++ == t τ = t τ T + = t τ + = L f τ 0 T e s τ d τ f τ T + 0 T e s τ T + d τ f τ + 0 T e s τ + d τ… ++ + = f τ f τ T + f τ + f τ nT + = = L f τ e 2sT + f τ 0 T e s τ d τ = 1aa 2 a 3 ++ + + 1 1a ----------- = L f τ f τ 0 T e s τ d τ ---------------------------------- = f0 ss t0 lim sF s s lim f 0
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Chapter 2 The Laplace Transformation 2 10 Signals and Systems with MATLAB
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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part7

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