Fourier transform

In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

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.^{[note 1]} 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.^{[note 2]}

The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional 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.^{[note 3]} Still further generalization is possible to functions on groups, which, besides the original Fourier transform on $R$ or $R n$ , 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 = $S 1$ , 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.

Real and imaginary parts of integrand for Fourier transform at 5 Hz

Magnitude of Fourier transform, with 3 and 5 Hz labeled.

The following figures provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The depicted function $f(t)=\cos(2\pi \ 3t)\ e^{-\pi t^{2}}$ oscillates at 3 Hz (if $t$ measures seconds) and tends quickly to 0. (The second factor in this equation is an envelope function that shapes the continuous sinusoid into a short pulse.). $f(t)$ was specially chosen to have a real Fourier transform that can be easily plotted. The first image is its graph. In order to calculate ${\widehat {f}}(3)$ we must integrate the product $f(t)e^{-i2\pi 3t}.$ The next 2 images are the real and imaginary parts of that product. The real part of the integrand has a non-negative average value, because the alternating signs of $f(t)$ and $\operatorname {Re} (e^{-i2\pi 3t})$ oscillate at the same rate and same phase, whereas $f(t)$ and $\operatorname {Im} (e^{-i2\pi 3t})$ are same rate but orthogonal phase. The result is that when you integrate the real part of the integrand you get a relatively large number (in this case ${\tfrac {1}{2}}$ ). Also, when you try to measure a frequency that is not present, as in the case when we look at ${\widehat {f}}(5),$ both real and imaginary component of the product vary rapidly between positive and negative values. Therefore, the integral is very small and the value for the Fourier transform for that frequency is nearly zero. The general situation is usually more complicated than this, but heuristically this is how the Fourier transform measures how much of an individual frequency is present in a function $f(t).$

To re-enforce an earlier point, the reason for the response at $\xi =-3$ Hz is because $\cos(2\pi 3t)$ and $\cos(2\pi (-3)t)$ are indistinguishable. The transform of $e^{i2\pi 3t}\cdot e^{-\pi t^{2}}$ would have just one response, whose amplitude is the integral of the smooth envelope: $e^{-\pi t^{2}},$ whereas $\operatorname {Re} (f(t)\cdot e^{-i2\pi 3t})$ (second graph above) is $e^{-\pi t^{2}}(1+\cos(2\pi 6t))/2.$

The transform of a real-valued function $(f_{_{RE}}+f_{_{RO}})$ is the function ${\hat {f}}_{RE}+i\ {\hat {f}}_{IO}.$ Conversely, a conjugate symmetric transform implies a real-valued time-domain.

conjugate symmetric

The transform of an imaginary-valued function $(i\ f_{_{IE}}+i\ f_{_{IO}})$ is the function ${\hat {f}}_{RO}+i\ {\hat {f}}_{IE},$ and the converse is true.

conjugate antisymmetric

The transform of an even-symmetric function $(f_{_{RE}}+i\ f_{_{IO}})$ is the real-valued function ${\hat {f}}_{RE}+{\hat {f}}_{RO},$ and the converse is true.

The transform of an odd-symmetric function $(f_{_{RO}}+i\ f_{_{IE}})$ is the imaginary-valued function $i\ {\hat {f}}_{IE}+i{\hat {f}}_{IO},$ and the converse is true.

Generalizations[edit]

Fourier–Stieltjes transform[edit]

The Fourier transform of a finite Borel measure $μ$ on $R n$ is given by:^[51] ${\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-i2\pi x\cdot \xi }\,d\mu .$

This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures.^[16] In the case that $dμ = f (x) dx$ , then the formula above reduces to the usual definition for the Fourier transform of $f$ . In the case that $μ$ is the probability distribution associated to a random variable $X$ , the Fourier–Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take $e iξx$ instead of $e - i 2π ξx$ .^[14] In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants.

The Fourier transform may be used to give a characterization of measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.^[16]

Furthermore, the Dirac delta function, although not a function, is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).

Alternatives[edit]

In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent.

As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform, fractional Fourier transform, Synchrosqueezing Fourier transform,^[54] or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.^[24]

Other notations[edit]

Other common notations for ${\hat {f}}(\xi )$ include: ${\tilde {f}}(\xi ),\ F(\xi ),\ {\mathcal {F}}\left(f\right)(\xi ),\ \left({\mathcal {F}}f\right)(\xi ),\ {\mathcal {F}}(f),\ {\mathcal {F}}\{f\},\ {\mathcal {F}}{\bigl (}f(t){\bigr )},\ {\mathcal {F}}{\bigl \{}f(t){\bigr \}}.$

In the sciences and engineering it is also common to make substitutions like these: $\xi \rightarrow f,\quad x\rightarrow t,\quad f\rightarrow x,\quad {\hat {f}}\rightarrow X.$

So the transform pair $f(x)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ {\hat {f}}(\xi )$ can become $x(t)\ {\stackrel {\mathcal {F}}{\Longleftrightarrow }}\ X(f)$

A disadvantage of the capital letter notation is when expressing a transform such as ${\widehat {f\cdot g}}$ or ${\widehat {f'}},$ which become the more awkward ${\mathcal {F}}\{f\cdot g\}$ and ${\mathcal {F}}\{f'\}.$

In some contexts such as particle physics, the same symbol $f$ may be used for both for a function as well as it Fourier transform, with the two only distinguished by their argument I.e. $f(k_{1}+k_{2})$ would refer to the Fourier transform because of the momentum argument, while $f(x_{0}+\pi {\vec {r}})$ would refer to the original function because of the positional argument. Although tildes may be used as in ${\tilde {f}}$ to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more Lorentz invariant form, such as ${\tilde {dk}}={\frac {dk}{(2\pi )^{3}2\omega }}$ , so care must be taken. Similarly, ${\hat {f}}$ often denotes the Hilbert transform of $f$ .

The interpretation of the complex function $f̂ (ξ)$ may be aided by expressing it in polar coordinate form ${\hat {f}}(\xi )=A(\xi )e^{i\varphi (\xi )}$ in terms of the two real functions $A (ξ)$ and $φ (ξ)$ where: $A(\xi )=\left|{\hat {f}}(\xi )\right|,$ is the amplitude and $\varphi (\xi )=\arg \left({\hat {f}}(\xi )\right),$ is the phase (see arg function).

Then the inverse transform can be written: $f(x)=\int _{-\infty }^{\infty }A(\xi )\ e^{i{\bigl (}2\pi \xi x+\varphi (\xi ){\bigr )}}\,d\xi ,$ which is a recombination of all the frequency components of $f (x)$ . Each component is a complex sinusoid of the form $e 2π ixξ$ whose amplitude is $A (ξ)$ and whose initial phase angle (at $x = 0$ ) is $φ (ξ)$ .

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted F and $F (f)$ is used to denote the Fourier transform of the function $f$ . This mapping is linear, which means that F can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function $f$ ) can be used to write $F f$ instead of $F (f)$ . Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value $ξ$ for its variable, and this is denoted either as $F f (ξ)$ or as $(F f)(ξ)$ . Notice that in the former case, it is implicitly understood that F is applied first to $f$ and then the resulting function is evaluated at $ξ$ , not the other way around.

In mathematics and various applied sciences, it is often necessary to distinguish between a function $f$ and the value of $f$ when its variable equals $x$ , denoted $f (x)$ . This means that a notation like $F (f (x))$ formally can be interpreted as the Fourier transform of the values of $f$ at $x$ . Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example, ${\mathcal {F}}{\bigl (}\operatorname {rect} (x){\bigr )}=\operatorname {sinc} (\xi )$ is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or ${\mathcal {F}}{\bigl (}f(x+x_{0}){\bigr )}={\mathcal {F}}{\bigl (}f(x){\bigr )}\,e^{i2\pi x_{0}\xi }$ is used to express the shift property of the Fourier transform.

Notice, that the last example is only correct under the assumption that the transformed function is a function of $x$ , not of $x 0$ .

As discussed above, the characteristic function of a random variable is the same as the Fourier–Stieltjes transform of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is defined $E\left(e^{it\cdot X}\right)=\int e^{it\cdot x}\,d\mu _{X}(x).$

As in the case of the "non-unitary angular frequency" convention above, the factor of 2 $π$ appears in neither the normalizing constant nor the exponent. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponent.