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Measurement
Published in Carl Hopkins, Sound Insulation, 2020
The integrated impulse response method involves generating an impulse, and recording or sampling the response of an acoustic system to this impulse. Due to its infinite height and infinitely narrow width, it is not possible to create a Dirac delta function in practice. However, it is possible to create an impulse of sufficiently short duration that can represent the Dirac delta function. On structures, it is possible to generate an impulse with a hammer blow. In rooms an impulse can be generated with a gunshot from a starting pistol, balloon bursts, handclaps, or noise bursts via a loudspeaker. Although these impulse sources have been used in rooms for many years, not all of them are omnidirectional and are able to generate a flat spectrum at a sufficiently high level whilst avoiding very high crest factors (ratio of peak to rms). The latter can cause problems due to the limitations of the detector in the analyser. All of these problems can be avoided when the system to be measured is linear and time-invariant (LTI). The required impulse response can then be measured with swept-sine signals (Müller and Massarani, 2001) or a signal referred to as a Maximum Length Sequence (MLS) (Section 3.9).
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
The Dirac delta-function δ(x) is not a genuine function in the ordinary sense, but it is a generalized function or distribution. Generalized functions were rigorously defined in 1936 by the Russian mathematician Sergei L'vovich Sobolev (1908-1989). Later in 1950 and 1951 , the French mathematician Laurent Schwartz 50 published two volumes of "Theore des Distributions," in which he presented the theory of distributions.
Laplace Transform
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
The Dirac delta-function δ(x) is not a genuine function in the ordinary sense, but it is a generalized function or distribution. Generalized functions were rigorously defined in 1936 by the Russian mathematician Sergei L'vovich Sobolev (1908–1989). Later in 1950 and 1951, the French mathematician Laurent Schwartz38 published two volumes of “Theore des Distributions,” in which he presented the theory of generalized functions.
Universal approximation with neural networks on function spaces
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Wataru Kumagai, Akiyoshi Sannai, Makoto Kawano
Thus, is exactly represented by a bounded affine map, if the Dirac delta function is allowed. However, the Dirac delta function is not a function but a generalised function. Here, the Dirac delta can be approximated by a smooth function called a mollifier, with any precision. Thus, instead of , we take as . We can verify that is approximated by a bounded affine map with precision.