Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part25

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part25

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Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition 6 3 Copyright © Orchard Publications The Impulse Response in Time Domain and Therefore, or (6.8) The current can now be computed from Thus, Using the sampling property of the delta function, we obtain (6.9) Example 6.2 For the circuit of Figure 6.3, compute the impulse response given that the initial conditions are zero, that is, , and . Figure 6.3. Circuit for Example 6.2 Solution: This is the same circuit as that of Example 5.10, Chapter 5, Page 5 22, where we found that and b1 R C = ht () v C t e tRC 1 RC ------- == 1 ------- e u 0 t = i C i C C dv C dt -------- = i C C d ----ht C d ---- 1 u 0 t ⎝⎠ ⎛⎞ 1 R 2 C ---------- e 1 R --- e δ t + = i C 1 R --- δ t 1 R 2 C = v C t = i L 0 0 = v C 0 0 = + R L + C 1 Ω δ t v C t = 14 H 43 F b 4 0 =
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Chapter 6 The Impulse Response and Convolution 6 4 Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition Copyright © Orchard Publications The impulse response is obtained from (6.5), Page 6 1, that is, then, (6.10) In Example 5.10, Chapter 5, Page 5 22, we defined and Then, or (6.11) Of course, this answer is not the same as that of Example 5.10, because the inputs and initial con- ditions were defined differently. 6.2 Even and Odd Functions of Time A function is an even function of time if the following relation holds. (6.12) that is, if in an even function we replace with , the function does not change. Thus, poly- nomials with even exponents only, and with or without constants, are even functions. For instance, the cosine function is an even function because it can be written as the power series Other examples of even functions are shown in Figure 6.4. e At 0.5 e t 1.5e 3t + 2 e t 2e + 3 8 --e t 3 8 1.5e t 0.5e = ht () xt = e bu 0 t = = x 1 x 2 0.5 e t 1.5e + 2 e t + 3 8 t 3 8 1.5e t 0.5e 4 0 u 0 t 2 e t 6e + 3 2 t 3 2 u 0 t == = x 1 i L = x 2 v C = x 2 v C t 1.5e t 1.5e = v C t 1.5 e t e ft = tt t cos 1 t 2 2 ! ---- t 4 4 ! + t 6 6 ! + =
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Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition 6 5 Copyright © Orchard Publications Even and Odd Functions of Time Figure 6.4. Examples of even functions A function is an odd function of time if the following relation holds.
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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_Part25

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