The Fourier transform is a transformation that maps the functions of a certain real variable. This operation is performed every time we perceive various sounds. The ear performs automatic “calculation”, which our consciousness is capable of performing only after studying the corresponding section of higher mathematics. The human hearing organ builds a transformation, as a result of which the sound (oscillatory movement of conditioned particles in an elastic medium that propagates in a wave form in a solid, liquid or gaseous medium) is provided as a spectrum of successive values ​​of the volume level of tones of different heights. After that, the brain turns this information into a familiar sound.
Mathematical Fourier Transform
The transformation of sound waves or other oscillatory processes (from light radiation and the ocean tide to stellar or solar activity cycles) can also be carried out using mathematical methods. So, using these techniques, you can expand the functions by representing the oscillatory processes as a set of sinusoidal components, that is, wave-like curves that go from minimum to maximum, then back to minimum, like a sea wave. The Fourier transform is a transformation whose function describes the phase or amplitude of each sinusoid corresponding to a specific 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 fairly complex equations that describe the dynamic processes that occur under the influence of light, heat or electric energy. In other cases, it allows one to determine the regular components in complex oscillatory signals, thanks to this, various experimental observations in chemistry, medicine, and astronomy can be correctly interpreted.
Historical reference
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 thermal conductivity. Fourier spent his entire 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, which distinguished himself during the construction of the road to Turin (under his leadership, more than 80 thousand square kilometers of malaria swamps were drained). However, all this active work did not prevent the scientist from doing mathematical analysis. In 1802, he derived an equation that describes the distribution 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 used a mathematical method to describe the mechanism of thermal conductivity. A convenient example, in which there is no difficulty with the calculation, is the distribution of thermal energy through an iron ring immersed in one part in a fire. To conduct the experiments, Fourier heated a red part of this ring and buried it in fine sand. After that, he took temperature measurements on the opposite side of it. Initially, the heat distribution is irregular: part of the ring is cold and the other is hot, a sharp temperature gradient can be observed between these zones. However, in the process of heat spreading over the entire metal surface, it becomes more uniform. So, soon this process takes the form of a sinusoid. At first, the graph gradually increases and decreases equally smoothly, just according to the laws of changing the function of the cosine or sine. The wave gradually equalizes 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 completely decomposed into a series of elementary sinusoids. Each of them will have its own phase (initial position) and its temperature maximum. In addition, each such component changes from minimum to maximum and vice versa at full revolution around the ring an integer number of times. A component having one period was called the main harmonic, and a value with two or more periods is 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 mathematically describe, to an easy-to-use tool - the cosine and sine rows, in total giving the initial distribution.
The essence of the analysis
Applying this analysis to the transformation of heat distribution over a solid object having a circular shape, the mathematician reasoned that increasing the periods of the sinusoidal component would lead to its rapid attenuation. This can be clearly seen at the fundamental and second harmonics. In the latter, the temperature reaches the maximum and minimum values ​​twice in one pass, and in the first - only once. It turns out that the distance covered by heat in the second harmonic will be half as much as in the main. In addition, the gradient in the second will also be twice as steeper than the first. Therefore, since a more intense heat flux travels a smaller widow distance, this harmonic will decay four times faster than the main one, as a function of time. In subsequent, this process will go even faster. The mathematician believed that this method allows us to calculate the process of the initial distribution of temperature over time.
Challenge Contemporaries
The Fourier transform algorithm became a challenge to the theoretical foundations of mathematics of that time. At the beginning of the nineteenth century, most prominent scientists, including Lagrange, Laplace, Poisson, Legendre and Biot, did not accept his assertion that the initial temperature distribution decomposes into components in the form of the fundamental 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 laws of heat conduction, as well as its comparison with physical experiments. In the Fourier approach, the main objection was caused by the fact that the discontinuous function is represented by the sum of several sinusoidal functions that are continuous. After all, they describe breaking straight and curved lines. Contemporaries of the scientist never faced a similar situation when discontinuous functions were described by a combination of continuous, such as quadratic, linear, sinusoid or exponent. In that case, if the mathematician was right in his statements, then the sum of an infinite series of trigonometric functions should be reduced to an exact stepwise one. At that time, such a statement seemed absurd. However, despite doubts, some researchers (for example, Claude Navier, Sophie Germain) expanded the scope of research and moved them beyond the analysis of the distribution of thermal energy. And mathematicians, meanwhile, continued to be tormented by the question of whether the sum of several sinusoidal functions can be reduced to an exact discontinuous representation.
200 year history
This theory has been developing for two centuries, today it is 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 many discrete individual changes. If an expression is obtained from values ​​that are determined by discrete intervals, then it can be divided into several sinusoidal expressions with discrete frequencies - from the lowest and then twice, triple, and so on above the main one. This amount is usually called the Fourier series. If the initial expression is given a value for each real number, then it can be decomposed into several sinusoidal all possible frequencies. It is commonly called the Fourier integral, and the solution implies integral transformations of the function. Regardless of the method of obtaining the conversion, two numbers should be indicated 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, analyze mechanical vibrations, study the mechanism of wave propagation, and more.
Fourier Transform Today
Nowadays, the study of this process mainly comes down to finding effective methods of transition 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 perform the direct Fourier transform, one can use mathematical methods, or it is possible analytical. Despite the fact that when using them in practice, certain difficulties arise, most of the integrals have already been found and entered into mathematical reference books. Using numerical methods, one can calculate expressions whose form is based on experimental data, or functions whose integrals are absent in the tables and are difficult to represent in analytical form.
Before the advent of computer technology, the calculations of such transformations were very tedious; they required the manual execution of a large number of arithmetic operations, which depended on the number of points describing the wave function. To facilitate the calculations, today there are special programs that have allowed the implementation of new analytical methods. So, in 1965, James Cooley and John Tukey created software known as the “fast Fourier transform”. It allows you to save time calculations by reducing the number of multiplications in the analysis of the curve. The Fast Fourier Transform method is based on dividing the curve by a large number of uniform sample values. Accordingly, the number of multiplications is halved with the same decrease in the number of points.
Application of Fourier Transform
This 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 "properties of the Fourier transform". Consider them.
1. The transformation of a function is a linear operator and, with appropriate normalization, is unitary. This property is known as the Parseval theorem, or in the general case the Plancherel theorem, or the Pontryagin dualism.
2. The conversion is reversible. Moreover, the opposite result has almost the same form as with the direct solution.
3. Sinusoidal basic expressions are their 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 elementary multiplication.
5. The discrete Fourier transform can be quickly calculated on a computer using the “fast” method.
Varieties of Fourier Transforms
1. Most often, this term is used to denote a continuous transformation that provides any quadratically integrable expression in the form of a sum of complex exponential expressions with specific angular frequencies and amplitudes. This view has several different forms, which may differ by constant coefficients. The continuous method includes a transformation table, which can be found in mathematical reference books. A general case is a fractional transformation, through which a given process can be raised to the necessary real power.
2. The continuous method 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 necessary to have functions that determine individual points on a discrete set, periodic or bounded domains instead of continuous Fourier integrals. The signal conversion in this case is presented as the sum of the sinusoids. Moreover, the use of the “fast” method allows the use of discrete solutions for any practical problems.
4. Window Fourier transform is a generalized form of the classical method. Unlike the standard solution, when the spectrum of the signal is used, which is taken in the full range of the existence of this variable, here, of particular interest is only the local frequency distribution, provided that the original variable is preserved (time).
5. Two-dimensional Fourier transform. This method is used to work with two-dimensional data arrays. In this case, the conversion is first performed in one direction, and then in the other.
Conclusion
Today, the Fourier method is firmly entrenched in various fields of science. For example, in 1962, a double DNA helix was discovered using Fourier analysis in combination with X-ray diffraction. The latter were focused on crystals of DNA fibers; as a result, the image obtained by radiation diffraction was fixed on the film. This picture gave information about the value of the amplitude when using the Fourier transform to a given crystal structure. Phase data was obtained by comparing the diffraction map of DNA with the maps obtained by analysis of similar chemical structures. As a result, biologists restored the crystal structure - the original function.
Fourier transforms play a huge role in the study of outer space, the physics of semiconductor materials and plasma, microwave acoustics, oceanography, radar, seismology and medical examinations.