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some notes on the Fourier transform - IGPP

some notes on the Fourier transform - IGPP

some notes on the Fourier transform -

FOURIER TRANSFORMSClass Notes by Bob Parker1. What is the Fourier Transform?For asuitable function f of a single real variable the definition of the Fouriertransform is:ˆ f (ν) = F [f ] =∞−∞ ∫ dt e−2π iν t f (t). (1.1)This operation takes a real function and generates from it another complex-valuedfunction, that is, afunction of a single variable (here ν) that has a real part andimaginary part. We can think of this as a kind of linear mapping: a function goes in,and another (complex) function comes out.It is easier to start with the idea of synthesizing a complex function f ratherthan a real one. What does the transform mean? In loose physical terms theFourier transform (we’ll write FT, from now on) produces a complex amplitude spectrumthat shows how large the amplitude of the cosine and sine functions must be ateach frequency ν, inadecomposition of the function into periodic parts, parts thateach have adefinite frequency. When the independent variable is a coordinate isspace, wavenumber, which wewill write as k, ismore appropriate. Imagine buildingup the function f from (an infinite) sum of sines and cosines; this may soundimplausible for an infinite interval, and indeed, not every function can be built likethis, but many can. Because we must allow every possible frequency of sine andcosine,the sum must be an integral, whichwewould write:∞f (t) =∫ dν e+2π iν t f ˆ (ν). (1.2)−∞Foraparticular frequency ν,the contribution to the sum is juste 2π iν t ˆ f (ν) = (cos 2πνt + isin 2πνt) ˆ f (ν). (1.3)Youcan see this function is a sine wave, but it has real and imaginary parts. Asafunction of t, the function in (1.3) repeats itself exactly, with period 1/ν; for functionsin space the corresponding quantity is of course the wavelength, λ = 1/k. Ifthe magnitude of the amplitude function | f ˆ (ν)| is large at ong>someong> frequency ν 0 comparedto that at any frequency, wewould expect f (t) built by (1.2) to approximate acomplex cosine wave with frequency ν 0 . Asanillustration, consider the function ĝwhichisaGaussian hump with its peak at ν 0 shown in Figure 1a:ĝ(ν) = e −(ν −ν 0) 2 /2σ 2 (1.4)where σ is a standard measure of the width. If we plug this into (1.2) we get (bymethods we shall describe later on)g(t) = √⎺⎺⎺2πσe −2π 2 σ 2 t 2 e 2π iν 0t . (1.5)

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