Fourier and the Early Days of Sound Analysis

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Fourier and the Early Days of Sound Analysis

By: 
Patrick Flandrin

Joseph Fourier’s methods (and their variants) are omnipresent in audio signal processing. However, it turns out that the underlying ideas took some time to penetrate the field of sound analysis and that different paths were first followed in the period immediately following Fourier’s pioneering work, with or without reference to him. This illustrates the interplay between mathematics and physics as well as the key role played by instrumentation, with notable inventions by outsiders to academia, such as Rudolph Koenig and Édouard-Léon Scott de Martinville.

Fourier analysis, Fourier series, (Fast) Fourier transform. Fourier has today something of a common name. If his presence is now ubiquitous in almost all fields of science and technology, the name of Fourier is especially unavoidable for all those interested in the theory and practice of signal processing. In particular, the methods he developed—and the attached fundamental concepts, such as that of spectral representation—are the cornerstone of audio signal processing (speech, music, and so on). This might suggest that they were developed in connection with the idea of analyzing and/or synthesizing sounds or at least that such an application was envisaged from the outset. This turned out not to be the case, the whole project of Fourier being devoted to a different physical problem, namely, the theory of heat, and to mathematical developments attached to it. Whereas many attempts had been made before Fourier (by Bernoulli, d’Alembert, Euler, Lagrange, and others) to solve the problem of vibrating strings and express solutions by means of sine/cosine expansions, Fourier himself seemed to have developed almost no interest in applying his results in this direction. Indeed, while his 1822 treatise on the analytical theory of heat is more than 600 pages long, there is only one sentence evoking such a possibility: “If we apply those principles to the question of the motion of vibrating strings, we shall overcome the difficulties first encountered in Daniel Bernoulli’s analysis.” It was only 20 years later that Fourier ideas entered explicitly the field of acoustics, thanks to Georg Simon Ohm (most famous for his law of electrical conductivity, established in 1827). This was, however, not a fully shared recognition, and, between theory and experiments, the following years witnessed a number of developments aimed at analyzing sounds, with or without a reference to Fourier. This is what this text is about. In complement to the immediate post-Fourier influences in acoustics discussed here, a comprehensive study of the (pre-Fourier) acoustics origins of harmonic analysis can be found in [1].

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