Fourier Transform

A Fourier Transform is a timeseries transform that decomposes a signal into its constituent frequencies.

• Context:
  • It can be mathematically defined as: For a function [math]\displaystyle{ f(t) }[/math] which is piecewise continuous and converges absolutely in the interval [math]\displaystyle{ (-\infty,\infty) }[/math], the fourier transform is [math]\displaystyle{ F[f(t)]=\int_{-\infty}^\infty f(t) e^{-iwt} dt }[/math].
  • It can range from being a Continuous Fourier Transform to being a Discrete Fourier Transform.
  • It is a Reversible Function.
  • …
• Example(s):
  • [math]\displaystyle{ F[e^{-at^2}]=\sqrt{\frac{\pi}{a}} e^{-\frac{w^2}{4a}}; }[/math] for [math]\displaystyle{ a\gt 0 }[/math].
  • …
• Counter-Example(s):
  • a Linear Transform.
  • a Short-Time Fourier Transform, such as: a Gabor Transform.
  • a Laplace Transform.
• See: Exponential Function, Transform Operation, Frequency Spectrum, Fourier Inversion Theorem, Phase (Waves), Analytic Signal, Periodic Function, Sine Wave, Fourier Series, Complex Exponential, Fast Fourier Transform, Integral Transform.

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References

2023

• (Mathworld, 2023) ⇒ http://mathworld.wolfram.com/FourierTransform.html Retrieved: 2023-5-14.
  • QUOTE: The Fourier transform is a generalization of the complex Fourier series in the limit as $L\to\infty$. Replace the discrete $A_n$ with the continuous $F(k)dk$ while letting $n/L\to k$. Then change the sum to an integral, and the equations become

    [math]\displaystyle{ f(x) =\displaystyle \int_{-\infty}^\infty F(k)e^{2\pi ikx}dk }[/math]

    [math]\displaystyle{ F(k)=\int_{-\infty}^\infty f(x)e^{-2\pi ikx}dx. }[/math]

    Here,

    [math]\displaystyle{ F(k) =F_x\left[f(x)\right](k)=\displaystyle\int_{-\infty}^\infty f(x)e^{-2\pi ikx}dx }[/math]

    is called the forward (-i) Fourier transform, and

    [math]\displaystyle{ f(x)=F_k^{-1}\left[F(k)\right](x)= \displaystyle \int_{-\infty}^\infty F(k)e^{2\pi ikx}dk }[/math]

    is called the inverse (+i) Fourier transform. The notation $F_x\left[f(x)\right](k)$ is introduced in Trott (2004, p. xxxiv), and $\hat{f}(k)$ and $\check{f}(x)$ are sometimes also used to denote the Fourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202).

2021

• (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Fourier_transform Retrieved:2021-3-16.
  • In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

    The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem. ... The Fourier transform decomposes a function into eigenfunctions for the group of translations. The imaginary part of ĝ (ω) is negated because a negative sign exponent has been used in the fourier transform, which is the default as derived from the fourier series, but the sign does not matter for a transform that is not going to be reversed.

    ... Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.

    Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

    The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 'position space' to a function of momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on ℝ or ℝⁿ (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = ℤ), the discrete Fourier transform (DFT, group = ℤ mod N) and the Fourier series or circular Fourier transform (group = S¹, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

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