Fourier Transforms

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ExampleFrom Example 1 of §3.2.1, it is known that the FT of the function fx exp−a|x|, where2a2aa is a real constant and a 0isf̂k . What is the FT of gx ?a 2 k 2 a 2 x 2Using the terminology of the question, we have gx f̂x and the result above states thatthe FT of f̂x is 2f−k. Thus the FT of gx is 2f−k, i.e.as |−k| |k|.ĝk 2 f−k 2 exp−a|−k| 2 exp−a|k|3.2.5 Raleigh / Plancheral TheoremsThese results were originally due to Raleigh and rigorously proved by Plancherel; they arethe FT equivalent of Parseval’s theorem in Fourier Series. Let fx and gx be real- orcomplex-valued functions and both absolutely integrable. Consider the integral of theproduct of fx and the complex conjugate gx, employing (3.2.1) and (3.2.9) gives− −−fx gx dx fx 1 2 gk e −ikx dk dx− 12− fx gk e ikx dk dx− 1 2 gk− fx e ikx dx dk−fx gx dx 1 2 f k gk dk (3.2.11)−A special case occurs when f g and (3.2.11) becomes−|fx| 2 dx 1 22 f k dk (3.2.12)−This relationship is referred to as the Conservation of Energy, whereas (3.2.11) sometimescalled the Power Relation.ExampleThe FT of the real-valued function fx exp−a|x| has already been shown in one of theexamples above to be 2a/a 2 k 2 . Applying (3.2.12) gives−|e −a|x| | 2 dx 12−2aa 2 k 22dk.The first integral can be evaluated10
−|e −a|x| | 2 dx 2 0e −2ax dx 1 aOn substitution and re-arrangement−1 dk a 2 k 2 2 2a 33.2.6 Alternative notationsIt has been stated previously that there are a number of definitions of the Fourier Transformin common use. The attractiveness of one alternative definition can be seen clearly from acomparison of (3.2.1) and (3.2.9), which shows that the inverse FT has a 1/2multiplicative factor, whereas the FT does not. For this reason, some texts employ a 1/ 2factor in transform and inverse; occasionally the 1/2 factor is used in the transform but noton the inverse. There is no standard practice.However, all are of the form f k C2−fx e iAkx dx fx |A|C − f k e −iAkx dkfor constants A and C. Some of the common combinations are given below.A C Fourier Transform f k Inverse Transform fx1 2 fx e ikx 1dx −2 − f̂k e −ikx dk−1 2 fx e −ikx 1dx −2 − f̂k e ikx dk1−1 1211 22 fx e −ikx dx − − fx e ikx 1dx −2−2 2 fx e −2ikx dx − − −f̂k e ikx dkf̂k e −ikx dkf̂k e 2ikx dk11
Page 2 and 3: −xfx dx f0 (3.1.1)or equivalentl
Page 4 and 5: fx Hx a 1 − Hx − aSignum (Sig
Page 6 and 7: f̂k a−a1 e ikx dxa 1ike ikx −a
Page 9: fx fx −2 12−f̂k e −ikx dk
Page 13 and 14: −−d 2 ydx 2d 2 y− 2 y e ikx
Page 15 and 16: uk,y Ak e|k|y Bk e −|k|y .Now ap
Page 17 and 18: ecause of the convolution structure
Page 19 and 20: complex FS of f on −L ≤ ≤
Page 21 and 22: 3.4 Fourier Transforms and Generali
Page 23 and 24: function and kgk 0, then gk is a c

Fourier Transforms

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