Fourier transform. Fast Fourier transform. Discrete Fourier Transform

The Fourier transform is a transformation that associates functions with a certain real variable. This operation is performed every time we hear different sounds. The ear produces an automatic "computation", which our consciousness is able to perform only after studying the corresponding section of higher mathematics. The human organ of hearing constructs a transformation, as a result of which sound (the vibrational motion 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 levels of loudness 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 vibrational processes (from light radiation and the ocean tide and to cycles of stellar or solar activity) can also be carried out using mathematical methods. So, using these techniques, you can expand the functions by representing the oscillatory processes by a set of sinusoidal components, that is, wave-like curves that move from a minimum to a maximum, then again to a minimum, like a sea wave. The Fourier transform is a transformation whose function describes the phase or amplitude of each sinusoid corresponding to a particular 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 complicated equations that describe the dynamic processes that occur under the influence of light, heat, or electric energy. In other cases, it allows us to determine the regular components in complex vibrational signals, thanks to this, it is possible to correctly interpret various experimental observations in chemistry, medicine, and astronomy.

Historical reference

The first person to apply this method was the French mathematician Jean Baptiste Fourier. The transformation, later named after it, was originally used to describe the mechanism of thermal conductivity. Fourier spent his entire adult life studying the properties of heat. He made a great contribution to the mathematical theory of determining the roots of algebraic equations. Fourier was professor of analysis at the Polytechnic School, secretary of the Institute of Egyptology, was on the imperial service, which distinguished himself during the construction of the road to Turin (under his leadership, was drained more than 80 thousand square kilometers of malaria bogs). However, all this active 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."

Heat conductivity analysis

The scientist used a mathematical method to describe the mechanism of thermal conductivity. A convenient example, in which there are no difficulties with calculation, is the propagation of thermal energy along an iron ring immersed by one part in a fire. To conduct the experiments, Fourier heated the red part of this ring and buried it in fine sand. After that he measured the temperature on the opposite side of it. Initially, the heat distribution is irregular: a 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 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 gradually increases and also decreases smoothly, exactly according to the laws of the change in the cosine or sine function. The wave gradually flattened and, as a result, the temperature becomes the same over 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 own temperature maximum. In this case, each such component changes from a minimum to a maximum and back on a full revolution around the ring an integer number of times. A component having one period was called the basic harmonic, and a value with two or more periods is the second and so on. Thus, the mathematical function that describes the temperature maximum, phase or position is called the Fourier transform of the distribution function. The scientist reduced the single component, which is difficult to be mathematically described, to a tool that is easy to use-cosine and sinus rows, which together give the initial distribution.

The essence of the analysis

Applying this analysis to the transformation of heat propagation through a solid object having an annular shape, the mathematician judged that increasing the periods of the sinusoidal component would lead to its rapid decay. This is well traced on the fundamental and second harmonics. In the latter, the temperature reaches the maximum and minimum values twice in one pass, and in the first one only once. It turns out that the distance overcome by heat in the second harmonic will be half that of the main one. In addition, the gradient in the second will also be twice steeper than the first. Consequently, since a more intense heat flow passes the widest distance, the given harmonic will decay four times faster than the fundamental, as a function of time. In the following, this process will take place even faster. The mathematician believed that this method allows us to calculate the process of the initial temperature distribution over time.

Challenge to 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 outstanding scientists, including Lagrange, Laplace, Poisson, Legendre and Bio, did not accept his assertion that the initial temperature distribution is decomposed into components in the form of fundamental harmonics 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 heat conduction laws, as well as comparing it 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 the breaking straight and curved lines. Contemporaries of the scientist never encountered a similar situation, when the discontinuous functions were described by a combination of continuous ones, such as a quadratic, linear, sinusoid or exponential. In the event that the mathematician was right in his statements, the sum of an infinite series of trigonometric function must be reduced to an exact step. At that time, such a statement seemed absurd. However, despite the 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 suffer the question of whether the sum of several sinusoidal functions can be reduced to an exact representation of a discontinuous one.

200-year history

This theory has developed over the course of two centuries, today it has finally been 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 applied in the case when the initial function is continuous, and the second - in the case when it is represented by a set of discrete individual changes. If the 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 frequency and then twice, three times, and so on, higher than the fundamental one. This amount is usually called the Fourier series. If the initial expression is given by a value for each real number, then it can be decomposed into several sinusoidal all possible frequencies. It is usually called the Fourier integral, and the solution implies the integral transformations of the function. Regardless of the method of obtaining the transformation, two numbers should be specified for each frequency: amplitude and frequency. These values are expressed as a single complex number. The theory of the expressions of complex variables in conjunction with the Fourier transform allowed us to perform calculations for the construction of various electrical circuits, the analysis of mechanical oscillations, the study of the mechanism of propagation of waves, and so on.

Fourier transform today

Nowadays the study of this process basically reduces to finding effective methods of transition from a function to its transformed form and back. This solution is called the direct and inverse Fourier transform. What does it mean? In order to determine the integral and perform a direct Fourier transform, one can use mathematical methods, or even analytical ones. Despite the fact that when using them in practice there are certain difficulties, most of the integrals have already been found and entered in the mathematical reference books. Using numerical methods, it is possible to calculate expressions, the form of which is based on experimental data, or functions whose integrals are absent in the tables and are difficult to present in analytical form.

Before the advent of computer technology 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 Tewki created software that became known as the "fast Fourier transform." It allows to save time of carrying out of calculations due to reduction of number of multiplications at the analysis of a 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 decrease in the number of points.

Application of the 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 have been called "properties of the Fourier transform." Consider them.

1. The transformation of a function is a linear operator and with a corresponding 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 transformation is reversible. And the reverse result has almost the same form, as well as with a direct solution.

3. Sinusoidal basic expressions are eigenfunctions. This means that such a representation modifies linear equations with a constant coefficient into ordinary algebraic equations .

4. According to the "convolution" theorem, this process transforms a complex operation into 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 quadratically integrable expression as a sum of complex exponential expressions with specific angular frequencies and amplitudes. This species has several different forms, which may differ in constant coefficients. A 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 necessary real power.

2. A continuous method is a generalization of the early technique of Fourier series defined for various periodic functions or expressions that exist in a bounded domain 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. For carrying out this type of calculation, it is required to have functions that determine on a discrete set individual points, periodic or bounded domains instead of continuous Fourier integrals. The signal conversion in this case is represented as the sum of the sinusoids. In this case, the use of the "fast" method allows us to apply discrete solutions for any practical tasks.

4. The windowed Fourier transform is a generalized form of the classical method. Unlike the standard solution, when the signal spectrum is used, which is taken in the full range of existence of a given variable, the local frequency distribution is of special interest only if the original variable (time) is preserved.

5. Two-dimensional Fourier transform. This method is used to work with two-dimensional data sets. In this case, first the transformation is done 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, the form of 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 of which the image obtained during the diffraction of radiation was fixed on the film. This picture gave information on the value of the amplitude when using the Fourier transform to the given crystal structure. Data on the phase were obtained by comparing the diffraction map of DNA with the maps obtained when analyzing 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 outer space, the physics of semiconductor materials and plasma, microwave acoustics, oceanography, radar, seismology and medical surveys.