Fourier transform is a transformation that compares the functions of some real variable. This operation is performed every time we perceive different sounds. The ear performs an automatic "calculation" that our consciousness can only perform after studying the appropriate section of higher mathematics. The human hearing organ builds a transformation, as a result of which sound (oscillatory motion of conditional particles in an elastic medium that propagate in a wave form in a solid, liquid or gaseous medium) is provided in the form of a spectrum of successive values of the volume level of tones of different heights. After that, the brain turns this information into a sound familiar to everyone.
Mathematical Fourier Transform
Transformation of sound waves or other oscillatory processes (from light radiation and ocean tide to cycles of stellar or solar activity) can also be carried out using mathematical methods. So, using these techniques, it is possible to decompose functions by representing oscillatory processes as a set of sinusoidal components, that is, wavy curves thatgo from low to high, then back to low, like a sea wave. Fourier transform - a transformation whose function describes the phase or amplitude of each sinusoid corresponding to a certain frequency. The phase is the starting point of the curve, and the amplitude is its height.
The Fourier transform (examples are shown in the photo) is a very powerful tool that is used in various fields of science. In some cases, it is used as a means of solving rather complex equations that describe dynamic processes that occur under the influence of light, thermal or electrical energy. In other cases, it allows you to determine the regular components in complex oscillatory signals, thanks to which you can correctly interpret various experimental observations in chemistry, medicine and astronomy.
Historical background
The first person to apply this method was the French mathematician Jean Baptiste Fourier. The transformation, later named after him, was originally used to describe the mechanism of heat conduction. Fourier spent his entire adult life studying the properties of heat. He made a huge contribution to the mathematical theory of determining the roots of algebraic equations. Fourier was a professor of analysis at the Polytechnic School, secretary of the Institute of Egyptology, was in the imperial service, where he distinguished himself during the construction of the road to Turin (under his leadership, more than 80 thousand square kilometers of malarialswamps). However, all this vigorous activity did not prevent the scientist from doing mathematical analysis. In 1802, he derived an equation that describes the propagation of heat in solids. In 1807, the scientist discovered a method for solving this equation, which was called the "Fourier transform".
Thermal Conductivity Analysis
The scientist applied a mathematical method to describe the mechanism of heat conduction. A convenient example, in which there are no difficulties in calculation, is the propagation of thermal energy through an iron ring, one part immersed in fire. To carry out experiments, Fourier heated a part of this ring red-hot and buried it in fine sand. After that, he took temperature measurements on the opposite side of it. Initially, the distribution of heat is irregular: part of the ring is cold and the other is hot, and a sharp temperature gradient can be observed between these zones. However, in the process of heat propagation over the entire surface of the metal, it becomes more uniform. So, soon this process takes the form of a sinusoid. At first, the graph smoothly increases and also decreases smoothly, exactly according to the laws of change of the cosine or sine function. The wave gradually levels off and as a result the temperature becomes the same on the entire surface of the ring.
The author of this method suggested that the initial irregular distribution can be decomposed into a number of elementary sinusoids. Each of them will have its own phase (initial position) and its own temperaturemaximum. In addition, each such component changes from a minimum to a maximum and back on a complete revolution around the ring an integer number of times. A component with one period was called the fundamental harmonic, and a value with two or more periods was called the second, and so on. So, the mathematical function that describes the temperature maximum, phase or position is called the Fourier transform of the distribution function. The scientist reduced a single component, which is difficult to describe mathematically, to an easy-to-use tool - the cosine and sine series, which sum up to give the original distribution.
The essence of the analysis
Applying this analysis to the transformation of the propagation of heat through a solid object that has an annular shape, the mathematician reasoned that increasing the periods of the sinusoidal component would lead to its rapid decay. This is clearly seen in the fundamental and second harmonics. In the latter, the temperature reaches the maximum and minimum values twice in one pass, and in the former, only once. It turns out that the distance covered by heat in the second harmonic will be half that in the main one. In addition, the gradient in the second one will also be twice as steep as in the first one. Therefore, since the more intense heat flow travels a distance twice as short, this harmonic will decay four times faster than the fundamental as a function of time. In the future, this process will be even faster. The mathematician believed that this method allows you to calculate the process of the initial distribution of temperature over time.
Challenge to contemporaries
The Fourier transform algorithm challenged the theoretical foundations of mathematics at the time. At the beginning of the nineteenth century, most prominent scientists, including Lagrange, Laplace, Poisson, Legendre and Biot, did not accept his statement that the initial temperature distribution is decomposed into components in the form of a fundamental harmonic and higher frequencies. However, the Academy of Sciences could not ignore the results obtained by the mathematician, and awarded him a prize for the theory of the laws of heat conduction, as well as comparing it with physical experiments. In Fourier's approach, the main objection was the fact that the discontinuous function is represented by the sum of several sinusoidal functions that are continuous. After all, they describe torn straight and curved lines. The contemporaries of the scientist never encountered a similar situation, when discontinuous functions were described by a combination of continuous ones, such as quadratic, linear, sinusoid or exponential. In the event that the mathematician was right in his statements, then the sum of an infinite series of a trigonometric function should be reduced to an exact stepwise one. At the time, such a statement seemed absurd. However, despite doubts, some researchers (eg Claude Navier, Sophie Germain) have expanded the scope of research and taken them beyond the analysis of the distribution of thermal energy. Meanwhile, mathematicians continued to struggle with the question of whether the sum of several sinusoidal functions can be reduced to an exact representation of a discontinuous one.
200 year oldhistory
This theory has evolved over two centuries, today it has finally formed. With its help, spatial or temporal functions are divided into sinusoidal components, which have their own frequency, phase and amplitude. This transformation is obtained by two different mathematical methods. The first of them is used when the original function is continuous, and the second - when it is represented by a set of discrete individual changes. If the expression is obtained from values that are defined by discrete intervals, then it can be divided into several sinusoidal expressions with discrete frequencies - from the lowest and then twice, three times and so on higher than the main one. Such a sum is called the Fourier series. If the initial expression is given a value for each real number, then it can be decomposed into several sinusoidal frequencies of all possible frequencies. It is commonly called the Fourier integral, and the solution implies integral transformations of the function. Regardless of how the conversion is obtained, two numbers must be specified for each frequency: amplitude and frequency. These values are expressed as a single complex number. The theory of expressions of complex variables, together with the Fourier transform, made it possible to carry out calculations in the design of various electrical circuits, the analysis of mechanical vibrations, the study of the mechanism of wave propagation, and more.
Fourier Transform today
Today, the study of this process is mainly reduced to finding effectivetransition methods from a function to its transformed form and vice versa. This solution is called the direct and inverse Fourier transform. What does it mean? In order to determine the integral and produce a direct Fourier transform, one can use mathematical methods, or analytical ones. Despite the fact that certain difficulties arise when using them in practice, most integrals have already been found and included in mathematical reference books. Numerical methods can be used to calculate expressions whose form is based on experimental data, or functions whose integrals are not available in tables and are difficult to present in analytical form.
Before the advent of computers, the calculations of such transformations were very tedious, they required manual execution of a large number of arithmetic operations, which depended on the number of points describing the wave function. To facilitate calculations, today there are special programs that have made it possible to implement new analytical methods. So, in 1965, James Cooley and John Tukey created software that became known as the "Fast Fourier Transform". It allows you to save time for calculations by reducing the number of multiplications in the analysis of the curve. The fast Fourier transform method is based on dividing the curve into a large number of uniform sample values. Accordingly, the number of multiplications is halved with the same reduction in the number of points.
Applying the Fourier transform
Thisthe process is used in various fields of science: in number theory, physics, signal processing, combinatorics, probability theory, cryptography, statistics, oceanology, optics, acoustics, geometry and others. The rich possibilities of its application are based on a number of useful features, which are called "Fourier transform properties". Consider them.
1. The function transformation is a linear operator and, with the appropriate normalization, is unitary. This property is known as Parseval's theorem, or in general the Plancherel theorem, or Pontryagin's dualism.
2. The transformation is reversible. Moreover, the reverse result has almost the same form as in the direct solution.
3. Sinusoidal base expressions are own differentiated functions. This means that such a representation changes linear equations with a constant coefficient into ordinary algebraic ones.
4. According to the "convolution" theorem, this process turns a complex operation into an elementary multiplication.
5. The discrete Fourier transform can be quickly calculated on a computer using the "fast" method.
Varieties of the Fourier transform
1. Most often, this term is used to denote a continuous transformation that provides any square-integrable expression as a sum of complex exponential expressions with specific angular frequencies and amplitudes. This species has several different forms, which candiffer by constant coefficients. The continuous method includes a conversion table, which can be found in mathematical reference books. A generalized case is a fractional transformation, by means of which the given process can be raised to the required real power.
2. The continuous mode is a generalization of the early technique of Fourier series defined for various periodic functions or expressions that exist in a limited area and represent them as series of sinusoids.
3. Discrete Fourier transform. This method is used in computer technology for scientific calculations and for digital signal processing. To carry out this type of calculation, it is required to have functions that define individual points, periodic or bounded areas on a discrete set instead of continuous Fourier integrals. The signal transformation in this case is represented as the sum of sinusoids. At the same time, the use of the “fast” method makes it possible to apply discrete solutions to any practical problems.
4. The windowed Fourier transform is a generalized form of the classical method. In contrast to the standard solution, when the signal spectrum is used, which is taken in the full range of the existence of a given variable, here only the local frequency distribution is of particular interest, provided that the original variable (time) is preserved.
5. Two-dimensional Fourier transform. This method is used to work with two-dimensional data arrays. In this case, first the transformation is performed in one direction, and then inother.
Conclusion
Today, the Fourier method is firmly entrenched in various fields of science. For example, in 1962 the DNA double helix shape was discovered using Fourier analysis combined with X-ray diffraction. The latter were focused on crystals of DNA fibers, as a result, the image that was obtained by diffraction of radiation was recorded on film. This picture gave information about the value of the amplitude when using the Fourier transform to a given crystal structure. Phase data were obtained by comparing the diffraction map of DNA with maps obtained from the analysis of similar chemical structures. As a result, biologists have restored the crystal structure - the original function.
Fourier transforms play a huge role in the study of space, semiconductor and plasma physics, microwave acoustics, oceanography, radar, seismology and medical surveys.
