Further Developments and Applications of the Finite Fourier Transform

Abstract

In actual applications, most mathematical methods have to deal with finite data sets. Thus it is not surprising that the finite Fourier transform is the main tool among transforms in applied research. Two topics in communication science have been selected to illustrate the use of the finite Fourier transform: signal filters and windows in Section 3.1 and signal detection in the presence of noise in Section 3.2. These make use of the operations of convolution and correlation. The implementations of these techniques would be impossible without present-day computers and an efficient algorithm for the numerical work. The fast Fourier transform (FFT) operating principles are given in Section 3.3. Finally, in Section 3.4 we let the dimension of the vector space grow without bound. In this way we arrive at the Fourier series and integral transforms which are the subjects of Parts II and III. The sections are mutually independent except for Section 3.2, which relies somewhat on concepts developed in Section 3.1. Otherwise, they can be read in any order. The References should be consulted if the reader wishes a wider picture of the applied technology.