Fourier Transform

The Fourier transform is a type of mathematical function that splits a waveform, which is a time function, into the type of frequencies that it is made of. The result generated by the Fourier transform is always a complex-valued frequency function. The Fourier transform’s absolute value shows the frequency value existing in the original function. Its complicated argument denotes the phase offset of the fundamental sinusoidal in that particular frequency.

The Fourier transform is also known as a generalisation of the Fourier series. The term can also be applied to both the mathematical function used and the frequency domain representation. The Fourier transform enables the Fourier series to extend to non-periodic functions. This allows for taking any function as a total of simple sinusoids.

Fourier Transform Definition

A function’s Fourier transform is a complex-valued function denoting the complex sinusoids that contain the original function. For every frequency, the magnitude of the complex value denotes the constituent complex sinusoid’s amplitude with that frequency, and the complex value’s argument denotes the phase offset of the complex sinusoid. If a frequency does not exist, the transform possesses a value of zero for that particular frequency. The Fourier transform is not all restricted to time functions, but the original function’s domain is generally regarded as the time domain. The Fourier inversion theorem gives a synthesis method that revives the original function from the representation of its frequency domain.

Functions that are localised in the domain of time have Fourier transforms that extend out across the domain of frequency and vice versa, a phenomenon called the uncertainty principle.

The critical scenario for the principle is known as the Gaussian function, which is of substantial significance in statistics and probability theory, as well as the analysis of physical phenomena showing normal distribution. The Fourier transform of a pure Gaussian function is always another Gaussian function. The mathematician Joseph Fourier put forward the transform in his heat transfer analysis, where Gaussian functions are seen as the heat equation solutions.

The video explains the basic nature of heat

The Fourier transform can also be stated as an improper Riemann integral, which means its an

integral transform. However, this statement is not ideal for numerous applications that need a more complex integration theory. For instance, various relatively basic applications utilise the Dirac delta function, which in turn can be handled formally as if it were a function; however, the justification needs a mathematically more complex viewpoint.

The Fourier transform can be generalised to functions of various variables on Euclidean space, forwarding a function of three-dimensional position space to a three-dimensional momentum function (or a space and time function to a 4-momentum function). This concept makes the spatial Fourier transform very normal in the analysis of waves, as well as in quantum physics, where it is necessary to be able to denote solutions of waves as functions of either momentum or position (or both).

There are various common conventions for representing the Fourier transform of an integrable function

The transform of function f(x) at frequency ξ. Calculating the equation for every value ξ generates the function of the frequency domain. The Fourier transform is represented here by placing a circumflex to the function symbol. When the independent variable denotes time, the transform variable represents frequency.

The Fourier transform is denoted here by adding a circumflex to the symbol of the function.

Fourier Transform Properties

The important properties of Fourier transform are duality, linear transform, modulation property, and Parseval’s theorem.

Duality: It shows that if h(t) possesses a Fourier transform H(f), then the Fourier transform related to H(t) is H(-f).

Linear transform: Fourier transform comes under the category of linear transform. Let g(t) and h(t) be two Fourier transforms, which are represented by G(f) and H(f), respectively. In this scenario, it is easy to find the Fourier transform of the linear combination of h and g.

Modulation property: As per the modulation property, functions are modulated by the other functions if they are multiplied in time.

Parseval’s theorem: As per this concept, the Fourier transform is unitary. This implies that the sum total of the squares of its Fourier transform, H(f), equals the sum of squares of the function h(t).

Who Pioneered the Fourier Transforms?

In 1822, Joseph Fourier proposed (The Analytic Theory of Heat) that any arbitrary function, whether discontinuous or continuous, can be extended into a series of sines. This primary work was updated and extended by other scholars to provide the base for the numerous forms of the Fourier transform that has been used since.

Fourier Series

A Fourier series is a way of depicting a periodic function as a sum of cosine and sine functions. The frequency of every wave in the sum, or harmonic, is always an integer multiple of the fundamental frequency of the periodic function. The amplitude and phase of each harmonic can be found by using harmonic analysis. A Fourier series could potentially possess an infinite number of harmonics. Not every harmonic in the Fourier series of a function generates an approximation to the given function. For instance, in the case of a square, using the initial few harmonics to the Fourier series results in an approximation of the square wave. It is observed that in the frequency domain, in order to process images in the frequency domain, it is first required to convert them using the frequency domain, and it is also required to take the inverse of the output to transform it back into the spatial domain.

The difference between the Fourier transform and the Fourier series is that the Fourier transform is applicable for non-periodic signals, while the Fourier series is applicable to periodic signals.

Uncertainty Principle and Fourier Transform

Generally, uncertainty principles point to a meta-theorem in Fourier study that describes that a non-zero function and its Fourier transform cannot be localised to arbitrary precision. The localisation of a function points to the behaviour of a function’s “size” near and away from zero. Any mathematical expression which multiplies a function that shows localisation with its Fourier transform could be an uncertainty principle. The most familiar application of Fourier uncertainty principles is as an illustration of the natural tradeoff between the measurability and stability of a system, specifically quantum mechanical systems.

The adjustment between the compaction of certain functions and their Fourier transform can be formalised in the setup of an uncertainty principle by taking a function and its Fourier transform in the form of conjugate variables relative to the symplectic form on the time-frequency domain. The Fourier transform (from the point of the linear canonical transformation’s view) is a rotation by 90° in the realm of the time-frequency domain and maintains the symplectic form.

Applications

The linear operations are done in one domain (frequency or time) and have respective operations existing in the other domain, which is occasionally much easier to perform. The differentiation operation existing in the time domain matches multiplication by the frequency, so some particular differential equations are comparatively easier to analyse in the domain of frequency. The convolution in the domain of time corresponds to normal multiplication in the frequency domain. After doing the required operations, the result’s transformation can be traced back to the time domain. Here, harmonic analysis is the methodical study of the correlation between time domains and frequency, including the types of operations or functions that are quite simpler and possess strong connections to various fields of modern mathematics.

Differential Equations Analysis

One of the most significant applications of Fourier transformation is to find the solution to partial differential equations. Various types of equations of the analytical physics (mathematics) of the 19th century can be handled this way.

Fourier Transform Spectroscopy

In the case of spectroscopy, the Fourier transform is used in infrared (FTIR), nuclear magnetic resonance (NMR), etc. In nuclear magnetic resonance, an exponentially moulded free induction decay signal is attained in the domain of time, and Fourier transformed into a Lorentzian line-shape in the domain of frequency. Also, the Fourier transform is used for mass spectrometry and magnetic resonance imaging (MRI).

Quantum Mechanics

The Fourier transform is extremely useful in quantum physics and its applications in two varied ways. Primarily, the fundamental structure of quantum physics predicts the existence of complementary variable pairs, bridged by the Heisenberg uncertainty principle. This transform can be employed to pass from one manner of denoting the particle state by a position wave function, to another manner of denoting the particle state using a wave function of momentum. Countless different polarisations are possible, and each is equally valid. It is very useful to be able to transform various states from one form to another.

Signal Processing

The Fourier transform is also used for time-series spectral analysis. However, the statistical signal processing doesn’t generally use the Fourier transformation to the whole signal. Even if a physical signal is certainly transient, it has been observed in practice (advisable) to model a signal by a function (or stochastic method) which is stationary in the perception that its fundamental properties are fixed over all time. Such a function’s Fourier transform doesn’t usually exist, and this transform has been observed to be more useful for signal analysis instead of considering the Fourier transform of its autocorrelation function.