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Unformatted text preview: way, given an arbitrary periodic waveform
can we uniquely determine the constants and appearing in Equation (340)? It turns out that we can. Incidentally, the decomposition of a periodic waveform into a linear superposition of sinusoidal waveforms is
commonly known as Fourier analysis.
The problem under investigation is as follows. Given a periodic waveform
all , we need to determine the constants and , where for in the expansion (341) where (342) It can be demonstrated that [cf., Equation (277)] (343) (344) (345) where and integrating over are positive integers. Thus, multiplying Equation (341) by
from 0 to , and then , we obtain (346) where use has been made of Equation (343)- (345), as well as Equation (278). Likewise, multiplying
Equation (341) by
, and then integrating over from 0 to , we obtain (347) Hence, we have uniquely determined the constants and in the expansion (341). These constants are generally known as Fourier coef f icient s, whereas the expansion itself is known as either a Fourier
ex pansion or a Fourier series. Figure 35: Fourier reconstruction of a periodic sawtooth waveform.
In principle, there is no restriction on the w...
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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 at Austin.
- Fall '08
- Waves And Optics