FourierTransform

FourierTransform[expr,t,ω]

给出 expr 的符号傅里叶变换.

FourierTransform[expr,{t1,t2,…},{ω1,ω2,…}]

给出 expr 的多维傅里叶变换.

更多信息和选项

  • The Fourier transform and its inverse are a way to transform between the time domain and the frequency domain.
  • Fourier transforms are typically used to reduce ordinary and partial differential equations to algebraic or ordinary differential equations, respectively. They are also used extensively in control theory and signal processing. Finally, they have applications in studying quantum mechanical phenomena, noise filtering, etc.
  • The Fourier transform of the time domain function is the frequency domain function :
  • 在缺省情况下，函数 的傅里叶变换的定义为 .
  • The multidimensional Fourier transform of a function is by default defined to be or when using vector notation .
  • 不同的定义选择可以使用选项 FourierParameters 指定.
  • The integral is computed using numerical methods if the third argument, , is given a numerical value.
  • The asymptotic Fourier transform can be computed using Asymptotic.
  • There are several related Fourier transformations:
  • FourierTransforminfinite continuous-time functions (FT)
    FourierSequenceTransforminfinite discrete-time functions (DTFT)
    FourierCoefficientfinite continuous-time functions (FS)
    Fourierfinite discrete-time functions (DFT)
  • The Fourier transform is an automorphism in the Schwartz vector space of functions whose derivatives are rapidly decreasing and thus induces an automorphism in its dual: the space of tempered distributions. These include absolutely integrable functions, well-behaved functions of polynomial growth and compactly supported distributions.
  • Hence, FourierTransform not only works with absolutely integrable functions, but it can also handle a variety of tempered distributions such as DiracDelta to enlarge the pool of functions or generalized functions it can effectively transform.
  • 可以给出下列选项：
  • AccuracyGoal Automaticdigits of absolute accuracy sought
    Assumptions $Assumptions所做的参数假定
    FourierParameters {0,1}定义的傅里叶转换的参数
    GenerateConditions False是否产生关于参数条件的结果
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoal Automaticdigits of precision sought
    WorkingPrecision Automaticthe precision used in internal computations
  • Common settings for FourierParameters include:
  • {0,1}default setting/physics
    {1,-1}systems engineering/mathematics
    {-1,1}classical physics
    {0,-2Pi}ordinary frequency
    {a,b}general setting
  • 在 TraditionalForm 中，FourierTransform 用 ℱ 输出. »

范例

打开所有单元关闭所有单元

基本范例  (6)

Compute the Fourier transform of a function:

Plot the function and its Fourier transform:

Fourier transform of :

For the systems engineering convention, change the parameters:

The Fourier transform of a Gaussian is another Gaussian:

Plot both Gaussians:

Compute the Fourier transform of a multivariate function:

Plot the result:

Compute the transform at a single point:

范围  (44)

Basic Uses  (4)

Fourier transform of a function for a symbolic parameter :

Fourier transforms of trigonometric functions:

Evaluate the Fourier transform for a numerical value of the parameter ω:

TraditionalForm formatting:

Elementary Functions  (8)

Fourier transform of a power function:

Polynomial:

Fourier transform of rational functions:

Plot the transform:

Plot the real and imaginary parts:

Reciprocal of square root:

Plot the transform:

Expressions involving trigonometric functions:

Ratio of sine and linear function:

Plot the transform:

Composition of elementary functions:

Plot the transform:

Logarithmic function:

Plot the transform:

Special Functions  (5)

Sinc function:

Plot the transform:

Expressions involving Bessel functions:

Plot the transform:

SinIntegral function:

Laguerre polynomial:

Airy function:

Plot the magnitude of the Fourier transform for complex :

Piecewise Functions and Distributions  (7)

Fourier transform of a piecewise function:

Absolute value using Sign function:

Restriction of a sine function to a half-period:

Triangular function:

Ramp:

UnitStep:

Product of UnitStep and cosine functions:

Plot the magnitude and phase:

Periodic Functions  (5)

Fourier transform of SquareWave:

TriangleWave:

SawtoothWave:

Full-wave-rectified function with period :

Rectified wave:

Generalized Functions  (5)

Fourier transform involving HeavisideTheta:

Plot the magnitude and phase:

DiracDelta:

Derivative of DiracDelta:

HeavisideLambda:

HeavisidePi:

Multivariate Functions  (5)

Bivariate Fourier transform of a constant:

Exponential function:

Trivariate cosine:

Product of power and exponential:

Fourier transform of a product of exponential and SquareWave functions:

Formal Properties  (3)

Fourier transform of a first-order derivative:

Fourier transform of a second-order derivative:

Fourier transform threads itself over equations:

Numerical Evaluation  (2)

Calculate the Fourier transform at a single point:

Alternatively, calculate the Fourier transform symbolically:

Then evaluate it for the specific value of :

选项  (7)

AccuracyGoal  (1)

The option AccuracyGoal sets the number of digits of accuracy:

With default settings:

Assumptions  (1)

Specify the range of a variable using Assumptions:

FourierParameters  (2)

Fourier transform for the unit box function and different parameters:

Set up your particular global choice of parameters to work with ordinary frequency once per session:

Restore defaults:

GenerateConditions  (1)

Use GenerateConditionsTrue to get parameter conditions for when a result is valid:

PrecisionGoal  (1)

The option PrecisionGoal sets the relative tolerance in the integration:

With default settings:

WorkingPrecision  (1)

If WorkingPrecision is specified, the computation is done at that working precision:

With default settings:

应用  (11)

Signals and Systems  (3)

Find the convolution of signals:

The product of their Fourier transforms:

Find the inverse transform:

Compare with Convolve:

Spectrum of the product of two signals, with one given in the frequency domain by:

The Fourier transform of the signal :

The Fourier transform of the product of with the original signal is the convolution of its transforms:

Its spectrum:

Frequency response of an LTI system defined by an ODE:

Apply the Fourier transform over the equation:

Solve for the Fourier transform of :

The frequency response of the LTI system is the ratio of the Fourier transforms of the output function over the input function :

Ordinary Differential Equations  (1)

Solve a differential equation using Fourier transforms:

Apply the Fourier transform over the equation:

Solve for the Fourier transform:

Find the inverse transform to get the solution:

Compare with DSolveValue:

Partial Differential Equations  (1)

Consider the heat equation: with initial condition :

Fourier transform with respect to :

With and , solve this ODE:

Compute the inverse Fourier transform:

And convolution to get the solution:

Consider the special case with initial condition and :

Compare with DSolveValue:

Plot the initial conditions and solutions for different values of :

Plot the solution over the - plane.

Evaluation of Integrals  (1)

Calculate the following definite integral:

Compute the Fourier transform with respect to and interchange the order of transform and integration:

Integrate over :

Use the inverse Fourier transform to get the result:

Compare with Integrate:

Other Applications  (5)

阻尼正弦曲线的幂频谱：

平面中径向对称函数的傅立叶变换可以表示为汉克尔变换. 验证由下面定义的函数的这种关系：

绘制函数：

计算它的傅立叶变换：

用 HankelTransform 得到同样的结果：

绘制傅立叶变换：

生成一组径向对称函数的傅立叶变换图集：

计算这些函数的汉克尔变换：

生成要求的傅立叶变换图集：

计算平稳 OrnsteinUhlenbeckProcess 的功率谱：

A quick look at the Heisenberg uncertainty principle:

Consider a fixed-area box function as the position space wavefunction of a particle. Its Fourier transform gives the momentum space wavefunction of the particle:

When is small, the height of the fixed-area box is big, and the position of the particle is almost guaranteed. The momentum space wavefunction is approximately for values between its two roots closest to zero, which makes it almost impossible to find its momentum. Similarly, vice versa, as seen here:

属性和关系  (6)

By default, the Fourier transform of is:

For , the definite integral becomes:

Compare with FourierTransform:

用 Asymptotic 计算渐近逼近：

FourierTransform 和 InverseFourierTransform 是互逆的：

对奇函数，FourierTransform 和 FourierCosTransform 是相等的：

对偶函数，FourierTransform 和 FourierSinTransform 的差异为 ：

可能存在的问题  (1)

一个逆傅里叶变换的结果可能和初始不相同：

巧妙范例  (2)

加权的 Hermite 多项式的傅里叶变换有非常简单的形式：

Create a table of basic Fourier transforms:

Wolfram Research (1999)，FourierTransform，Wolfram 语言函数，https://reference.wolfram.com/language/ref/FourierTransform.html （更新于 2025 年）.

文本

Wolfram Research (1999)，FourierTransform，Wolfram 语言函数，https://reference.wolfram.com/language/ref/FourierTransform.html （更新于 2025 年）.

CMS

Wolfram 语言. 1999. "FourierTransform." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2025. https://reference.wolfram.com/language/ref/FourierTransform.html.

APA

Wolfram 语言. (1999). FourierTransform. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/FourierTransform.html 年

BibTeX

@misc{reference.wolfram_2025_fouriertransform, author="Wolfram Research", title="{FourierTransform}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/FourierTransform.html}", note=[Accessed: 10-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fouriertransform, organization={Wolfram Research}, title={FourierTransform}, year={2025}, url={https://reference.wolfram.com/language/ref/FourierTransform.html}, note=[Accessed: 10-March-2025 ]}