Fourier’s theorem is an elegant mathematical transformation that has demonstrated its significant worth after being applied to countless physical scenarios. The examples in Chapter 2 explored the Fourier series expansion of a mathematical function f(t) that might represent a time-varying “analog” signal. In the laboratory or in the field, signals are produced by physical systems, and the goal may be to learn something about the physical system by exploring the signal’s frequency content. Instead of describing the signal with a mathematical function, the signal’s state is measured at selected points in time, such as with transducers and data-acquisition equipment. The result is a sequence of measurements where each value represents a sample of the continuous signal of interest at a given time; such data is referred to as time-series data. In order to explore the frequency content of the underlying signal based on the sample data, instead of finding the frequency content of a mathematical function, as in Chapter 2, one must perform an analogous operation with a series of discretely sampled points. Fourier transform methods can be adapted to analyze the frequency content of time-series data.
Fourier Transforms for Discretely Sampled Data