Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part4

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part4

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Chapter 1 Elementary Signals 1 10 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 1.3 The Unit Ramp Function The unit ramp function , denoted as , is defined as (1.23) where is a dummy variable. We can evaluate the integral of (1.23) by considering the area under the unit step function from as shown in Figure 1.18. Figure 1.18. Area under the unit step function from Therefore, we define as (1.24) Since is the integral of , then must be the derivative of , i.e., (1.25) Higher order functions of can be generated by repeated integration of the unit step function. For example, integrating twice and multiplying by , we define as (1.26) Similarly, (1.27) and in general, (1.28) Also, u 1 t () u 1 t u 1 t u 0 τ ()τ d t = τ u 0 t to t Area 1 τ ×τ t == = 1 τ t u 1 t u 1 t 0t 0 < tt 0 = u 1 t u 0 t u 0 t u 1 t d dt ----u 1 t u 0 t = t u 0 t 2u 2 t u 2 t 0 < t 2 t0 = or u 2 t 1 τ d t = u 3 t 0 < t 3 = u 3 t 3u 2 τ d t = u n t 0 < t n = u n t n1 τ d t =
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 1 11 Copyright © Orchard Publications The Delta Function (1.29) Example 1.7 In the network of Figure 1.19, the switch is closed at time and for . Express the inductor current in terms of the unit step function. Figure 1.19. Network for Example 1.7 Solution: The voltage across the inductor is (1.30) and since the switch closes at , (1.31) Therefore, we can write (1.30) as (1.32) But, as we know, is constant ( or ) for all time except at where it is discontinuous. Since the derivative of any constant is zero, the derivative of the unit step has a non zero value only at . The derivative of the unit step function is defined in the next section. 1.4 The Delta Function The unit impulse or delta function , denoted as , is the derivative of the unit step . It is also defined as (1.33) and (1.34) u n1 t () 1 n -- d dt ----u n t = t0 = i L t 0 = < i L t R i S = L v L t i L t + v L t L di L ------- = = i L t i S u 0 t = v L t Li S d 0 t = u 0 t 01 = u 0 t = δ t δ t u 0 t δτ ()τ d t u 0 t = δ t 0 for all t 0 =
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Chapter 1 Elementary Signals 1 12 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications To better understand the delta function , let us represent the unit step as shown in Fig- ure 1.20 (a).
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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_Part4

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