Fourier Transforms

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fx Hx a 1 − Hx − aSignum (Sign) Function. Thisisdefinedbysgnx 1 x 0 0 x 0 − 1 x 0 (3.1.6)and simply identifies the sign of the argument.ExampleConsider the function fx |x 3 | which can also be written as fx x 3 sgnx.4
3.2 Fourier Transforms3.2.1 The basic conceptsLet fx be a function such that fx is of total bounded variation and absolutely integrable,i.e. the integral of |fx| exists and is finite, then the Fourier Transform (FT) of fx isdefined by f k −fx e ikx dx . (3.2.1)This is now a function of the real variable k and sometimes also called the Complex FourierTransform. It should be noted that this not the only definition and there are others incommon use. Note that fx can be real- or complex-valued and that f k will generally becomplex-valued. It can be shown that if fx is a good function, then f k is also a goodfunction.The Fourier Cosine and Sine Transforms are defined by f ck 0fx coskx dx , f sk 0fx sinkx dx . (3.2.2)and they play the same roles for odd and even functions as in Fourier Series.Example 1Determine the FT of the function fx exp−a|x| where a is a real constant and a 0.We havef̂k −e −a|x| e ikx dx ,0− 1a ike ax e ikx dx 0e −ax e ikx dx 1a − ik2aa 2 k 2 .Example 2Determine the FT of the function fx 1 when |x| a and fx 0 when |x| a. Fromthe previous section, we know that this function can also be represented in terms of theHeaviside Unit Function.We have5
Page 2 and 3: −xfx dx f0 (3.1.1)or equivalentl
Page 6 and 7: f̂k a−a1 e ikx dxa 1ike ikx −a
Page 9 and 10: fx fx −2 12−f̂k e −ikx dk
Page 11 and 12: −|e −a|x| | 2 dx 2 0e −2ax d
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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