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].
As mentioned in the preceding, the scientific work of Fourier culminated in his Théorie Analytique de la Chaleur (Analytical Theory of Heat) [2] that was eventually published in its final form in 1822, i.e., 11 years after having been first presented as a memoir to the French Academy of Sciences. Although its value was recognized at that time by awarding Fourier a prize, this contribution was received by the examiners (including Lagrange) with some reservations concerning rigor, raising convergence issues that were eventually resolved in full generality by Dirichlet and others. Fourier’s seminal work established, nevertheless, the foundations of modern harmonic analysis, a branch of mathematics that flourished in the 19th and 20th centuries and proved to be of utmost importance in numerous applications. Starting from a problem in physics and considering that [2, p. 13] “the profound study of nature is the most fertile source of mathematical discovery,” Fourier is generally considered the creator of mathematical physics [3]. Eager to solve physics problems by giving solutions based on firm mathematical grounds, Fourier was also deeply concerned with effective calculations, claiming explicitly that [2, p. 11] “the [proposed] method does not leave anything vague and indefinite in its solutions; it drives them to their ultimate numerical applications, necessary condition for any research, and without which we would only end up with useless transformations.” This focus on what we would now call algorithmic efficiency also makes of Fourier an actual father of signal processing.
Jean-Baptiste Joseph Fourier (1768–1830) was a French mathematician, physicist, and political figure who has had more than one life (Figure 1). Orphaned at the age of nine and spotted for his intellectual abilities, he was taken in charge by a religious educational institution, where he developed a particular interest in mathematics. He thus became a teacher in various domains and finally in mathematics. After having taken an active part in the French Revolution, for which he was imprisoned twice, he was selected as one of the first students of the newly created École Normale, where he quickly became an assistant professor before succeeding Joseph-Louis Lagrange as a professor at École Polytechnique, in 1797. One year later, he was designated to join the Egyptian expedition of Napoléon Bonaparte and became secretary of the Institut d’Égypte, conducting there scientific and political activities until the British victory. Back in France, in 1801, he thought of resuming his academic position but was appointed governor of Isère by Napoléon. While supervising various road and sewerage works, it was during this period that he began his masterwork on the analytical theory of heat. He also played a key role in the creation of the University of Grenoble and became a mentor and close friend of Jean-François Champollion, whom he encouraged in his research to decipher hieroglyphs. Subjected to the vicissitudes of Napoleon’s resignation in 1814 and attempt to return to power in 1815, Fourier was reassigned as governor from Grenoble to Lyon, but he resigned before the battle of Waterloo and went to Paris in June 1815, having no position at all. Being eventually elected a member of the Académie des sciences, in 1817 (and secrétaire perpétuel, in 1822), he devoted entirely the final period of his life to his scientific activities. (An authoritative presentation of the life and works of Fourier can be found in, e.g., [3].)
Figure 1. Joseph Fourier.
Whereas Fourier theory is now central in acoustics, speech, and signal processing, it seems that its first explicit use in sound studies was due to Georg Simon Ohm, who claimed, in his 1843 seminal paper aimed at defining what a “tone” is, that he used [4, p. 519] “Fourier’s theorem, which has become famous through its multiple and important applications.” Ohm’s paper was devoted to specific sound systems, namely, sirens, whose physical construction clearly departed from more classical vibrating strings (for which sine/cosine descriptions were well accepted) and whose understanding was an open question. Sirens had been previously investigated by August Seebeck, who conducted a number of experiments, ending up with puzzling questions (combination tones, missing fundamental, and so on). Ohm proposed to interpret Seebeck’s findings in Fourier terms, but Seebeck raised objections, and a controversy followed [5]. Ohm’s approach was essentially mathematical and disconnected from hearing issues (Ohm even claimed to have an “unmusical ear” [6]). Seebeck, on the contrary, noted contradictions between Ohm’s predictions and actual perceptions by a trained ear. After Ohm lost interest in those questions, the controversy stopped, in 1849, when Seebeck passed away, and the fundamental question of confronting mathematical descriptions with physical realities resurfaced, in 1856, with Hermann von Helmholtz [7], who made acoustics fully enter experimental physics while taking into account physiological considerations. Helmholtz gave credit to Ohm for his introduction of Fourier methods in acoustics, and he followed him in proposing to consider the inner ear a Fourier analyzer sensitive to the intensities of Fourier components, or proper modes (what he referred to as “Ohm’s law”). As a side note, it is worth mentioning that the question of whether Fourier proper modes (or their variations) are a physical reality or a mathematical construct has been recurrent since then. One can quote, in this respect, Louis de Broglie, who once claimed that [8] “if we consider a quantity that can be represented in the manner of Fourier, by a superposition of monochromatic components, it is the superposition that has a physical meaning, and not the Fourier components considered in isolation,” or refer to [9] for data-driven versus model-based approaches to beating phenomena.
In parallel, Helmholtz conducted experiments with high-quality instruments—precise tuning forks for the production of well-controlled “pure” tones and resonators made of cavities of different sizes for identifying frequency components within complex sounds. Helmholtz’s contributions have been of primary importance in the development of modern acoustics and psychoacoustics. His approach was also emblematic of the key role played by instruments in addressing scientific questions and challenging theories (as once said by the philosopher Gaston Bachelard, “Instruments are reified theories”). To this end, he was in close contact with a gifted instrument maker settled in Paris: Rudolph Koenig (Figure 2).
Born in Königsberg (Prussia), Karl Rudolph Koenig settled in Paris in 1852 and died there in 1901. While developing a special interest in acoustics, Koenig was not part of any academic institution, but he was a prolific inventor—with 272 items in his 1889 catalog [10]—and a successful businessman who manufactured and sold his own products all around the world. He was especially famous for the quality of his tuning forks, and he contributed, with his experiments, to the debates and disputes about beats and combination tones [11]. Koenig happened to be, for a long time, the main maker of Helmholtz’s instruments, and his workshop in Paris was a busy meeting place, where the ideas of Helmholtz were popularized and spread in Parisian scientific circles, maintaining a vivid relationship with his native Germany [12]. In particular, exploiting the potentialities of Helmholtz resonators and combining them with his own invention of “manometric flames,” he designed a “sound analyzer” [10], [13] that allowed for a visualization of the frequency content of a sound. To visualize sound waves, he first designed, in 1862, an apparatus—the so-called manometric flame—that consists of a flexible membrane encapsulated in a chamber. When exposed to a sound, the vibration of the membrane modulates the flow of a flammable gas passed to a Bunsen burner, and the size of the flame is, in turn, modulated accordingly. The final visualization is made possible in a stroboscopic way, thanks to a four-faceted rotating mirror. Koenig later designed, in 1867, a more complete “sound analyzer” by plugging such capsules at the output of a family of Helmholtz resonators (i.e., cavities tuned to specific frequencies) playing the role of a filter bank (Figure 3). The overall system permits, therefore, a Fourier-like frequency analysis and, in cases where the impinging sound is time varying, a time-frequency analysis.
Figure 2. Rudolph Koenig.
Figure 3. Koenig’s sound analyzer [10].
One can remark that Koenig’s apparatus has very much the flavor of an electromechanical system that would appear almost one century later: the so-called sound spectrograph [14]. This instrument—which, ironically, would be due to another Koenig—shares with the “sound analyzer” the idea of visualizing the intensity of frequency components. In the sound analyzer, such intensities are evaluated acoustically and in parallel (with all resonators acting simultaneously for selecting frequencies), and they are visualized by the modulations of the manometric flames. In the sound spectrograph, the acoustic signal is first recorded on a magnetic tape, and the frequency intensities are evaluated electrically and sequentially, thanks to a heterodyne filtering that acts in a synchronous manner with the rotation of the disk on which the tape is fixed and that of a drum on which the intensity is reported by a stylus on an electrically sensitive paper (Figure 4). Today, these acoustic or electromechanical devices have been replaced by computers to perform time-frequency analysis of digitized data, with routine techniques, such as short-time Fourier analysis, operating in time rather than frequency. The basic principles of these modern approaches are nonetheless similar, the difference being essentially one of implementation.
Figure 4. Koenig’s sound spectrograph [14].
Whereas Helmholtz elaborated on the findings of Ohm, who himself referred to Fourier, no explicit reference to Fourier can be found in the written productions of Koenig nor in the description of his sound analyzer [10]. Indeed, the motivation of Koenig was elsewhere, far from the rooting of his instruments on mathematical bases. It turns out that this was an attitude shared by most physicists of the immediate post-Fourier period (say, 1822–1850). Fourier is now perceived as the pioneer of modern mathematical physics, but in the years that followed its main publication, Fourier’s treatise attracted mostly the attention of mathematicians who substantially contributed to consolidating and extending Fourier’s seminal work, and, with the notable exception of Ohm, its importance for physics seems to have escaped physicists [15]. As an acoustician and an instrument-maker, Koenig was, in fact, primarily interested in visualizing sounds and in achieving this program through a more experimental and empirical approach. This leads to another cornerstone on the way to sound analysis. It involved Koenig too but preceded the invention of his sound analyzer.
If we think of visualizing sounds, there are at least two possibilities. The first one, which corresponds to Koenig’s approach with his sound analyzer, is indirect in the sense that what is given to see are features resulting from some transformation upon the waveform (namely, intensities of Fourier modes). Another, more direct, approach could, however, be imagined, which would deliver a graphical representation of the waveform itself. Such a track was indeed followed in the middle of the 19th century by another outsider of academia: Édouard-Léon Scott de Martinville (Figure 5).
Figure 5. Édouard-Léon Scott de Martinville.
Édouard-Léon Scott de Martinville (1817–1879) was a French inventor who, by profession, was a typographer. In the early 1850s, he conceived the idea of drastically improving upon stenography for keeping track of spoken words and other sounds by developing a system that would solve, in his own words, “the problem of speech writing itself.” This question obsessed Scott until his final days [16], and it is worth quoting his agenda, as reported in the sealed manuscript he sent to the French Academy of Sciences, in 1857 [17] (English translation by P. Feaster [18]): “Is there a possibility of reaching in the case of sound a result analogous to that attained at present for light by photographic processes? Can one hope that the day is near when the musical phrase, escaped from the singer’s lips, will be written by itself and as if without the musician’s knowledge on a docile paper and leave an imperishable trace of those fugitive melodies which the memory no longer finds when it seeks them? Will one be able to have placed between two men brought together in a silent room an automatic stenographer that preserves the discussion in its minutest details while adapting to the speed of the conversation? Will one be able to preserve for the future generation some features of the diction of one of those eminent actors, those grand artists who die without leaving behind them the faintest trace of their genius?” To achieve this ambitious goal, Scott took his inspiration from the hearing process and proposed to make use of a membrane (mimicking the eardrum) at the output of a horn designed to collect and concentrate the sounds to be analyzed. The vibrations of the membrane were transmitted to a stylus attached to it, whose movements were inscribed as tracings on a sliding lampblacked glass plate. Scott gave to his invention the name “phonautograph” and started making experiments in 1853–1854, waiting until 1857 to submit it to the French Academy of Sciences [17] and patenting it [19]. Looking at those first attempts (with either speech or guitar sounds [17]), one must admit that the tracings he recorded are extremely erratic and unlikely to be interpretable. Scott’s first phonautograph was the work of an amateur, and to hope that it could be turned into a reliable instrument required the professionalism of an expert. The best expert one could think of in this regard at that time was Koenig, and it was only natural that Scott approached him to perfect his device. Their collaboration resulted in a second-generation phonautograph (Figure 6) with far better performance than the initial prototype. Koenig replaced the sliding plate with a rotating cylinder, allowing for much longer recordings. He also supplemented the sound recording with the reference trace of a tuning fork, thus making it easier to read the unavoidable irregularities in the rotation of the hand-cranked cylinder.
Figure 6. Scott’s phonautograph [10].
In 1860, reasonably neat tracings were obtained this way, leading to the fundamental issue: How to interpret them? The very purpose of Scott was to consider phonautograms as graphical representations of sounds and to uncover from their reading the content of what had been recorded. He was, thus, interested in finding specific features in the tracings, which could be considered elementary recognizable components of speech or natural sounds. This quest was not without echoes of some recent advances in signal processing, with representations built upon “waveform dictionaries” [20]. If we consider, for instance, the commented hand-drawn tracings of Figure 7, we see that Scott tried to identify elementary sounds from their graphical representation, that we would today refer to as tones with low/high frequency (“la voix grave/la voix aiguë”), down-going/upgoing chirps (“une voix aiguë descendant au grave/une voix grave montant à l’aigu”), different amplitudes (“une voix intense/moyenne/faible”), a plosive (“l’explosion de la voix”).
Figure 7. Scott’s “waveform dictionary” [19].
Reading tracings from complicated sounds, such as speech or songs, proved, however, to be tricky, and, short of support and encouragement, Scott abandoned, in the early 1860s, his project, becoming a librarian after having warranted Koenig an exclusive license to manufacture and sell the phonautograph. Far from Scott’s dream of “speech writing itself,” Koenig advocated the use of the phonautograph as a less ambitious scientific instrument aimed at mostly analyzing tuning forks or organ pipes, before leaving it and turning to his sound analyzer. Scott’s interest in his phonautograph was rekindled, however, in 1878, when Edison’s phonograph was demonstrated at a memorable session of the French Academy of Sciences. Having heard of it, Scott could not help but find elements in the phonograph—such as the system of recording by means of a membrane, a stylus, and a rotating drum—that seemed to him to be directly inspired by his phonautograph, without making any reference to it. Bitter about this lack of recognition as well as the contrast between the enthusiastic reception of Edison’s invention and the poor interest that his own invention had received 20 years earlier, Scott self-edited a long plea for his rights and his vision of speech analysis just before his death the following year [16]. Of course, one of the main reasons why the phonograph attracted so much more attention than the phonautograph was that the former allowed for the replay of recorded sounds, which the latter did not. As Scott had said many times, reproducing sound was not part of his program at all, his only goal being to decipher phonautograms. One can imagine that he would have found little interest in regenerating real sounds from his phonautograms, and yet this is what happened … in 2008.
Indeed, before deciding not to pursue his project any further, Scott deposited a number of annotated phonautograms with the French Academy of Sciences, in 1861 [21]. These recordings were properly archived and preserved, yet forgotten until 2007, when David Giovannoni—a historian specializing in old recordings, who had learned of their existence—had the idea of transforming them into truly audible sounds. This was made possible—within the First Sounds project [22] and thanks, in particular, to Patrick Feaster—by getting high-quality scans of the tracings and transforming them into digital files through modern signal processing techniques. This is how the folk song “Au Clair de la Lune” (“By the Light of the Moon”) (Figure 8), registered by Scott himself on 9 April 1860, can now be heard [23], the first recording of a human voice, 17 years before Edison.
Figure 8. Scott’s phonautogram of the folk song “Au Clair de la Lune” [19]. (a) The complete recording and (b) an enlargement showing (on three successive revolutions of the drum) plots of the recorded voice and the reference oscillation given by the tuning fork.
The purpose of this text was not to discuss Fourier’s achievements per se: this can be found in many textbooks, from different perspectives (see, e.g., [24] for a classical introduction, [25] for a more mathematically oriented treatise, or [26] for a modern treatment, including recent variations). What was at stake was to see how Fourier’s ideas, which today seem indissociable from sound analysis, were not immediately adopted in this context and that parallel pathways, based on different approaches, have been followed. It is striking to observe that some of the options considered then are still relevant today. For instance, an approach à la Koenig is based, in a first step, on the extraction of some features (in his case, the Fourier modes, even if not named as such) upon which the analysis is performed in a second step, whereas an approach à la Scott bypasses such a preprocessing and relies directly on the raw data, as can be the case in modern end-to-end recognition systems. Another important issue is the quest for interpretability when confronting experimental results with formal descriptions, i.e., physics with mathematics (this was at the heart of the Ohm–Seebeck dispute). Yet, under different forms extended to algorithms and computational issues, such a question of understanding is today of paramount importance, e.g., in deep neural networks when one wants to go beyond a black box.
We choose to close the piece of history that has been outlined here in 1878, when Edison opened a new chapter. Many things happened in the following years, with progressive and more and more pervasive use of Fourier techniques in many domains. In the same year, 1878, Lord Kelvin constructed his harmonic analyzer [27] that proved instrumental during decades for analyzing and predicting tides. Other mechanical [28], electromechanical [14], and, later, electronical [29] systems followed, eventually giving Fourier the full credit he deserves in the information era, but this is another story.
Patrick Flandrin (flandrin@ens-lyon.fr) received his Ph.D. degree from INP Grenoble, France, in 1982. He is currently a CNRS emeritus research director, working within the Department of Physics, ENS De Lyon, Lyon, France, since 1991. His research interests include nonstationary signal processing, time-frequency/wavelet methods, scaling stochastic processes, and complex systems. He was awarded the SPIE Wavelet Pioneer Award (2001), the CNRS Silver Medal (2010), and a Technical Achievement Award from the IEEE Signal Processing Society (2017) and European Association for Signal Processing (EURASIP) (2023). He was elected to the French Academy of Sciences in 2010 and served as its president in 2021–2022. He is a Fellow of IEEE (2002) and EURASIP (2009).
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Digital Object Identifier 10.1109/MSP.2023.3297313