Fourier series and transforms

Article By:

Bracewell, Ronald N. Formerly, Electrical Engineering Department, Stanford University, Stanford, California.

Last reviewed:June 2020

DOI:https://doi.org/10.1036/1097-8542.270600

Mathematical tools for the analysis of functions through decomposition into sinusoids. In 1822, J. Fourier proposed that ordinary mathematical functions could be represented by a sum of sinusoids, even though at first sight it might seem that the blandness and repetitive nature of the sinusoid ill suited it to accommodate the variety of functions in general. The advantage of such a representation is this: A differential equation that is difficult to solve under the untidy given external conditions that often arise in technology may very well be soluble under an external condition that is simply sinusoidal. If solutions can be obtained for each of the constituent sinusoids that make up the given external function, then perhaps the required solution will be the sum of the separate simple solutions. This was indeed so in the case studied by Fourier, who was interested in the differential equation governing the diffusion of heat in homogeneous solids; in fact, it is often the case that solving for sinusoids permits synthesis of the required solution. When this does happen, the solution is itself composed of sinusoids and there is said to be a sinusoidal response to sinusoidal input; the differential equation will also exhibit linearity combined with shift invariance. See also: Differential equation; Linearity; Spectrum analyzer; Trigonometry

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