Fourier Transforms

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made to the signal form by sampling at a finite number of chosen points and using atruncated series.20
3.4 Fourier Transforms and Generalised Functions3.4.1 The Dirac Delta FunctionWe know from (3.1.1.) that the Dirac Delta Function is defined by−xgx dx g0 ,for any piecewise continuous function gx. Ifwetakegx expikx, whichiscontinuous, then−x e ikx dx 1 k . (3.4.1)Thus the FT of x is given by k 1 provided that the definition of the FT in (3.2.1)remains applicable.The two conditions for the existence of the FT is that fx is of total bounded variation andabsolutely integrable. The function x satisfies the appropriate conditions as |x| → butmay not be bounded. Conversely, k 1 is a bounded function but not absolutelyintegrable.Suppose that the Fourier Theorem (3.2.10) holds for x, thenx 12 x 12−−−u e iku−x due −ikx dk 12−dke ikx dk (3.4.2)and this is a real-valued integral, since the imaginary part is zero. If this representation canbe justified, which the requires the use of generalised functions, it also means that x canbe represented as an improper integral whose integrand is bounded but is not absolutelyintegrable.By interchanging k and x in (3.4.2), we obtaink 12−e ikx dx , (3.4.3)and the resulting integral is real-valued and an even function of k, as it is easily shown thatits imaginary part is zero. A consequence of (3.4.3) is that k can be considered to be theFT of the function fx 1/2.The same integral appears in (3.4.2) and (3.4.3) and does not exist in a conventional sense.Denote this integral by Ik and adopt a slightly different perspective. Define two functionsI 1 k and I 2 k by21
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 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: complex FS of f on −L ≤ ≤
Page 23 and 24: function and kgk 0, then gk is a c

Fourier Transforms

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