Fourier analysisIn mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.
Fourier transformIn physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.
Discrete Fourier transformIn mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.
Discrete-time Fourier transformIn mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.
Fourier seriesA Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.
Non-uniform discrete Fourier transformIn applied mathematics, the nonuniform discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the shifted DFT. It has important applications in signal processing, magnetic resonance imaging, and the numerical solution of partial differential equations.
Fast Fourier transformA fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical.
Discrete Fourier transform over a ringIn mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring. Let R be any ring, let be an integer, and let be a principal nth root of unity, defined by: The discrete Fourier transform maps an n-tuple of elements of R to another n-tuple of elements of R according to the following formula: By convention, the tuple is said to be in the time domain and the index j is called time.
Infrared spectroscopyInfrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or functional groups in solid, liquid, or gaseous forms. It can be used to characterize new materials or identify and verify known and unknown samples. The method or technique of infrared spectroscopy is conducted with an instrument called an infrared spectrometer (or spectrophotometer) which produces an infrared spectrum.
SpectroscopySpectroscopy is the field of study that measures and interprets the electromagnetic spectra that result from the interaction between electromagnetic radiation and matter as a function of the wavelength or frequency of the radiation. Matter waves and acoustic waves can also be considered forms of radiative energy, and recently gravitational waves have been associated with a spectral signature in the context of the Laser Interferometer Gravitational-Wave Observatory (LIGO).
Fourier-transform spectroscopyFourier-transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of the radiation, electromagnetic or not. It can be applied to a variety of types of spectroscopy including optical spectroscopy, infrared spectroscopy (FTIR, FT-NIRS), nuclear magnetic resonance (NMR) and magnetic resonance spectroscopic imaging (MRSI), mass spectrometry and electron spin resonance spectroscopy.
Near-infrared spectroscopyNear-infrared spectroscopy (NIRS) is a spectroscopic method that uses the near-infrared region of the electromagnetic spectrum (from 780 nm to 2500 nm). Typical applications include medical and physiological diagnostics and research including blood sugar, pulse oximetry, functional neuroimaging, sports medicine, elite sports training, ergonomics, rehabilitation, neonatal research, brain computer interface, urology (bladder contraction), and neurology (neurovascular coupling).
Harmonic analysisHarmonic 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 the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis.
Digital signal processingDigital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor.
Fractional Fourier transformIn mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.
Signal processingSignal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing signals, such as sound, , potential fields, seismic signals, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. According to Alan V. Oppenheim and Ronald W.
SinIn a religious context, 'sin' is a transgression against divine law or a law of God. Each culture has its own interpretation of what it means to commit a sin. While sins are generally considered actions, any thought, word, or act considered immoral, selfish, shameful, harmful, or alienating might be termed "sinful".
Joseph FourierJean-Baptiste Joseph Fourier (ˈfʊrieɪ,_-iər; fuʁje; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.
Multiplier (Fourier analysis)In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol. Occasionally, the term multiplier operator itself is shortened simply to multiplier. In simple terms, the multiplier reshapes the frequencies involved in any function.
Mössbauer spectroscopyMössbauer spectroscopy is a spectroscopic technique based on the Mössbauer effect. This effect, discovered by Rudolf Mössbauer (sometimes written "Moessbauer", German: "Mößbauer") in 1958, consists of the nearly recoil-free emission and absorption of nuclear gamma rays in solids. The consequent nuclear spectroscopy method is exquisitely sensitive to small changes in the chemical environment of certain nuclei.
