Fourier transform

Fourier transform
Type	Integral transform
Key terms	inversion formula; convolution theorem; tempered distributions
Related	Fourier series; Laplace transform; Partial differential equations
Applications	PDEs; signal analysis; imaging
Domain	Mathematics; Signal processing
Examples	continuous FT; discrete-time FT; FFT
Wikidata	Q6520159

The Fourier transform is a mathematical operation that takes a function (often a signal in time or space) and represents it in terms of its frequency content. In essence, it breaks down a signal into a continuum of sinusoidal components of different frequencies. The result is a new function $F(\omega )$ (often of angular frequency $\omega$ or ordinary frequency $f$ ) that describes how much of each frequency is present in the original signal. Crucially, the Fourier transform is invertible: from $F(\omega )$ one can reconstruct the original signal. This duality between a time-domain (or space-domain) function and its frequency-domain representation is at the heart of Fourier analysis. Because of its broad applicability and elegant properties (for example, it turns convolution into multiplication), the Fourier transform is a cornerstone of mathematics, physics, and engineering, finding use in solving differential equations, analyzing signals and images, and much more.

Historical Background

The Fourier transform is named after the French mathematician Joseph Fourier (1768–1830), who studied heat flow in the early 19th century. Fourier’s key insight, in his work Théorie analytique de la chaleur (1822), was that a periodic function could be expressed as an infinite sum of sine and cosine waves (what we now call a Fourier series). He showed that this decomposition could solve the heat equation for a heated rod by analyzing how each sinusoidal component diffused. Fourier’s ideas were initially controversial but eventually gained acceptance because of their power in solving physical problems.

In the late 19th and early 20th centuries, mathematicians generalized Fourier’s idea of expanding periodic functions into a more general integral transform suitable for non-periodic functions. The resulting “Fourier integral” or Fourier transform allows us to analyze arbitrary (even non-repeating) signals by integrating them against complex exponentials $e^{-i\omega t}$ . This continuous transform was developed and refined by many mathematicians (including Dirichlet, Plancherel, and Lebesgue), who established the conditions under which the transform exists and is invertible. In the mid-20th century, the theory was further extended to distributions (generalized functions) by Laurent Schwartz, making the transform applicable even to objects like impulses or slowly growing functions.

On the computational side, a major milestone came in 1965 with the Cooley–Tukey Fast Fourier Transform (FFT) algorithm. This clever divide-and-conquer method computes the discrete Fourier transform of a sequence much faster than the obvious method. The FFT made it practical to apply Fourier analysis to large datasets and real-time signal processing, revolutionizing fields such as audio and image processing, communications, and many others.

Today, Fourier transforms are taught in mathematics and engineering classrooms around the world. They form the basis of harmonic analysis, and their ideas extend into many domains (for example, distributions, wavelets, and abstract harmonic analysis). The following sections define the Fourier transform precisely, explain its key properties (especially the convolution theorem), describe how it extends to distributions, and survey important applications in differential equations and signal processing.

Definition and Inversion

Informally, the Fourier transform of a function $f(t)$ describes its composition into different frequency components. More formally, for a suitable function $f(t)$ defined on the real line ( $-\infty <t<\infty$ ), its Fourier transform $F(\omega )$ is defined by the integral.

$F(\omega )\;=\;\int _{-\infty }^{\infty }f(t)\,e^{-i\omega t}\,dt,$

where $i$ is the imaginary unit and $\omega$ is the angular frequency. This integral (when it converges) produces, for each real $\omega$ , a complex value $F(\omega )$ . The real and imaginary parts of $F(\omega )$ encode the amplitude and phase of the sine-wave component of frequency $\omega$ in the original function $f(t)$ . If one instead uses ordinary frequency $f$ in cycles per second, a similar formula involves $e^{-2\pi ift}$ .

The transform $F(\omega )$ is often called the frequency-domain representation or spectrum of $f(t)$ . The pair of functions $f(t)$ (in the “time domain”) and $F(\omega )$ (in the “frequency domain”) are said to be a Fourier transform pair. Because $F(\omega )$ typically oscillates or decays as $\omega \to \pm \infty$ , the integral defining the inverse transform usually converges.

The Fourier transform is invertible, meaning we can recover $f(t)$ from $F(\omega )$ . Under mild conditions (for example, if $f$ is absolutely integrable or has certain regularity and decay), the inverse Fourier transform is given by.

$f(t)\;=\;{\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\,e^{i\omega t}\,d\omega .$

Combining both definitions: a function is reconstructed by superposing all its frequency components $e^{i\omega t}$ , each weighted by $F(\omega )/(2\pi )$ . In practice many authors use alternative normalizations (such as a factor $1/{\sqrt {2\pi }}$ in both forward and inverse transforms), but the exact constants do not change the fundamental nature of the transform—they just shift factors of $2\pi$ between the two formulas.

Linearity is a key property: if $f$ and $g$ are two functions and $a$ and $b$ are constants, then. ${\mathcal {F}}\{af+bg\}(\omega )=a\,{\mathcal {F}}\{f\}(\omega )+b\,{\mathcal {F}}\{g\}(\omega ),$ where ${\mathcal {F}}\{f\}$ denotes the Fourier transform of $f$ . Thus the transform of a linear combination of signals is the same linear combination of their transforms. This property follows directly from the linearity of the integral.

Another important idea is the time-frequency duality: properties in the time-domain often map to complementary properties in the frequency-domain. For example, if $f(t)$ is shifted in time (i.e.\ replaced by $f(t-a)$ for some $a$ ), then its Fourier transform acquires a phase factor $e^{-i\omega a}$ . Conversely, modulation in time (multiplying $f(t)$ by $e^{i\omega _{0}t}$ ) produces a shift in frequency. These relationships are fundamental in signal processing (one can delay a signal in time or modulate it into a carrier frequency, affecting the spectrum in predictable ways).

Another key property is differentiation: if $f'(t)$ is the derivative of $f(t)$ , then. ${\mathcal {F}}\{f'(t)\}=i\omega \,F(\omega ).$ In words, differentiating a signal in the time domain corresponds to multiplying its Fourier transform by $i\omega$ in the frequency domain. More generally, the $n$ -th derivative corresponds to multiplying by $(i\omega )^{n}$ . This is extremely useful in solving linear differential equations (see the section below on PDEs), because it turns differentiation (a complicated operation in time) into simple multiplication by a polynomial in $\omega$ . In physical terms, higher derivatives accentuate the high-frequency components (since $\omega$ grows with frequency).

Lastly, some simple examples can illustrate the Fourier transform. If $f(t)$ is a Gaussian function (a bell-shaped curve), its Fourier transform is also a (possibly scaled) Gaussian. This “self-similarity” of Gaussians under Fourier is a unique and useful property. Another example: a pure cosine wave $f(t)=\cos(\omega _{0}t)$ transforms into two impulses (delta spikes) at frequencies $\pm \omega _{0}$ . These examples show how a time-domain signal’s shape determines its frequency content: a single-tone sine wave corresponds to sharp lines in frequency, while a broad pulse corresponds to a broad, smooth spectrum.

Convolution Theorem

One of the most useful properties of the Fourier transform is the convolution theorem. Convolution is an operation on two functions defined (for functions $f$ and $g$ ) as. $(f*g)(t)\;=\;\int _{-\infty }^{\infty }f(\tau )\,g(t-\tau )\,d\tau .$ Intuitively, convolution “mixes” the shape of $f$ and $g$ by sliding one over the other, multiplying and integrating at each shift. Convolution arises naturally when dealing with linear time-invariant systems in signal processing (for example, applying an impulse response to an input signal).

The convolution theorem states that the Fourier transform converts convolution in the time domain into multiplication in the frequency domain. Specifically, if $F(\omega )$ and $G(\omega )$ are the Fourier transforms of $f(t)$ and $g(t)$ respectively, then. ${\mathcal {F}}\{f*g\}(\omega )=F(\omega )\cdot G(\omega ).$ In other words, convolving two signals and then transforming is the same as transforming each and then multiplying their spectra. This is extremely powerful because multiplication is often much simpler to deal with than convolution (which involves an integral). Moreover, the reverse is also true: multiplication in time corresponds to convolution in frequency. That is, ${\mathcal {F}}\{f(t)\,g(t)\}(\omega )=(F*G)(\omega ),$ (up to factors of $2\pi$ depending on convention).

Applications of the convolution theorem are widespread: for example, filtering a signal is often done by convolving the signal $f$ with a filter kernel $g$ . Using the convolution theorem, one can perform this by multiplying the spectrum $F(\omega )$ by the spectrum $G(\omega )$ of the filter, which is computationally efficient especially when using FFT algorithms. In imaging, blurring or sharpening an image is a convolution operation on the image with a small filter kernel; the Fourier approach explains why such operations affect certain frequencies. The convolution theorem also underpins many algorithms: for instance, an efficient way to compute a large convolution is to take FFTs of the two signals, multiply them pointwise, and inverse-transform back.

Another way to see convolution is to note that broad, slow-changing functions tend to convolve to still broad outputs, whereas convolution with a very peaky function (like an impulse) can reproduce or slightly shift the other function. The delta function $\delta (t)$ acts as an identity for convolution: $f*\delta =f$ . In fact, the Fourier transform of the delta is a constant (showing that an impulse contains all frequencies equally), which is the Fourier counterpart of the fact that multiplying by a constant (in frequency) connotes convolving with a delta (in time).

In summary, the convolution theorem is:

Convolution to Multiplication: ${\mathcal {F}}\{f*g\}=F(\omega )\,G(\omega )$ .
Multiplication to Convolution: ${\mathcal {F}}\{f\,g\}=F*G$ .

This dual property is key to many signal-processing techniques and to solving integral equations in engineering and physics.

Fourier Transform and Distributions

The basic definition of the Fourier transform (as an integral) requires the function $f(t)$ to be well-behaved enough that the integral converges. For many practical signals this is true, but in both theory and applications we often encounter “functions” that are not ordinary. For example, the Dirac delta distribution $\delta (t)$ (an idealized unit impulse) is not a function in the usual sense (it is “infinite” at $t=0$ and zero elsewhere) and does not have an ordinary integral. Nevertheless, we still want to understand its frequency content. To handle such cases, mathematicians developed the theory of distributions or generalized functions.

A distribution can be thought of as a rule that assigns a number to each test function (a smooth, rapidly decaying function). Distributions include objects like the delta “function”, step functions, and even things that grow like polynomials, as long as they can be paired with test functions in a linear continuous way. Crucially, the Fourier transform extends naturally to distributions: if $f$ is a distribution and $\varphi$ is a test function, one defines ${\mathcal {F}}\{f\}$ by how it acts on the Fourier transform of $\varphi$ . The result is another distribution.

Some examples help illustrate why this is useful:

The Fourier transform of the Dirac delta $\delta (t)$ is a constant function (in frequency). Formally, one often writes ${\mathcal {F}}\{\delta (t)\}(\omega )=1$ . In words, an impulse in time contains all frequencies equally. This matches the intuition: since $\delta$ is “all at one point” in time, its spectrum is completely flat.

Conversely, a constant function in time (say $f(t)=1$ ) has all its energy concentrated at zero frequency. Indeed, its Fourier transform is $2\pi \delta (\omega )$ . This means $f(t)=1$ oscillates only at frequency $\omega =0$ (no oscillation). Here the transform of a non-decaying function is a delta function in frequency, which is again a distribution.

The Heaviside step function $H(t)$ (which is 0 for $t<0$ and 1 for $t>0$ ) is another example. Its Fourier transform (in distributional sense) involves a delta at zero frequency plus a principal-value term $1/(i\omega )$ . This shows that even discontinuous or non-decaying signals have well-defined Fourier behavior in the sense of distributions.

More generally, one often works with tempered distributions, which are distributions that grow at most polynomially at infinity. The Fourier transform is especially well-behaved on tempered distributions. For instance, any reasonable polynomially bounded function (like $f(t)=e^{at}$ , which does not have an ordinary transform unless interpreted damped or shifted) can be manipulated in terms of distributions.

In practical terms, the distribution perspective ensures that the Fourier transform can handle impulses, step signals, and other idealized objects used in engineering. For example, in circuit theory or mechanical vibrations, impulses are natural concepts. The ability to assign a Fourier transform to them (often as a constant or a delta) gives meaning to spectral analysis in these cases. In physics, the momentum representation of a particle’s wavefunction is essentially a Fourier transform of its position wavefunction; delta-like states in one domain translate to flat or delta-like states in the other.

Key takeaway: By extending the Fourier transform to distributions, one allows it to apply to a much wider class of “signals,” including impulses and signals that do not vanish at infinity. This generality underlies much of modern analysis and physics, where idealizations like point sources (deltas) are common.

Applications in Partial Differential Equations (PDEs)

The Fourier transform is a fundamental tool for solving linear partial differential equations, especially with constant coefficients. The key idea is that a spatial or temporal derivative becomes multiplication by $(i\omega )$ in the Fourier domain. This turns differential operators into algebraic ones, often greatly simplifying the problem.

A classic example is the heat equation on the real line: ${\frac {\partial u}{\partial t}}(x,t)=D{\frac {\partial ^{2}u}{\partial x^{2}}}(x,t),\quad -\infty <x<\infty ,\;t>0,$ where $D$ is the diffusion constant. If the initial temperature distribution is $u(x,0)=f(x)$ , one wants $u(x,t)$ for $t>0$ . Taking the Fourier transform in the spatial variable $x$ (but not in time), we let. ${\hat {u}}(\omega ,t)=\int _{-\infty }^{\infty }u(x,t)\,e^{-i\omega x}\,dx,$ and similarly $F(\omega )={\hat {u}}(\omega ,0)$ is the transform of the initial data $f(x)$ . Since differentiation in $x$ becomes multiplication by $i\omega$ , the heat equation in the frequency domain becomes an ordinary differential equation in time: ${\frac {\partial {\hat {u}}}{\partial t}}(\omega ,t)=-D\,\omega ^{2}\,{\hat {u}}(\omega ,t).$ This is now a simple first-order ODE in $t$ . Its solution is. ${\hat {u}}(\omega ,t)=F(\omega )\,e^{-D\omega ^{2}t}.$ Thus in the Fourier domain, each frequency component decays like $e^{-D\omega ^{2}t}$ . Plugging back, the inverse transform gives the solution in physical space: $u(x,t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }F(\omega )\,e^{-D\omega ^{2}t}\,e^{i\omega x}\,d\omega .$ In fact, one can recognize this as a convolution of the initial data with a Gaussian kernel: $u(x,t)=\int _{-\infty }^{\infty }f(y)\,{\frac {1}{\sqrt {4\pi Dt}}}e^{-(x-y)^{2}/(4Dt)}\,dy.$ This Gaussian is the fundamental solution of the heat equation. The Fourier method gave this result succinctly, illustrating that heat diffusion blurs out the initial data with a Gaussian whose width grows over time.

Similarly, the wave equation can be attacked. For example, the one-dimensional wave equation. $\partial ^{2}u/\partial t^{2}=c^{2}\partial ^{2}u/\partial x^{2}$ transforms to $\partial ^{2}{\hat {u}}/\partial t^{2}=-c^{2}\omega ^{2}{\hat {u}}$ , which is an ordinary harmonic oscillator equation for each $\omega$ . Its solutions are combinations of $\cos(c|\omega |t)$ and $\sin(c|\omega |t)$ , showing that each frequency component oscillates at its own rate. Inverting back yields d’Alembert’s solution or the Kirchhoff formula for the wave.

One can apply Fourier transforms in multiple spatial dimensions as well. For example, solving the heat or wave equation in two or three dimensions, or solving Poisson’s equation $\nabla ^{2}u=f$ , can be handled by taking the multidimensional Fourier transform in all spatial directions. Then $\nabla ^{2}$ becomes $-|\mathbf {k} |^{2}$ (with $\mathbf {k}$ being the frequency vector), turning the PDE into an algebraic rational function equation in $\mathbf {k}$ .

The Fourier approach is particularly useful for problems on infinite or periodic domains. On a finite interval or with specific boundary conditions, one often uses Fourier series (discrete sums) rather than the transform (integral). But conceptually, Fourier series are a special case of the Fourier transform for periodic signals.

In summary, the Fourier transform turns differentiation into multiplication, converting many linear PDEs into simpler equations. This method is often taught as the method of Fourier transforms or Fourier analysis for PDEs. It is widely used not only in mathematical theory but also in engineering—for instance, in heat transfer, acoustics, electromagnetics, and quantum mechanics (where the Schrödinger equation is a PDE whose Fourier treatment gives the momentum-space wavefunction).

Applications in Signal Processing

In signal processing, the Fourier transform is used to analyze and manipulate signals in the frequency domain. A signal here means any time-varying or spatially varying function, such as an audio waveform, an image, a radio transmission, or a time series of data. The basic idea is that many tasks – filtering, compression, or analysis – are more natural when viewed in terms of frequency.

For a continuous-time signal $s(t)$ , the Fourier transform $S(\omega )$ tells us which frequencies are present and with what amplitude and phase. For example, in audio engineering, a musical note is typically a combination of a fundamental frequency and harmonics; its Fourier spectrum $S(\omega )$ shows spikes at those frequencies. Sound experts can manipulate these frequencies: boosting bass (low frequencies) or treble (high frequencies) by applying filters that modify $S(\omega )$ . In practice, one might take the Fourier transform, multiply $S(\omega )$ by a human-chosen filter function $H(\omega )$ , and then inverse transform to get the filtered time signal. Without the Fourier viewpoint, designing such filters would be much harder.

Another common task is noise reduction. If unwanted noise occupies a certain frequency band, one can transform the signal, attenuate (reduce) the noise-band frequencies in $S(\omega )$ , and then inverse-transform to get a cleaner signal. Equivalently, this is convolution with a smoothing kernel in time, but doing it in the frequency domain is often more intuitive and computationally efficient.

In electrical engineering, the Fourier transform is essential for understanding how circuits respond to signals. A linear circuit’s behavior with any input can be determined by knowing its response to sinusoidal inputs (this is the idea of frequency response). Fourier analysis justifies this: since any input can be treated as a sum of sinusoids, and the circuit’s response to each sinusoid is easily computed (by multiplying by the circuit’s transfer function at that frequency), one can rebuild the full output. Radio engineers, for example, use this to design antennas and filters that pick out desired frequency bands (channels) and reject others.

Modern digital signal processing often uses the Discrete Fourier Transform (DFT). Real-world signals are sampled and quantized, giving a sequence of numbers. The DFT treats a finite sequence as one period of an underlying periodic signal and computes its spectrum at a discrete set of frequency points. The Fast Fourier Transform (FFT) algorithm allows this to be done extremely quickly, in time roughly proportional to $N\log N$ for an $N$ -point sequence. This real-time analysis and processing is the foundation of many technologies: MP3 audio compression uses a modified discrete cosine transform (closely related to Fourier), JPEG image compression uses a block-wise DCT, and modern wireless standards use orthogonal frequency-division multiplexing (OFDM) which splits a channel into many narrowband Fourier sub-channels.

Spectral analysis is another major application. Scientists and engineers often look at the power spectrum $|F(\omega )|^{2}$ of a signal to detect periodicities, resonances, or hidden structures. For example, in seismology, the frequency content of vibrations can indicate underground structures. In medical imaging (MRI, for example), data is collected in a frequency-like domain (k-space) and transformed to produce an image.

The two-dimensional Fourier transform is used in image processing. An image can be seen as a function $f(x,y)$ on the plane. Its 2D Fourier transform $F(k_{x},k_{y})$ tells us about spatial frequencies: patterns, edges, and textures of the image. Filters in the frequency plane can sharpen or blur images, detect edges, or remove periodic noise. Many lenses perform an optical Fourier transform: a lens can take the light field of an object and focus it so that the resulting intensity pattern on a screen is (up to a scaling) the magnitude squared of the Fourier transform of the object’s transparency. This is the principle behind optical image processing and Fourier spectroscopy.

In communications, the Fourier transform describes bandwidth. A signal of limited bandwidth means $F(\omega )$ is essentially zero outside some range. The celebrated Shannon sampling theorem shows that a bandlimited signal can be perfectly reconstructed from discrete samples, with the Fourier transform playing a central role in the proof. In practice, when we convert audio to digital, we rely on Fourier ideas to avoid aliasing (which occurs when a higher frequency signal looks like a lower frequency in sampled data if sampling isn’t fast enough).

Electrical power systems, acoustics, optics, and many other fields rely on Fourier analysis. Any time one needs to analyze vibrations, waves, or cycles, the Fourier transform is a likely tool. In data analysis, Fourier techniques can reveal hidden periodic components in time series (climate data, stock prices, etc.), and they underpin methods like Fourier-based trend filtering.

Key points in signal processing:.

Filtering: Multiplying the spectrum by a design function to enhance or suppress frequencies.
Convolution (in time): Represented as filtering in frequency via multiplication (and vice versa).
Modulation: Signals can be moved to different frequency bands by modulation, explained easily by Fourier (modulation in time is shift in frequency).
Feature extraction: Identifying peaks in the frequency domain corresponds to dominant oscillatory patterns.
Compression: Many compression schemes work by transforming to frequency domain and discarding small high-frequency components (which often correspond to details imperceptible to humans).

Overall, the Fourier transform is a universal language of signals. By switching to the frequency domain, engineers and scientists can use powerful linear-algebraic techniques to solve problems that are hard in the original domain.

Computational Aspects: Fast Fourier Transform

In practical applications, the Fourier transform often needs to be computed numerically. For data on a computer, this means using a discrete version of the transform. The Discrete Fourier Transform (DFT) applies to a sequence of numbers $x_{0},x_{1},\ldots ,x_{N-1}$ . Its formula is. $X_{k}=\sum _{n=0}^{N-1}x_{n}\,\exp {\bigl (}-2\pi i\,kn/N{\bigr )},\quad k=0,1,\dots ,N-1.$ The DFT $X_{k}$ gives the frequency content of the sequence, assuming periodicity. The naive computation of the DFT requires $O(N^{2})$ operations, which is slow for large $N$ .

The breakthrough came with the Fast Fourier Transform (FFT), an efficient algorithm to compute the DFT in $O(N\log N)$ operations. First popularized by Cooley and Tukey in 1965, the FFT exploits symmetries in the DFT formula to break it into smaller transforms recursively. Today, the FFT is a standard library function in almost every computing environment. Because of the FFT, tasks like real-time audio equalization, digital radar processing, and medical imaging (MRI, CT) are computationally feasible.

For continuous Fourier transforms (as discussed earlier), one often samples the continuous signal at a fine grid and uses the FFT to approximate the transform. Care must be taken about sampling rate and windowing (avoiding artifacts like aliasing or spectral leakage), but the core idea is the same: digital computation allows harnessing the Fourier transform even for analog signals.

The success of the FFT means that even though the continuous Fourier transform is defined by an integral, in practice we almost always compute it with discrete sums (DFT/FFT) for large datasets. Nevertheless, understanding the continuous theory guides how we sample and interpret the discrete computations.

Significance and Extensions

The Fourier transform’s significance in applied mathematics and engineering cannot be overstated. It provides a lens—the frequency domain—through which many problems become simpler. Because of its linearity and well-understood behavior, it forms the basis of spectral methods in numerical solutions of PDEs, Fourier optics in physics, and harmonic analysis in pure math.

Over time, many extensions and related ideas have grown up around the Fourier transform. These include windowed Fourier transforms and wavelet transforms (which address the analysis of nonstationary signals whose frequency content changes over time), Fourier series (for periodic signals), discrete transforms (for computational use), and multi-dimensional transforms (for images and volumes). Concepts of duality, orthonormal bases, and distributions in Fourier analysis also influenced quantum mechanics (where the position-momentum duality is mathematically a Fourier pair) and number theory (e.g. Poisson summation formula).

While the Fourier transform itself is a classical idea, research continues in areas like compressive sensing (which uses Fourier measurements sparingly), the study of Fourier transforms on groups and manifolds in pure mathematics, and applications in data science. Moreover, practical refinements such as sparse FFT algorithms for signals with special structure, or Fourier-based machine learning kernels, represent ongoing development.