Generalized characteristics of the signal. Generalized structure of the communication channel. Geometric representation of signals and their characteristics

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ROSTOV TECHNOLOGICAL INSTITUTE

SERVICE AND TOURISM

________________________________________________________________

Department of Radioelectronics

Lazarenko S.V.

LECTURE No. 1

in the discipline “Radio Engineering Circuits and Signals”

Rostov-on-Don

2010

LECTURE 1

INTRODUCTION MAIN CHARACTERISTICS OF SIGNALS

In the discipline RADIO ENGINEERING CIRCUITS AND SIGNALS

Time: 2 hours

Questions studied: 1. Subject, purpose and objectives of the course

2. Brief overview course, connection with other disciplines

3. Brief history of the development of the discipline

4. General methodology for working on the course, types of classes,

reporting forms, educational literature

5 Energy characteristics of the signal

6 Correlation characteristics of deterministic signals

7 Geometric methods in signal theory

8 Theory of orthogonal signals. Generalized Fourier series

This lecture implements the following elements of the qualification characteristics:

The student must know the basic laws, principles and methods of electrical circuit analysis, as well as methods for modeling electrical circuits, diagrams and devices.

The student must master the techniques of performing circuit calculations in steady-state and transient modes.

1. SUBJECT AND OBJECTIVES OF THE COURSE

The subject of study of the discipline RADIO ENGINEERING CIRCUITS AND SIGNALS is electromagnetic processes in linear and nonlinear radio circuits, methods for calculating circuits in steady-state and transient modes, continuous and discrete signals and their characteristics.

The discipline takes objects of research from practice - typical circuits and signals, from physics - her laws of the electromagnetic field, from mathematics - research apparatus.

The purpose of studying the discipline is to instill in students the skill of calculating the simplest radio circuits and familiarize them with modern algorithms for optimal signal processing.

As a result of studying the discipline, each student must

HAVE AN INTRODUCTION:

On modern algorithms for optimal signal processing;

On trends in the development of the theory of radio circuits and signals,

KNOW:

Classification of radio signals;

Temporal and spectral characteristics of deterministic signals;

Random signals, their characteristics, correlation and spectral analysis random signals;

Discrete signals and their characteristics;

Digital signal processing algorithms,

BE ABLE TO USE:

Methods for analytical and numerical solution of problems of signal transmission through linear and non-linear linear circuits;

Methods of spectral and correlation analysis of deterministic and random signals,

OWN:

Techniques for measuring the basic parameters and characteristics of radio circuits and signals;

Techniques for analyzing the passage of signals through circuits,

HAVE EXPERIENCE:

Studies of the passage of deterministic signals through linear stationary circuits, nonlinear and parametric circuits;

Calculation of the simplest radio circuits.

The operational focus of training in the discipline is ensured by conducting a laboratory workshop, during which each student is imparted with practical skills:

Work with electrical and radio measuring instruments;

Carrying out express analysis of emergency situations in the operation of fragments of radio circuits based on measurement results.

2 BRIEF OVERVIEW OF THE COURSE, CONNECTION WITH OTHER DISCIPLINES

The discipline "Radio Engineering Circuits and Signals" is based on knowledge And yakh "Mathematics", "Physics", "Informatics", and ensures the assimilation of art at dents of general scientific and special disciplines, "Metrology and radioism e rhenium", "Devices for generating and forming radio signals", "Devices for receiving and processing signals", "Fundamentals of television and video O technology", "Statistical theory of radio engineering systems", "Radio engineering And logical systems", coursework and diploma projects to titration.

Studying the discipline “Radio Engineering Circuits and Signals” develops engineering thinking in students and prepares them for mastering special disciplines.

Teaching the discipline is aimed at:

For in-depth study by students of the basic laws, principles and methods of analysis of electrical circuits, the physical essence of electromagnetic processes in radio electronics devices;

To develop solid skills in the analysis of steady-state and transient processes in circuits, as well as in conducting experiments to determine the characteristics and parameters of electrical circuits.

The discipline consists of 5 sections:

1 Signals;

2 Passage of signals through linear circuits;

3 Nonlinear and parametric circuits;

4 Chains with feedback and self-oscillating circuits

5 Principles digital filtering signals

3. A BRIEF HISTORY OF THE DEVELOPMENT OF THE DISCIPLINE

The emergence of the theory of electrical and radio circuits is inextricably linked with practice: with the formation of electrical engineering, radio engineering and radio electronics. Many domestic and foreign scientists contributed to the development of these areas and their theories.

The phenomena of electricity and magnetism have been known to man for a long time. However, in the second half of the 18th century, they began to be studied seriously, and the aura of mystery and supernaturalism began to be stripped from them.

Already Mikhail Vasilievich Lomonosov (1711 - 1765) assumed that in nature there is only electricity and that electrical and magnetic phenomena are organically related to each other. Russian academician Frans Epinus made a great contribution to the science of electricity (1724 - 1802).

The rapid development of the doctrine of electromagnetic phenomena occurred in XIX century, caused by the intensive development of machine production. At this time, humanity invents for its practical needs the TELEGRAPH, TELEPHONE, ELECTRIC LIGHTING, WELDING OF METAL, ELECTRICAL MACHINE GENERATORS and ELECTRIC MOTORS.

Let us indicate in chronological order the most striking stages in the development of the doctrine of electromagnetism.

In 1785 French physicist Charles Coulomb Answer (1736 - 1806) established the law of mechanical interaction electric charges(Coulomb's law).

In 1819 Dane Ørsted Hans Christian (1777 - 1851) discovered the effect of electric current on a magnetic needle, and in 1820 French physicist Ampere Andre Marie (1775 - 1836) established a quantitative measure (force) acting from the magnetic field on a section of a conductor (Ampere’s law).

In 1827 German physicist Ohm Georg Simon (1787 - 1854) obtained experimentally the relationship between tone and voltage for a section of a metal conductor (Ohm's law).

In 1831 English physicist Michael Faraday (1791 - 1867) established the law of electromagnetic induction, and in 1832 Russian physicist Lenz Emilius Christianovich (1804 - 1865) formulated the principle of generality and reversibility of electrical and magnetic phenomena.

In 1873 year, based on a generalization of experimental data on electricity and magnetism, the English scientist J. C. Maxwell put forward the hypothesis of the existence of electromagnetic waves and developed a theory to describe them.

In 1888 German physicist Hertz Heinrich Rudolf (1857 - 1894) experimentally proved the existence of radiation of electromagnetic waves.

Practical use radio waves were first realized by the Russian scientist Alexander Stepanovich Popov(1859 - 1905), who May 7, 1895 demonstrated at a meeting of the Russian Physics - Chemical Society transmitter (spark device) and receiver of electromagnetic waves (lightning detector) .

At the end of the XIX centuries, famous engineers and scientists worked in Russia Lodygin Alexander Nikolaevich (1847 - 1923), created the world's first incandescent lamp (1873); Yablochkov Pavel Nikolaevich (1847 - 1894), who developed the electric candle (1876); Dolivo-Dobrovolsky Mikhail Osipovich (1861 - 1919), who created a three-phase current system (1889) and founded modern energy.

In the XIX century, the analysis of electrical circuits was one of the tasks of electrical engineering. Electric circuits were studied and calculated according to purely physical laws that describe their behavior under the influence of electric charges, voltages and currents. These physical laws formed the basis of the theory of electrical and radio circuits.

In 1893 - 1894 years, through the works of C. Steinmetz and A. Kennelly, the so-called symbolic method was developed, which was first applied to mechanical vibrations in physics, and then transferred to electrical engineering, where complex quantities began to be used for a generalized representation of the amplitude-phase picture of a steady sinusoidal oscillation.

Based on the work of Hertz(1888) and then Pupina (1892) by resonance and tuning RLC circuits and related oscillatory systems, problems arose in determining transfer characteristics chains.

In 1889 year A. Kennelly developed formally - mathematical method of equivalent transformation of electrical circuits.

In the second half XIX century, Maxwell and Helmholtz developed methods of loop currents and nodal voltages (potentials), which formed the basis for matrix and topological methods of analysis of later times. Very important was Helmholtz’s definition of the principle of SUPERPOSITION, i.e. separate consideration of several simple processes in the same circuit, followed by algebraic summation of these processes into a more complex electrical phenomenon in the same circuit. The superposition method made it possible to theoretically solve a large range of problems that were previously considered unsolvable and could only be examined empirically.

The next significant step in the development of the theory of electrical and radio circuits was the introduction to 1899 the concept of complex resistance of an electrical circuit to alternating current.

An important step the formation of the theory of electrical and radio circuits was a study frequency characteristics chains. The first ideas in this direction are also associated with the name of Helmholtz, who used the principle of superposition and the method of harmonic analysis for analysis, i.e. applied the expansion of the function in a Fourier series.

At the end of the XIX century, the concepts of T- and P-shaped circuits were introduced (they began to be called quadripoles). Almost simultaneously with this, the concept of electrical filters arose.

The foundation of the modern theory of radio circuits and radio engineering in general was laid by our compatriots M.B. Shuleikin, B.A. Vedensky, A.I. Berg, A.L. Mints, V.A. Kotelnikov, A.N. Mandelstamm, N.D. .Papalexi and many others.

4 GENERAL METHODS OF WORK ON THE COURSE, TYPES OF CLASSES, REPORTING FORMS, TRAINING LITERATURE

The discipline is studied through lectures, laboratory and practical exercises.

Lectures are one of the most important types of educational activities and with O form the basis of theoretical training. They provide a systematized foundation of scientific knowledge in the discipline, focus the attention of teaching e on the most complex and key issues, stimulate their active cognitive activity, and form creative thinking.

At the lectures, along with fundamentality, the need for And May degree of practical orientation of training. The presentation of the material is linked to military practice, specific objects of special equipment in which electrical circuits are used.

Laboratory classes are aimed at teaching students methods of With perimental and scientific research, instill skills in scientific analysis and generalization of the results obtained, skills in working with laboratory equipment O mining, instrumentation and computing x no one.

When preparing for laboratory classes, students independently or (if necessary) at targeted consultations study the compliance yu general theoretical material, the general procedure for conducting research, prepare report forms (draw a diagram of the laboratory setup, the necessary tables).

The experiment is the main part of laboratory work and real And is learned by each student independently in accordance with the laboratory manual. Before the experiment is carried out n troll survey in the form of a meeting, the purpose of which is to check the quality of training O preparing students for laboratory work. In this case, it is necessary to pay attention to knowledge of theoretical material, the order of work, and the nature of the expected results. When receiving reports, you should consider: To accuracy of registration, students’ compliance with ESKD requirements, cash And and the correctness of the necessary conclusions.

Practical classes are conducted with the aim of developing skills in solving e research of problems, production of calculations. Their main content is the right To tical work of each student. Butts are brought out for practical lessons A chi, having an applied nature. Increasing the level of computer software d cooking is carried out in practical classes by performing calculations e com using programmable microcalculators or personal computers. At the beginning of each lesson, a quiz is conducted, the purpose of which is O rogo - checking students’ preparedness for the lesson, as well as - activating A tion of their cognitive activity.

In the process of mastering the content of the discipline, students systematically And Methodological skills and independent work skills are formally developed. Students are instilled with the ability to correctly ask a question, put a O the simplest task, report the essence of the work done, use it to With Coy and visual aids.

To instill primary skills in preparing and conducting training sessions, it is envisaged to involve students as assistant supervisors of laboratory classes.

Among the most important areas for enhancing cognitive development I Problem-based learning is a part of students' activity. To implement it with O problem situations are presented for the course as a whole, for individual topics and in O questions that are being implemented:

By introducing new problematic concepts, showing how they emerged historically and how they are applied;

By confronting the student with contradictions between new phenomena e niami and old concepts;

With the need to choose necessary information;

Using contradictions between existing knowledge on p e the results of the decision and the requirements of practice;

Presentation of facts and phenomena that are inexplicable at first glance

using known laws;

By identifying interdisciplinary connections and connections between phenomena.

In the process of studying the discipline, control of the assimilation of the material is provided for in all practical types of classes in the form of flights, and for topics 1 and 2 in the form of a two-hour test.

To determine the quality of training as a whole in the discipline, conduct T Xia exam. Students who have fulfilled all requirements are allowed to take the exam. curriculum, who reported on all laboratory work, get V highest positive ratings for course work. Exams are conducted in-house T formal form with the necessary written explanations on the blackboard (formulas, graphs, etc.). Each student is given no more than 30 minutes to prepare. To prepare for the answer, students can use O provide methodological and reference materials authorized by the head of the department e rials. Preparation for the answer can be done in writing. The head of the department may exempt students from taking the exam who have demonstrated T personal knowledge based on the results of current control, with a rating given to them nki "excellent".

Thus, the discipline "Radio Engineering Circuits and Signals" is I is provided by a system of concentrated and at the same time quite complete and A perfect knowledge allowing the radio engineer to freely navigate critical issues operation of special radio devices and systems.

BASIC LITERATURE:

1. Baskakov S.I. Radio engineering circuits and signals. 3rd edition. M.: Higher School, 2000.

ADDITIONAL READING

2. Baskakov S.I. Radio engineering circuits and signals. Guide to solving problems: Proc. manual for radio engineering. specialist. universities - 2nd edition. M.: Higher school o la, 2002.

3. POPOV V.P. Basics of circuit theory. Textbook for universities.-3rd ed. M.: Higher school o la, 2000.

5 SIGNAL ENERGY CHARACTERISTICS

The main energy characteristics of a real signal are:

1) instantaneous power, defined as the square of the instantaneous value of the signal

If voltage or current, then instantaneous power released across the resistance and 1 Ohm.

Instantaneous power is not additive, i.e. the instantaneous power of the sum of signals is not equal to the sum of their instantaneous powers:

2) energy over a time interval is expressed as an integral of instantaneous power

3) the average power over an interval is determined by the value of the signal energy over this interval per unit time

Where.

If the signal is given over an infinite time interval, then the average power is determined as follows:

Information transmission systems are designed so that information is transmitted with less than specified distortions with minimal energy and signal power.

The energy and power of signals determined over an arbitrary time interval can be additive if the signals over this time interval are orthogonal. Let's consider two signals and, which are specified on the time interval. The energy and power of the sum of these signals are expressed as follows:

, (1)

. (2)

Here, and, energy and power of the first and second signals, — mutual energy and mutual power of these signals (or energy and power of their interaction). If the conditions are met

then the signals on the time interval are called orthogonal, and the expressions(1) and (2) take the form

The concept of orthogonality of signals is necessarily associated with the interval of their definition.

In relation to complex signals, the concepts of instantaneous power, energy and average power are also used. These quantities are introduced so that the energy characteristics of the complex signal are real quantities.

1. Instantaneous power is determined by the product of the complex signalto a complex conjugate signal

2. Signal energyover a time interval is, by definition, equal to

3. Signal strengthon the interval is defined as

Two complex signals and, given over a time interval, are orthogonal if their mutual power (or energy) is zero.

6 CORRELATION CHARACTERISTICS OF DETERMINISTIC SIGNALS

One of the most important temporal characteristics of a signal is the autocorrelation function (ACF), which allows one to judge the degree of connection (correlation) of the signal with its time-shifted copy.

For a real signal specified over a time intervaland limited in energy, the correlation function is determined by the following expression:

, (3)

Where - the amount of signal time shift.

For each value, the autocorrelation function is expressed by a certain numerical value.

From (3) it follows that the ACF is an even function of the time shift. Indeed, replacing in (3) variable on, we get

When the similarity of the signal with its unshifted copy is greatest, the functionreaches a maximum value equal to the total signal energy

With an increase, the function of all signals, except periodic ones, decreases (not necessarily monotonically) and with a relative shift of the signals and by an amount exceeding the duration of the signal, it becomes zero.

The autocorrelation function of a periodic signal is itself a periodic function with the same period.

To assess the degree of similarity of two signals, the cross-correlation function (MCF) is used, which is determined by the expression

Here and signals given over an infinite time intervaland having finite energy.

The value does not change if instead of the delay of the signal we consider the advance of the first signal.

The autocorrelation function is a special case of the VCF, when the signals and are the same.

In contrast, the function in the general case is not relatively even and can reach a maximum of three at any time.

The value determines the mutual energy of the signals and

7 GEOMETRIC METHODS IN SIGNAL THEORY

When solving many theoretical and applied problems in radio engineering, the following questions arise: 1) in what sense can we talk about the magnitude of a signal, saying, for example, that one signal is significantly superior to another; 2) Is it possible to objectively assess how “similar” two unequal signals are to each other?

In XX V. functional analysis was created — a branch of mathematics that summarizes our intuitive ideas about the geometric structure of space. It turned out that the ideas of functional analysis make it possible to create a coherent theory of signals, which is based on the concept of a signal as a vector in a specially constructed infinite-dimensional space.

Linear space of signals. Let -many signals. The reason for combining these objects — the presence of some properties common to all elements of the set.

The study of the properties of signals that form such sets becomes especially fruitful when it is possible to express some elements of the set through other elements. It is commonly said that many signals are endowed with a certain structure. The choice of one structure or another should be dictated by physical considerations. Thus, in relation to electrical oscillations, it is known that they can be added and also multiplied by an arbitrary scale factor. This makes it possible to introduce the structure of linear space in sets of signals.

The set of signals forms a real linear space if the following axioms are true:

1. Any signal takes only real values ​​at any value.

2. For any and there is their sum, and is also contained in. The summation operation is commutative: and associative: .

3. For any signal and any real number a signal is defined=.

4. The set M contains a special zero element , such that  for everyone.

If mathematical models signals take complex values, then, assuming in the axiom 3 multiplication by a complex number, we arrive at the concept of a complex linear space.

The introduction of the structure of linear space is the first step towards a geometric interpretation of signals. Elements of linear spaces are often called vectors, emphasizing the similarity between the properties of these objects and ordinary three-dimensional vectors.

The restrictions imposed by the axioms of linear space are very strict. Not every set of signals turns out to be a linear space.

The concept of a coordinate basis. As usual three-dimensional space, in the linear space of signals we can select a special subset that plays the role of coordinate axes.

It is said that the collection of vectors (}, belonging is linearly independent if the equality

is possible only in the case of simultaneous vanishing of all numerical coefficients.

A system of linearly independent vectors forms a coordinate basis in linear space. If a decomposition of some signal is given in the form

then numbers() are projections of the signal relative to the selected basis.

In problems of signal theory, the number of basis vectors is, as a rule, unlimitedly large. Such linear spaces are called infinite-dimensional. Naturally, the theory of these spaces cannot be embedded in the formal scheme of linear algebra, where the number of basis vectors is always finite.

Normalized linear space. Signal energy. In order to continue and deepen the geometric interpretation of the theory of signals, it is necessary to introduce a new concept, which in its meaning corresponds to the length of the vector. This will not only give the exact meaning of a statement like “the first signal is greater than the second,” but also indicate how much greater it is.

The length of a vector in mathematics is called its norm. The linear space of signals is normalized if each vector is uniquely associated with a number — the norm of this vector, and the following axioms of a normed space are satisfied:

1. The norm is non-negative, i.e.. Normal if and only if .

2. For any number the equality is true.

3. If and are two vectors from , then the triangle inequality holds: .

Can you suggest different ways introduction of signal norms. In radio engineering it is most often believed that real analog signals have the norm

(4)

(from two possible values ​​of the root, the positive one is chosen). For complex signals the norm is

where * symbol for a complex conjugate quantity. The square of the norm is called the signal energy

It is this energy that is released in a resistor with a resistance 1 Ohm, if there is voltage at its terminals.

Determine the signal norm using the formula (4) advisable for the following reasons:

1. In radio engineering, the magnitude of a signal is often judged based on the total energy effect, for example, the amount of heat generated in a resistor.

2. The energy norm turns out to be “insensitive” to changes in the signal shape, perhaps significant, but occurring over short periods of time.

Linear normed space with a finite norm of the form (1.15) is called the space of functions with square integrable and is briefly denoted.

8 THEORY OF ORTHOGONAL SIGNALS. GENERALIZED FOURIER SERIES

Having introduced the structure of linear space in a variety of signals, defining the norm and metric, we are nevertheless deprived of the opportunity to calculate such a characteristic as the angle between two vectors. This can be done by formulating the important concept of the scalar product of elements of a linear space.

Dot product of signals. Recall that if two vectors and are known in ordinary three-dimensional space, then the squared modulus of their sum

where is the scalar product of these vectors, depending on the angle between them.

Using analogy, we calculate the energy of the sum of two signals and:

. (5)

Unlike the signals themselves, their energies are non-additive - the energy of the total signal contains the so-called mutual energy

. (6)

Comparing formulas(5) and (6), Let's define the scalar product of real signals and:

The scalar product has the following properties:

1. , where is a real number;

A linear space with such a scalar product, complete in the sense that it contains all the limit points of any convergent sequences of vectors from this space, is called a real Hilbert space.

The fundamental Cauchy inequality is true Bunyakovsky

If the signals take complex values, then we can define a complex Hilbert space by introducing the scalar product in it using the formula

such that.

Orthogonal signals and generalized Fourier series. Two signals are called orthogonal if their scalar product, and therefore their mutual energy, is equal to zero:

Let Hilbert space of signals with finite energy. These signals are defined over a period of time, finite or infinite. Let us assume that an infinite system of functions is given on the same segment, orthogonal to each other and having unit norms:

They say that in this case an orthonormal basis is given in the space of signals.

Let's expand an arbitrary signal into a series:

(7)

Performance (7) is called the generalized Fourier series of the signal in the chosen basis.

The coefficients of this series are found as follows. Let's take a basis function with an arbitrary number and multiply both sides of the equality by it (7) and then integrate the results over time:

. (8)

Due to the orthonormality of the basis on the right side of the equality (8) only the sum term with number will remain, so

The possibility of representing signals using generalized Fourier series is a fact of great fundamental importance. Instead of studying the functional dependence at an uncountable set of points, we are able to characterize these signals with a countable (but, generally speaking, infinite) system of coefficients of the generalized Fourier series.

The energy of the signal, represented in the form of a generalized Fourier series. Let us consider some signal expanded into a series according to an orthonormal basis system:

and calculate its energy by directly substituting this series into the corresponding integral:

(9)

Since the basis system of functions is orthonormal, in total (9) Only members with numbers will be different from zero. This gives a wonderful result:

The meaning of this formula is as follows: the signal energy is the sum of the energies of all components that make up the generalized Fourier series.

Senior lecturer of the Department of Radioelectronics S. Lazarenko

The transmission speed of measurement information determines the efficiency of the communication system included in the measurement system.

Simplified diagram measuring system shown in Fig. 175.

Typically, the primary measuring transducer converts the measured quantity into an electrical signal X (t), which needs to be sent by communication channel. Depending on what the communication channel is (electrical wire or cable, light guide, water medium, air or airless space), the carriers of measurement information can be electric current, a beam of light, sound vibrations, radio waves, etc. Selecting a carrier is the first step in matching the signal with the channel.

The generalized characteristics of the communication channel are time T to, in during which it is provided for transmitting measurement information, bandwidth F to and dynamic range N to, which is understood as the ratio of the permissible power in the channel to the power of the interference inevitably present in the channel, expressed in decibels. Work

called channel capacity.

Similar generalized signal characteristics are time T s, during which measurement information is transmitted, spectrum width Fc and dynamic range Nc is the ratio of the highest signal power to the lowest power that must be distinguished from zero for a given transmission quality, expressed in decibels. Work

called signal volume.

The geometric interpretation of the introduced concepts is shown in Fig. 176.

The condition for matching a signal with a channel that ensures the transmission of measurement information without loss and distortion in the presence of interference is the fulfillment of the inequality

when the signal volume completely “fits” into the channel capacity. However, the condition for matching the signal with the channel can be satisfied even when some (but not all) of the last inequalities are not satisfied. In this case, there is a need for the so-called exchange transactions, in which there is a kind of “exchange” of the duration of the signal for the width of its spectrum, or the width of the spectrum for the dynamic range of the signal, etc.

Example 82. A signal having a spectral width of 3 kHz must be transmitted over a channel whose bandwidth is 300 Hz. This can be done by first recording it on magnetic tape and playing it back during transmission at a speed 10 times lower than the recording speed. In this case, all frequencies of the original signal will decrease by 10 times, and the transmission time will increase by the same amount. The received signal will also need to be recorded on magnetic tape. By then playing it back at 10 times the speed, it will be possible to reproduce the original signal.

In a similar way you can short time transmit a long signal if the channel bandwidth is wider than the signal spectrum.

In channels with additive uncorrelated interference

where P c and P p are the signal and interference powers, respectively. When transmitting electrical signals, the ratio

can be considered as the number of signal quantization levels that ensure error-free transmission. Indeed, with the selected quantization step, a signal of any level cannot be mistaken for a signal of an adjacent level due to the influence of interference. If we now imagine the signal as a set of instantaneous values ​​taken in accordance with V.A.’s theorem. Kotelnikov at intervals D t= ,

then at each of these moments in time it will correspond to one of the levels, i.e. may have one of n equally probable values, which corresponds to entropy

After the receiving device registers one of the levels at a fixed point in time, entropy (a posteriori) will be equal to 0, and the quantum of information (the amount of information transmitted at a discrete point in time)

Since the entire signal is transmitted N= 2 F c T with quanta, then the amount of information contained in it

directly proportional to the volume of the signal. To transmit this information in time Tk, it is necessary to ensure the transmission speed

If the signal and channel are consistent and T c = T c; F c = F k, then

This K. Shannon's formula for the maximum channel capacity.It sets the maximum speed for error-free information transfer. At T c< T к скорость может быть меньшей, а при Т с >Errors are possible.

Limit dependence bandwidth channel from the signal-to-noise ratio for several values ​​of bandwidth is shown in Fig. 177. The nature of this dependence is different for large and small ratios

those. The dependence of the channel capacity on the signal/noise ratio is logarithmic.

If “1, then despite the fact that R p » R c , error-free transmission is still possible, but with very low speed. In this case, the expansion is valid

in which we can restrict ourselves to the first term. Taking into account that log e = 1.443, we get

Thus, for small signal-to-noise ratios, the dependence of throughput on the signal-to-noise ratio is linear.

The dependence of throughput on channel bandwidth in real systems is more complex than just linear. The power of noise interference at the input of the receiving device depends on the channel bandwidth. If the interference spectrum is uniform, then

where G is the spectral power density of the interference, i.e. interference power per unit frequency band. Then

The signal power can be expressed in terms of the same spectral density if we take into account equivalent frequency band F e:

Dividing both sides of this expression by F e, we get:

The nature of this dependence is shown in Fig. 178. It is important to note that as the channel bandwidth increases, its capacity does not increase indefinitely, but tends to a certain limit. This is explained by increased noise in the channel and a deterioration in the signal-to-noise ratio at the input of the receiving device. The limit to which c tends with increasing Fk can be determined by using the already known expansion for large Fk logarithmic function in a row. Then if

Thus, maximum value, to which the maximum channel capacity tends as its bandwidth increases, is proportional to the ratio of the signal power to the interference power per unit frequency band. This obviously leads to the following practical conclusion: to increase the maximum channel capacity, you need to increase the power of the transmitting device and use receiver with minimal noise level at the input.

Along with efficiency, the second most important indicator of the quality of a communication system is noise immunity. When transmitting measurement information in analog form, it is evaluated by deviation received signal from the transmitted. The noise immunity of discrete communication channels is characterized by probability of error Rosh (the ratio of the number of erroneously accepted characters to total number transmitted) and is associated with dependence

If, for example, Рosh = 10 -5, then æ = 5; if Rosh = 10 -6, then æ = 6.

An effective way to increase noise immunity when transmitting measurement information in analog form and uncorrelated interference is accumulation. The signal is transmitted several times and with the coherent addition of all received implementations, its values ​​at the corresponding times are summed up, while the interference at these times, being random, is partially compensated. As a result, the signal-to-noise ratio increases and noise immunity increases. Similarly, the idea of ​​accumulation is implemented when transmitting measuring information over a discrete channel.

Example 83. Let the nature of the interference be such that it can be mistaken for a signal (i.e., 0 can be mistaken for 1). When transmitted by Baudot code, the combination 01001 is received three times in the form:

If the adder is a device that does not operate when at least one zero appears in the column, then the combination will be accepted correctly provided that each zero was accepted correctly at least once.

If during one transmission the probability of independent errors is denoted by Posh, then after N- If the transmission is repeated multiple times, it will be equal to Rosh. Therefore, noise immunity after N retransmissions

where æ - Noise immunity during single transmission. Thus, the noise immunity during accumulation increases by the number of repetitions.

One of the ways to increase noise immunity is also application of correction codes.

Increasing noise immunity is achieved by increasing redundancy, and more generally, by increasing the signal volume with the same amount of measurement information. In this case, the condition of matching the signal with the channel must be maintained. If this condition is met and T c = T k; Н с = Н к transmission of measuring information using amplitude-modulated high-frequency oscillations is more noise-resistant than direct signal transmission, because in the case of, for example, tone modulation it occupies twice the frequency band. In turn, the use of deep frequency or phase modulation, due to spectrum expansion, further increases the noise immunity of the communication system. In this sense, it is promising to use not simple signals that

F c T c ≈ 1,

A complex, for which

These include pulse signals with high-frequency filling and frequency modulation or phase shift keying of carrier oscillations, etc.

The requirements for efficiency and noise immunity of communication systems are contradictory. They encourage, on the one hand, to reduce, and on the other hand, to increase the volume of the signal, without violating the conditions of its coordination with the channel and without changing the amount of information contained in it. Satisfying these requirements involves the synthesis of optimal technical solutions.

When studying the generalized theory of signals, the following questions are considered.

1. Basic characteristics and methods of analyzing signals used in radio engineering to transmit information.

2. The main types of signal transformations in the process of building channels.

3. Methods for constructing and methods for analyzing radio circuits through which operations are performed on the signal.

Radio engineering signals can be defined as signals that are used in radio engineering. According to their purpose, radio signals are divided into signals:

radio broadcasting,

television,

telegraph,

radar,

radio navigation,

telemetry, etc.

All radio signals are modulated. When generating modulated signals, primary low-frequency signals (analog, discrete, digital) are used.

Analog signal repeats the law of change in the transmitted message.

Discrete signal – the message source transmits information at certain time intervals (for example, about the weather), in addition, a discrete source can be obtained as a result of time sampling of an analog signal.

Digital signal is the display of a message in digital form. Example: we encode a text message into a digital signal.

All message characters can be encoded into binary, hexadecimal and other codes. Encoding is carried out automatically using an encoder. Thus, the code symbols are converted into standard signals.

The advantage of digital data transmission is its high noise immunity. The reverse conversion is carried out using a digital-to-analog converter.

Mathematical models of signals

When studying the general properties of signals, one usually abstracts from their physical nature and purpose, replacing them with a mathematical model.

Mathematical model – the chosen method of mathematical description of the signal, reflecting the most essential properties of the signal. Based on a mathematical model, it is possible to classify signals in order to determine their common properties and fundamental differences.

Radio signals are usually divided into two classes:

deterministic signals,

random signals.

Deterministic signal is a signal whose value at any time is a known quantity or can be calculated in advance.

Random signal is a signal whose instantaneous value is random variable(for example, a beep).

Mathematical models of deterministic signals

Deterministic signals are divided into two classes:

periodic,

non-periodic.

Let s ( t ) – deterministic signal. Periodic signals are described by a periodic function of time:

and repeat after a period T . Approximately t >> T . The remaining signals are non-periodic.

A pulse is a signal whose value is different from zero for a limited time interval (pulse duration ).

However, when describing a mathematical model, functions defined over an infinite time interval are used. The concept of effective (practical) pulse duration is introduced:

.

Exponential momentum.

For example: defining the effective duration of an exponential pulse as the time interval during which the signal value decreases by a factor of 10. Determine the effective pulse duration for the pattern:

Energy characteristics of the signal . Instantaneous power is the signal power at a resistance of 1 ohm:

.

For a non-periodic signal, we introduce the concept of energy at a resistance of 1 Ohm:

.

For a periodic signal, the concept of average power is introduced:

The dynamic range of a signal is defined as the ratio of the maximum P ( t ) to that minimum P ( t ) , which allows you to ensure a given transmission quality (usually expressed in dB):

.

The calm speech of a speaker has a dynamic range of approximately 25...30 dB, for a symphony orchestra up to 90 dB. Selecting a value P min related to the level of interference:
.

Messages and their corresponding signals can be continuous or discrete in structure.

Continuous signals are defined by an infinite set of values ​​over a finite time interval. Such signals are described over some sufficiently large time interval by continuous functions of time. A typical example of a continuous signal would be a telephone signal indicating speech, music, temperature changes, etc. (Figure 1.2).

Discrete signals are those characterized by a finite number of values ​​over the time interval of their existence. An example of a discrete signal is telegraph communication signals that display letters of the alphabet and signs with certain combinations of discrete signal states (Fig. 1.3).

Rice. 1.2. Telephone signal Fig. 1.3. Telegraph signals

It should be noted that any continuous signal for transmitting a message can be sampled with a certain accuracy. This possibility is based on the fact that all real signals have limited frequency spectra, that is, they are described by functions with a finite set of values ​​over a finite time interval.

Functions describing communication signals can be periodic or non-periodic functions of time. From the course of the theory of radio signals it is known that a signal (function) of any kind can be decomposed into harmonic components: periodic signals - using Fourier series, non-periodic - using the Fourier integral.

The set of amplitudes of harmonic components is called the amplitude spectrum or simply the signal spectrum.

To analyze signals, it is more convenient to use incomplete analytical descriptions signals (the full implementation of which is not always possible), but by some generalized indicators or parameters.

Such generalized physical parameters of the signal are:

– signal duration ;

– frequency spectrum width;

– dynamic range;

Duration characterizes the lifetime of the signal and, therefore, the time for which it is necessary to provide a communication channel for transmitting the signal.

The frequency spectrum width characterizes the shape of the signal and the channel bandwidth that is necessary to transmit the signal through the channel.

The dynamic range of the signal D characterizes the excess of the signal power over the power of the corresponding interference signal, written in logarithmic form:

More precisely, the dynamic range of a signal should be considered the logarithm of the ratio of its highest instantaneous power and lowest instantaneous power. But since in a communication channel the minimum signal power must always exceed the interference power, the excess of the signal over the interference was chosen as a generalized parameter.

Primary telephone signals

Telephone signals are the result of conversion of voice messages. They represent a continuously varying current (voltage), which unambiguously reflects sound vibrations. Primary telephone signals refer to continuous, non-periodic signals.

Human speech contains sound vibrations in the range from 80 to 10,000 Hz, and the hearing aid is capable of perceiving sound vibrations in the range from 16 to 20,000 Hz. It has also been established that the main part of the average signal power, which ensures the loudness of speech, is concentrated in the range from 300 to 600 Hz; the remaining frequency components of the spectrum provide the color of the sound.

The average energy spectrum of the speech signal is shown in Fig. 1.4.

Rice. 1.4. Average spectrum of a speech signal

Based on the given digital data, in military communications technology, in order to obtain sufficient volume and speech intelligibility, the signal is limited to a band of 300...3400 Hz. This band is standard for military radio communications and has special name EPFC is an efficiently transmitted frequency band.

In radio broadcasting on long, medium and short waves, the EPFR will be 50...4500 Hz, in the VHF range - 30...10000 Hz, and for the transmission of stereophonic programs - 20...20000 Hz.

The first characteristic of the signal is its duration T s.

The second characteristic is the dynamic range of the signal, showing the excess of the signal over the noise (threshold)

For example, speech (when transitioning from a whisper to a scream) has a dynamic range D = 50 dB.

The third indicator is the signal spectrum width

, (1.46)

where are the upper and lower boundaries of the signal spectrum.

A generalized characteristic of a signal is the volume of the signal, determined by the expression

V c =T c *D c * F c . (1.47)

To resolve the issue of the possibility of signal transmission over the communication channel, the same characteristics are introduced for the channel T to, D to, F to, V to

V to =T to *D to *F to, (1.48)

where Tk is the time of channel use,

Dk - dynamic range of the channel (ability to transmit different levels),

F to - bandwidth of frequencies transmitted by the channel,

Vc - channel capacity.

Transfer is possible subject to the following conditions:

T to T s; D to D with; F to F s. (1.49)

This requirement can be relaxed by writing

V to V c (1.50)

The process of modifying the characteristics of a signal to ensure the possibility of transmitting it over a given channel is called matching the signal with the communication channel.

Signal modulation

Classification of modulation types

Message submitted electrical signal, must be transmitted over a certain distance (including a fairly large one). For this purpose, carrier signals are used. The energy of the carriers must be sufficient to transmit over a given distance.

Thus, the transformation of signals during transmission consists of influencing the carrier, changing one or another of its parameters. This effect is called modulation.

Different types of modulation are characterized various types carriers, as well as a number of parameters subject to change.

According to the type of carriers, they are distinguished:

§ modulation of sinusoidal (harmonic) signals;

§ modulation of pulse signals.

According to the changeable parameters there are:

§ amplitude modulation;

§ frequency modulation;

§ phase modulation;

§ code modulation, etc.

In cases where a continuous message is transmitted in discrete (digital) form, a preliminary conversion of the continuous message into a discrete one is carried out, including sampling (quantization) in time and level.

Amplitude modulation

Amplitude modulation is characterized by a change in the amplitude A 0 of the carrier according to the law of the signal of the transmitted message

, (1.51)

where is the largest change in amplitude during modulation,

f(t) is a function expressing the law of change in time of the transmitted message.

Then the amplitude-modulated harmonic signal will have the following form

, (1.52)

where is the depth amplitude modulation.

In the case when

Signal spectrum width.

If the modulation of a harmonic signal is carried out according to a more complex law, and the spectrum of the amplitude envelope is in the frequency range from (lower) to (upper), then it can be shown that in the spectrum of an amplitude-modulated (AM) signal instead of two side frequencies there will be two side frequency bands : bottom and top . For undistorted transmission of such a signal, the communication channel must have a frequency bandwidth equal to , i.e. the bandwidth should be twice as large highest frequency spectrum of the modulating signal.

To reduce the frequency band of the modulated signal, and, consequently, the required bandwidth of the communication channel, so-called single-sideband transmission is used. With such transmission, filters are used to suppress the carrier frequency and the frequencies of one of the side bands, i.e. transmission is carried out in the frequency band - .

With amplitude modulation, the condition must be met. Failure to comply with this condition leads to specific distortions - the so-called overmodulation.

Frequency modulation

With frequency modulation, the frequency of the signal changes according to the law

, (1.54)

where is the constant frequency component;

Frequency modulation depth;

The largest change in frequency during modulation (frequency deviation);

Law of modulation.

A frequency-modulated signal can be represented as

where is the current phase of the harmonic signal;

Initial phase.

In the special case when frequency modulation is carried out according to the harmonic law

, (1.56)

where is the frequency modulation index.

A signal of this type is represented as follows:

where is the zero-order Bessel function;

Bessel function of kth order.

The amplitude spectrum of the frequency-modulated signal is shown in Fig. a) b). The amplitude spectrum of a frequency-modulated signal is discrete and consists of side frequencies. The distribution of the amplitudes of the harmonic components depends on - the frequency modulation index.

When very small, the FM spectrum does not differ from the AM spectrum, i.e. contains , , . With growth, the weight of the side components increases and, accordingly, the required bandwidth of the communication channel increases.

If we limit ourselves in the spectrum to only components whose amplitude is at least 5-10% of before modulation, then the spectrum width of these components will be .

At large (>>1), the spectrum width is practically equal to twice the frequency deviation and does not depend on the frequency of the modulating signal.

Phase modulation

With phase modulation, the phase of the harmonic signal changes according to the law of the modulating signal

, (1.57)

where is the phase modulation depth ,

The largest phase change during modulation (phase modulation index).

where is the frequency deviation of the FM signal.

A phase-modulated signal can be represented as

Since the instantaneous frequency value is the derivative of the phase with respect to time

then with phase modulation it can be represented in the form

. (1.59)

Thus, an FM signal is equivalent to an FM signal with a modulating function.

For the frequency case, when the modulating signal is harmonic, the total phase of the PM signal is determined by the equality

and the FM oscillation is described by the expression

Instantaneous frequency value of the FM signal

where is the frequency deviation of the FM signal. Comparison of expressions (1.61) and (1.56) shows that with a harmonic modulating signal, the FM and PM signals differ only in the phase of the harmonic function, which determines the change in the total phase of the RF. fluctuations. Those. By appearance signal cannot be concluded that it is FM or FM?

However, when changing the modulation frequency, the difference between FM and PM appears.

In FM, the magnitude of the frequency deviation depends only on the amplitude of the modulating signal and does not depend on the modulation frequency. The value of the index decreases with increasing frequency.