Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part9

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part9

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ------------------------------- =
Background image of page 2
Chapter 2 The Laplace Transformation 2 26 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 8

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part9

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online