Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part29

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part29

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 5 Copyright © Orchard Publications Evaluation of the Coefficients Figure 7.5. Graphical proof of Figure 7.6. Graphical proof of To evaluate any coefficient in (7.10), say , we multiply both sides of (7.10) by . Then, Next, we multiply both sides of the above expression by , and we integrate over the period to . Then, (7.11) x sin x 2 sin mt sin () 2 t d 0 2 π π = x cos x 2 cos m cos t 2 t d 0 2 π π = b 2 2t sin ft sin 1 2 --a 0 sin a 1 t2 t sin cos a 2 sin cos a 3 3t sin a 4 4t sin cos + cos ++ + + = b 1 t sin sin b 2 sin 2 b 3 sin b 4 sin sin + sin + dt 0 2 π sin t d 0 2 π 1 2 0 sin t d 0 2 π a 1 t sin cos t d 0 2 π a 2 sin cos t d 0 2 π = + a 3 sin cos t d 0 2 π + + b 1 t sin sin t d 0 2 π b 2 sin 2 t d 0 2 π b 3 sin sin t + d 0 2 π
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Chapter 7 Fourier Series 7 6 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications We observe that every term on the right side of (7.11) except the term is zero as we found in (7.6) and (7.7). Therefore, (7.11) reduces to or and thus we can evaluate this integral for any given function . The remaining coefficients can be evaluated similarly. The coefficients , , and are found from the following relations. (7.12) (7.13) (7.14) The integral of (7.12) yields the average ( ) value of . 7.3 Symmetry in Trigonometric Fourier Series With a few exceptions such as the waveform of the half rectified waveform, Page 7 17, the most common waveforms that are used in science and engineering, do not have the average, cosine, and sine terms all present. Some waveforms have cosine terms only, while others have sine terms only. Still other waveforms have or have not components. Fortunately, it is possible to pre- dict which terms will be present in the trigonometric Fourier series, by observing whether or not the given waveform possesses some kind of symmetry. b 2 2t sin () 2 t d 0 2 π ft sin t d 0 2 π b 2 sin 2 t d 0 2 π b 2 π == b 2 1 π -- sin t d 0 2 π = a 0 a n b n 1 2 --a 0 1 2 π ------ t d 0 2 π = a n 1 π nt t d cos 0 2 π = b n 1 π d sin 0 2 π = DC f t
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 7 Copyright © Orchard Publications Symmetry in Trigonometric Fourier Series We will discuss three types of symmetry * that can be used to facilitate the computation of the trigonometric Fourier series form. These are: 1 . Odd symmetry If a waveform has odd symmetry, that is, if it is an odd function, the series will consist of sine terms only. In other words, if is an odd function, all the coefficients including , will be zero. 2 . Even symmetry If a waveform has even symmetry, that is, if it is an even function, the series will consist of cosine terms only, and may or may not be zero. In other words, if is an even function, all the coefficients will be zero.
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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_Part29

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