(25) Fourier Analysis

# To put it another way given an arbitrary periodic

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: 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...
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 at Austin.

Ask a homework question - tutors are online