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fourier analysis: signals and filters 237power spectrum and provides you with an immediate view of the amount of poweror strength in each component.If the Fourier transform and its inverse are consistent with each other, we shouldbe able to substitute (10.17) into (10.18) and obtain an identity:Y (ω)=∫ +∞dt e−iωt−∞√2π∫ +∞−∞dω ′ eiω′ t√2πY (ω ′ )=∫ +∞−∞dω ′ { ∫ +∞−∞}−ω)tdt ei(ω′ Y (ω ′ ).2πFor this to be an identity, the term in braces must be the Dirac delta function:∫ +∞−∞dt e i(ω′ −ω)t =2πδ(ω ′ − ω). (10.19)While the delta function is one of the most common and useful functions in theoreticalphysics, it is not well behaved in a mathematical sense and misbehaves terriblyin a computational sense. While it is possible to create numerical approximations toδ(ω ′ − ω), they may well be borderline pathological. It is certainly better for you todo the delta function part of an integration analytically and leave the nonsingularleftovers to the computer.10.4.1 Discrete Fourier Transform AlgorithmIf y(t) or Y (ω) is known analytically or numerically, the integral (10.17) or (10.18)can be evaluated using the integration techniques studied earlier. In practice, thesignal y(t) is measured at just a finite number N of times t, and these are what wemust use to approximate the transform. The resultant discrete Fourier transform isan approximation both because the signal is not known for all times and becausewe integrate numerically. 4 Once we have a discrete set of transforms, they can beused to reconstruct the signal for any value of the time. In this way the DFT can bethought of as a technique for interpolating, compressing and extrapolating data.We assume that the signal y(t) is sampled at (N +1) discrete times (N timeintervals), with a constant spacing h between times:y kdef= y(t k ), k =0, 1, 2,...,N, (10.20)t kdef= kh, h =∆t. (10.21)In other words, we measure the signal y(t) once every hth of a second for a totaltime T . This corresponds to period T and sampling rate s:T def= Nh, s= N T = 1 h . (10.22)4 More discussion can be found in [B&H 95], which is devoted to just this topic.−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 237