Fourier Transforms

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