The Logic of Filtering by Melle Jan Kromhout;

Author:Melle Jan Kromhout;
Language: eng
Format: epub
Publisher: OUP Premium
Published: 2021-07-15T00:00:00+00:00

The Plane of the Ideal Filter

With Ohm’s application of Fourier’s theorem to the analysis of sound, and Helmholtz’s expansion and experimental verification of its principles, the sine wave came to be defined as the elemental tone: a pure frequency with no overtones and no timbral characteristics of its own. As with the graphs showing the “seemingly sharp contours of surfaces” produced by Fourier’s analysis of heat propagation, the purity and clarity of the sine wave are properties of a conceptual limit case—the most extreme limit of a phenomenon—drawing seemingly sharp contours around infinitesimally fuzzy sound waves. A fixed and indivisible standard of sonic purity, the sine wave is an ideal form toward which all physical sounds seem to tend. The relation between this idealized mathematical object (a sine wave) and the physical phenomenon it represents (a simple sound wave) can therefore be described as the relation between symbol and signal.34

Produced by the strictly symbolic operations of mathematical analysis, a sine wave is not a physical signal, but an analytical symbol. Its symbolic clarity depends upon a prior, conceptual act of noise reduction that suppresses all reference to its material carriers (transmission channels). “The mathematician,” Serres explains, “does not see any difficulty on this point,” for the mathematical manipulation of written signs already serves “to isolate an ideal form [and] render it independent of the empirical domain and of noise.”35 The mathematical production of an ideal symbol, in other words, entails the removal of any trace of its material production as signal. Ultimately, this entails denying its physical production and transmission as signal, and thus the complete symbolic reduction of noise. To function mathematically, the sine wave requires a process of abstraction that separates its “pure” symbol from its physicality as a contingent signal. In subsequently being physically produced as an actual acoustic object, the sine wave becomes what we might call an “idealized signal,” discursively positioned in between purely symbolic mathematical analysis (the plane of the ideal filter) and physical acoustics (the domain of technical filters).

So, the production of a perfect sine wave—a single frequency—would require complete noise reduction. This indicates that Fourier analysis and the concept of the sine wave are subject to an uncertainty principle. The ideal sine wave presupposes the analytical filtering out of all material channels; and because it represents a signal as a series of such sine waves, the operations of Fourier analysis can be interpreted as an ideal—that is, infinitely accurate—spectral filter.36 Following the uncertainty principle described earlier, the more a physical filter comes to resemble this ideal filter, the narrower its frequency range becomes and the more time it will need to complete the operation. At the analytical limit of this process, the filter will be attuned to a single frequency and its response time will tend mathematically to infinity. At that point, the physical filter would become an ideal filter and the physical signal an ideal signal: a pure sine wave. In this way, the infinity of the sine wave correlates directly

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