Transformée de Fourier has many Chinese translations, and the common ones are Fourier transform, Fourier transform, Fourier transform, Fourier transform, Fourier transform and so on. For convenience, this paper will write "Fourier transform" in a unified way.
App application
Fourier transform is widely used in physics, number theory, combinatorial mathematics, signal processing, probability theory, statistics, cryptography, acoustics, optics, oceanography, structural dynamics and other fields (for example, in signal processing, the typical use of Fourier transform is to decompose signals into amplitude components and frequency components).
brief introduction
* Fourier transform can express functions that meet certain conditions as trigonometric functions (sine and/or cosine functions) or linear combinations of their integrals. In different research fields, Fourier transform has many different variants, such as continuous Fourier transform and discrete Fourier transform. First, Fourier analysis was put forward as a tool for thermal process analysis (see: Applied Mathematics of Deterministic Problems in Natural Science by Lin Jiaqiao and Siegel, Science Press, Beijing). The original title is C.C. Lin&; L. A. Segel, Mathematical Application of Deterministic Problems in Natural Science, Macmillan Company, new york, 1974).
* Fourier transform belongs to harmonic analysis.
* The inverse transform of Fourier transform is easy to find, and the form is very similar to the forward transform;
* Sine basis function is the intrinsic function of differential operation, which transforms the solution of linear differential equation 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.
* convolution theorem points out that Fourier transform can transform complex convolution operation into simple product operation, thus providing a simple means to calculate convolution;
* Discrete Fourier transform can be calculated quickly by digital computer (its algorithm is called FFT).
Basic attribute
Linear property
The Fourier transform of the sum of two functions is equal to the sum of their respective transforms. The mathematical description is: if Fourier transforms \mathcal[f] and \mathcal[g] of functions f \left( x\right) and g \left(x \right) exist, and α and β are arbitrary constant coefficients, then \ mathcal [\ alphaf+\ betag] = \ alpha. Fourier Transform Operators \ Mathematics can be standardized into unitary operators;
Frequency shift characteristic
If the function f \left( x\right) has Fourier transform, the function f (x) e {i \ omega _ x} also has Fourier transform for any real number ω0, and \ mathcal [f (x) e {i \ omega _ x}] = f (\ omega+\ where \.
Differential relation
If the limit of function f \left( x\right) is 0 when |x|\rightarrow\infty, and the Fourier transform of its derivative function f'(x) exists, then \ mathcal [f' (x)] =-i \ omega \ mathcal [f (x)], that is. 6? 1 iω. More generally, if f (\ pm \ infty) = f' (\ pm \ infty) = \ l dots = f {(k-1)} (\ pm \ infty) = 0, and \ mathcal. 1 iω)k .
Convolution characteristic
If the functions f \left( x\right) and g \left( x\right) are absolutely integrable on (-\infty, +\infty), then the convolution function f * g = \ int _ {-\ infty} {+\ infty} f (x-\) convolves. Omega)] = \ mathcal [f (\ omega)] * \ mathcal [g (\ omega)], that is, the inverse Fourier transform of the product of two functions is
parseval's theorem
If the function f \left( x\right) is integrable and square is integrable, then \ int _ {-\ infty} {+\ infty} F2 (x) dx = \ frac {2 \ pi} \ int _ {-\ infty} where
Different variants of Fourier transform
continuous fourier transform
Main project: continuous Fourier transform
Generally speaking, if the word "Fourier transform" is not preceded by any qualifier, it means "continuous Fourier transform". "Continuous Fourier Transform" expresses the square integrable function f(t) as an integral or series form of a complex exponential function.
f(t)= \mathcal^[f(\omega)]= \ frac { \ sqrt { 2 \ pi } } \int\limits_{-\infty}^\infty f(\ω)e^{i\omega t } \，d \ω。
The above formula actually represents the inverse transformation of continuous Fourier transform, that is, the function f(t) in time domain is expressed as the integral of function F(ω) in frequency domain. On the contrary, its forward transformation happens to be the integral form of the function F(ω) in frequency domain as the function f(t) in time domain. General function f(t) can be called original function, function F(ω) is called image function of Fourier transform, and original function and image function form Fourier transform pair.
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 (? 6? 1ω) = F(ω)* holds.