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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 53 Copyright © Orchard Publications Exercises 7.16 Exercises 1 . Compute the first 5 components of the trigonometric Fourier series for the waveform shown below. Assume . 2 . Compute the first 5 components of the trigonometric Fourier series for the waveform shown below. Assume . 3 . Compute the first 5 components of the exponential Fourier series for the waveform shown below. Assume . 4 .Compute the first 5 components of the exponential Fourier series for the waveform shown below. Assume . ω 1 = 0 ω t A ft () ω 1 = 0 ω t A ω 1 = 0 ω t A ω 1 = 0 ω t A2 A 2

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Chapter 7 Fourier Series 7 54 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications 5 . Compute the first 5 components of the exponential Fourier series for the waveform shown below. Assume . 6. Compute the first 5 components of the exponential Fourier series for the waveform shown below. Assume . ω 1 = 0 ω t A ft () ω 1 = 0 ω t A A
Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 55 Copyright © Orchard Publications Solutions to End of Chapter Exercises 7.17 Solutions to End of Chapter Exercises 1 . This is an even function; therefore, the series consists of cosine terms only. There is no half wave symmetry and the average ( component) is not zero. We will integrate from to and multiply by . Then, (1) From tables of integrals, and thus (1) becomes and since for all integer , (2) We cannot evaluate the average from (2); we must use (1). Then, for , or We observe from (2) that for , . Then, 0 ω t A ft () A π ---t 2 π π π 2 π DC 0 π 2 a n 2 π -- A π --- tn t cos t d 0 π 2A π 2 ------- t cos t d 0 π == xa x cos x d 1 a 2 ---- ax cos x a a sin x + = a n π 2 1 n 2 nt cos t n sin +   0 π π 2 1 n 2 n π cos t n π 1 n 2 0 sin + π sin 0 = n a n π 2 1 n 2 n π cos 1 n 2 n 2 π 2 ---------- n π 1 cos 12 a 0 n0 = 1 2 --a 0 2 π 2 -------- tt d 0 π A π 2 ----- t 2 2 0 π A π 2 π 2 2 = a 0 A2 = ne v e n = a v

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