Radio engineering circuits and signals of Baskaks. Radio signals and circuits
Ministry of Education and Science of the Russian Federation
Federal State Budgetary Educational Institution
higher professional education
National Mineral Resources University "Mining"
S. I. Malinin Radio engineering circuits and signals
lecture notes
ST. PETERSBURG
UDC 621.396.6:681.3
BBK z 844.1я73-5
Lecture notes have been developed in accordance with the requirements of the state educational standard for higher professional education.
The purpose of studying the discipline is to equip students with knowledge in the field of synthesis and analysis of various radio circuits and mastering the principles of ensuring noise immunity when transmitting, receiving and reproducing signals.
The main objective of the discipline is to study the principles of generation, amplification, radiation and reception of electromagnetic waves related to the radio range; practical use of these waves for the purposes of transmitting, storing and converting information.
Lecture notes are intended for students of specialty 210601.65 - radio-electronic systems and complexes.
Malinin S.I.
D 33. Radio engineering circuits and signals/. National Mineral Resources University "Mining". St. Petersburg, 2013, 226 p.
UDC 621396.6:681.3.
BBK z 844.1я73-5
© National Mineral Resources
University "Mining", 2013
Introduction
The purpose of studying the discipline is to equip students with knowledge in the field of synthesis and analysis of various radio circuits and mastering the principles of ensuring noise immunity when transmitting, receiving and reproducing signals, the principles of generation, amplification, emission and reception of electromagnetic waves related to the radio range; practical use of these waves for the purposes of transmitting, storing and converting information.
The discipline “Radio Engineering Circuits and Signals” fully complies with the curriculum, belongs to the OPD.F.07 cycle and is studied by students of specialty 210302.65 of all forms of study in two semesters.
Improving lectures on the discipline “Radio Engineering Circuits and Signals” has been and remains an urgent task in connection with the development of radio engineering in general. The novelty of the proposed text of lectures lies in the updated methodology for presenting the material, associated with a deeper coverage of those issues that, in the author’s opinion, were not sufficiently covered in previous editions, and have become relevant to the present time.
This text of lectures does not replace the classical textbooks given in the list of references, but allows you to systematize the material being studied and makes it possible to more freely navigate a large volume of literature.
1. Deterministic radio signals
1.1. The main tasks solved by radio engineering
Radio engineering is a field of science and technology that deals with the study and application of electromagnetic oscillations and radio frequency waves. The radio range includes frequencies below the infrared range (3 THz, which is equal to 3 × 10 12 Hz).
Radio engineering solves many problems, the main one of which is transmitting information over a distance using radio waves.
Radio waves - This electromagnetic waves with frequencies up to 3 THz, propagating in space without artificial guide lines. The development of radio engineering began with the invention of a device for receiving electromagnetic waves (Popov, Marconi).
Radio communication is communication between objects via radio waves. Radio communication happens:
one-sided,
double-sided,
between two objects
between several objects,
between moving or stationary objects
The range of applications of radio technology is constantly expanding. Currently, radio technology provides not only the transmission, but also the receipt of information about the environment. Radio engineering provides impact on natural or technical objects:
radar,
radio navigation,
radio control,
radiotelemetry, etc.
Radar solves the problem of detecting and recognizing various objects, as well as determining their coordinates and movement parameters using radio waves (ships, planes, missiles, structures on the ground, clouds, precipitation, etc.). Radar allows you to accurately measure the distance from the Earth to the Moon and other planets.
Radio navigation solves problems of control and movement along optimal trajectories of various objects. The main tasks solved by radio navigation are determining the optimal course and geographic coordinates of the object.
Radio control provides automatic control of objects at a distance using radio waves, radio engineering methods and funds. Ensures the movement of aircraft in automatic mode (artificial earth satellites, weather probes, etc.).
Radiotelemetry solves problems of measuring at a distance using radio waves, for example, on hard-to-reach objects - radiosondes, earth satellites, etc.
Radio engineering is widely used in medicine; radio engineering methods and devices are widely used in all areas of science and technology.
Information is a collection of information about events, phenomena, objects, intended for transmission, reception, storage, and use. All applications of radio engineering are related to the transmission of information.
To convey information you need to present it in some form. Information presented in this form is called message . There are audio messages, text messages and
A message (information) can be transmitted over a distance using a specific material medium. Various signals act as carriers.
Signals – these are physical processes whose parameter values reflect transmitted messages (electrical oscillations and electromagnetic oscillations and waves).
Radio channel ensures the transmission of a message from one point to another. Basic elements of a radio channel:
transmitter,
receiver,
the physical environment in which radio waves propagate.
Processes that ensure the functioning of a radio channel using the example of a radio communication channel:
1 – Source of message (person, audio cassette...).
2 – Converter of a message into an electrical signal (microphone, tape recorder...). The output of the converter receives message signals. These signals are generally low frequency, and they are not used to excite radio waves, since the size of the antenna must be commensurate with the wavelength. To transmit information, it is modulated. Modulation consists of changing the parameters of high-frequency (secondary signal) oscillations in accordance with the low-frequency (primary) signal. A high frequency signal modified to match a low frequency signal is called a modulated signal.
3 – Radio transmitter (modulator).
High frequency vibrations are emitted by the transmitting antenna. The radio wave becomes the material carrier of the message. Some of the energy is captured by the receiving antenna.
4 – Radio receiver. It is used to receive signals and convert them to their original form (the process of converting a high-frequency modulated signal into a low-frequency signal is called demodulation, or detection). At the output of receiver 4 a low-frequency signal appears, close to the transmitted signal. The low-frequency signal is partially distorted by interference, etc. The receiver is designed in such a way that it attenuates interference as much as possible.
5 – Converter of electrical signal into message.
6 – Recipient of the message.
Basically, all these processes are associated with various signal transformations. The conversion is carried out through radio circuits.
Textbook. - M.: Higher School, 1983. - 536 pp.: illus. The textbook contains a systematic presentation of the sections of theoretical radio engineering included in the course program "Radio Engineering Circuits and Signals".
Issues of the general theory of signals and their spectral representations are considered. Elements of statistical radio engineering and methods for analyzing the passage of signals through linear, nonlinear and parametric systems are presented. The theories of circuits with feedback, self-oscillating systems devices digital processing signals, optimal linear filters.
For students of radio engineering specialties at universities. Can be used by radio engineers and persons improving their qualifications in the field of theoretical radio engineering. Preface
IntroductionRadio signals
Elements of the general theory of radio signals
Classification of radio signals.
Dynamic presentation of signals.
Geometric methods of signal theory.
Theory of orthogonal signals.
Spectral representations of signals
Periodic signals and Fourier series.
Spectral analysis of non-periodic signals. Fourier transform.
Basic properties of the Fourier transform.
Spectral densities of non-integrable signals.
Laplace transform.
Basic properties of the Laplace transform.
Energy spectra of signals. Principles correlation analysis
Mutual spectral density of signals. Energy spectrum.
Correlation analysis of signals.
Autocorrelation function of discrete signals.
Cross correlation function of two signals.
Modulated signals
L Signals with amplitude modulation.
Angle modulated signals.
Signals with intrapulse frequency modulation.
Limited Spectrum Signals
Some mathematical models of signals with a limited spectrum and their properties.
Kotelnikov's theorem.
Narrowband signals.
Analytical signal and Hilbert transform.
Basics of the theory of random signals
Random variables and their characteristics.
Statistical characteristics of systems of random variables.
Random processes.
Correlation theory of random processes
Spectral representations of stationary random processes.
Differentiation and integration of random processes.
Narrowband random processes. Radio circuits, devices and systems
Impact of deterministic signals on linear stationary systems
Physical systems and their mathematical models.
Pulse, transient and frequency characteristics of linear stationary systems.
Linear dynamic systems.
Spectral method.
Operator method.
Impact of deterministic signals on frequency-selective systems
Models of frequency-selective circuits.
Frequency-selective circuits under broadband input influences.
Frequency-selective circuits for narrow-band input influences.
Impact of random signals in linear stationary circuits
Spectral method for analyzing the passage of random signals through linear stationary circuits.
Sources of fluctuation noise in radio engineering devices.
Signal conversion in nonlinear radio circuits
Inertia-free nonlinear transformations.
Spectral composition of the current in an inertia-free nonlinear element under harmonic external influence.
Nonlinear resonant amplifiers and frequency multipliers.
Inertia-free nonlinear transformations of the sum of harmonic signals.
Amplitude modulation. Detection of AM signals.
Impact of stationary random signals on inertia-free nonlinear circuits.
Signal conversions in linear parametric circuits
Passage of signals through resistive parametric circuits.
Energy relationships in parametric reactive circuit elements.
Principles of parametric amplification.
Non-stationary dynamic systems.
Impact of harmonic signals on parametric systems with random characteristics.
Basic theory of synthesis of linear radio circuits
Analytical properties of the input resistance of a linear passive two-terminal network.
Synthesis of passive two-terminal networks.
Frequency characteristics of quadripoles.
Low pass filters.
Implementation of filters.
Active feedback circuits and self-oscillating systems
Transfer function of a linear feedback system.
Stability of feedback circuits.
Active RC filters.
Autogenerators of harmonic oscillations. Small signal mode.
Autogenerators of harmonic oscillations. Large signal mode.
Discrete signals. Principles of digital filtering
Discrete pulse sequences.
Sampling of periodic signals.
Z-transform theory.
Digital filters.
Implementation of digital filtering algorithms.
Synthesis of linear digital filters.
Optimal linear signal filtering
Optimal linear filtering of signals of known shape.
Implementation of matched filters.
Optimal filtering of random signals. Applications
Recommended reading
Subject index
A signal is a physical process that is a function of certain parameters and is used as an information carrier. In radio engineering, two groups of electrical signals are studied: deterministic and random.
The information contained in the signal is displayed by the law of its change in time S (t). If this law is known and predetermined in advance, then the signal is called deterministic (from the Latin determinatio - determination). An example of such a signal is a cosine oscillation described by the function
where S m is the signal amplitude; u=2рf - circular frequency of the signal; c - initial phase of the signal.
For deterministic signals, the value s (t) is known in advance at any time t for given amplitude values, circular frequency and the initial phase.
If the law of signal change s (t) is not predetermined, then it is not known in advance what value it will have at one time or another. The values of such signals at different times are random. That's why they are called random.
The classification of signals is carried out on the basis of the essential features of the corresponding mathematical models signals. All signals are divided into two independent groups: deterministic and random.
Deterministic signals are divided into periodic and non-periodic (pulse). Pulse signal- this is a signal of finite energy, significantly different from zero during a limited time interval commensurate with the time of completion of the transient process in the system to which this signal is intended to influence. Periodic signals can be harmonic, that is, containing only one harmonic, and polyharmonic, the spectrum of which consists of many harmonic components. Harmonic signals include signals described by a sine or cosine function. All other signals are called polyharmonic.
Random signals are signals whose instantaneous values at any time are unknown and cannot be predicted with a probability equal to one. Paradoxical as it may seem at first glance, only a random signal can be a signal carrying useful information. The information in it is contained in a variety of amplitude, frequency (phase) or code changes in the transmitted signal. In practice, any radio signal that contains helpful information, should be considered random.
Most radio signals used in practice are classified as random for two reasons. Firstly, any signal that carries information must be considered random. Secondly, in devices that “work” with signals, there is almost always noise or interference that is superimposed on the useful signal. Therefore, in any communication channel, the useful signal is distorted during transmission and the message on the receiving side is reproduced with some error.
There is no insurmountable boundary between deterministic and random signals. Under conditions of a large useful signal to noise ratio, i.e. in the case when the level of interference is significantly less than the level of the useful signal, the deterministic model of the signal is adequate to the real situation. In this case, it is possible to apply methods for analyzing non-random signals.
In the process of transmitting information, signals can be subjected to one or another transformation. This is usually reflected in their name: signals modulated, demodulated (detected), encoded (decoded), amplified, delayed, sampled, quantized, etc.
According to the purpose that signals have during the modulation process, they can be divided into modulating (the primary signal that modulates the carrier wave) or modulated (carrier wave).
Radio circuits
Radio engineering circuits are a set of passive and active elements connected in a certain way, ensuring the passage and functional transformation of signals.
An electrical circuit arises if sufficiently narrow paths are created in space for electric current, placing along these paths conductors made of materials with high electrical conductivity, surrounded by a well-insulating environment. It is also possible to place circuit elements along the chain, i.e. conductive devices limited in volume (resistors, vacuum tubes, semiconductors), or similarly limited in volume devices with local concentrators of electric and magnetic fields (capacitors, inductors).
The main passive (i.e. those without energy sources inside) elements are:
A) Active resistance R is the element in which irreversible loss occurs electrical energy, i.e. Ohm's law also holds for alternating currents;
b) Capacitance - an element in which the flow of current is accompanied by the accumulation of charges on the plates, and the energy from EMF sources is converted into the energy of the electric field between the plates.
c) Inductance is an element in which the flow of current is accompanied by the transition of electrical energy into the energy of a magnetic field.
Based on the nature of signal conversion in them, circuits are divided into linear with constant parameters, linear-parametric and nonlinear circuits.
Linear circuits are circuits in which all elements are linear, i.e. parameters do not depend on voltage and current values. If these parameters do not change over time, then the circuits are called linear with constant parameters.
Linear-parametric - circuits that contain elements that depend on time due to control external influence, but independent of current and voltage.
Nonlinear - circuits containing at least one nonlinear element, the parameters of which depend on the processes occurring in them (current and voltage levels). Nonlinear circuits are described by nonlinear differential equations.
Basic radio engineering processes
Converting the original message into an electrical signal.
Generation of high-frequency oscillations.
Oscillation control (modulation).
Gain weak signals in the receiver.
Isolating a message from a high-frequency oscillation (detection and decoding).
Radio circuits and methods
their analysis
Circuit classification
And the elements used to carry out the listed transformations of signals and oscillations can be divided into the following main classes:
Linear circuits with constant parameters;
Linear circuits with variable parameters;
Nonlinear circuits.
^
Linear circuits with constant parameters
We can proceed from the following definitions:
A circuit is linear if its elements do not depend on the external force (voltage, current) acting on the circuit.
A linear circuit obeys the principle of superposition (overlay).
Where L is an operator characterizing the effect of the circuit on the input signal.
When several external forces act on a linear circuit, the behavior of the circuit (current, voltage) can be determined by superposing (superposition) the solutions found for each of the forces separately.
Otherwise: in a linear chain, the sum of the effects of individual influences coincides with the effect of the sum of influences.
For any, no matter how complex, influence in a linear circuit with constant parameters, no oscillations of new frequencies arise.
^ Linear circuits with variable parameters
This refers to circuits in which one or more parameters change over time (but do not depend on the input signal). Such circuits are often called linear parametric.
Properties 1 and 2 from the previous paragraph are also valid for these circuits. However, even the simplest harmonic effect creates a complex oscillation in a linear circuit with variable parameters, which has a frequency spectrum.
^ Nonlinear circuits
A radio circuit is nonlinear if it includes one or more elements whose parameters depend on the level of the input signal. The simplest nonlinear element is a diode.
Basic properties of nonlinear circuits:
To nonlinear circuits (and elements) the superposition principle does not apply.
Important property nonlinear circuit is the transformation of the signal spectrum.
^ Signal classification
WITH information point In terms of vision, signals can be divided into deterministic and random.
Deterministic call any signal whose instantaneous value at any time can be predicted with probability one.
TO random refer to signals whose instantaneous values are unknown in advance and can be predicted only with a certain probability less than one.
Along with useful random signals, in theory and practice we have to deal with random interference - noise. Useful random signals, as well as interference, are often combined under the term random fluctuations or random processes.
Signals in a radio communication channel are often divided into control signals and on radio signals; The former are understood as modulating, and the latter as modulated oscillations.
Signals used in modern radio electronics can be divided into the following classes:
Arbitrary in magnitude and continuous in time (analog);
Arbitrary in size and discrete in time (discrete);
Quantized in magnitude and continuous in time (quantized);
Quantized in magnitude and discrete in time (digital).
^
Characteristics of deterministic
signals
Energy characteristics
Main energy characteristics of a real signal s(t) are its power and energy.
Instantaneous power is defined as the square of the instantaneous value s(t):
The signal energy over the interval t 2, t 1 is defined as the integral of the instantaneous power:
.
Attitude
Means the average signal power over the interval t 2, t 1.
^
Arbitrary Signal Representation
as a sum of elementary vibrations
For the theory of signals and their processing, the expansion of a given function f(x) into various orthogonal systems of functions j n (x) is important. Any signal can be represented as a generalized Fourier series:
,
Where C i are weight coefficients,
J i - orthogonal expansion functions (basis functions).
For basic functions the following condition must be met:
If the signal is defined in the interval from t 1 to t 2, then
Norm of the basis function.
If the function is not orthonormal, then it can be reduced in this way. As n increases, Cn decreases.
Let us assume that a set of basis functions (j n ) is given. When specifying a set of basis functions and a fixed number of terms in the generalized Fourier series, the Fourier series gives an approximation of the original function that has a minimum root-mean-square error in the definition of the original function. The generalized Fourier series gives
Such a series gives a minimum on average error (error).
There are 2 problems of decomposing a signal into simple functions:
^ Exact decomposition into simplest orthogonal functions (analytical model of the signal, analysis of signal behavior).
^ Approximation of process signals and characteristics , when it is necessary to minimize the number of terms of a generalized series. These include: Chebyshev, Hermite, and Legendre polynomials.
^ Harmonic analysis of periodic signals
When expanding a periodic signal s(t) into a Fourier series trigonometric functions take as the orthogonal system
The orthogonality interval is determined by the norm of the function
The average value of the function over the period.
- basic formula for
definitions of Fourier series
Modulus is an even function, phase is an odd function.
Consider a pair for the kth term
- Fourier series expansion
^
Examples of spectra of periodic signals
Square wave. This kind of fluctuation, often called meander(Meander is a Greek word meaning “ornament”) and is especially widely used in measuring technology.
Let the signal s(t) be given in the form of some function different from zero in the interval (t 1 ,t 2). This signal must be integrable.
Let's take an infinite period of time T, including the interval (t 1,t 2). Then . The spectrum of a non-periodic signal is continuous. A given signal can be represented as a Fourier series 
, Where
Based on this we get:
Since T®µ, the sum can be replaced by integration, and W 1 by dW and nW 1 by W. Thus we move on to the double Fourier integral
,
where is the spectral density of the signal. When the interval (t 1 ,t 2) is not specified, the integral has infinite limits. This is the inverse and forward Fourier transform, respectively.
If we compare the expressions for the envelope of the continuous spectrum (modulus of spectral density) of a non-periodic signal and the envelope of the line spectrum of a periodic signal, we will see that they coincide in shape, but differ in scale 
.
Consequently, the spectral density S(W) has all the basic properties of the complex Fourier series. That is, we can write where
, A 
.
Spectral density module 
is an odd function and can be considered as an amplitude-frequency characteristic. Argument 
- odd function considered as a phase-frequency characteristic.
Based on this, the signal can be expressed as follows
From the evenness of the module and the oddness of the phase, it follows that the integrand in the first case is even, and in the second case it is odd with respect to W. Therefore, the second integral is equal to zero (an odd function in even limits) and finally .
Note that at W=0 the expression for the spectral density is equal to the area under the curve s(t)
.
^
Properties of the Fourier transform
Signal time shift
Let a signal s 1 (t) of an arbitrary shape have a spectral density S 1 (W). When this signal is delayed for a time t 0, we obtain a new time function s 2 (t)=s 1 (t-t 0). The spectral density of the signal s 2 (t) will be as follows 
. Let's introduce a new variable. From here 
.
Each signal has its own spectral density. A shift of the signal along the time axis leads to a change in its phase, and the magnitude of this signal does not depend on the position of the signal on the time axis.
^ Changing the time scale
Let the signal s 1 (t) be compressed in time. The new signal s 2 (t) is associated with the original relation.
The pulse duration s 2 (t) is n times less than the initial one. Spectral density of compressed pulse 
. Let's introduce a new variable. We'll get it.
When a signal is compressed n times, its spectrum expands by the same amount. The modulus of the spectral density will decrease by n times. When the signal is stretched in time, the spectrum narrows and the spectral density modulus increases.
^ Vibration spectrum shift
Let's multiply the signal s(t) by the harmonic signal cos(w 0 t+q 0). The spectrum of such a signal
Let's split it into 2 integrals.
The resulting relationship can be written in the following form
Thus, multiplying the function s(t) by a harmonic oscillation leads to splitting the spectrum into 2 parts, shifted by ±w 0.
^ Signal differentiation and integration
Let a signal s 1 (t) with spectral density S 1 (W) be given. Differentiation of this signal 
gives the ratio 
. Integration 
leads to the expression 
.
^ Signal addition
When adding signals s 1 (t) and s 2 (t) having spectra S 1 (W) and S 2 (W), the total signal s 1 (t) + s 2 (t) corresponds to the spectrum S 1 (W) + S 2 (W) (since the Fourier transform is a linear operation).
^ Product of two signals
Let . This signal corresponds to the spectrum
Let's represent the functions in the form of Fourier integrals.
Substituting the second integral into the expression for S(W) we obtain
Hence 
.
That is, the spectrum of the product of two time functions is equal to the convolution of their spectra (with a coefficient of 1/2p).
If 
, then the signal spectrum will be 
.
^ Mutual reversibility of frequency and time
in Fourier transform
Let s(t) be an even function with respect to time.
Assuming that s(t) is an even function. Let us write s(t) in the form
. Let's replace W by t and t by W, we get 
. If the spectrum has the shape of a signal, then the signal corresponding to this spectrum repeats the shape of the spectrum of a similar signal.
^
Energy distribution in the spectrum of a non-periodic signal
Consider the expression in which f(t)=g(t)=s(t). In this case, this integral is equal to . This relation is called Parseval's equality.
Energy calculation of bandwidth: 
, Where 
, A 
.
^
Examples of spectra of non-periodic signals
Square pulse
Defined by the expression
Let's find the spectral density
.
As the pulse lengthens (stretches), the distance between the zeros decreases, and the value of S(0) increases. The modulus of the function can be considered as the frequency response, and the argument as the phase response of the spectrum of a rectangular pulse. Each sign change takes into account the phase increment by p.
When counting time not from the middle of the pulse, but from the front, the phase response of the pulse spectrum must be supplemented with a term that takes into account the shift of the pulse by time (the resulting phase response is shown by the dotted line).
Bell-shaped (Gaussian) pulse
Determined by the expression . The constant a has the meaning of half the pulse duration, determined at the level e -1/2 of the pulse amplitude. Thus, the total pulse duration is .
Signal spectral density 
.
For convenience, we add the exponent to the square of the sum 
, where the value d is determined from the condition 
, where . Thus, the expression for the spectral density can be reduced to the form 
.
Moving to a new variable 
we get 
. Considering that the integral included in this expression is equal to , we finally obtain 
, Where 
.
Pulse spectrum width
The Gaussian pulse and its spectrum are expressed by identical functions and have the property of symmetry. For it, the ratio of pulse duration and bandwidth is optimal, i.e., for a given pulse duration, a Gaussian pulse has a minimum bandwidth.
delta pulse (single pulse)
The signal is given by the relation 
. It can be obtained from the above impulses by tending t and to zero.
It is known that, therefore, the spectrum of such a signal will be constant (this is the pulse area, equal to one).
To create such a pulse, all harmonics are needed.
Exponential momentum
Signal of the form , c>0.
The signal spectrum is found as follows
Let's write the signal in another form 
.
If, then. This means that we will get a single jump. At 
we obtain the following expression for the signal spectrum 
.
Hence the module
Radio signals
Modulation
Let a signal be given, in which A(t) is amplitude modulation, w(t) is frequency modulation, j(t) is phase modulation. The last two form angular modulation. The frequency w must be large compared to highest frequency signal spectrum W (spectrum width occupied by the message).
A modulated oscillation has a spectrum, the structure of which depends both on the spectrum of the transmitted message and on the type of modulation.
There are several types of modulation possible: continuous, pulse, pulse code.
^
Amplitude modulation
The general expression for amplitude modulated oscillation is as follows
The nature of the envelope A(t) is determined by the type of message being transmitted.
If the signal is a message, then the envelope of the modulated oscillation can be represented as . Where W is the modulation frequency, g is the initial phase of the envelope, k is the proportionality coefficient, DA m is the absolute change in amplitude. Attitude 
- modulation coefficient. Based on this, we can write . Then the amplitude-modulated oscillation will be written in the following form.
With undistorted modulation (M £ 1), the amplitude of the oscillation varies from 
before 
.
The maximum value corresponds to peak power. The average power over the modulation period is .
The power for transmitting an amplitude-modulated signal is greater than for transmitting simple signal.
Spectrum of amplitude-modulated signal
Let the modulated oscillation be defined by the expression
Let's transform this expression
The first term is the original unmodulated oscillation. The second and third are oscillations that appear during the modulation process. The frequencies of these oscillations (w 0 ±W) are called side modulation frequencies. Spectrum width 2W.
In the case when the signal is a sum 
, where , a . Moreover, where 
.
From here we get
Each of the components of the modulating signal spectrum independently forms two side frequencies (left and right). The spectrum width in this case is 2W 2 = 2W max 2 maximum frequency of the modulating signal.
In the vector diagram, the time axis rotates clockwise with angular frequency w 0 (counting from horizontal axis) . The amplitudes and phases of the side lobes are always equal to each other, so their resulting vector DF will always be directed along the OD line. The final vector OF changes only in amplitude without changing its angular position.
Let there be a signal. Let's write it in another form.
The signal corresponds to the spectrum 
, where , and S A is the spectral density of the envelope. This gives us the final expression for the spectrum
This is explained by the gating effect of the d-function, i.e. all components are equal to zero except for the frequencies w±w n (these are the values at which the d-function is equal to zero). Even if the spectrum is not discrete, there are still side components.
^
Frequency modulation
Let there be an oscillation with frequency modulation. However, frequency is a derivative of phase. If you change the phase, then current frequency will change too.
Frequency modulation
,
Where represents the amplitude of the frequency deviation. For brevity, in what follows we will call frequency deviation or simply deviation.
Where w 0 t is the current phase change; - angular modulation index.
Let's assume where 
.
,
Where m is the modulation coefficient.
Thus, harmonic phase modulation with index is equivalent to frequency modulation with deviation.
With a harmonic modulating signal, the difference between FM and PM can only be detected changing the modulation frequency.
At the World Cup deviation W.
With FM the value proportional to the amplitude of the modulating voltage and does not depend on the modulation frequencyW.
For a monochromatic modulating signal, phase and frequency modulation are indistinguishable.
^
Signal spectrum with angle modulation
Let the oscillation be given
There are two amplitude modulated signals. Such components that differ in are called quadrature components.
Let . This coincides with . Here q 0 =0, g=0.
Cos and sin are periodic functions and can be expanded in a Fourier series
J(m) - Bessel function of the 1st kind.
The spectrum with angular modulation is infinitely large, in contrast to the spectrum with amplitude modulation.
With angular modulation, the spectrum of a frequency-modulated oscillation, even when modulated by 1 frequency, consists of an innumerable number of harmonics grouped around the carrier frequency.
Flaws: the spectrum is very wide.
Advantages: most noise resistant.
Consider the case when m<< 1.
If m is very small, then only 2 side frequencies are present in the spectrum.
Spectrum width (m<< 1) будет равна 2W.
If m=0.5¸1, then the second pair of side frequencies w±2W appears. The spectrum width is 4W.
If m=1¸2, then the third and fourth harmonics w±3W, w±4W appear.
Spectral width at m very large
ShS=2mW=2w d
If the modulation coefficient is significantly less than unity, then such modulation is called fast, then w d<< W.
If m >> 1, then this slow modulation, then w d >> W.
^
Frequency modulated radio pulse spectrum
filling
, Where
Where , 
The main parameter of a linearly frequency modulated signal (chirp) or the base of the chirp signal.
B can be both positive and negative.
Let's assume that b>0
The signal spectrum consists of 2 components:
1 - burst near frequency w o;
2 - surge near frequency -w o.
When determining the spectral density in the region of positive frequencies, the second term can be discarded.
Let's add the exponential to a complete square
, where C(x) and S(x) are Fresnel integrals
Chirp signal spectral density module
Phase of the spectral density of the chirp signal
The larger m, the closer the spectrum shape is to rectangular with spectrum width . The phase dependence is quadratic.
As m tends to large values, the shape of the frequency response tends to be rectangular, and the phase consists of two parts:
1). gives a parabola
2). strives for
For large m and : 
Then the module value is: .
Mixed amplitude-frequency modulation
Spectral density of cosine quadrature wave 
at =0 it will be
When determining the spectrum of a sine quadrature oscillation 
the phase angle should be equal to -90°. Hence,
Thus, the final spectral density of vibration is determined by the expression
Passing to the variable, we get
.
The structure of the signal spectrum with mixed amplitude-frequency modulation depends on the ratio and type of functions A(t) and q(t).
With frequency modulation, the phases of odd harmonics change by 180°. Simultaneous modulation of both frequency and amplitude at certain ratios A(t) and q(t) leads to a violation of the symmetry of the spectrum, not only in phase, but also in amplitude.
If q(t) is an odd function of t, then for any A(t) the spectrum of the output signal is asymmetrical.
Let A(t) be an even function, then A c (t) is even, A s (t) is odd, is purely real, symmetric with respect to W, even, and is purely imaginary, asymmetric with respect to W and odd.
Taking into account the factor j, the spectrum of the output oscillation is real. As a result, the spectrum is asymmetrical, but with respect to w = 0 it is symmetrical. The same result can be obtained with an odd function A(t). In this case, the spectrum is purely imaginary and odd.
For symmetry of the output spectrum, parity of q(t) is required, provided that A(t) was either even or odd with respect to t. If A(t) is the sum of even and odd functions, then the output spectrum is asymmetric under any conditions.
The chirp has an even phase and even amplitude. 
Moreover 
The output spectrum turned out to be symmetrical.
A(t) = even function + odd function, and q(t) is an even function.
.
The spectrum turned out to be asymmetrical.
Narrowband signal
It is understood as any signal in which the frequency band occupied by the signal is significantly less than the carrier frequency: .
Where A s (t) is the in-phase amplitude, B s (t) is the quadrature amplitude.
Complex amplitude of a narrowband signal 
.
,
Where is the rotation operator.
The simplest vibration 
can be represented in the form 
, Where . In this expression, the envelope A(t), unlike A o, is a function of time, which can be determined from the condition of preserving the given function a(t)
From this expression it is clear that new feature A(t) is not essentially an “envelope” in the conventional sense, since it may intersect the a(t) curve (instead of touching at the points where A(t) is at its maximum value). That is, we did not correctly determine the envelope and frequency. There is a method of instantaneous frequency - Hilbert's method for determining frequency.
If there is a signal, then
Total signal phase 
, and the instantaneous frequency 
Physical envelope 
.
Let us assume that we have chosen the reference frequency not w o, but w o + Dw, then
, Where .
First
The modulus of the complex envelope is equal to the physical envelope and is constant, independent of the choice of frequency.
Second complex envelope property:
The magnitude of the signal s(t) is always less than or equal to u s (t). Equality occurs when cos w o t = 1. At these moments, the derivative of the signal and the derivative of the envelope are equal.
The physical envelope coincides with maximum value signal amplitude.
Knowing the complex envelope, you can find its spectrum, and through it the signal itself.
,
.
Knowing G(w) we find U s (t).
Multiply by (-b-jt) and get the real and imaginary parts, respectively 
, 
. Hence the amplitude will be 
.
^
Analytical signal
Let there be a signal s(t) defined as 
. Let's divide it into two components 
.
In that expression 
–– analytical signal. If you enter a variable then . That is, we received 
. Real signal There is 
, Hilbert conjugate signal 
. There is an analytical signal 
.
, 
–– direct and inverse Hilbert transforms.
Determination of carrier and envelope using Hilbert's method
Signal amplitude 
, its phase 
. Instantaneous frequency value 
.
Example: . .
–– precise definition of the envelope. Using the Hilbert method allows you to give unambiguous and absolutely reliable values of the envelope and instantaneous frequency of the signal.
–– any signal can be expanded into a Fourier series.
– Hilbert conjugate signal.
If the signal is represented not by a Fourier series, but by a Fourier integral, then the following relations are valid: 
, 
.
^
Analytical Signal Properties
The product of the analytical signal z s (t) and its associated signal z s * (t) is equal to the square of the envelope of the original (physical) signal s (t).
Otherwise, where.
Hilbert transform for narrowband process
Let , then the Hilbert conjugate signal 
.
Based on this we get
Properties of Hilbert transforms
––Hilbert transform, where H() is the transformation operator. 
Example. Signal s(t) is an ideal low-frequency signal.
Frequency and time characteristics
radio circuits
Let there be a linear active four-port network.
1. Transfer function 
. Characterizes the change in the output signal relative to the input signal. The module is called amplitude- frequency response or simply frequency response. The argument is the phase-frequency characteristic or simply phase.
2. Impulse response 
–– the reaction of the circuit to a single impulse. Characterizes the change in signal over time. The connection with the transfer function is carried out through the inverse and direct Fourier transform (respectively) 
. Or through the Laplace transform 
.
3. Transition function – the reaction of the circuit to a single step. This is the accumulation of a signal over time t.
^
Aperiodic amplifier
Equivalent circuit of the simplest aperiodic amplifier. The amplifying device is presented in the form of a current source SE 1 with internal conductivity G i =1/R i . Capacitance C includes the interelectrode capacitance of the active element and the capacitance of the external circuit that shunts the load resistor R n.
The transfer function of such an amplifier
,
where S is the slope of the active element, E 1 is the input voltage.
Maximum gain (at ) 
. From here 
, where is the delay time.
Transfer characteristic module 
–– frequency response. That is, this amplifier passes a signal only in a certain frequency band. FFC –– 
.
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Preface | |
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SECTION I. TASKS AND EXERCISES |
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Topic 1. General theory of radio signals | |
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Topic 2. Spectral representations of signals | |
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Topic 3. Energy spectra of signals. Principles of correlation analysis | |
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Topic 4. Modulated signals | |
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Topic 5. Signals with limited spectrum | |
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Topic 6. Basics of the theory of random signals | |
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Topic 7. Correlation theory of random processes | |
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Topic 8. Impact of deterministic signals on linear stationary systems | |
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Topic 9. Impact of deterministic signals on frequency-selective systems | |
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Topic 10. Impact of random signals on linear stationary circuits | |
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Topic 11. Signal conversions in nonlinear radio circuits | |
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Topic 12. Signal conversion in linear parametric circuits | |
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Topic 13. Fundamentals of the theory of synthesis of linear radio circuits | |
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Topic 14. Active feedback circuits and self-oscillating systems | |
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Topic 15. Discrete signals. Principles of digital filtering | |
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Topic 16. Optimal linear filtering of signals | |
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SECTION II. Directions | |
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SECTION III. Solutions | |
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SECTION IV. Answers | |
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Applications | |
PREFACE TO THE SECOND EDITION
"An example is sometimes more useful than a rule" I. Newton
From his own experience, the reader certainly knows that an integral part of the process of studying exact sciences - first of all, mathematics and physics, as well as many natural sciences - is problem solving. Having first encountered tasks during school years, we then become so accustomed to them that we do not bother ourselves with questions about what the task as such is, what its cognitive role is. Moreover, some students view tasks as a necessary evil that must simply be endured patiently. In this regard, it is useful to note that European science and pedagogy, whose history goes back more than one millennium, only by the end of the 17th century came to the conclusion that teaching based on rote memorization of theoretical principles was extremely ineffective. Newton's words from his textbook "Algebra", taken as an epigraph, successfully emphasize a principle that is unlikely to become outdated - the key to successful learning is the active cognitive creativity of the student, who gets the opportunity to see the theory in action through his own experience.
The educational tasks are, by their nature, close to chess etudes, or rather to those scales and arpeggios that no aspiring musician can do without. A well-written problem carries all the features of a small scientific and pedagogical essay - its scientific topic is strictly delineated and, most importantly, to successfully solve the problem, you need to independently construct the mental algorithm that is known in advance to the teacher and which the student must demonstrate.
Like everything in the world, the method of teaching through problem solving has its own internal limitation: the formulation of the problem is inevitably poorer than the reality to which this problem relates. This circumstance must certainly be taken into account when correlating the conclusions of theory with practice.
How to learn to solve problems? Many serious books have been written on this subject. Without in any way pretending to generalize, we emphasize the following.
Firstly, you should develop an attitude towards this activity as an exciting work that allows you to widely
reveal a person's intellectual abilities. The techniques are varied - having successfully solved a problem, think about what other similar problems can be solved using the method you found. Don't forget to praise yourself if your work is going well. And most importantly, do not become discouraged if the task stubbornly “does not want to be solved.” After resting, get to work again - persistence in achieving a goal is an indispensable personality trait of a true professional. If you were unable to cope with a difficulty on your own and have to turn to a teacher, do not prioritize the “prescription” side of the matter - after all, the goal is not just to get the right answer, but to understand as deeply as possible why you need to act this way and not otherwise.
Secondly, after opening a textbook, you should not reduce the matter to looking for a formula that will immediately give the desired answer. Formal knowledge of the theory is necessary, but by no means a sufficient condition for successfully solving a problem. The most important mental procedure has always been a certain guess, and this is, in fact, the beginning of any creativity. If it is immediately clear how to solve a particular problem, it still should not be neglected. Accurate completion of all calculations and calculations is very important for developing the skill of independent work.
I take this opportunity to express my gratitude to the reviewer of the book, Prof. M.P. Demina for useful tips and favorable criticism.
FROM THE PREFACE TO THE FIRST EDITION
This book contains material for exercises in the course “Radio Engineering Circuits and Signals”.