Search results
Results From The WOW.Com Content Network
In the study of heat conduction, the Fourier number, is the ratio of time, , to a characteristic time scale for heat diffusion, . This dimensionless group is named in honor of J.B.J. Fourier , who formulated the modern understanding of heat conduction. [ 1 ]
The substitution = into Eq.1 produces this convention, where function ^ is relabeled ^: ^ () = ^ (), = ^ (). Unlike the Eq.1 definition, the Fourier transform is no longer a unitary transformation, and there is less symmetry between the formulas for the transform and its inverse.
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula () =, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation .
But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact.
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.
Fourier series, a weighted sum of sinusoids having a common period, the result of Fourier analysis of a periodic function; Fourier analysis, the description of functions as sums of sinusoids; Fourier transform, the type of linear canonical transform that is the generalization of the Fourier series
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
Left: A continuous function (top) and its Fourier transform (bottom). Center-left: Periodic summation of the original function (top). Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right: Original function is discretized (multiplied by a Dirac comb) (top).