(25) Fourier Analysis

# The annoying oscillations in the reconstructed

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

## 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.

Ask a homework question - tutors are online