Bản chất vật lý buộc các nhà khoa học, các kỹ sư phải thực hiện trong những khung giới hạn của đối tượng vật lý cần miêu tả. Tuỳ thuộc vào các hiện tượng vật lý khác nhau của mỗi chuyên ngành kỹ thuật phải xem xét mà chúng ta sử dụng những công cụ toán học cho phù hợp nhằm mô tả hiện tượng, ước lượng và tối ưu hoa chúng trên ý nghĩa kỹ thuật. | Poisson s summation formula If y x is integrable over oo oc there exists a relationship between the function and its Fourier transform commonly called Poisson s summation We begin by inventing a periodic function g x defined by g i y 27T . 3 fc 03 Because g x is a periodic function of 2tf it can be represented by the complex Fourier series 52 c einI n oo or DU oo z 0 Ỉ2 2 52 c - fc oo n OG Computing cn we find that 1 fir If 2 Q . cn f- lị -1 dx - 52 Hx dx 3 1 ê r f dx f- r dx 3 F n . 2tt where F w is the Fourier transform of a - Substituting into we obtain oo . oo 52 2 f- 52 r n Z7T Jfc -oc n -oo 64 or INVERSION OF FOURIER TRANSFORMS Having focused on the Fourier transform in the previous sections we consider in detail the inverse Fourier transform in this section Recall that the improper integral 31-6 defines the inverse Consequently one method of inversion is direct integration. Another method for inverting Fourier transforms is rewriting the Fourier transform using partial fractions so that we can use transform tables The following example illustrates this technique Although we may find the inverse by direct integration or partial fractions in many instances the Fourier transform does not lend itself to these techniques. On the other hand if we view the inverse Fourier transform as a line integral along the real axis in the complex u -plane then perhaps some of the techniques that we developed in Chapter 1 might be applicable to this problem. To this end we rewrite the inversion integral as 0 FW di i c r e d c . 3-4 9 where c denotes a closed contour consisting of the entire real axis phis a new contour Cr that joins the point oo 0 to 00 0 There are countless possibilities for Cr. For example it could be the loop oo 0 to oc 7 to -oo R to 00 0 with R 0. However any choice of Cr must be such that we can compute fc F z eits dz. When wo take that constraint into account the number of acceptable
