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fourier analysis: signals and filters 233expansions are possible because we have linear operators and, subsequently, theprinciple of superposition: Ify 1 (t) and y 2 (t) are solutions of some linear equation,then α 1 y 1 (t)+α 2 y 2 (t) is also a solution. The principle of linear superposition doesnot hold when we solve nonlinear problems. Nevertheless, it is always possible toexpand a periodic solution of a nonlinear problem in terms of trigonometric functionswith frequencies that are integer multiples of the true frequency of the nonlinearoscillator. 1 This is a consequence of Fourier’s theorem being applicable to any singlevaluedperiodic function with only a finite number of discontinuities. We assumewe know the period T , that is, thatThis tells us the true frequency ω:y(t + T )=y(t). (10.5)ω ≡ ω 1 = 2πT . (10.6)A periodic function (usually called the signal) can be expanded as a series ofharmonic functions with frequencies that are multiples of the true frequency:y(t)= a ∞ 02 + ∑(a n cos nωt + b n sin nωt) . (10.7)n=1This equation represents the signal y(t) as the simultaneous sum of pure tones offrequency nω. The coefficients a n and b n are measures of the amount of cos nωtand sin nωt present in y(t), specifically, the intensity or power at each frequency isproportional to a 2 n + b 2 n.The Fourier series (10.7) is a best fit in the least-squares sense of Chapter 8,“Solving Systems of Equations with Matrices; Data Fitting,” because it minimizes∑i [y(t i) − y i ] 2 , where i denotes different measurements of the signal. This meansthat the series converges to the average behavior of the function but misses thefunction at discontinuities (at which points it converges to the mean) or at sharpcorners (where it overshoots). A general function y(t) may contain an infinite numberof Fourier components, although low-accuracy reproduction is usually possiblewith a small number of harmonics.The coefficients a n and b n are determined by the standard techniques for functionexpansion. To find them, multiply both sides of (10.7) by cos nωt or sin nωt, integrateover one period, and project a single a n or b n :( )an= 2 ∫ Tdtb n T 0( cos nωtsin nωt)y(t),ω def= 2πT . (10.8)1 We remind the reader that every periodic system by definition has a period T and consequentlya true frequency ω. Nonetheless, this does not imply that the system behaves likesin ωt. Only harmonic oscillators do that.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 233