Fourier analysis
The subject of Fourier analysis encompasses a vast spectrum of mathematics.
One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis.
In mathematics, the term Fourier analysis often refers to the study of both operations.
Different approaches have been developed for analyzing unequally spaced data, notably the least-squares spectral analysis (LSSA) methods that use a least squares fit of sinusoids to data samples, similar to Fourier analysis.
[2][3] Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in long gapped records; LSSA mitigates such problems.
[4] Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, digital image processing, probability theory, statistics, forensics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis, and other areas.
This wide applicability stems from many useful properties of the transforms: In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum.
The FT method is used to decode the measured signals and record the wavelength data.
Fourier transforms are not limited to functions of time, and temporal frequencies.
They can equally be applied to analyze spatial frequencies, and indeed for nearly any function domain.
This justifies their use in such diverse branches as image processing, heat conduction, and automatic control.
[10] Some examples include: Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, and it produces a continuous function of frequency, known as a frequency distribution.
can be represented as a recombination of complex exponentials of all possible frequencies: which is the inverse transform formula.
becomes a Dirac comb function, modulated by a sequence of complex coefficients: The inverse transform, known as Fourier series, is a representation of
in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients: Any
Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function: which is known as the DTFT.
sequence is also the Fourier transform of the modulated Dirac comb function.
[B] The Fourier series coefficients (and inverse transform), are defined by: Parameter
corresponds to the sampling interval, and this Fourier series can now be recognized as a form of the Poisson summation formula.
, becomes a Dirac comb function, modulated by a sequence of complex coefficients (see DTFT § Periodic data): The
causes overlap (adding) in the time-domain (analogous to aliasing), which corresponds to decimation in the frequency domain.
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.
It is common in practice for the duration of s(•) to be limited to the period, P or N. But these formulas do not require that condition.
And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:[11] From this, various relationships are apparent, for example: An early form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions).
In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit,[16] which has been described as the first formula for the DFT,[17] and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.
[17] Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits.
[18] Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.
A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation,[20] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series, introducing the Fourier series.
[17] The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.
The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithm is more often attributed to its modern rediscoverers Cooley and Tukey.
