# Signals.and.Systems. - Chapter 3 The Inverse Laplace Transformation nient to make the coefficient of s unity thus we rewrite F s as n 1 b s m b m 1

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Chapter 3 The Inverse Laplace Transformation 3 2 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications nient to make the coefficient of unity; thus, we rewrite as (3.4) The zeros and poles of (3.4) can be real and distinct, repeated, complex conjugates, or combina- tions of real and complex conjugates. However, we are mostly interested in the nature of the poles, so we will consider each case separately, as indicated in Subsections 3.2.1 through 3.2.3 below. 3.2.1 Distinct Poles If all the poles of are distinct (different from each another), we can factor the denominator of in the form (3.5) where is distinct from all other poles. Next, using the partial fraction expansion method, * we can express (3.5) as (3.6) where are the residues , and are the poles of . To evaluate the residue , we multiply both sides of (3.6) by ; then, we let , that is, (3.7) Example 3.1 Use the partial fraction expansion method to simplify of (3.8) below, and find the time domain function corresponding to . *T h e p a r t i a l f r a c t i o n e x p ansion method applies only to proper rational functions. It is used extensively in integration, and in finding the inverses of the Laplace transform, the Fourier transform, and the z-transform. This method allows us to decom- pose a rational polynomial into smaller rational polynomials with simpler denominators from which we can easily recognize their integrals and inverse transformations. This method is also being taught in intermediate algebra and introductory cal- culus courses. s n Fs () Ns Ds ----------- 1 a n ---- b m s m b m1 s b m2 s b 1 sb 0 +++ + + s n a n1 a n -----------s a n2 a n a 1 a n ----s a 0 a n ---- + + ------------------------------------------------------------------------------------------------------------------------------ == p 1 p 2 p 3 p n ,,,, sp 1 2 3 n ⋅⋅⋅ ------------------------------------------------------------------------------------------------- = p k r 1 1 ----------------- r 2 2 r 3 3 r n n ------------------ + = r 1 r 2 r 3 r n ,,, , p 1 p 2 p 3 p n r k k k r k k k lim k k = F 1 s f 1 t F 1 s

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 3 3 Copyright © Orchard Publications Partial Fraction Expansion (3.8) Solution: Using (3.6), we obtain (3.9) The residues are (3.10) and (3.11) Therefore, we express (3.9) as (3.12) and from Table 2.2, Chapter 2, Page 2 22, we find that (3.13) Therefore, (3.14) The residues and poles of a rational function of polynomials such as (3.8), can be found easily using the MATLAB residue(a,b) function. For this example, we use the script Ns = [3, 2]; Ds = [1, 3, 2]; [r, p, k] = residue(Ns, Ds) and MATLAB returns the values r = 4 -1 p = -2 -1 k = [] F 1 s () 3s 2 + s 2 2 ++ ------------------------- = F 1 s 2 + s 2 2 2 + s1 + s2 + -------------------------------- r 1 + ---------------- r 2 + + == = r 1 + Fs lim 2 + + = 1 = r 2 + lim 2 + + = 4 = F 1 s 2 + s 2 2 1 + 4 + + e at u 0 t 1 sa + ---------- F 1 s 1 + 4 + + = e t 4e 2t + u 0 t f 1 t =
Chapter 3 The Inverse Laplace Transformation 3 4

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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. - Chapter 3 The Inverse Laplace Transformation nient to make the coefficient of s unity thus we rewrite F s as n 1 b s m b m 1

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