It can be demonstrated from equations 346 347 357

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Unformatted text preview: for , with for . In fact, only the odd- Fourier harmonics are non- zero. Figure 36 shows a Fourier reconstruction of the ``tent'' waveform using the first 1, 2, 4, and 8 terms (in addition to the term) in the Fourier series (these cases correspond to the top- left, top- right, bottom- left, and bottomright panels, respectively). The reconstruction becomes increasingly accurate as the number of terms in the series increases. Moreover, in this example, there is no sign of Gibbs' phenomena, because the tent waveform is completely continuous. In our first example- - that is, the sawtooth waveform- - all of the Fourier coefficients are zero, whereas in our second example- - that is, the tent waveform- - all of the coefficients are zero. This occurs because the sawtooth waveform is odd in - - that is, for all for all - - whereas the tent waveform is even- - that is, . It is a general rule that waveforms that are even in only have cosines in their Fourier series, whereas waveforms that are odd only have sines (Riley1974). Waveforms that are neither even nor odd in have both cosines and sines in their Fourier series. Fourier series...
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