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In
mathematics, Fourier analysis () is the study of the way general
functions may be represented or approximated by sums of simpler
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
. 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
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...
.
The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into
oscillatory
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component
frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. 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.
The decomposition process itself is called a
Fourier transformation
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
. Its output, the
Fourier transform, is often given a more specific name, which depends on the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as
harmonic analysis. Each
transform
Transform may refer to:
Arts and entertainment
* Transform (scratch), a type of scratch used by turntablists
* ''Transform'' (Alva Noto album), 2001
* ''Transform'' (Howard Jones album) or the title song, 2019
* ''Transform'' (Powerman 5000 album ...
used for analysis (see
list of Fourier-related transforms) has a corresponding
inverse transform that can be used for synthesis.
To use Fourier analysis, data must be equally spaced. Different approaches have been developed for analyzing unequally spaced data, notably the
least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
(LSSA) methods that use a
least squares fit of
sinusoids to data samples, similar to Fourier analysis.
Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in long gapped records; LSSA mitigates such problems.
Applications
Fourier analysis has many scientific applications – in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
partial differential equations,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
,
combinatorics,
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
,
digital image processing
Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
,
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
,
statistics,
forensics,
option pricing
In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the imple ...
,
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
,
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
,
acoustics,
oceanography,
sonar
Sonar (sound navigation and ranging or sonic navigation and ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigate, measure distances (ranging), communicate with or detect objects on o ...
,
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
,
diffraction,
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
,
protein
Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, res ...
structure analysis, and other areas.
This wide applicability stems from many useful properties of the transforms:
* The transforms are
linear operators and, with proper normalization, are
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
as well (a property known as
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates ...
or, more generally, as the
Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integ ...
, and most generally via
Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
).
[
* The transforms are usually invertible.
* The ]exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
s are eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s of differentiation, which means that this representation transforms linear differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s with constant coefficients
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
into ordinary algebraic ones.[ Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency independently.
* By the ]convolution theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
, Fourier transforms turn the complicated convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as signal filtering, polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
multiplication, and multiplying large numbers.[
* The ]discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms.[
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. And by using a computer, these Fourier calculations are rapidly carried out, so that in a matter of seconds, a computer-operated FT-IR instrument can produce an infrared absorption pattern comparable to that of a prism instrument.][
Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation ( discrete cosine transform) of small square pieces of a digital image. The Fourier components of each square are rounded to lower ]arithmetic precision
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expre ...
, and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce an approximation of the original image.
In signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, the Fourier transform often takes a time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Ex ...
or a function of continuous time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
, and maps it into a frequency spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
. That is, it takes a function from the time domain into the frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
domain; it is a decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is e ...
of a function into sinusoids
A capillary is a small blood vessel from 5 to 10 micrometres (μm) in diameter. Capillaries are composed of only the tunica intima, consisting of a thin wall of simple squamous endothelial cells. They are the smallest blood vessels in the body: ...
of different frequencies; in the case of a Fourier series or discrete Fourier transform
In 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 comple ...
, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.
When a function is a function of time and represents a physical signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of the resulting complex-valued function at frequency represents the amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of a frequency component whose initial phase is given by the angle of (polar coordinates).
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
Automation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines ...
.
When processing signals, such as audio
Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to:
Sound
* Audio signal, an electrical representation of sound
*Audio frequency, a frequency in the audio spectrum
* Digital audio, representation of sou ...
, radio waves, light waves, seismic wave
A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic energy ...
s, and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.[
Some examples include:
* Equalization of audio recordings with a series of ]bandpass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-p ...
s;
* Digital radio reception without a superheterodyne
A superheterodyne receiver, often shortened to superhet, is a type of radio receiver that uses frequency mixing to convert a received signal to a fixed intermediate frequency (IF) which can be more conveniently processed than the original carri ...
circuit, as in a modern cell phone or radio scanner
A scanner (also referred to as a radio scanner) is a radio receiver that can automatically tune, or ''scan'', two or more discrete frequencies, stopping when it finds a signal on one of them and then continuing to scan other frequencies when the ...
;
* Image processing to remove periodic or anisotropic artifacts such as jaggies
"Jaggies" is the informal name for artifacts in raster images, most frequently from aliasing, which in turn is often caused by non-linear mixing effects producing high-frequency components, or missing or poor anti-aliasing filtering prior to samp ...
from interlaced video
Interlaced video (also known as interlaced scan) is a technique for doubling the perceived frame rate of a video display without consuming extra bandwidth. The interlaced signal contains two fields of a video frame captured consecutively. Thi ...
, strip artifacts from strip aerial photography, or wave patterns from radio frequency interference
Electromagnetic interference (EMI), also called radio-frequency interference (RFI) when in the radio frequency spectrum, is a disturbance generated by an external source that affects an electrical circuit by electromagnetic induction, electrost ...
in a digital camera;
* Cross correlation of similar images for co-alignment;
* X-ray crystallography
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
to reconstruct a crystal structure from its diffraction pattern;
* Fourier-transform ion cyclotron resonance
Fourier-transform ion cyclotron resonance mass spectrometry is a type of mass analyzer (or mass spectrometer) for determining the mass-to-charge ratio (''m''/''z'') of ions based on the cyclotron frequency of the ions in a fixed magnetic field. T ...
mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field;
* Many other forms of spectroscopy, including infrared
Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from around ...
and nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
spectroscopies;
* Generation of sound spectrogram
A spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies with time.
When applied to an audio signal, spectrograms are sometimes called sonographs, voiceprints, or voicegrams. When the data are represen ...
s used to analyze sounds;
* Passive sonar
Sonar (sound navigation and ranging or sonic navigation and ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigate, measure distances (ranging), communicate with or detect objects on o ...
used to classify targets based on machinery noise.
Variants of Fourier analysis
(Continuous) Fourier transform
Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
argument, and it produces a continuous function of frequency, known as a ''frequency distribution''. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time (), and the domain of the output (final) function is ordinary frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, the transform of function at frequency is given by the complex number:
:
Evaluating this quantity for all values of produces the ''frequency-domain'' function. Then can be represented as a recombination of complex exponentials
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
of all possible frequencies:
:
which is the inverse transform formula. The complex number, , conveys both amplitude and phase of frequency .
See Fourier transform for much more information, including:
* conventions for amplitude normalization and frequency scaling/units
* transform properties
* tabulated transforms of specific functions
* an extension/generalization for functions of multiple dimensions, such as images.
Fourier series
The Fourier transform of a periodic function, , with period , becomes a Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
function, modulated by a sequence of complex coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
:
: (where is the integral over any interval of length ''P'').
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
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
functions, each with an amplitude and phase specified by one of the coefficients:
:
Any can be expressed as a periodic summation
In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
of another function, :
:
and the coefficients are proportional to samples of at discrete intervals of :
:
Note that any whose transform has the same discrete sample values can be used in the periodic summation. A sufficient condition for recovering (and therefore ) from just these samples (i.e. from the Fourier series) is that the non-zero portion of be confined to a known interval of duration , which is the frequency domain dual of the Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that per ...
.
See Fourier series for more information, including the historical development.
Discrete-time Fourier transform (DTFT)
The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent periodic summation
In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called p ...
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. Thus the DTFT of the sequence is also the Fourier transform of the modulated Dirac comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula
\operatorname_(t) \ := \sum_^ \delta(t - k T)
for some given period T. Here ''t'' is a real variable and th ...
function.
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
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of ...
. Thus we have the important result that when a discrete data sequence, , is proportional to samples of an underlying continuous function, , one can observe a periodic summation of the continuous Fourier transform, . Note that any with the same discrete sample values produces the same DTFT But under certain idealized conditions one can theoretically recover and exactly. A sufficient condition for perfect recovery is that the non-zero portion of be confined to a known frequency interval of width . When that interval is , the applicable reconstruction formula is the Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker i ...
. This is a cornerstone in the foundation of digital signal processing.
Another reason to be interested in is that it often provides insight into the amount of aliasing
In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when ...
caused by the sampling process.
Applications of the DTFT are not limited to sampled functions. See Discrete-time Fourier transform for more information on this and other topics, including:
* normalized frequency units
* windowing (finite-length sequences)
* transform properties
* tabulated transforms of specific functions
Discrete Fourier transform (DFT)
Similar to a Fourier series, the DTFT of a periodic sequence,