Essentials of Geophysical Data Processing

The Discrete Fourier Transform (DFT) is one of the most important tools in geophysical data processing and in many other fields. The DFT may be understood from a number of viewpoints, but here we emphasize that it is a Fourier series representing a uniformly sampled time series as a sum of sampled complex sinusoids. We refer to the time series as being in the time domain while the set of its complex-valued sinusoidal coefficients computed using the DFT is in the frequency domain. The inverse DFT (IDFT) computes time series values by adding together Fourier frequency sinusoids, each scaled by a frequency domain sinusoidal coefficient. We develop the DFT by converting the ordinary Fourier series to complex form, transitioning to sampled time series, and finally explaining standard normalization and the usual (and often baffling) frequency and time ordering conventions. The DFT came into widespread use only in the 1960s after the development of Fast Fourier Transform (FFT) algorithms. The speed of FFT algorithms has led to many important applications. Those presented here include the interpolation and computation of analytic signals for real-valued time series. In later chapters we show the important roles of the DFT in linear filtering and spectral analysis.