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Chapter 2 The Laplace Transformation 2 24 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Figure 2.2. Waveform for a linear segment We must first derive the equation of the linear segment. This is shown in Figure 2.3. Figure 2.3. Waveform for a linear segment with the equation that describes it Next, we express the given waveform in terms of the unit step function as follows: From Table 2.1, Page 2 13, and from Table 2.2, Page 2 22, Therefore, the Laplace transform of the linear segment of Figure 2.2 is (2.68) 2.4.3 The Laplace Transform of a Triangular Waveform The waveform of a triangular waveform, denoted as , is shown in Figure 2.4. Figure 2.4. Triangular waveform The equations of the linear segments are shown in Figure 2.5. 1 t 0 1 2 f L t () 1 t 0 1 2 f L t t1 f L t u 0 = ft a u 0 ta e as Fs tu 0 t 1 s 2 ---- u 0 e s 1 s 2 ---- f T t 1 t 0 1 2 f T t

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 2 25 Copyright © Orchard Publications The Laplace Transform of Common Waveforms Figure 2.5. Triangular waveform with the equations of the linear segments Next, we express the given waveform in terms of the unit step function. Collecting like terms, we obtain From Table 2.1, Page 2 13, and from Table 2.2, Page 2 22, Then, or Therefore, the Laplace transform of the triangular waveform of Figure 2.4 is (2.69) 2.4.4 The Laplace Transform of a Rectangular Periodic Waveform The waveform of a rectangular periodic waveform, denoted as , is shown in Figure 2.6. This is a periodic waveform with period , and we can apply the time periodicity property 1 t 0 1 2 f T t () t –2 + t f T t tu 0 t u 0 t1 [] t 2 + u 0 u 0 t2 + = 0 t 0 0 2u 0 0 0 ++ = f T t 0 t 2t 1 u 0 u 0 + = ft a u 0 ta e as Fs 0 t 1 s 2 ---- 0 t u 0 u 0 + 1 s 2 2 e s 1 s 2 e 2s 1 s 2 + 0 t u 0 u 0 + 1 s 2 12 e s e + f T t 1 s 2 1e s 2 f R t T2 a = L f τ {} f τ 0 T e s τ d τ sT ------------------------------- =
Chapter 2 The Laplace Transformation 2 26 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications

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