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CHAPTER EIGHT FOURIER ANALYSIS In this chapter, Fourier analysis will be discussed. Topics covered are Fourier series expansion, Fourier transform, discrete Fourier transform, and fast Fourier transform. Some applications of Fourier analysis, using MATLAB, will also be discussed. 8.1 If a function FOURIER SERIES g (t ) is periodic with period Tp , i.e., (8.1) g (t ) = g (t ± Tp ) and in any finite interval g ( t ) has at most a finite number of discontinuities and a finite number of maxima and minima (Dirichlets conditions), and in addition, Tp ∫ g(t )dt . | Attia John Okyere. Fourier Analysis. Electronics and Circuit Analysis using MATLAB. Ed. John Okyere Attia Boca Raton CRC Press LLC 1999 1999 by CRC PRESS LLC CHAPTER EIGHT FOURIER ANALYSIS In this chapter Fourier analysis will be discussed. Topics covered are Fourier series expansion Fourier transform discrete Fourier transform and fast Fourier transform. Some applications of Fourier analysis using MATLAB will also be discussed. 8.1 FOURIER SERIES If a function g t is periodic with period Tp i.e. g t g t Tp 8.1 and in any finite interval g t has at most a finite number of discontinuities and a finite number of maxima and minima Dirichlets conditions and in addition T p j g t dt 0 8.2 then g t can be expressed with series of sinusoids. That is g t y an cos nw01 bn sin nw01 8.3 2 n 1 where 2n w 0 p 8.4 and the Fourier coefficients an and bn are determined by the following equations. an T jg t cos nwot dt n 0 1 2 . 8.5 TP to 1999 CRC Press LLC bn T Jg t sin nw0 dt n 0 1 2 . 8.6 Tp to _. . _ a0 . Equation 8.3 is called the trigonometric Fourier series. The term in Equation 8.3 is the dc component of the series and is the average value of g t over a period. The term an cos nw01 bn sin nw01 is called the nth harmonic. The first harmonic is obtained when n 1. The latter is also called the fundamental with the fundamental frequency of o . When n 2 we have the second harmonic and so on. Equation 8.3 can be rewritten as g t 0 X An c0s nwo t 0 n 2 n 1 where 2 bn and 0 n tan-1 bn 1 l n J P - 1 P The total power in g t is given by the Parseval s equation tn Tn J O P 2 J g t d Al X A T t n 1 2 lo n 1 where A2 Adc 8.7 8.8 8.9 8.10 8.11 2 0 I 2 J The following example shows the synthesis of a square wave using Fourier series expansion. 1999 CRC Press .
