Unformatted text preview: are known as Gibbs' phenomena, and are the inevitable
consequence of trying to represent a discontinuous waveform as a Fourier series (Riley 1974). In fact, it can be
demonstrated mathematically that, no matter how many terms are retained in the series, the Gibbs' phenomena
never entirely go away (Zygmund 1955).
We can slightly generalize the Fourier series (341) by including an term. In other words, (354) which allows the waveform to have a non- zero average. There is no term involving
when , because . It can be demonstrated that (355) (356) where , and is a positive integer. Making use of the preceding expressions, as well as Equations (343)- (345), we can show that (357) and also that Equations (346) and (347) still hold for . Figure 36: Fourier reconstruction of a periodic ``tent'' waveform.
As an example, consider the periodic ``tent'' waveform (358) where for all a peak value at . (See Figure 36.) This waveform rises linearly from zero at , falls linearly, becomes zero again at , reaches , and repeats ad infinitum. Moreover, the waveform has a non- zero average. It can be demonstrated, from Equations (346), (347), (357), and (358),
that (359) and (360)...
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This note was uploaded on 08/25/2013 for the course PHY 315 taught by Professor Staff during the Fall '08 term at University of Texas.
- Fall '08
- Waves And Optics