Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part3

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part3

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Chapter 1 Elementary Signals 1 2 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications We can express (1.3) by the waveform shown in Figure 1.2. Figure 1.2. Waveform for as defined in relation (1.3) The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be dis- continuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without taking on any intermediate values. 1.2 The Unit Step Function A well known discontinuous function is the unit step function * which is defined as (1.4) It is also represented by the waveform of Figure 1.3. Figure 1.3. Waveform for In the waveform of Figure 1.3, the unit step function changes abruptly from to at . But if it changes at instead, it is denoted as . In this case, its waveform and definition are as shown in Figure 1.4 and relation (1.5) respectively. Figure 1.4. Waveform for * In some books, the unit step function is denoted as , that is, without the subscript 0. In this text, however, we will reserve the designation for any input when we will discuss state variables in Chapter 5. 0 v out v S t v u 0 t () u 0 t ut u 0 t 0t 0 < 1t 0 > = u 0 t 0 1 t u 0 t u 0 t 01 t0 = tt 0 = u 0 0 1 t 0 0 u 0 0 t u 0 0
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 1 3 Copyright © Orchard Publications The Unit Step Function (1.5) If the unit step function changes abruptly from to at , it is denoted as . In this case, its waveform and definition are as shown in Figure 1.5 and relation (1.6) respectively. Figure 1.5. Waveform for (1.6) Example 1.1 Consider the network of Figure 1.6, where the switch is closed at time . Figure 1.6. Network for Example 1.1 Express the output voltage as a function of the unit step function, and sketch the appropriate waveform. Solution: For this example, the output voltage for , and for . Therefore, (1.7) and the waveform is shown in Figure 1.7. u 0 tt 0 () 0t t 0 < 1t t 0 > = 01t t 0 = u 0 0 + t t 0 0 1 u 0 0 + u 0 0 + u 0 0 + t 0 < t 0 > = tT = + + v out v S = R open terminals v v 0 = < v v S = > v v S u 0 =
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Chapter 1 Elementary Signals 1 4 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications Figure 1.7. Waveform for Example 1.1 Other forms of the unit step function are shown in Figure 1.8.
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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_Part3

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