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Unformatted text preview: aveform in the previous analysis, other than the requirement that it be periodic in time. In other words, we ought to be able to Fourier analyze any periodic waveform. Let us see
how this works. Consider the periodic sawtooth waveform
(348) with for all
to a final value at . (See Figure 35.) This waveform rises linearly from an initial value at , discontinuously jumps back to its initial value, and then repeats ad infinitum. According to Equations (346) and (347), the Fourier harmonics of the waveform are (349) (350) where . Integration by parts (Riley 1974) yields (351)
(352) Hence, the Fourier reconstruction of the waveform is written (353) Given that the Fourier coefficients fall off like , as increases, it seems plausible that the preceding series can be truncated after a finite number of terms without unduly affecting the reconstructed waveform.
Figure 35 shows the result of truncating the series after 4, 8, 16, and 32 terms (these cases correspond the topleft, top- right, bottom- left, and bottom- right panels, respectively). It can be seen that the reconstruction becomes
increasingly accurate as the number of terms retained in the series increases. The annoying oscillations in the
reconstructed waveform at
, , and...
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- Fall '08
- Waves And Optics