The Fourier series is written as:

, where k is the harmonic number.

The Fourier coefficients for this series are found using the formulas:

Periodic signals are represented by a Fourier series in the form:

, where is the fundamental frequency;

Here the coefficients are calculated using the formulas:

Another form of writing the Fourier series is often used:

, Where:

– amplitude k th harmonics; - initial phase

For convenience of calculations, the Fourier series is written in complex form:

Graphic time and frequency display

Spectrum of a periodic signal

temporary image

(f)

ASF frequency image

Similar to PSF, only taking into account that the phases can also be negative.

Such a spectrum is called discrete or line, it is characteristic of a periodic signal.

Spectrum of a sequence of rectangular pulses

Consider the symmetrical arrangement of pulses

, where is the duty cycle.

Let's find the zero points of the sine:

The first zero point is the most important for the spectrum of the rectangular pulse sequence.

ASF sequence of rectangular pulses:

ω 1 ω 2 2π/t u 4π/t u

The main share of energy is carried by harmonics located from 0 to the first zero point (about 90% of the energy). This frequency region, where 90% of the signal energy is concentrated, is called the spectral width of the (frequency) signal.

For a rectangular pulse, the spectrum width is .

Any digital signal transmission requires more spectrum than simple analog transmission.

PSF sequence of rectangular pulses:

if sun(x)>0, then Ψ k =0

if sin(x)<0, то Ψ k = π

The influence of pulse duration and period on the type of spectrum

If the duration decreases, the fundamental frequency will not change, the zero points will move to the right. More components reach the first zero point, where the main energy is concentrated. Technically they note that the spectrum is expanding.

If the pulse duration increases, then the spectrum narrows.

If the repetition period increases, the fundamental frequency decreases. If the repetition period decreases, the fundamental frequency increases.

Changing the pulse position or reference point

This does not affect the ASF; only the phase spectrum changes. This can be reflected based on the delay theorem:

Phase spectrum of the shifted signal at N=4:

The concept of calculating circuits with periodic signals

Calculation method:

1. The complex spectrum of the periodic signal is determined;

2. The spectrum is evaluated, leaving the most significant harmonics (first criterion: all that are less than 0.1 of the maximum harmonic amplitude are cut off);

Currents and voltages from each component are calculated separately. You can use a complex calculation method.

I 0 =0

The non-harmonic function can be estimated by its effective value, i.e. root mean square for the period:

The concept of the spectrum of a non-periodic signal

Non-periodic signals are the most important because they carry information. Periodic signals are service signals for transmitting information, and do not carry new information. Therefore, the question of spectra of non-periodic signals arises. You can try to obtain them by passing to the limit from periodic signals, directing the period to infinity (). A single signal remains. Let us find the complex amplitude of the spectrum of a single signal: at .

A non-periodic signal can be divided into an infinite sum of harmonic components with infinitely small amplitudes and differing in frequency by infinitesimal values - This is called a continuous spectrum of a non-periodic signal, and not a discrete one. For calculations, the concept of non-complex amplitudes and complex spectral density of amplitudes is used - the amplitude value per unit frequency.

This is a direct Fourier transform (two-way).

APPLICATION OF FOURIER SERIES FOR FORECASTING AND OPTIMIZING SUPPLY TO A WHOLESALE TRADE ENTERPRISE IN THE ASPECT OF MANAGING OWN AND RENTED TRANSPORT

Gorlach Boris Alekseevich 1, Shigaeva Natalya Valerievna 2
1 Samara State Aerospace University named after Academician S.P. Koroleva (National Research University), Doctor of Technical Sciences, Professor
2 Samara State Aerospace University named after Academician S.P. Queen (NIU)

Annotation
The paper examines the mechanism for modeling a random process (for statistical data about an enterprise) using the apparatus of harmonic analysis. The problem of rational distribution of raw material supply volumes between own and leased transport has been solved in order to reduce the cost of storing products.

THE FOURIER SERIES APPLICATION FOR PREDICTION AND OPTIMIZATION OF DELIVERY COSTS

Gorlach Boris Alekseevich 1 , Shigaeva Nathalie Valerievna 2
1 Samara State Aerospace University, doctor of technical Sciences, Professor
2 Samara State Aerospace University

Abstract
The mechanism of simulation of a random process is considered (for the enterprise data). Harmonic analysis is widely adopted in modeling of enterprise costs. The problem of rational distribution of the raw materials deliveries between own transport and rented transport is solved.

Bibliographic link to the article:
Gorlach B.A., Shigaeva N.V. Application of Fourier series for forecasting and optimization of enterprise supplies wholesale trade in the aspect of managing own and leased transport // Economics and management of innovative technologies. 2014. No. 7 [Electronic resource]..02.2019).

Introduction. The enterprise's costs for creating a goods storage system create the need for rational distribution of supplies. Solving the supply management problem is associated with changes in the enterprise's needs for raw materials. To develop a rational distribution model, processing of enterprise statistical data on the demand for raw materials was carried out.

The article consists of the following parts: building a model of a random process, optimizing supplies using the example of a simplified model and using real data as an example.

Part one. Construction of a mathematical model of a random process.

In the retrospective period, statistical data on resource storage in a warehouse is as follows (Table 1). It is assumed that a set of statistical data Y i =Y(t i) is given in the form of a time series.

Table 1 – Statistical data on resource demand

As a rule, mathematical models of time series of economic processes are presented as a set of 4 components: seasonal S, cyclical C, random ξ and trend U. These components form an additive model of statistical data.

The component U - trend - is selected in such a way that it does not contradict the main trend of change in the function under study and does not complicate its analysis. In this work, trend selection is carried out using Excel functions, as well as manually using the “normal equations” method.

After performing the procedure for selecting the most adequate trend, the function is normalized, allowing for modeling of the oscillatory component. In this study, the oscillatory component is selected using a model representing a trigonometric Fourier series:

The coefficients of the Fourier series are defined as follows :

After conducting a search in 6 iterations using Excel tools, the following function of the oscillatory component was identified :

S(t) = -0.215sinπt/6 – 0.077cos πt/6 -0.085sin πt/3-0.013cos πt/3+0.001 sin πt/2+0.023cosπt/2-0.035 sin2πt/3+0.055cos 2πt/3 +0.003 sin 5πt/6+0.054cos 5πt/6+0.056cos πt

The dynamics of supply and storage of the resource in the warehouse, as well as the functional dependence of the volume of the resource after normalization are presented in Figure 1.

Figure 1 – Oscillatory component for real data

Let's calculate the coefficient of determination for the resulting function.

The coefficient of determination for the resulting function is 0.75. Consequently, the trend describes the statistical data by 75 percent, and the probability that the resulting function does not correspond to the real statistical values is 0.25.

Part two. Delivery Process Optimization

When drawing up the proportion of raw material supplies, several factors that influence the economic efficiency of supplies should be taken into account:

Timeliness and frequency of deliveries

Delivery cost

Acceptable shelf life of raw materials

Providing the enterprise with storage facilities

Other factors.

Let's look at the supply optimization process using a simplified diagram. Let us single out one harmonic in the normalized trend (one term of the harmonic series) and limit ourselves to considering one period. The result is the following simplified supply function:

In this work, we will consider three supply options.

1. Supplies are provided only by our own transport at level y=1, which corresponds to the value s(t)=0.

The process of accumulation of resources in the first half of the year and consumption in the second half of the year is determined by the formula of the integral of the function in the area under consideration.

The accumulated resources are completely spent in the next half of the year. The problem is that the storage volume in the warehouse varies too much over time and needs to be optimized.

2. Own transport ensures supplies corresponding to the minimum intensity of resource consumption. This option is suitable for a company if the company has less capital and for other reasons cannot afford transport more than the minimum level of resource requirements, it looks like this. The enterprise receives less resources in an amount equal to the area of the integral between s(t) and the straight line characterizing the minimum level of supplies.

Suppose that the company decides to rent transport at the level of maximum resource requirements in the first half of the year, then the savings are completely spent in the second half of the year.

3. Own transport ensures supplies at the level -h. The lack of resources is compensated by renting transport.

Calculating the supply level h from the condition of equality of accumulation and consumption areas:

With the obtained value h The lack of resources without rent looks like this:

Summarizing the results obtained, a general accumulation/consumption graph has been compiled, which shows how the optimal plan differs in the minimum amount of warehouse resources (Figure 2).

Figure 2 – Minimization of warehouse resources

Based on the schedule, the use of rented transport when optimizing storage in a warehouse makes it possible to reduce the specific storage volume in a warehouse up to 10 times, since the amplitude of the accumulation function values has decreased from 10 units to 1.

Part 3. Supply optimization using real data as an example

Supply optimization begins with identifying the period of the oscillatory component (in our example, t i ϵ 11..23) and searching for the intersection points of the function s(t) with the Ox axis.

An illustration of the dynamics of resource receipt and consumption at an enterprise in which transport lease is not provided is presented in Figure 3.

Figure 3 – Accumulation/expense for real data without rental

The function of the oscillatory component is as follows:

S(t) = -0.215 sin πt/6-0.077cos πt/6 -0.085 sin πt/3-0.013cos πt/3+0.001 sin πt/2+0.023cos πt/2-0.035 sin 2πt/3+0.055cos 2πt/3+0.003 sin 5πt/6+0.054cos 5πt/6+0.056cos πt

Accumulation function:

Q = ∫S = (1/π)(0.215 *6* cos (πt/6)-0.077*6*sin (πt/6) +0.085*3*cos πt/3 – 0.013*3*sin πt/3 – 0.0013*2*cos πt/2+(0.023*2*sin πt/2+0.0349*6/4 cos 2πt/3+(0.0552*6/4)sin 2πt/3 – (0.0032*6/5) cos 5πt/6 + (0.0538*6/5)sin 5πt/6 + (0.0559*sin π t)

Let us determine the maximum areas of stock and consumption for the supply function, provided that the supply intensity s(t) is equal to zero.

Table 2 – Determination of reserve areas and resource consumption

Thus, Q max =0.9078 is the maximum possible amount of resources stored in the warehouse. Resources accumulated in the first half of the year are completely spent in the second, because trigonometric functions have the property of symmetry.

Optimization using rented transport is an effective way to reduce the costs of storing resources in a warehouse. The level of supply of an enterprise by its own transport is specified by the value Y(t)=1-h, or S(t)=-h from the condition of equality of accumulation and consumption areas by half year (Figure 4).

Figure 4 – Determination of the level of supply by leased transport

In this case, the need for the resource will remain in the volume determined by the area of the rectangle with the height h and the basis, constituting the entire interval of consideration, equal (from the properties of symmetry) to the area of the integral of the cyclic component above the straight line of the level of supply by own transport. The company rents transport for part of the considered interval. The level of supply by rented transport will be determined from the equality of the areas of resource shortage (2) and the rental volume (1), shown in Figure 4.

Search for levels h is carried out iteratively. In the option of using rented vehicles, the maximum level of inventory storage in the warehouse is equal to:

Upper level h* we find from the condition of equality of the areas of unmet demand (1) for resources and the volume of supplies (2), indicated in Figure 4. The rental level is determined by the value h*=0.144.

After optimization, the flow and reserve area was found:

The total area of reserves decreased from 0.9 to 0.5:

Q max2 =0.2016+ 0.3137=0.515

Thus, optimization of the supply process using rental vehicles resulted in a reduction in warehouse costs by 44%, which indicates the successful completion of the optimization task.

Results and conclusions. The proposed algorithm for the rational distribution of supplies between the enterprise’s own transport and the leased one in the course of modeling the cost function with a Fourier series is based on the characteristic features of a normalized trend graph, takes into account the limitations of warehouse space, the shelf life of raw materials, and ensures a reduction in warehouse costs (the level of storage of resources in the warehouse) up to 50% times for the considered supply function data. Thus, using rented transport is an effective way to reduce warehouse and storage costs at a high cost of renting and maintaining warehouse space.

Bibliography

Savelyev G.L. The problem of optimizing enterprise resources in conditions of cyclical changes in demand. – Samara: SSAU, 2010. – 30 pages.
Chuikova Yu.S. Optimization of material flow in the problem of enterprise inventory management / Collection of scientific articles “Management of organizational and economic systems”. – Samara: SSAU, 2009. – p. 25-30.
Rardin R.L. Optimization in Operations Research. Prentice Hall, 1998.

Number of views of the publication: Please wait functions. This transformation is of great importance because it can be used to solve many practical problems. Fourier series are used not only by mathematicians, but also by specialists in other sciences.

The expansion of functions into a Fourier series is a mathematical technique that can be observed in nature if you use a device that senses sinusoidal functions.

This process occurs when a person hears a sound. The human ear is designed in such a way that it can sense individual sinusoidal fluctuations in air pressure of different frequencies, which, in turn, allows a person to recognize speech and listen to music.

The human ear does not perceive sound as a whole, but through its Fourier series components. The strings of a musical instrument produce sounds that are sinusoidal vibrations of various frequencies. The reality of the Fourier series expansion of light is represented by a rainbow. Human vision perceives light through some of its components of different frequencies of electromagnetic oscillations.

The Fourier transform is a function that describes the phase and amplitude of sinusoids of a certain frequency. This transformation is used to solve equations that describe dynamic processes that arise under the influence of energy. Fourier series solve the problem of identifying constant components in complex oscillatory signals, which made it possible to correctly interpret the data obtained from experiments, observations in medicine, chemistry and astronomy.

The discovery of this transformation belongs to the French mathematician Jean Baptiste Joseph Fourier. In honor of whom the Fourier series was subsequently named. Initially, the scientist found application of his method in studying and explaining the mechanisms of thermal conductivity. It was suggested that the initial irregular distribution of heat can be represented in the form of simple sinusoids. For each of which the temperature minimum, maximum and phase will be determined. The function that describes the upper and lower peaks of the curve, the phase of each harmonic is called the Fourier transform from the expression of the temperature distribution. The author of the transformation proposed a method for decomposing a complex function as a sum of periodic functions cosine, sine.

The purpose of the course work is to study the Fourier series and the relevance of the practical application of this transformation.

To achieve this goal, the following tasks were formulated:

1) give the concept of a trigonometric Fourier series;

2) determine the conditions for the decomposability of a function in a Fourier series;

3) consider the Fourier series expansion of even and odd functions;

4) consider the Fourier series expansion of a non-periodic function;

5) reveal the practical application of the Fourier series.

Object of study: expansion of functions in Fourier series.

Subject of study: Fourier series.

Research methods: analysis, synthesis, comparison, axiomatic method.

1.5. Fourier series for even and odd functions

Consider the symmetric integral

where is continuous or piecewise continuous on. Let's make a change in the first integral. We believe. Then

Therefore, if the function is even, then (i.e. the graph of the even function is symmetrical about the and axis

If is an odd function, then (i.e. the graph of an odd function is symmetrical about the origin) and

Those. the symmetric integral of an even function is equal to twice the integral over half the integration interval, and the symmetric integral of an odd function is equal to zero.

Note the following two properties of even and odd functions:

1) the product of an even function and an odd one is an odd function;

2) the product of two even (odd) functions is an even function.

Let be an even function defined on and expandable on this segment into a trigonometric Fourier series. Using the results obtained above, we find that the coefficients of this series will have the form:

If is an odd function defined on a segment and expands on this segment into a trigonometric Fourier series, then the coefficients of this series will have the form:

Consequently, the trigonometric Fourier series on the segment will have the form

for an even function:

(16)

for odd function:

Series (16) does not contain sines of multiple angles, that is, the Fourier series of an even function includes only even functions and an independent term. Series (17) does not contain cosines of multiple angles, that is, the Fourier series of an odd function includes only odd functions.

Definition. Rows
are parts of a complete Fourier series and are called incompletetrigonometric Fourier series.

If a function is expanded into an incomplete trigonometric series (16) (or (17)), then it is said to beexpands into a trigonometric Fourier series in cosines (or sines).

1.6. Fourier series expansion of a non-periodic function

1.6.1. Fourier series expansion of functions on

Let a function be given on an interval and satisfy the conditions of the Dirichlet theorem on this interval. Let's perform a variable change. Let where we select so that the resulting argument function is defined on. Therefore, we believe that

The resulting function can be expanded into a Fourier series:

Where

Let's make a reverse replacement⇒ We get

Where

(19)

Series (18) – Fourier series in the basic trigonometric system of functions

Thus, we found that if a function is given on an interval and satisfies the conditions of the Dirichlet theorem on this interval, then it can be expanded into a trigonometric Fourier series (18) according to the trigonometric system of functions (20).

The trigonometric Fourier series for an even function defined on will have the form

Where

for odd function

Where

Comment! In some problems, it is required to expand a function into a trigonometric Fourier series according to the system of functions (20) not on a segment, but on a segment. In this case, you just need to change the limits of integration in formulas (19) ((15), if, that is, in this case

(23)

or if

(24)

The sum of a trigonometric Fourier series is a periodic function with a period, which is a periodic continuation of a given function. And for a periodic function equality (4) is true.

1.6.2. Fourier series expansion of functions on

Let the function be given on and satisfy the conditions of the Dirichlet theorem on this interval. Such a function can also be expanded into a Fourier series. To do this, the function must be extended to the interval and the resulting function expanded into a Fourier series on the interval. In this case, the resulting series should be considered only on the segment on which the function is specified. For convenience of calculations, we will define the function in an even and odd way.

1) Let us extend the function into the interval in an even manner, that is, we will construct a new even function that coincides with the function on the interval. Consequently, the graph of this function is symmetrical about the axis and coincides with the graph on the segment. Using formulas (21), we find the coefficients of the Fourier series for the function and write the Fourier series itself. The sum of the Fourier series for is a periodic function, with a period. It will coincide with the function on at all points of continuity.

2) Let us extend the function to the interval in an odd way, that is, we will construct a new odd function that coincides with the function. The graph of such a function is symmetrical about the origin of coordinates and coincides with the graph on the segment. Using formulas (22), we find the coefficients of the Fourier series for the function and write the Fourier series itself. The sum of the Fourier series for is a periodic function with a period. It will coincide with the function on at all points of continuity.

Notes!

1) Similarly, you can expand a function defined on the interval into a Fourier series

2) Since the expansion of a function on a segment presupposes its continuation onto the segment in an arbitrary manner, the Fourier series for the function will not be unique.

1.6.3. Fourier series expansion of functions on

Let the function be given on an arbitrary segment of length and satisfy the conditions of the Dirichlet theorem on it.

Then this function can be expanded into a Fourier series. To do this, the function must be periodically (with a period) continued along the entire number line and the resulting function must be expanded into a Fourier series, which should be considered only on the segment. Due to property (3) of periodic functions, we have

Therefore, the Fourier coefficients for the resulting continuation of the function can be found using the formulas

(25)

2. Practical application of Fourier series

2.1. Problems involving the expansion of functions in Fourier series and their solution

It is required to expand into a trigonometric Fourier series a function that is a periodic continuation of a function specified on an interval. To do this, it is necessary to use an algorithm for expanding a periodic function into a Fourier series.

Algorithm for expanding a periodic function into a Fourier series:

1) Construct a graph of a given function and its periodic continuation;

2) Set the period of the given function;

3) Determine whether the function is even, odd or general;

4) Check the feasibility of the conditions of the Dirichlet theorem;

5) Create a formal representation of the Fourier series generated by this function;

6) Calculate Fourier coefficients;

7) Write down the Fourier series for a given function, using the coefficients of the Fourier series (item 4).

Example 1. Expand the function into a Fourier series on the interval.

Solution:

1) Let's construct a graph of the given function and its periodic continuation.

2) Period of expansion of the function.

3) The function is odd.

4) The function is continuous and monotonic on, i.e. the function satisfies the Dirichlet conditions.

5) Let's calculate the coefficients of the Fourier series.

6) Write the Fourier series by substituting the Fourier coefficients into the formula

Answer:

Example 2. Let us expand a function with an arbitrary period into a Fourier series.

Solution: the function is defined on the half-interval (-3;3]. Period of expansion of the function, half-period. Let us expand the function into a Fourier series

At the origin, the function is discontinuous, so we will represent each Fourier coefficient as a sum of two integrals.

Let us write the Fourier series by substituting the found coefficients of the Fourier series into the formula.

Example 3. Expand a functionin betweenin the Fourier series in cosines. Construct a graph of the sum of the series.

Solution: we extend the function into the interval in an even way, that is, we construct a new even function that coincides with the function on the interval. Let's find the coefficients of the Fourier series for the function and write the Fourier series. The sum of the Fourier series for is a periodic function, with a period. It will coincide with the function on at all points of continuity.

The trigonometric Fourier series for the function will have the form

Let's find the coefficients of the Fourier series

Thus, when the coefficients are found, we can write the Fourier series

Let's plot the sum of the series

Example 4. Given a function defined on the segment. Find out whether the function can be expanded into a Fourier series. Write the expansion of the function in a Fourier series.

Solution:

1) construct a graph of the function on .

2) the function is continuous and monotonic on , that is, according to Dirichlet’s theorem, it can be expanded into a trigonometric Fourier series.

3) calculate the Fourier coefficients using formulas (1.19).

4) write the Fourier series using the found coefficients.

2.2. Examples of the application of Fourier series in various fields of human activity

Mathematics is one of the sciences that has wide application in practice. Any production and technological process is based on mathematical laws. The use of various mathematical tools makes it possible to design devices and automated units capable of performing operations, complex calculations and calculations in the design of buildings and structures.

Fourier series are used by mathematicians in geometry whensolving problems in spherical geometry; in mathematic physics atsolving problems on small vibrations of elastic media. But besides mathematics, Fourier series have found their application in other fields of science.

Every day people use various devices. And often these devices do not work properly. For example, the sound is difficult to hear due to a lot of noise, or the image received by fax is unclear. A person can determine the cause of a malfunction by sound. The computer can also diagnose whether the device is damaged. Excess noise can be removed using computer signal processing. The signal is represented as a sequence of digital values, which are then entered into a computer. After performing certain calculations, the coefficients of the Fourier series are obtained.

Changing the spectrum of the signal allows you to clear the recording of noise, compensate for signal distortion by various recording devices, change the timbres of instruments, and focus listeners’ attention on individual parts.

In digital image processing, the use of Fourier series allows for the following effects: blurring, edge highlighting, image restoration, artistic effects (embossing)

Fourier series expansion is used in architecture in the study of oscillatory processes. For example, when creating a project for various types of structures, the strength, rigidity and stability of structural elements are calculated.

In medicine, to conduct a medical examination using cardiograms and an ultrasound machine, a mathematical apparatus is used, which is based on the theory of Fourier series.

Large computational problems of assessing the statistical characteristics of signals and filtering noise arise when recording and processing continuous seabed data. When making measurements and recording them, holographic methods using Fourier series are promising. That is, Fourier series are also used in such a science as oceanology.

Elements of mathematics are found in production at almost every step, so it is important for specialists to know and be well oriented in the field of application of certain analysis and calculation tools.

Conclusion

The topic of the course work is devoted to the study of the Fourier series. An arbitrary function can be expanded into simpler ones, that is, it can be expanded into a Fourier series. The scope of the course work does not allow us to reveal in detail all aspects of the series expansion of a function. However, from the tasks posed, it seemed possible to reveal the basic theory about Fourier series.

The course work reveals the concept of the trigonometric Fourier series. Conditions for the decomposability of a function in a Fourier series are determined. Fourier series expansions of even and odd functions are considered; non-periodic functions.

The second chapter provides only some examples of the expansion of functions given on various intervals into Fourier series. The areas of science where this transformation is used are described.

There is also a complex form of representation of the Fourier series, which could not be considered because the volume of the course work does not allow. The complex form of the series is algebraically simple. Therefore, it is often used in physics and applied calculations.

The importance of the topic of the course work is due to the fact that it is widely used not only in mathematics, but in other sciences: physics, mechanics, medicine, chemistry and many others.

References

1. Bari, N.K. Trigonometric series. [text]/ N.K. Bari. - Moscow, 1961. - 936 s.

2. Bermant, A.F. A short course in mathematical analysis: a textbook for universities[text]/ A.F. Bermant, I.G. Aramanovic. – 11th ed., erased. – St. Petersburg: Publishing House “Lan”, 2005. – 736 p.

3. Bugrov, Ya. S. Higher mathematics: Textbook for universities: In 3 volumes.[text]/ Ya. S. Bugrov, S. M. Nikolsky; Ed. V. A. Sadovnichy. - 6th ed., stereotype. - M.: Bustard, 2004. -512 p.

4. Vinogradova, I. A. Problems and exercises in mathematical analysis: a manual for universities, pedagogical. universities: At 2 o'clock.[text]/ I. A. Vinogradova, S. N. Olehnik, V. A. Sadovnichy; edited by V.A. Sadovnichigo. – 3rd ed., rev. – M.: Bustard, 2001. – 712 p.

5. Gusak, A.A. Higher mathematics. In 2 volumes. T. 2. Textbook for university students.[text]/ A. A. Gusak.– 5th ed. – Minsk: TetraSystems, 2004.

6. Danko, P.E. Higher mathematics in exercises and problems: textbook for universities: 2 hours.[text]/ P.E. Danko, A.G. Popov, T.Ya. Kozhevnikova. Moscow: ONIX: Peace and Education, 2003. – 306 p.

7. Lukin, A. Introduction to digital signal processing (mathematical foundations) [text]/ A. Lukin. - M., 2007. - 54 p.

8. Piskunov, N. S. Differential and integral calculus for colleges, vol. 2: Textbook for colleges.[text]/ N. S. Piskunov. - 13th ed. - M.: Nauka, 1985. - 432 p.

9. Rudin, U. Fundamentals of mathematical analysis.[text]/ U. Rudin. - 2nd ed., Trans. from English .- M.: Mir, 1976 .- 206 p.

10. Fikhtengolts, G. M. Fundamentals of mathematical analysis. Part 2.[text]/ G. M. Fikhtengolts. -6th ed., erased. - St. Petersburg: Lan Publishing House, 2005. – 464 p.

Orenburg, 2015

Fourier series and their application in communications technology

Parameter name	Meaning
Article topic:	Fourier series and their application in communications technology
Rubric (thematic category)	Education

Decomposition of a continuous signal into orthogonal series

Lecture 6. Continuous channel

Restoration quality criteria.

The following criteria exist:

1) Criterion for the largest deviation

where: permissible reconstruction error, - max value - current approximation error.

At the same time, there is confidence that any changes in the original signal, including short-term emissions, will be recorded.

2) SKZ criterion. where: - additional CS approximation error, - CS approximation error.

3) Integral criterion

The max average value for the sampling period is determined.

4) Probabilistic criterion

The permissible level is set, the value P is the probability that the current approximation error does not depend on some specific value.

Purpose of the lecture: familiarization with the continuous channel

a) decomposition of a continuous signal into orthogonal series;

b) Fourier series and their application in communications technology;

c) Kotelnikov’s theorem (Shannon’s Fundamental Theorem);

d) capacity of a continuous channel;

e) NKS model.

In communication theory, two special cases of expansion of functions into orthogonal series are widely used to represent signals: expansion in trigonometric functions and expansion in functions of the form sinx/x. In the first case, we obtain a spectral representation of the signal in the form of an ordinary Fourier series, and in the second case, a temporal representation in the form of a V.A. series. Kotelnikov.

The simplest form of signal expression from a practical point of view is a linear combination of some elementary functions

In general, the signal is a complex oscillation, which makes it extremely important to represent a complex function s(t), defining the signal through simple functions.

When studying linear systems, this representation of the signal is very convenient. It allows the solution of many problems to be divided into parts using the principle of superposition. For example, to determine the signal at the output of a linear system, the system’s response to each elementary effect ψ k (t) is calculated, and then the results multiplied by the corresponding coefficients a k were easily calculated and did not depend on the number of terms of the sum. These requirements are most fully satisfied by a set of orthogonal functions.

Functions ψ 1 (t), ψ 2 (t), . . . . , ψ n (t) . (6.2)

Given on an interval are called orthogonal,

if at. (6.3)

The basis of spectral analysis of signals is the representation of time functions in the form of a Fourier series or integral. Any periodic signal s(t) that satisfies the Dirichlet condition must be represented as a series in trigonometric functions

The quantity a 0, expressing the average value of the signal over a period, is usually called the constant component. It is calculated by the formula

The complex form of writing the Fourier series is very convenient

Magnitude A k is a complex amplitude, it is found by the formula

Relations (6.8) and (6.9) constitute a pair of discrete Fourier transforms. It should be noted that the Fourier series can represent not only a periodic signal, but also any signal of finite duration. In the latter case, the signal S(t) is assumed to be periodically extended along the entire time axis. In this case, equality (6.4) or (6.8) represents the signal only in the interval of its duration (- T/2,T/2). A random signal (or noise) specified over an interval (- T/2,T/2), must also be represented by the Fourier series

Where a k And b k are random variables (for fluctuation noise - independent random with normal distribution).

Fourier series and their application in communication technology - concept and types. Classification and features of the category "Fourier series and their application in communications technology" 2017, 2018.

This series can also be written as:

(2),
where , k-th complex amplitude.

The relationship between coefficients (1) and (3) is expressed by the following formulas:

Note that all these three representations of the Fourier series are completely equivalent. Sometimes, when working with Fourier series, it is more convenient to use exponents of the imaginary argument instead of sines and cosines, that is, use the Fourier transform in complex form. But it is convenient for us to use formula (1), where the Fourier series is presented as a sum of cosines with the corresponding amplitudes and phases. In any case, it is incorrect to say that the Fourier transform of a real signal will result in complex harmonic amplitudes. As Wiki correctly says, “The Fourier transform (?) is an operation that associates one function of a real variable with another function, also a real variable.”

Total:
The mathematical basis for spectral analysis of signals is the Fourier transform.

The Fourier transform allows you to represent a continuous function f(x) (signal), defined on the segment (0, T) as the sum of an infinite number (infinite series) of trigonometric functions (sine and/or cosine) with certain amplitudes and phases, also considered on the segment (0, T). Such a series is called a Fourier series.

Let us note some more points, the understanding of which is required for the correct application of the Fourier transform to signal analysis. If we consider the Fourier series (the sum of sinusoids) on the entire X-axis, we can see that outside the segment (0, T) the function represented by the Fourier series will periodically repeat our function.

For example, in the graph of Fig. 7, the original function is defined on the segment (-T\2, +T\2), and the Fourier series represents a periodic function defined on the entire x-axis.

This happens because sinusoids themselves are periodic functions, and accordingly their sum will be a periodic function.

Fig.7 Representation of a non-periodic original function by a Fourier series

Thus:

Our original function is continuous, non-periodic, defined on a certain segment of length T.
The spectrum of this function is discrete, that is, it is presented in the form of an infinite series of harmonic components - the Fourier series.
In fact, the Fourier series defines a certain periodic function that coincides with ours on the segment (0, T), but for us this periodicity is not significant.

The periods of the harmonic components are multiples of the value of the segment (0, T) on which the original function f(x) is defined. In other words, the harmonic periods are multiples of the duration of the signal measurement. For example, the period of the first harmonic of the Fourier series is equal to the interval T on which the function f(x) is defined. The period of the second harmonic of the Fourier series is equal to the interval T/2. And so on (see Fig. 8).

Fig.8 Periods (frequencies) of the harmonic components of the Fourier series (here T = 2?)

Accordingly, the frequencies of the harmonic components are multiples of 1/T. That is, the frequencies of the harmonic components Fk are equal to Fk = k\T, where k ranges from 0 to?, for example k = 0 F0 = 0; k=1 F1=1\T; k=2 F2=2\T; k=3 F3=3\T;… Fk= k\T (at zero frequency - constant component).

Let our original function be a signal recorded during T=1 sec. Then the period of the first harmonic will be equal to the duration of our signal T1=T=1 sec and the harmonic frequency will be 1 Hz. The period of the second harmonic will be equal to the signal duration divided by 2 (T2=T/2=0.5 sec) and the frequency will be 2 Hz. For the third harmonic T3=T/3 sec and the frequency is 3 Hz. And so on.

The step between harmonics in this case is 1 Hz.

Thus, a signal with a duration of 1 second can be decomposed into harmonic components (obtaining a spectrum) with a frequency resolution of 1 Hz.
To increase the resolution by 2 times to 0.5 Hz, you need to increase the measurement duration by 2 times - up to 2 seconds. A signal lasting 10 seconds can be decomposed into harmonic components (to obtain a spectrum) with a frequency resolution of 0.1 Hz. There are no other ways to increase frequency resolution.

There is a way to artificially increase the duration of a signal by adding zeros to the array of samples. But it does not increase the actual frequency resolution.

3. Discrete signals and discrete Fourier transform

With the development of digital technology, the methods of storing measurement data (signals) have also changed. If previously a signal could be recorded on a tape recorder and stored on tape in analog form, now signals are digitized and stored in files in computer memory as a set of numbers (samples).

The usual scheme for measuring and digitizing a signal is as follows.

Fig.9 Diagram of the measuring channel

The signal from the measuring transducer arrives at the ADC during a period of time T. The signal samples (sampling) obtained during the time T are transmitted to the computer and stored in memory.

Fig. 10 Digitized signal - N samples received during time T

What are the requirements for signal digitization parameters? A device that converts an input analog signal into a discrete code (digital signal) is called an analog-to-digital converter (ADC) (Wiki).

One of the main parameters of the ADC is the maximum sampling frequency (or sampling rate, English sample rate) - the sampling rate of a time-continuous signal when sampling it. It is measured in hertz. ((Wiki))

According to Kotelnikov’s theorem, if a continuous signal has a spectrum limited by the frequency Fmax, then it can be completely and unambiguously reconstructed from its discrete samples taken at time intervals , i.e. with frequency Fd? 2*Fmax, where Fd is the sampling frequency; Fmax - maximum frequency of the signal spectrum. In other words, the signal digitization frequency (ADC sampling frequency) must be at least 2 times higher than the maximum frequency of the signal that we want to measure.

What will happen if we take samples with a lower frequency than required by Kotelnikov’s theorem?

In this case, the “aliasing” effect occurs (also known as the stroboscopic effect, moiré effect), in which a high-frequency signal, after digitization, turns into a low-frequency signal, which actually does not exist. In Fig. 5 red high frequency sine wave is a real signal. A blue sinusoid of a lower frequency is a fictitious signal that arises due to the fact that during the sampling time more than half a period of the high-frequency signal has time to pass.

Rice. 11. The appearance of a false low-frequency signal at an insufficiently high sampling rate

To avoid the aliasing effect, a special anti-aliasing filter is placed in front of the ADC - a low-pass filter (LPF), which passes frequencies below half the ADC sampling frequency, and cuts off higher frequencies.

In order to calculate the spectrum of a signal from its discrete samples, the discrete Fourier transform (DFT) is used. Let us note once again that the spectrum of a discrete signal “by definition” is limited by the frequency Fmax, which is less than half the sampling frequency Fd. Therefore, the spectrum of a discrete signal can be represented by the sum of a finite number of harmonics, in contrast to the infinite sum for the Fourier series of a continuous signal, the spectrum of which can be unlimited. According to Kotelnikov's theorem, the maximum frequency of a harmonic must be such that it accounts for at least two samples, therefore the number of harmonics is equal to half the number of samples of a discrete signal. That is, if there are N samples in the sample, then the number of harmonics in the spectrum will be equal to N/2.

Let us now consider the discrete Fourier transform (DFT).

Comparing with Fourier series

We see that they coincide, except that time in the DFT is discrete in nature and the number of harmonics is limited by N/2 - half the number of samples.

DFT formulas are written in dimensionless integer variables k, s, where k are the numbers of signal samples, s are the numbers of spectral components.
The value s shows the number of complete harmonic oscillations over period T (duration of signal measurement). The discrete Fourier transform is used to find the amplitudes and phases of harmonics using a numerical method, i.e. "on the computer"

Returning to the results obtained at the beginning. As mentioned above, when expanding a non-periodic function (our signal) into a Fourier series, the resulting Fourier series actually corresponds to a periodic function with period T (Fig. 12).

Fig. 12 Periodic function f(x) with period T0, with measurement period T>T0

As can be seen in Fig. 12, the function f(x) is periodic with period T0. However, due to the fact that the duration of the measurement sample T does not coincide with the period of the function T0, the function obtained as a Fourier series has a discontinuity at point T. As a result, the spectrum of this function will contain large number high frequency harmonics. If the duration of the measurement sample T coincided with the period of the function T0, then the spectrum obtained after the Fourier transform would contain only the first harmonic (sinusoid with a period equal to the sampling duration), since the function f(x) is a sinusoid.

In other words, the DFT program “does not know” that our signal is a “piece of a sinusoid”, but tries to represent a periodic function in the form of a series, which has a discontinuity due to the inconsistency of individual pieces of the sinusoid.

As a result, harmonics appear in the spectrum, which should sum up the shape of the function, including this discontinuity.

Thus, in order to obtain the “correct” spectrum of a signal that is the sum of several sinusoids with different periods, it is necessary that the signal measurement period contains an integer number of periods of each sinusoid. In practice, this condition can be met for a sufficiently long duration of signal measurement.

Fig. 13 Example of the function and spectrum of the gearbox kinematic error signal

With a shorter duration, the picture will look “worse”:

Fig. 14 Example of the function and spectrum of a rotor vibration signal

In practice, it can be difficult to understand where are the “real components” and where are the “artifacts” caused by the non-multiple periods of the components and the duration of the signal sampling or “jumps and breaks” in the signal shape. Of course, the words “real components” and “artifacts” are put in quotation marks for a reason. The presence of many harmonics on the spectrum graph does not mean that our signal actually “consists” of them. This is the same as thinking that the number 7 “consists” of the numbers 3 and 4. The number 7 can be represented as the sum of the numbers 3 and 4 - this is correct.

So our signal... or rather not even “our signal”, but a periodic function composed by repeating our signal (sampling) can be represented as a sum of harmonics (sine waves) with certain amplitudes and phases. But in many cases that are important for practice (see the figures above), it is indeed possible to relate the harmonics obtained in the spectrum to real processes, which are cyclic in nature and make a significant contribution to the signal shape.

Some results

1. A real measured signal with a duration of T seconds, digitized by an ADC, that is, represented by a set of discrete samples (N pieces), has a discrete non-periodic spectrum, represented by a set of harmonics (N/2 pieces).

2. The signal is represented by a set of real values and its spectrum is represented by a set of real values. Harmonic frequencies are positive. The fact that it is more convenient for mathematicians to represent the spectrum in complex form using negative frequencies does not mean that “this is correct” and “this should always be done.”

3. A signal measured over a time interval T is determined only over a time interval T. What happened before we started measuring the signal, and what will happen after that, is unknown to science. And in our case, it’s not interesting. The DFT of a time-limited signal gives its “true” spectrum, in the sense that, under certain conditions, it allows one to calculate the amplitude and frequency of its components.

Materials used and other useful materials.