Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part43

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part43

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 57 Copyright © Orchard Publications Solutions to End of Chapter Exercises fprintf(' \n'); fprintf('atan(6*pi) = %4.2f \t', atan(6*pi)); fprintf('atan(2*pi) = %4.2f', atan(2*pi)) atan(6*pi) = 1.52 atan(2*pi) = 1.41 and thus Therefore, the percentage of the input appearing at the output is 10 . First, we compute the total energy of the pulse in terms of . and since is an even function, Next, we denote the energy in the frequency interval as in the frequency domain we obtain W out 9 2 π ------ 1 2 -- ω 2 --- atan 4 π 12 π 9 4 π 12 π 2 --------- atan 4 π 2 atan   9 4 π 6 π atan 2 π atan () == = W 9 4 π ------ 1.52 1.41 0.08 J W W in ----------- 100 × 0.08 2.25 3 . 5 6 % Au 0 tT + u 0 [] 2AT ω T sin ω T --------------- 0 ω T 2 π 2 π π −π A T T t 0 ft F ω W total 2 t d A 2 t d T T W 2A 2 t d 0 T 2 t 0 T 2 T = π T rad ωπ T ≤≤ W W 1 2 π F ω 2 ω d 1 2 π ω T sin ω T 2 ω d π T π T
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Chapter 8 The Fourier Transform 8 58 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications and since is an even function, (1) For simplicity, we let . Then, and . Also, when , , and when or , . With these substitutions we express (1) as (2) But the last integral in (2) is an improper integral and does not appear in tables of integrals. * Therefore, we will attempt to simplify (2) using integration by parts. We start with the familiar relation from which or Letting and , it follows that and . With these substitutions (2) is written as (3) The last integral in (3) is also an improper integral. Fortunately, some handbooks of mathe- matical tables include numerical values of the integral * It is shown in Advanced Calculus textbooks that if the upper limit is , then but for other finite limits are not equal. F ω () W out 2 1 2 π ------ 4A 2 T 2 ω T 2 sin ω T 2 ----------------- ω d 0 π T 2 T 2 π ω T 2 sin ω T 2 ω d 0 π T == ω Ty = ω yT = ω d1 T dy = ω 0 = y0 = ωπ T = ω T π = y π = W 2 T 2 π y 2 sin T 2 -------------------------- y d 0 π 2 T 2 π T y 2 sin y 2 ------------ y d 0 π 2 T π ------------- y 2 sin y 2 d 0 π = x sin x ---------- x d 0 x 2 sin x 2 ------------ x d 0 π 2 -- duv udv vdu + = d uv u dv + = u = uy 2 sin = 1 y 2 = du 2 y y sin cos 2y sin v1 y = W 2 T π y 2 sin y ------------ 0 π 1 y ----- sin y d 0 π 2 T π ------------- 0 sin y ------------- y d 0 π + 2 2 T π sin d 0 π 8A 2 T π sin d 0 π = = x sin x d 0 π
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 8 59 Copyright © Orchard Publications Solutions to End of Chapter Exercises for arguments of in the interval . Then, replacing
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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_Part43

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