The fast Fourier transform for experimentalists. Part I. Concepts

The inverse transform reverses the process, converting frequency data into time-domain data. Such transformations can be applied in a wide variety of fields, from geophysics to astronomy, from the analysis of sound signals to CO 2 concentrations in the atmosphere. Over the course of three articles, our goal is to provide a convenient summary that the experimental practitioner will find useful. In the first two parts of this article, we'll discuss concepts associated with the fast Fourier transform (FFT), an implementation of the DFT. In the third part, we'll analyze two applications: a bat chirp and atmospheric sea-level pressure differences in the Pacific Ocean. The FFT provides an efficient algorithm for implementing the DFT and, as such, we'll focus on it. This transform is easily executed; indeed, almost every available mathematical software package includes it as a built-in function. Some books are devoted solely to the FFT, 1–3 while others on signal processing, 4–6 time series, 7, 8 or numerical methods 9,10 include major sections on Fourier analysis and the FFT. We draw together here some of the basic elements that users need to apply and interpret the FFT and its inverse (IFFT). We will avoid descriptions of the Fourier matrix , which lies at the heart of the DFT process, 11 and the parsing of the Cooley-Tukey algorithm 12 (or any of several other comparable algorithms), which provides a means for transforming the discrete into the fast Fourier transform. The Cooley-Tukey algorithm makes the FFT extremely useful by reducing the number of computations from something on the order of n 2 to n log(n), which obviously provides an enormous reduction in computation time. It's so useful, in fact, that the FFT made Computing in Science & Engineering's list of the top 10 algorithms in an article that noted the algorithm is, " perhaps, the most ubiquitous algorithm in use today. " 13 The interlaced decomposition method used in the Cooley-Tukey algorithm can be applied to other orthogonal transformations such as the Hadamard, Hartley, and Haar. However, in this article, we concentrate on the FFT's application and interpretation. As a rule, data to be transformed consists of N uniformly spaced points x j = x(t j), where N = 2 n with n an integer, and t j = j × ⌬t where j ranges from 0 to N – 1. (Some FFT implementations don't require that N be a …