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Function () (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform S ( f ) {\displaystyle S(f)} is a frequency-domain representation that reveals the amplitudes of the summed sine waves.
An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30. Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.
[note 3] Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or R n, notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S 1, the unit circle ≈ ...
In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes. When any two sine waves of the same frequency (but arbitrary phase ) are linearly combined , the result is another sine wave of the same frequency; this property is ...
The Fourier transform converts the function's time-domain representation, shown in red, to the function's frequency-domain representation, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.
List of Fourier-related transforms; Fourier transform on finite groups; Fractional Fourier transform; Continuous Fourier transform; Fourier operator; Fourier inversion theorem; Sine and cosine transforms; Parseval's theorem; Paley–Wiener theorem; Projection-slice theorem; Frequency spectrum
In mathematical analysis, Parseval's identity, named after Marc-Antoine Parseval, is a fundamental result on the summability of the Fourier series of a function. The identity asserts the equality of the energy of a periodic signal (given as the integral of the squared amplitude of the signal) and the energy of its frequency domain representation (given as the sum of squares of the amplitudes).
Fourier series of a 33. 3 % pulse wave, first fifty harmonics (summation in red) The Fourier series expansion for a rectangular pulse wave with period T {\displaystyle T} , amplitude A {\displaystyle A} and pulse length τ {\displaystyle \tau } is [ 1 ]