Fourier Analysis of Signals

Abstract

As we have seen in the last chapter, music signals are generally complex sound mixtures that consist of a multitude of different sound components. Because of this complexity, the extraction of musically relevant information from a waveform constitutes a difficult problem. A first step in better understanding a given signal is to decompose it into building blocks that are more accessible for the subsequent processing steps. In the case that these building blocks consist of sinusoidal functions, such a process is also called Fourier analysis. Sinusoidal functions are special in the sense that they possess an explicit physical meaning in terms of frequency. As a consequence, the resulting decomposition unfolds the frequency spectrum of the signal—similar to a prism that can be used to break light up into its constituent spectral colors. The Fourier transform converts a signal that depends on time into a representation that depends on frequency. Being one of the most important tools in signal processing, we will encounter the Fourier transform in a variety of music processing tasks.