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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 13 Copyright © Orchard Publications Trigonometric Form of Fourier Series for Common Waveforms Figure 7.13. Square waveform as even function Since the waveform has half wave symmetry and is an even function, it will suffice to integrate from to , and multiply the integral by . The coefficients are found from (7.25) We observe that for , all coefficients are zero, and thus all even harmonics are zero as expected. Also, by inspection, the average ( ) value is zero. For , we observe from (7.25) that , will alternate between and depending on the odd integer assigned to . Thus, (7.26) For , and so on, (7.26) becomes and for , and so on, it becomes Then, the trigonometric Fourier series for the square waveform with even symmetry is (7.27) The trigonometric series of (7.27) can also be derived as follows: 0 π / 2 2 π T ω t A A π 3π / 2 0 π 2 4a n a n 4 1 π -- ft () nt cos t d 0 π 2 4 π An t cos t d 0 π 2 4A n π ------- sin 0 π 2 n π n π 2 sin   == = = ne v e n = a n DC no d d = n π 2 sin +1 1 n a n n π ± = n1 5 9 1 3 ,,, = a n n π = n3 7 1 1 1 5 ,, , = a n n π ---------- = π ω cos t 1 3 3 ω t 1 5 5 ω t cos + cos π 1 2 ---------------- 1 n n cos ω t n odd =

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