Fourier harmonics. Fourier row

Fourier and Hartley transforms transform the time functions in the frequency function containing amplitude information and phase. Below are graphs of continuous function. g.(t.) and discrete g.(τ), where t. And τ - moments of time.

Both functions begin in zero, the jump reaches a positive value and exponentially fade. By definition, Fourier transform for a continuous function is an integral throughout the real axis, F.(f.), and for a discrete function - the amount by the final set of references, F.(ν):

where f., ν - frequency values, n. - the number of selective values \u200b\u200bof the function, and i.\u003d √ -1 - imaginary unit. The integral representation is more suitable for theoretical studies, and the representation in the form of a finite amount is for calculating the computer. Integral and discrete conversion of Hartley are defined in the same way:

Although the only difference in the notation between Fourier and Hartley definitions is the presence of a multiplier in front of sinus, the fact that the Fourier transform has both valid and the imaginary part, makes the presentations of these two transformations completely different. Discrete transformations Fourier and Hartley are essentially the same form as their continuous analogues.

Although graphics look different, from Fourier and Hartley transformations can be displayed, as shown below, the same information about amplitude and phase.

Fourier amplitude is determined by square root from the sum of the squares of valid and imaginary parts. Hartley amplitude is determined by square root from the sum of squares H.(-Ν) and H.(ν). Fourier phase is determined by the Arctangent of the imaginary part divided by the actual part, and the Hartley phase is determined by the amount of 45 ° and Arctangent from H.(-Ν) divided by H.(ν).

Decomposition of periodic non-velocidal functions

General definitions

Part 1. The theory of linear chains (continued)

Electrical Engineering

THEORETICAL BASIS

Tutorial For students of electric power specialties

T. Electrical chains of periodic nonsenseoidal current

As is well known, a sinusoidal form is adopted in the electric power industry as a standard form for currents and stresses. However, in real conditions, the shape of curves and voltages can be different from sinusoidal to one way or another. Distortion of forms of curves of these functions in receivers leads to additional energy losses and a decrease in their efficiency. The sinusoidalness of the shape of the voltage curve of the generator is one of the quality indicators of electrical energy as a product.

The following reasons for the distortion of the shape of the curves of currents and stresses in the complex chain are possible:

1) the presence in the electrical circuit of nonlinear elements, the parameters of which depend on the instantaneous values \u200b\u200bof the current and voltage [ R, L, C \u003d F(u, I.)], (for example, rectifier devices, electric welding units, etc.);

2) the presence in the electrical circuit of parametric elements whose parameters change over time [ R, L, C \u003d F(t.)];

3) the source of electrical energy (three-phase generator) due to the structural features can not provide the perfect sinusoidal form of the output voltage;

4) influence in the complex listed above factors.

Nonlinear and parametric chains are discussed in separate chapters of the TOE Course. This chapter examines the behavior of linear electrical circuits when exposed to energy sources on them with a non-centered curve.

From the course of mathematics it is known that any periodic time function f.(t.), satisfying the conditions of Dirichle, can be represented by the harmonic Near Fourier:

Here BUT 0 - constant component - k.- Harmonic component or abbreviated k.- Harmonica. The 1st harmonic is called the main, and all subsequent - higher.

The amplitudes of individual harmonics A K. do not depend on the method of decomposition of the function f.(t.) In Fourier series, at the same time, the initial phases of individual harmonics depend on the selection of the start of the timing (the start of the coordinates).

Separate harmonics of the Fourier series can be represented as the sum of the sinus and cosine components:

Then the whole range of Fourier will receive a view:

The ratios between the coefficients of the two forms of the Fourier series have the form:

If a k.- The harmonic and its sinus and cosine components are replaced by complex numbers, the relationship between the coefficients of the Fourier series can be represented in a comprehensive form:

If the periodic inconsistency of the time function is set (or may be expressed) analytically in the form of a mathematical equation, then the coefficients of the Fourier series are determined by formulas known from the math rate:

In practice, the investigated non-conjunctional function f.(t.) Usually specified as a graphic diagram (graphically) (Fig. 118) or in the form of a table coordinate table (table) in the interval of one period (Table 1). To perform a harmonic analysis of such a function according to the equations given above, it must be previously replaced with a mathematical expression. Replacing the function specified graphically or table with a mathematical equation, obtained the name of the approximation of the function.

Home\u003e Law

Chains of non-censoroidal current

Until now, we studied the sinusoidal current chains, however, the law of changes in time may differ from the sinusoidal. In this case, there is a chain of a non-conjuncidal current. All non-censoroidal currents are divided into three groups: periodic, i.e. having a period T. (Fig.6.1, a), non-periodic (Fig.6.1, b) and almost periodic having a periodically changing envelope ( T. o) and the period of the impulses ( T. and) (Fig.6.1, B). There are three ways to obtain non-centered currents: a) in the chain there is a non-monosoidal EMF; b) the chain acts a sinusoidal EMF, but one or more elements of the chain are nonlinear; c) in the chain there is a sinusoidal EMF, but the parameters of one or more chain elements are periodically changed over time. In practice, the method b is most often used). The greatest propagation of non-novinosoidal currents received in devices of radio engineering, automation, telemechanics and computing equipment, where the impulses of the most diverse form are often found. There are inconsistency and electric power industry. We will consider only periodic nonsense voltages and currents that can be decomposed on harmonic components.

Decomposition of periodic non-centered curves in the trigonometric row of Fourier

The phenomena occurring in linear circuits at periodic non-centered stresses and currents are the easiest possible to calculate and research if the unnsuitsoid curves lay out in trigonometric series Fourier. From mathematics it is known that periodic function f (ωt)satisfying Dirichle's conditions, i.e. Having a finite number of gaps of only the first kind and a finite number of highs and minima, can be decomposed into Trigonometric Fourier series.

f (ωt) \u003d a o. +
sinωt +.
sin2ωt +.
sin3ωt + ··· +
cosωt +.
cos2ωt +.
cos3ωt + ··· \u003d

A. o. +
.

Here: A. o. - constant component or zero harmonic;
- the amplitude of the sinus component k.Harmonics;
- the amplitude of the cosine component k.-d harmonics. They are determined by the following formulas

Since where as follows from the vector diagram (Fig.6.2), then we get

.

The components included in this expression are called harmonics. Distinguish even ( k. - Even) and odd harmonics. The first harmonic is called the main, and the rest are higher. The last form of the Fourier series is convenient when it is necessary to know the percentage of each harmonic. The same form of a series of Fourier is used in calculating the chains of non-censoroidal current. Although theoretically, the Fourier series contains an infinitely large number of components, but it usually converges quickly. A convergent nearby can express a given function with any degree of accuracy. In practice, it is enough to take a small number of harmonics (3-5) to obtain accuracy of calculations of several percent.

Features of decomposition in a row of Fourier curves, possessing symmetry

1. Curves, mean for the period of whose value is zero, do not contain a constant component (zero harmonics). 2.
f (ωt) \u003d - f (ωt + π), it is called symmetrical relative to the abscissa axis. This type of symmetry is easy to determine the type of curve: if it is shifted to the x-axis on the abscissa axis, it is mirroring to mirror and at the same time it rushes with a source curve (Fig.6.3), then symmetry is available. When decomposing such a curve in a Fourier series, there is no constant component and all even harmonics, since they do not satisfy the condition f (ωt) \u003d - f (ωt + π).

f (ωt) \u003d sin (ωt + ψ 1 ) + sin (3ωt + ψ 3 )+
sin (5ωt + ψ 5 )+···.

3
. If the function satisfies the condition f (ωt) \u003d f (-ωt), it is called symmetrical relative to the ordinate axis (even). This type of symmetry is easy to determine the type of curve: if the curve, the left axis of the ordinate, mirroring and it is solved with the source curve, then the symmetry is available (Fig.6.4). When decomposing such a curve in a Fourier series in the latter there will be no sinus components of all harmonics ( = f (ωt) \u003d f (-ωt).Consequently, for such curves

f (ωt) \u003d a about +
cosωt +.
cos2ωt +.
cos3ωt + ···.

4
. If the function satisfies the condition f (ωt) \u003d - f (--ωt), it is called symmetrical on the origin of the coordinates (odd). The presence of this type of symmetry is easy to determine the view of the curve: if the curve lying to the left axis of the ordinate to deploy relatively points The origin of the coordinates and it is solved with the source curve, then the symmetry is available (Fig.6.5). When decomposing such a curve in a Fourier series, there will be no cosine components of all harmonics (
= 0) because they do not satisfy the condition f (ωt) \u003d - f (---ωt).Consequently, for such curves

f (ωt) \u003d
sinωt +.
sin2ωt +.
sin3ωt + ···.

In the presence of any symmetry in the formulas for and You can take the integral for the half period, but the result is doubled, i.e. Use expressions

In curves there are several types of symmetry at the same time. To facilitate the issue of harmonious components in this case, fill in the table

Type of symmetry	Analytical expression
1. Axis abscissa	f (ωt) \u003d - f (ωt + π)		Only odd
2. The axes of the ordinate	f (ωt) \u003d f (-ωt)
3. The beginning of the coordinates	f (ωt) \u003d - f (--ωt)
4. Absissal axis and ordinate axes	f (ωt) \u003d - f (ωt + π) \u003d f (-ωt)			Odd
5. The axis of the abscissa and the beginning of the coordinates	f (ωt) \u003d - f (ωt + π) \u003d - f (-ωt)		Odd

By laying out the curve in the Fourier row, it should be first clarified, whether it does not have any kind of symmetry, the presence of which allows you to predict in advance which harmonics will be in a number of Fourier and do not fulfill extra work.

Grafanalytic decomposition of curves in a row of Fourier

When the non-censoroidal curve is set by a graph or table and does not have an analytical expression, to determine its harmonics resort to grafoanalytic decomposition. It is based on the replacement of a certain integral of the sum of the final number of the terms. For this purpose, the function period f (ωt)broken on n. equal parts Δ. ωt \u003d.2π / n.(Fig.6.6). Then for zero harmonics

where: r - Current index (area number) that accepts values \u200b\u200bfrom 1 to n.; f. r (ωt) -meaning function f (ωt)for ωt \u003d p ·Δ ωt.(see Fig. 6.6) . For amplitude of the sinus component k.Harmonica

For the amplitude of the cosine component k.Harmonica

Here sin. p. kωt. and cos. p. kωt.- values sinkωt.and coskΩt.for ωt \u003d p ·. In practical calculations usually take n.\u003d 18 (Δ ωt \u003d.20˚) or n.\u003d 24 (Δ ωt \u003d.fifteen). In case of grafoanalytic decomposition of curves in a Fourier series even more important than when analytical find out whether it does not have any kind of symmetry, the presence of which significantly reduces the volume computational work. So, formulas for and In the presence of symmetry takes the view

When building a harmonic in general graph, it is necessary to take into account that the scale of the abscissa axis for k.Harmonics B. k.once more than for the first.

Maximum, average and actant values \u200b\u200bof nonsense

Periodic non-monosoidal values, in addition to their harmonic components, are characterized by the maximum, average and acting values. Maximum value BUT M is the value of the function module for the period largest (Fig.6.7). The average value of the module is determined so

.

If the curve is symmetrical relative to the abscissa axis and during the half-period never changes the sign, then the average module value is equal to the average value for the half

,

and in this case, the start of time counting should be selected so that f (0)= 0.If the function for the entire period never changes the sign, then its average modulo is equal to the constant component. In the chains of non-censusoidal current under the values \u200b\u200bof EDC, stresses or currents understand their valid values \u200b\u200bdetermined by the formula

.

If the curve is decomposed in a Fourier series, its acting value can be determined as follows.

Let us clarify the result. The product of the sinusoid of different frequency ( kΩ. and iω.) It is a harmonic function, and the integral for a period of any harmonic function is zero. The integral that is a sign of the first amount was defined in the sinusoidal circuits and its value was shown there. Hence,

.

From this expression, it follows that the active value of periodic non-censusoidal values \u200b\u200bdepends only on the valid values \u200b\u200bof its harmonics and does not depend on their initial phases ψ k. . Let us give an example. Let be u.=120
sIN (314. t.+ 45˚) -50Sin (3 · 314 t.-75˚) B.. Its valid

There are cases when the average for the module and the active values \u200b\u200bof non-intrusive values \u200b\u200bcan be calculated based on the integration of the analytical expression of the function and then there is no need to lay the curve in a row of Fourier. In power industry, where the curves are predominantly symmetrical with respect to the abscissa axis, a number of coefficients are used to characterize their form. Three of them got the greatest use: amplitude coefficient k. a, form coefficient k. F and distortion ratio k. and. They are determined like this: k. A \u003d. A. m / A.; /A. cf; k. and \u003d. A. 1 /A. For sinusoids, they have the following values: k. a \u003d; k. F \u003d π. A. M. / 2A. m ≈1.11; 1. D. the rectangular curve (Fig.6.8, a) coefficients are as follows: k. a \u003d 1; k. F \u003d 1; k. and \u003d 1.26 /. For a crooked (pico-shaped) form (Fig.6.8, b) the values \u200b\u200bof the coefficients are as follows: k. And the higher the more pico-shaped is its form; k. Ф\u003e 1.11 and the higher what a pointed curve; k. and<1 и чем более заостренная кривая, тем меньше. Как видим рассмотренные коэффициенты в определенной степени характеризуют форму кривой. Уsee one of the practical applications of the distortion coefficient. Industrial network voltage curves usually differ from perfect sinusoids. In the power industry, the concept of a practically sinusoidal curve is introduced. According to GOST voltage of industrial networks, it is considered practically sinusoidal if the most difference between the corresponding ordents of the true curve and its first harmonics does not exceed 5% of the amplitude of the main harmonic (Fig.6.9). Measurement of non-substantial values \u200b\u200bof devices of various systems gives unequal results. Amplitude electronic voltmeters measure maximum values. Magnetoelectric devices react only to the constant component of the measured values. Magnetoelectric devices with rectifier measured the average value of the module. Instruments of all other systems measure valid values.

Calculation of non-conjuncoidal circuits

If one or more sources with non-sinusoidal EDC are operating in the chain, then its calculation disintegrates into three stages. 1. Determination of EMF sources for harmonic components. How to do this is discussed above. 2. The use of the principle of overlay and the calculation of currents and stresses in the chain from the action of each component of the EDC separately. 3. Joint consideration (summation) of decisions obtained in paragraph 2. The summation of the components in general is most often difficult and not always necessary, since, on the basis of harmonic components, it is possible to judge both the form of the curve and the main values \u200b\u200bcharacterizing it. ABOUT
the second stage is the second. If the non-sinusoidal EMF is represented next to Fourier, then such a source can be considered as a sequential connection of the source of the constant EDC and sources of sinusoidal EDC with different frequencies (Fig. 6.10). Using the principle of overlay and considering the effect of each EDC separately, you can determine the current components in all branches of the chain. Let be E. o Creates I. o, e. 1 - i. 1 , e. 2 - i. 2, etc. Then the actual current i.=I. O +. i. 1 +i. 2 +··· . Consequently, the calculation of the incinerate circuit is reduced to solving one task with a constant EDC and a number of tasks with sinusoidal EDC. When solving each of these tasks, it is necessary to take into account that for different frequencies, inductive and capacitive resistance of unequal. Inductive resistance is directly proportional to the frequency, so it is for k.Harmonics x. LK \u003d. kΩL.=kX. L1, i.e. for k.-Y harmonica it is in k.once more than for the first. Capacitive resistance is inversely proportional to the frequency, so it is for k.Harmonics x. CK \u003d 1 / kΩS=x. C1 / k.. for k.-Y harmonica it is in k.once smaller than for the first. The active resistance in principle also depends on the frequency due to the surface effect, but with small sections of the conductors and at low frequencies, the surface effect is practically absent and permissible to assume that the active resistance for all harmonics is equally. If the non-sinusoidal voltage is connected directly to the tank, then for k.Harmonics Toka

C. we are higher than the harmonic number, the smaller for it the resistance of the container. Therefore, even if the voltage amplitude of high order harmonics is a minor share from the amplitude of the first harmonic, it can still cause a current commensurate with the current of the main harmonic or exceeding it. In this regard, even at a voltage, close to the sinusoidal current in the container may be sharply non-censor (Fig. 6.11). On this occasion, it is said that the capacity emphasizes the currents of high harmonics. If the non-sinusoidal voltage is connected directly to inductance, then for k.Harmonics Toka

.

FROM
Increasing the order of harmonic increases inductive resistance. Therefore, in the current through the inductance, the highest harmonics are presented to a lesser extent than in the voltage on its clips. Even with sharply unsinkidal voltage, the current curve in inductance is often approaching sinusoid (Fig. 6.12). Therefore, it is said that inductance brings the current curve to the sinusoid. When calculating each harmonic component of the current, it is possible to use the complex method and build vector diagrams, but it is unacceptable to produce geometric summation of vectors and the addition of voltage complexes or currents of different harmonics. Indeed, vectors depicting the currents of the first and third harmonics rotate with different speeds (Fig.6.13). Therefore, the geometric sum of these vectors gives the instantaneous value of their sum only ω t.\u003d 0 and in the general case does not make sense.

Power of non-censoroidal current

As well as in the chains of sinusoidal current, we are talking about the facilities consumed by a passive two-pole. Under active power, also understand the average for the period of instantaneous power.

Let the voltage and current at the entrance of the two-pole will be represented by the Fourier rows

Substitute meaning u. and i. in the formula R

The result was obtained taking into account that the integral for the period from the product of the sinusoid of different frequencies is zero, and the integral for the period from the product of the sinusoid of the same frequency was determined in the section of the sinusoidal circuits. Thus, the active power of the nonsense is equal to the sum of the active capacities of all harmonics. It's clear that R k. can be determined by any known formulas. By analogy with the sinusoidal current, the concept of complete power is introduced for the non-sinusoidal, as a product of active voltage and current values, i.e. S \u003d UI.. Attitude R to S. called power coefficient and equivalent to a cosine of some conditional angle θ . cos. θ =P / S.. In practice, very often non-replacement voltages and currents are replaced by equivalent sinusoids. At the same time, you need to perform two conditions: 1) the active value of the equivalent sinusoid should be equal to the current value of the value of the replaceable value; 2) angle between equivalent voltage and current sinusoids θ should be so that UIcos. θ equal to active power R. Hence, θ - This is the angle between the equivalent sinusoids of voltage and current. Typically, the current equivalent sinusoid value is close to the current values \u200b\u200bof the main harmonics. By analogy with a sinusoidal current for a non-reactive, the concept of reactive power is introduced, defined as the amount of the reactive capacity of all harmonics

For non-censoroidal current, unlike sinusoidal S. 2 ≠P. 2 +Q. 2. Therefore, the concept of distortion power is introduced here. T.characterizing the difference of forms of voltage and current curves and is determined so

Higher harmonics in three-phase systems

In three-phase systems, the voltage curves in the phases of the phases are usually reproduced by the phase and shift to a third of the period. So, if u. A \u003d. f (ωt)T. u. In \u003d. f (ωt-2π/ 3), but u. C \u003d. f (Ωt +2π/ 3). Suppose the phase voltages of non-censoroidal and are decomposed in the Fourier series. Then consider k."Harmonica in all three phases." Let be u. AK \u003d. U. KM SIN ( kωt + ψ k.), then get u. Ink \u003d U. KM SIN ( kωt + ψ k. -K.2π/ 3) I. u. Ck \u003d. U. KM SIN ( kωt + ψ k. + K.2π/ 3). Cutting these expressions at different values k., notice that for harmonics, multiple three ( k.=3n., n. - Natural number of numbers, starting with 0) in all phases of voltage at any time have the same value and direction, i.e. Form the zero sequence system. For k.=3n +.1 harmonics form a voltage system, the sequence of which coincides with the sequence of actual stresses, i.e. They form a direct sequence system. For k.=3n-1 harmonics form a voltage system, the sequence of which is opposite to the sequence of actual stresses, i.e. They form a reverse sequence system. In practice, it is most often missing both the constant component and all even harmonics, therefore, in the future, we will limit ourselves to the consideration of only odd harmonics. Then the nearest harmonics forming the reverse sequence is the fifth. In electric motors, it causes the greatest harm, so it is with her a merciless struggle. Consider the features of the work of three-phase systems caused by the presence of harmonics, multiple three. one . When connecting the generator or transformer windings into a triangle (Fig. 6.14), the branches of the latter flow the currents of harmonics, multiple three, even in the absence of external load. Indeed, the algebraic amount of EDC harmonic, multiple three ( E. 3 , E. 6, etc.), in the triangle has a tripled value, in contrast to the rest of the harmonics, for which this amount is zero. If the phase winding resistance for the third harmonic Z. 3, then the current of the third harmonics in the circuit of the triangle will be I. 3 =E. 3 /Z. 3. Similar to the current sixth harmonics I. 6 =E. 6 /Z. 6, etc. The active value of the current flowing on the windings will be
. Since the resistance of the generator windings are small, the current can reach large quantities. Therefore, if there are harmonics, multiple three, winding the generator or transformer in the phase EMPS, do not connect. 2. . If you connect the winding of the generator or transformer into an open triangle (Fig.6.155, then voltage will be valid on its clips equal to the amount of EMF harmonic, multiple three, i.e. u. BX \u003d 3. E. 3m sin (3 ωt + ψ 3)+3E. 6m sin (6 ωt + ψ 6)+3E. 9m sin (9 ωt + ψ 9)+···. Its valid

.

The open triangle is usually used before connecting the generator windings to a conventional triangle to check the possibility of trouble-free realization of the latter. 3. Linear voltages, regardless of the scheme for connecting the generator windings or a transformer, harmonic, multiple three, do not contain. When the triangle is connected, phase EDCs containing harmonics, multiple three, are compensated by a voltage drop on the inner resistance of the generator phase. Indeed, according to the second law of Kirchoff for the third, for example, harmonics for the scheme Fig.6.14 can be recorded U. AB3 +. I. 3 Z. 3 =E. 3, where we get U. AB3 \u003d 0. Similar to any of the harmonics, multiple three. When connecting to a star, linear stresses are equal to the difference between the corresponding phase EDS. For harmonics, multiple three, when drawing up these differences, phase EMFs are destroyed because they form a zero sequence system. Thus, the components of all harmonics may be present in phase voltages and their valid value. In the linear voltages of harmonics, multiple of three are missing, therefore their valid value. In this regard, in the presence of harmonics, multiple three, U. l / U. F.<
. 4. In the schemes without zero wire, the currents of harmonics, multiple three, cannot be closed, as they form a zero sequence system and can only be closed in the presence of the latter. At the same time, there is even a voltage between the zero dots of the receiver and the source, even in the case of a symmetric load, a voltage equal to the amount of EMF harmonic, multiple three, is to easily be convinced by the equation of the second law of Kirchhoff, taking into account the fact that these harmonics are missing. Instant value of this voltage u. 0 1 0 =E. 3m sin (3 ωt + ψ 3)+E. 6m sin (6 ωt + ψ 6)+E. 9m sin (9 ωt + ψ 9)+···. Its valid
. 5. In the star-star scheme with zero wire (Fig. 6.16), the currents of the harmonic, multiple three, even in the case of a symmetrical load, if the phase EDS contains the specified harmonics. Considering that harmonics, multiple three, form a zero sequence system, can be recorded

General descriptions

French Mathematics Fourier (J. B. J. Fourier 1768-1830) The procurement said a hypothesis brave enough for his time. According to this hypothesis, there is no function that could not be decomposed into the trigonometric row. However, unfortunately, at that time such an idea was not perceived seriously. And it is natural. Fourier himself could not lead convincing evidence, and it is very difficult to intuitively in the Fourier hypothesis. Especially hard to imagine the fact that when adding simple functions similar to trigonometric, the functions are reproduced, not similar to them. But if we assume that the Fourier hypothesis is true, then the periodic signal of any form can be decomposed on the sinusoids of different frequencies, or vice versa, by means of the corresponding addition to the sinusoid with different frequencies it is possible to synthesize the signal of any form. Therefore, if this theory is true, its role in signal processing can be very large. In this chapter, first will try to illustrate the correctness of Fourier's hypothesis.

Consider a function

f (T) \u003d2sin. T -sin. 2T.

Simple trigonometric series

The function is the sum of trigonometric functions, in other words, is represented as a trigonometric series of two members. Add one category and create a new row of three members

After adding several terms, we get a new trigonometric row of ten members:

The coefficients of this trigonometric series are denoted as b. K. , where k. - whole numbers. If you carefully look at the last ratio, it can be seen that the coefficients can be described by the following expression:

Then the function f (t) can be represented as follows:

Factors b. K. - these are amplitudes sinusoid with an angular frequency to.In other words, they specify the amount of frequency components.

Considered the case when the upper index toequal to 10, i.e. M \u003d10. Enlargement M.up to 100, we get a function f (T).

This function, being trigonometric nearby, in shape is approaching a saw-shaped signal. And, it seems, Fourier's hypothesis is quite true in relation to the physical signals that we are dealing with. In addition, in this example, the signal form is not smooth, but includes the gap points. And the fact that the function is reproduced even at the break points looks like a promising.

In the physical world there are really many phenomena that can be represented as the amount of oscillations of different frequencies. A typical example of these phenomena is the light. It is an amount of electromagnetic waves with a wavelength of 8,000 to 4000 angstrom (from the red color of the purple). Of course, you know that if the white light is skipped through the prism, then a spectrum of seven pure colors will appear. This is because the refractive index of the glass, from which the prism is made, varies depending on the length of the electromagnetic wave. It is just evidence that white light is the sum of light waves of different lengths. So, skipping the light through the prism and having received its spectrum, we can analyze the properties of the light, exploring the color combinations. Like this, through the decomposition of the received signal to various frequency components, we can find out how the initial signal originated, on which path he followed or finally, which external influence he was subjected. In short, we can get information to clarify the origin of the signal.

Such an analysis method is called spectral analysisor analysis of Fourier.

Consider the following system of orthonormal functions:

Function f (T)you can decompose on this system of functions on the segment [-π, π] as follows:

The coefficients α. k,β k, as shown earlier, can be expressed through scalar works:

In general, the function f (T) can be represented as follows:

The coefficients α. 0 , α k,β k called fourier coefficientsand this presentation of the function is called decomposition in a series of Fourier.Sometimes such a representation is called validdecomposition in a row of Fourier, and coefficients - the valid Fourier coefficients. The term "valid" is introduced in order to distinguish the decomposition of decomposition in a row of Fourier in a comprehensive form.

As mentioned earlier, an arbitrary function can be decomposed on the system of orthogonal functions, even if the functions from this system are not submitted in the form of a trigonometric series. Usually under decomposition in the Fourier series, the decomposition into the trigonometric series is implied. If Fourier coefficients express through α 0 , α k,β k We get:

Since when K. \u003d 0 Coskt.\u003d 1, then constant a 0/2expresses a general view of the coefficient a K.for k.= 0.

In relation (5.1), the fluctuation of the largest period represented by the sum cos.t I. sin.t, called oscillation of the main frequency or first harmonic.Oscillation with a period equal to half of the main period called the second harmonic.Oscillation with a period of 1/3 of the main period called third harmonicetc. As can be seen from the ratio (5.1) a. 0 is a permanent value expressing the average function F (T). If the function f (T)represents an electrical signal then a 0.represents its constant component. Consequently, all other Fourier coefficients express its variable components.

In fig. 5.2 shows the signal and its decomposition in the Fourier range: on the constant component and harmonics of different frequencies. In the time domain, where the variable value is time, the signal is expressed by the function f (T), And in the frequency domain, where the variable value is the frequency, the signal appears to Fourier coefficients (A k, b k).

The first harmonic is a periodic function with a period 2 π. The first harmonics also have a period of multiple 2 π . Based on this, in the formation of a signal from the components of the Fourier series, we naturally obtain a periodic function with a period 2 π. And if so, the decomposition in the Fourier series is, in fact, the method of representing periodic functions.

Spread in a row Fourier signal of a frequently found species. For example, consider the previously mentioned sawn curve (Fig. 5.3). Signal of such a form on the segment - π < t < π is expressed by the function f ( t)= t.Therefore, Fourier coefficients can be expressed as follows:

Example 1.

Decomposition in a series of Fourier signal of a saws

f (t) \u003d t,

In the previous chapter, we got acquainted with another point of view on the fluctuating system. We saw that in the string there are various own harmonics and that any private oscillation, which can only be obtained from the initial conditions, can be viewed as a combination of several simultaneously oscillating their own harmonics in the proper proportion. For the string, we found that our own harmonics have frequencies ω 0, 2ω 0, zω 0, .... Therefore, the most common movement of the string is made of sinusoidal oscillations of the main frequency ω 0, then the second harmonic 2ω 0, then the third harmonic of Zω 0, etc. The main harmonic is repeated through each period T 1 \u003d 2π / ω 0, the second harmonic - through each period T 2 \u003d 2π / 2ω 0; She repeats alsoand through every period T. 1 \u003d 2T. 2 , i.e. after twohis periods. Exactly the same way T. 1 the third harmonic is repeated. Three periods are stacked in this segment. And again we understand why the string is condemned after the period T. 1 fully repeats the form of its movement. So it turns out a musical sound.

So far, we talked about the movement of the string. but sound,which is the movement of the air caused by the movement of the string, should also consist of the same harmonics, although here we can no longer talk about our own air harmonics. In addition, the relative strength of various harmonics in the air can be completely different than in the string, especially if the string is "connected" with air through a "sounding board". Different harmonics are differently connected with air.

If a music tone feature f.(t.) represents the air pressure depending on the time (let's say, such as FIG. 50.1.6), then we can expect that f.(t.) it is written in the form of a certain number of simple harmonic functions from time (similar cos ω t.) for each of the various harmonic frequencies. If the period of oscillations is equal T,then the main angular frequency will be ω \u003d 2π / t, and the following harmonics will be 2Ω, zω, etc.

A little complication appears here. We are not right to expect that for each frequency, the initial phases will definitely be equal to each other. Therefore, you need to use the functions of type COS (ωt + φ) - instead, however, it is easier to use for eachfrequencies both sinus and cosine. Recall that

and since φ is constant, then anysinusoidal oscillations with the CO frequency can be recorded in the form of the amount of members, to one of which includes SIN ωT, and in the other - COS ωT.

So, we come to the conclusion that anyperiodic function f.(t.) with a period T.mathematically can be recorded in the form

where Ω \u003d 2π / T, but but and b. - Numeric constants indicating what weight each oscillation component is included in the overall oscillation f.(t.). For greater generality, we added a member to our formula with zero frequency A 0, although usually for musical tones it is zero. This is simply a shift of the average size of sound pressure (i.e., the shift "zero" level). With this member, our formula is true for any occasion. Equation (50.2) schematically shown in FIG. 50.2. Amplitudes of harmonic functions but N. and b. N. selected by special rule. In the figure, they are shown only schematically without complying. [Row (50.2) called near Fourierfor functions f.(t.).]

We said that anyoneperiodic function can be written in this form. It should be made a small amendment and emphasize that in such a number you can decompose any sound wave or any function with which we are confronted in physics. Mathematics, of course, can come up with such a function that it cannot be made up of simple harmonic (for example, a function that "worst" back, so for some values t. it has two meanings!). However, here we should not worry about such functions.