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The basic idea of Fourier transform was first put forward by the French scholar Fourier system, so it was named after it to commemorate it.
From the point of view of modern mathematics, Fourier transform is a special integral transform. It can represent a function satisfying certain conditions as a linear combination or integral of sine basis functions. In different research fields, Fourier transform has many different variants, such as continuous Fourier transform and discrete Fourier transform.
Fourier transform belongs to harmonic analysis. The word "analysis" can be interpreted as in-depth research. Literally, the word "analysis" is actually "piecemeal analysis". It realizes in-depth understanding and research of complex functions through "piecemeal analysis" of functions. Philosophically, "Analyticism" and "Reductionism" aim to improve the understanding of the essence of things through proper analysis. Modern atomism, for example, tries to analyze the origin of all substances in the world as atoms, but there are only a few hundred atoms. Compared with the infinite richness of the material world, this analysis and classification undoubtedly provide a good means to understand the various attributes of things.
In the field of mathematics, the same is true. Although Fourier analysis was originally used as an analytical tool of thermal process, its thinking method still has the characteristics of typical reductionism and analytical theory. Any function can be expressed as a linear combination of sine functions through a certain decomposition, and sine functions are relatively simple functions that have been fully studied in physics. How similar this idea is to that of atomism in chemistry! Strangely, modern mathematics has found that Fourier transform has very good properties, which makes it so easy to use and useful that people have to sigh the magic of creation:
1. Fourier transform is a linear operator, and it is also a unitary operator if a proper norm is given;
2. The inverse transform of Fourier transform is easy to find, and the form is very similar to the forward transform;
3. Sine basis function is the inherent function of differential operation, so that the solution of linear differential equation is transformed into the solution of algebraic equation with constant coefficient. In a linear time-invariant physical system, frequency is an invariable property, so the response of the system to complex excitation can be obtained by combining its responses to sinusoidal signals with different frequencies.
4. The famous convolution theorem points out that Fourier transform can transform complex convolution operation into simple product operation, thus providing a simple means to calculate convolution;
5. Discrete Fourier transform can be quickly calculated by digital computer (its algorithm is called FFT).
Because of the above good properties, Fourier transform is widely used in physics, number theory, combinatorial mathematics, signal processing, probability, statistics, cryptography, acoustics, optics and other fields.
Fourier transform is being translated. Welcome everyone to actively translate and modify.
Directory [hidden]
Different variants of 1 Fourier transform
1. 1 continuous Fourier transform
1.2 Fourier series
1.3 discrete Fourier transform
1.4 Other display methods
1.5 Fourier transform family
See also.
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Different variants of Fourier transform
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continuous fourier transform
Generally speaking, if the word "Fourier transform" is not preceded by any modifiers, it can be considered as "continuous Fourier transform". "Continuous Fourier Transform" expresses the square integrable function f(t) as an integral or series form of a complex exponential function.
In fact, the above formula represents the inverse transformation of continuous Fourier transform, that is, the function f(t) in time domain is expressed as the integral of the function F(ω) in frequency domain. On the contrary, its forward transformation is the integral form of function F(ω) in frequency domain. The general function f(t) can be called the original function, while the function F(ω) is called the mirror function of Fourier transform.
Please refer to the entry of continuous Fourier transform for further understanding. An extension of continuous Fourier transform is called fractional Fourier transform.
When f(t) is odd function (or even function), other chord (or sine) components will die out, and the transformation at this time can be called cosine transformation or sine transformation.
Another noteworthy property is that when f(t) is a purely real function, f (? ω) = F(ω)* holds.
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Fourier series
Fourier transform in continuous form is actually a generalization of Fourier series, because integral is actually a summation operator in limit form. For periodic functions, the Fourier series exists:
Fn is the complex amplitude. For real functions, the Fourier series of the function can be written as:
An and bn are that amplitude of real frequency components.
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dft
In order to use computers for Fourier transform in the fields of scientific calculation and digital signal processing, the function xk must be defined in the discrete domain instead of the continuous domain, and the continuous domain is also fine or periodic. In this case, the discrete Fourier transform (DFT) is used to express the function xk as the following summation form:
Fj is the amplitude of Fourier. Although the complexity of directly using this formula is O(n2), the complexity of the algorithm can be increased to O(n log n) by using the fast Fourier transform (FFT) algorithm, which makes Fourier a very practical and important processing method in the computer field.
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Other display methods
DFT is a special case of discrete-time Fourier transform (DTFT) (sometimes used as its approximation), in which xk is defined on a discrete but infinite domain, so the spectrum is continuous and periodic. DTFT is essentially the inverse of Fourier series.
These variations of Fourier transform can be more uniformly expressed as Fourier transform on any locally compact Abelian topological group, which belongs to the category of harmonic analysis. In harmonic analysis, transformation is from a group to its dual group. This processing also allows the general formula of convolution theorem, which involves Fourier transform and convolution. See Pontryagin duality to understand the generalized basis of Fourier transform.
Time-frequency analysis transforms, such as short-time Fourier transform, wavelet transform, LFM pulse transform and fractal Fourier transform, try to obtain the frequency information of signals, which is a function of time (or any other independent variable), although the ability to analyze frequency and time at the same time (mathematically) is limited by the uncertainty principle.
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Fourier transform family
The following table lists the members of the Fourier transform series. It is easy to find that the dispersion of a function in time (frequency) domain corresponds to the periodicity of its image function in time (frequency) domain. On the contrary, continuity means that the signal is aperiodic in the corresponding domain.
Transform time frequency
Continuous Fourier transform continuous, aperiodic continuous, aperiodic
Fourier series are continuous, periodic, discrete and aperiodic.
Discrete-time Fourier transform discrete, aperiodic continuous, periodic
Discrete Fourier Transform Discrete, Periodic Discrete, Periodic
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see
Orthogonal transformation
Fourier series
continuous fourier transform
Discrete time Fourier transform
Sketch (short for draft)
laplace transform
wavelet transformation
Pasval theorem
Fourier Transform is an unfinished small work related to mathematics. You are welcome to actively edit or modify and expand its content.
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