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Chapter 3 The Inverse Laplace Transformation 3 18 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications and since , it follows that . To obtain the third residue , we equate the constant terms of (3.70). Then, or , or . Then, by substitution into (3.68), we obtain (3.72) as before. The remaining steps are the same as in Example 3.3, and thus is the same as , that is, r 1 25 = r 2 = r 3 38 r 1 r 3 + = 2 5 × r 3 + = r 3 15 = F 8 s () s1 + ---------------- 1 5 -- 2s 1 + s 2 4s 8 ++ ------------------------------ = f 8 t f 3 t f 8 t f 3 t 2 5 --e t 2 5 2t 3 10 -----e sin + cos   u 0 t = =

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 3 19 Copyright © Orchard Publications Summary 3.5 Summary The Inverse Laplace Transform Integral defined as is difficult to evaluate because it requires contour integration using complex variables theory. For most engineering problems we can refer to Tables of Properties, and Common Laplace transform pairs to lookup the Inverse Laplace transform. The partial fraction expansion method offers a convenient means of expressing Laplace trans- forms in a recognizable form from which we can obtain the equivalent time domain functions. If the highest power of the numerator is less than the highest power of the denomi- nator , i.e., , is said to be expressed as a proper rational function. If , is an improper rational function. The Laplace transform must be expressed as a proper rational function before applying the partial fraction expansion. If is an improper rational function, that is, if , we must first divide the numerator by the denominator to obtain an expression of the form In a proper rational function, the roots of numerator are called the zeros of and the roots of the denominator are called the poles of . The partial fraction expansion method can be applied whether the poles of are distinct, complex conjugates, repeated, or a combination of these. When is expressed as are called the residues and are the poles of . The residues and poles of a rational function of polynomials can be found easily using the MATLAB residue(a,b) function. The direct term is always empty (has no value) whenever is a proper rational function.
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