This preview shows page 1. Sign up to view the full content.
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...
View Full
Document
 Fall '08
 Staff
 Waves And Optics

Click to edit the document details