Before embarking on the study of any phenomena, processes or objects, science always strives to classify them according to the largest possible number of signs. Let's make a similar attempt in relation to radio signals and interference.
Basic concepts, terms and definitions in the field of radio technical signals are established by the state standard “Radio signals. Terms and Definitions". Radiotechnical signals are very diverse. They can be classified according to a variety of characteristics.
1. It is convenient to consider radio-technical signals in the form of mathematical functions given in time and physical coordinates. From this point of view, the signals are divided into one-dimensional and multidimensional... In practice, one-dimensional signals are most common. They are usually functions of time. Multidimensional signals consist of many one-dimensional signals, and in addition, they reflect their position in n- dimensional space. For example, signals that carry information about the image of an object, nature, man or animal, are functions of both time and position on the plane.
2. According to the peculiarities of the structure of the temporal representation, all radio technical signals are subdivided into analog, discrete and digital... In lecture number 1, their main features and differences from each other have already been considered.
3. According to the degree of availability of a priori information, it is customary to divide the entire variety of radio-technical signals into two main groups: deterministic(regular) and random signals. Radio-technical signals are called deterministic, the instantaneous values of which are reliably known at any time. An example of a deterministic radio engineering signal is a harmonic (sinusoidal) oscillation, a sequence or burst of pulses, the shape, amplitude, and temporal position of which are known in advance. In fact, a deterministic signal does not carry any information and almost all of its parameters can be transmitted via a radio communication channel with one or more code values. In other words, deterministic signals (messages) essentially do not contain information, and there is no point in transmitting them. They are usually used to test communication systems, radio channels or individual devices.
Deterministic signals are subdivided into periodic and non-periodic (impulse). An impulse signal is a signal of final energy that is significantly different from zero for a limited time interval commensurate with the time of completion of the transient in the system for which this signal is intended to act. Periodic signals are harmonic, that is, containing only one harmonic, and polyharmonic, the spectrum of which consists of many harmonic components. Harmonic signals are signals described by a sine or cosine function. All other signals are called polyharmonic.
Random signals- these are signals whose instantaneous values at any time instant 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 embedded in a variety of amplitude, frequency (phase) or code changes in the transmitted signal. In practice, any radio signal containing useful information should be considered random.
4. 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), coded (decoded), reinforced, detainees, discretized, quantized and etc.
5. According to the purpose, which the signals have in the process of modulation, they can be divided into modulating(the primary signal that modulates the carrier waveform) or modulated(bearing vibration).
6. By belonging to one or another type of information transmission systems are distinguished telephone, telegraph, broadcasting, television, radar, managers, measuring and other signals.
Let us now consider the classification of radio-technical interference. Under radio interference they understand a random signal that is homogeneous with a useful one and acts simultaneously with it. For radio communication systems, interference is any accidental effect on a useful signal that impairs the fidelity of the transmitted messages. Classification of radio-technical interference is also possible according to a number of signs.
1. At the place of occurrence, the interference is divided into external and internal... Their main types have already been discussed in lecture number 1.
2. Depending on the nature of the interaction of the interference with the signal, one distinguishes additive and multiplicative interference. Interference is called additive, which is added to the signal. Interference is called multiplicative and is multiplied with the signal. In real communication channels, both additive and multiplicative interference usually take place.
3. According to its main properties, additive noise can be divided into three classes: spectrum-lumped(narrowband interference), impulse noise(centered in time) and fluctuation noise(fluctuation noise), not limited either in time or in the spectrum. Spectrum-centered interference is called interference, the bulk of the power of which is located in separate parts of the frequency range, less than the bandwidth of the radio engineering system. Pulse noise is a regular or chaotic sequence of pulsed signals that are homogeneous with a useful signal. Sources of such interference are digital and switching elements of radio circuits or devices operating near them. Pulsed and lumped disturbances are often referred to as tips.
There is no fundamental difference between signal and interference. Moreover, they exist in unity, although they are opposite in their action.
Random processes
As mentioned above, a distinctive feature of a random signal is that its instantaneous values are not predictable in advance. Almost all real random signals and noises are chaotic functions of time, the mathematical models of which are random processes studied in the discipline of statistical radio engineering. By a random process it is customary to call a random function of an argument t, where t the current time. A random process is denoted by the capital letters of the Greek alphabet,,. Another designation is also acceptable if it is agreed in advance. A specific type of random process that is observed during an experiment, for example, on an oscilloscope, is called implementation this random process. Type of specific implementation x (t) can be specified by a certain functional dependence of the argument t or schedule.
Depending on whether continuous or discrete values take an argument t and implementation NS, there are five main types of random processes. Let us explain these types with examples.
A continuous random process is characterized by the fact that t and NS are continuous quantities (Fig. 2.1, a). Such a process, for example, is noise at the output of a radio receiver.
A discrete random process is characterized by the fact that t is a continuous quantity, and NS- discrete (Fig. 2.1, b). The transition from to occurs at any time. An example of such a process is a process that characterizes the state of a queuing system when the system jumps at arbitrary times t passes from one state to another. Another example is the result of quantizing a continuous process by level only.
A random sequence is characterized by the fact that t is discrete and NS- continuous quantities (Fig. 2.1, c). As an example, you can point to time samples at specific points in time from a continuous process.
A discrete random sequence is characterized by the fact that t and NS are discrete quantities (Fig. 2.1, d). Such a process can be obtained by level quantization and time sampling. These are the signals in digital communication systems.
A random stream is a sequence of points, delta functions or events (Fig. 2.1, e, g) at random times. This process is widely used in the theory of reliability, when the flow of faults in electronic equipment is considered as a random process.
From an informational point of view, signals can be divided into deterministic and random.
Any signal is called deterministic, the instantaneous value of which at any moment of time can be predicted with a probability of one. Examples of deterministic signals are pulses or bursts of pulses, the shape, amplitude and position in time of which are known, as well as a continuous signal with specified amplitude and phase relationships within its spectrum.
Random signals include signals whose instantaneous values are not known in advance and can be predicted only with a certain probability less than one. Such signals are, for example, an electric voltage corresponding to speech, music, a sequence of characters of a telegraph code when transmitting a non-repeating text. Random signals also include a sequence of radio pulses at the input of a radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, target position, and some other reasons. There are many other examples of random signals. Essentially, any signal that carries information should be considered random.
The deterministic signals listed above, "fully known", no longer contain information. In the following, such signals will often be referred to as wobble.
Along with useful random signals, in theory and practice, one has to deal with random interference - noise. The noise level is the main factor limiting the information transfer rate for a given signal.
Rice. 1.2. Signals are arbitrary in magnitude and time (a), arbitrary in magnitude and discrete in time (b), quantized in magnitude and continuous in time (c), quantized in magnitude and discrete in time (d)
Therefore, the study of random signals is inseparable from the study of noise. Useful random signals, as well as noise, are often combined with the term random fluctuations or random processes.
Further subdivision of signals can be associated with their nature: you can talk about the signal as a physical process or as encoded, for example, in a binary code, numbers.
In the first case, a signal is understood as any electrical quantity that changes over time (voltage, current, charge, etc.), in a certain way associated with the transmitted message.
In the second case, the same message is contained in a sequence of binary-coded numbers.
The signals generated in radio transmitting devices and emitted into space, as well as entering the receiving device, where they are amplified and some transformations, are physical processes.
In the previous paragraph, it was indicated that modulated oscillations are used to transmit messages over a distance. In this regard, the signals in the radio communication channel are often subdivided into control signals and radio signals; the former are understood to be modulating, and the latter are modulated oscillations.
Signal processing in the form of physical processes is carried out using analog electronic circuits (amplifiers, filters, etc.).
Digitally encoded signals are processed using computer technology.
Shown in Fig. 1.1 and the block diagram of the communication channel described in § 1.2 does not contain instructions on the type of signal used to transmit the message and the structure of individual devices.
Meanwhile, signals from the source of messages, as well as after the detector (Fig. 1.1) can be both continuous and discrete (digital). In this regard, the signals used in modern radio electronics can be divided into the following classes:
arbitrary in size and continuous in time (Fig. 1.2, a);
arbitrary in size and discrete in time (Fig. 1.2, b);
quantized in magnitude and continuous in time (Fig. 1.2, c);
quantized in magnitude and discrete in time (Fig. 1.2, d).
Signals of the first class (Fig. 1.2, a) are sometimes called analog, since they can be interpreted as electrical models of physical quantities, or continuous, since they are set along the time axis at an uncountable set of points. Taki? sets are called continuous. In this case, the signals along the ordinate axis can take any value within a certain interval. Since these signals can have discontinuities, as in Fig. 1.2, a, then, in order to avoid incorrectness in the description, it is better to designate such signals with the term continuous.
So, the continuous signal s (t) is a function of the continuous variable t, and the discrete signal s (x) is a function of the discrete variable x, which takes only fixed values. Discrete signals can be generated directly by a source of information (for example, discrete sensors in control or telemetry systems) or be generated as a result of discretization of continuous signals.
In fig. 1.2, b shows a signal given at discrete values of time t (on a countable set of points); the magnitude of the signal at these points can take any value within a certain interval along the ordinate (as in Fig. 1.2, a). Thus, the term discrete does not characterize the signal itself, but the way it is specified on the time axis.
The signal in Fig. 1.2, в is set on the entire time axis, but its value can only take discrete values. In such cases, one speaks of a level-quantized signal.
In what follows, the term discrete will only be applied to time sampling; discreteness in terms of level will be denoted by the term quantization.
Quantization is used when representing signals in digital form using digital coding, since the levels can be numbered with numbers with a finite number of digits. Therefore, the signal discrete in time and quantized in level (Fig. 1.2, d) will be referred to as digital in the following.
Thus, we can distinguish between continuous (Fig. 1.2, a), discrete (Fig. 1.2, b), quantized (Fig. 1.2, c) and digital (Fig. 1.2, d) signals.
Each of these signal classes can be assigned to analog, discrete or digital circuits. The relationship between the type of signal and the type of circuit is shown in the functional diagram (Fig. 1.3).
When processing a continuous signal using an analog circuit, no additional signal conversions are required. When processing a continuous signal using a discrete circuit, two transformations are required: time sampling of the signal at the input of the discrete circuit and the inverse transformation, i.e., restoration of the continual structure of the signal at the output of the discrete circuit.
Rice. 1.3. Signal types and corresponding circuits
Finally, digital processing of a continuous signal requires two more additional transformations: analog-to-digital, i.e., quantization and digital coding at the input of a digital circuit, and inverse digital-to-analog conversion, i.e. decoding at the output of a digital circuit.
The signal sampling procedure and especially the analog-to-digital conversion require very high performance of the corresponding electronic devices. These requirements increase with increasing frequency of the continuous signal. Therefore, digital technology is most widely used in signal processing at relatively low frequencies (audio and video frequencies). However, advances in microelectronics contribute to a rapid increase in the upper limit of the processed frequencies.
Thus, a signal is a physical process whose parameters contain information (message) and which is suitable for processing and transmission over a distance.
One-dimensional and multidimensional signals. A typical signal for radio engineering is the voltage at the terminals of a circuit or the current in a branch. Such a signal, described by one function of time, is usually called one-dimensional.
However, it is sometimes convenient to introduce into consideration multidimensional, or vector, signals of the form
formed by some set of one-dimensional signals. The integer N is called the dimension of such a signal.
Note that a multidimensional signal is an ordered collection of one-dimensional signals. Therefore, in the general case, signals with different order of components are not equal to each other.
Analog, discrete and digital signals... Concluding a brief overview of the principles of classification of radio engineering signals, we note the following. Often the physical process that generates a signal develops in time in such a way that the signal values can be measured at any time. Signals of this class are usually called analog (continuous). The term "analog signal" emphasizes that such a signal is "analogous", completely similar to the physical process that generates it.
A one-dimensional analog signal is clearly represented by its graph (oscillogram), which can be either continuous or with break points.
.
Multivariate signal models are especially useful in cases where the functioning of complex systems is analyzed using a computer.
Deterministic and random signals. Another principle of classification of radio engineering signals is based on the possibility or impossibility of accurate prediction of their instantaneous values at any time.
If the mathematical model of the signal allows such a prediction, then the signal is called deterministic. The methods of its assignment can be varied - a mathematical formula, a computational algorithm, and finally, a verbal description.
Analog (continuous), discrete and digital signals... Often the physical process that generates a signal develops in time in such a way that the signal values can be measured at any time. Signals of this class are usually called analog (continuous). The term "analog signal" emphasizes that such a signal is "analogous", completely similar to the physical process that generates it.
A one-dimensional analog signal is clearly represented by its graph (oscillogram), which can be either continuous or with break points.
Initially, signals of an exclusively analog type were used in radio engineering. Such signals made it possible to successfully solve relatively simple technical problems (radio communication, television, etc.). Analog signals were easy to generate, receive, and process using the tools available in those years.
The increased requirements for radio engineering systems, a variety of applications forced the search for new principles for their construction. In some cases, analogue systems have been replaced by impulse systems, the operation of which is based on the use of discrete signals. The simplest mathematical model of a discrete signal is a countable set of points (is an integer) on the time axis, at each of which the reference value of the signal is determined. Typically, the sampling rate for each signal is constant.
One of the advantages of discrete signals over analog signals is that there is no need to reproduce the signal continuously at all times. Due to this, it becomes possible to transmit messages from different sources over the same radio link, organizing multichannel communication with time division of channels.
Intuitively, fast time-varying analog signals require small steps to sample.
A special kind of discrete signals are digital signals. They are characterized by the fact that the reading values are presented in the form of numbers. For reasons of technical convenience of implementation and processing, binary numbers with a limited and usually not too large number of digits are usually used. Recently, there has been a trend towards widespread adoption of systems with digital signals. This is due to the significant advances made by microelectronics and integrated circuitry.
It should be borne in mind that in essence any discrete or digital signal (we are talking about a signal - a physical process, not a mathematical model) is an analog signal.
Strictly speaking, deterministic signals, as well as deterministic processes corresponding to them, do not exist. The inevitable interaction of the system with the surrounding physical objects, the presence of chaotic thermal fluctuations and simply incomplete knowledge about the initial state of the system - all this forces us to consider real signals as random functions of time.
In radio engineering, random signals often manifest themselves as interference, preventing the extraction of information from the received waveform. The problem of countering interference, increasing the noise immunity of radio reception is one of the central problems of radio engineering.
It may seem that the concept of "random signal" is controversial. However, it is not. For example, the signal at the output of a radio telescope receiver directed at a source of cosmic radiation is chaotic oscillations, which, however, carry a variety of information about a natural object.
There is no insurmountable boundary between deterministic and random signals. Very often, in conditions when the level of interference is much less than the level of a useful signal with a known shape, a simpler deterministic model turns out to be quite adequate to the task at hand.
Lecture number 2 Radio signals
Signal theory. Classification. Main characteristics of signals
The change over time of voltage, current, charge or power in electrical circuits is called electrical oscillation. The electrical vibration used to transmit information is a signal. The complexity of processes in electrical circuits depends on the complexity of the original signals. Therefore, it is advisable to use the spectrum of signals. Fourier series and transformations are known from mathematics, with the help of which it is possible to represent signals as a set of harmonic components. In practice, a characteristic analysis is useful to give an idea of the rate of change and duration of the signal. This can be achieved using correlation analysis.
2.1. General information about radio signals
Traditionally, electrical (and now optical) signals related to the radio range are considered radio engineering. From a mathematical point of view, any radio engineering signal can be represented by some
the function of time u (t), which characterizes the change in its instantaneous values of voltage (such a representation is used most often), current, charge or power. Each class of signals has its own characteristics and requires specific methods of description and analysis. Analysis is one of the key components of signal presentation and processing. The main purpose of the analysis is to compare signals with each other to identify their similarities and differences. There are three main components of the analysis of electrical signals:
Measurement of numerical parameters of signals (energy, average power and root mean square value);
Decomposition of the signal into elementary components either for their consideration separately, or for comparing the properties of different signals; such an expansion is carried out using series and integral transformations, the most important of which are the series and the Fourier transform;
Quantitative measurement of the degree of "similarity" of various signals, their parameters and characteristics; such a measurement is performed using a correlation analysis apparatus.
In order to make signals objects of study and calculations, one should indicate the method of their mathematical description, i.e. create a mathematical model of the signal under study. In radio engineering, each class of signals has its own mathematical representation, its own mathematical model, and the same mathematical model can almost always adequately describe voltage, current, charge, power, electromagnetic field strength, etc. The most common ways of representing (describing) signals are temporal, spectral, analytical, statistical, vector, graphic and geometric. Functions describing signals can take both real and complex values. Therefore, in what follows in the book, we will often talk about real and complex signals. A part of the brief classification of signals based on a number of features is shown in Figure 2.1.
Figure 2.1. Classification of radio engineering signals
It is convenient to consider radio engineering signals in the form of mathematical functions given in time and physical coordinates. From this point of view, signals are usually described by one (one-dimensional signal; n = 1), two (two-dimensional signal; n = 2) or more (multidimensional signal n> 2) independent variables. One-dimensional signals are functions of only time, and multidimensional signals, in addition, reflect the position in the "-dimensional space. For the sake of clarity and simplicity, we will mainly consider one-dimensional time-dependent signals, multidimensional case, when the signal is represented as a finite or infinite collection of points, for example in space, the position of which depends on time. In television systems, a black-and-white image signal can be considered as a function f (x, y, f) of two spatial coordinates and time, representing the radiation intensity at a point (x, y) at a time t at the cathode. When transmitting a color television signal, we have three functions f (x, y, t), g (x, y, t), h (x, y, t), defined on a three-dimensional set (these three functions can also be considered as components of a three-dimensional vector field). In addition, various types of television signals can occur when a television image is transmitted together with sound. A multidimensional signal is an ordered collection of one-dimensional signals. A multidimensional signal is created, for example, by a system of voltages at the terminals of a multipole (Fig. 2.2).
Rice. 2.2. Multipole voltage system.
Multidimensional signals are described by complex functions, and their processing is more often possible in digital form. Therefore, multidimensional signal models are especially useful in cases where the functioning of complex systems is analyzed using computers. So, multidimensional, or vector, signals consist of many one-dimensional signals
where n - an integer, the dimension of the signal. According to the features of the structure of the temporal representation (Fig. 2.3), all radio technical signals are divided into analog ( analog), discrete (discrete - time; from Latin discretus - split, intermittent) and digital ( digital ). If the physical process generating a one-dimensional signal can be represented by a continuous function of time u (t) (Fig. 2.3, a), then such a signal is called analog (continuous). An example of an analog signal is some voltage that is applied to the input of an oscilloscope, resulting in a continuous curve on the screen as a function of time. A discrete signal is obtained from an analog signal by a special conversion. The process of converting an analog signal to a sequence of samples is called sampling, and the result of this conversion is called a discrete series. The simplest mathematical model of a discrete signal U n (t) is a sequence of points on the time axis, taken, as a rule, at equal time intervals T = ∆t, called the sampling period (or interval, sampling time; sample time), and in each of which the values of the corresponding continuous signal are set (Fig. . 2.3, b). The reciprocal of the sampling period is called the sampling frequency: f D = 1 / T (another designation f D f D = 1 / ∆t). The corresponding angular (circular) frequency is determined as follows: ω D = 2π / ∆t.
Rice. 2.3. Radio-technical signals: a - analog; b - discrete; в - quantized; d - digital
A kind of discrete signals is a digital signal ( digital signal ), In the process of converting discrete samples of the signal into digital form (usually into binary numbers), it is quantized by the level ( quantization ) voltage ∆. In this case, the values of the signal levels can be numbered with binary numbers with a finite required number of digits. A signal that is discrete in time and quantized in level is called a digital signal. In a digital signal, the discrete values of the signal u T (t) is first quantized according to the level (Fig. 2.3, c) and then the quantized samples of the discrete signal are replaced by the numbers u C (t), most often implemented in a binary code, which is represented by high (one) and low (zero) levels of voltage potentials - short pulses of duration τ (Fig. 2.3, d). This code is called unipolar. When presenting a signal, rounding inevitably occurs. The resulting round-off errors are called quantization errors (or noise) ( quantization error, quantization noise ). The sequence of numbers representing a digitally processed signal is a discrete series. One of the main features by which signals are distinguished is the predictability of the signal (its values) over time. Radio-technical signals are called deterministic, the instantaneous values of which are reliably known at any time. The simplest examples of a deterministic signal are harmonic oscillations with a known initial phase, high-frequency oscillations modulated according to a known law. A deterministic signal cannot be a carrier of information. Deterministic signals are divided into periodic and non-periodic.(pulse). A final energy signal that is substantially different from zero for a limited time interval commensurate with the time of completion of the transient in the system for which it is intended to be influenced is called a pulse signal.
Signals are called random if their instantaneous values are not known at any moment and cannot be predicted with a probability equal to one. Only a random signal can be a signal carrying useful information.
Random processes, the parameters and properties of which can be determined from one random realization (sample) are called ergodic, they have certain properties.
Often, when describing and analyzing certain types of signals (primarily narrowband), it is convenient to use a complex form of their presentation.
where - respectively, the modulus and phase of the complex quantity
The complex function u (t) can also be represented as
where Re, Im - real and imaginary parts of the complex function. From both formulas we get:
In vector representation, a complex signal is a vector on a complex plane with a real axis - the abscissa axis and an imaginary axis - the ordinate axis (Fig. 2.5). The vector on the plane rotates in the positive direction (counterclockwise) with the speed ω 0 ... The length of the vector is equal to the modulus of the complex signal, the angle between the vector and the abscissa axis is equal to the argument φ 0 ... The vector projections on the coordinate axes are equal, respectively, to the real and imaginary parts of the complex quantity.
Chapter 1 Elements of the General Theory of Radio Engineering Signals
The term "signal" is often found not only in scientific and technical issues, but also in everyday life. Sometimes, without thinking about the severity of terminology, we identify concepts such as signal, message, information. Usually this does not lead to misunderstandings, since the word "signal" comes from the Latin term "signum" - "sign", which has a wide semantic range.
Nevertheless, starting a systematic study of theoretical radio engineering, it is necessary, if possible, to clarify the meaning of the concept of "signal". In accordance with the accepted tradition, a signal is called a process of changing in time the physical state of an object, which serves to display, register and transmit messages. In the practice of human activity, messages are inextricably linked with the information contained in them.
The range of issues based on the concepts of "message" and "information" is very wide. It is the object of close attention of engineers, mathematicians, linguists, philosophers. In the 40s, K. Shannon completed the initial stage of developing a deep scientific direction - information theory.
It should be said that the problems mentioned here, as a rule, go far beyond the scope of the course "Radio circuits and signals". Therefore, this book will not describe the relationship that exists between the physical appearance of the signal and the meaning of the message contained in it. Moreover, the question of the value of the information contained in the message and, ultimately, in the signal will not be discussed.
1.1. Classification of radio engineering signals
When starting to study any new objects or phenomena, science always strives to carry out their preliminary classification. Below, such an attempt is made in relation to signals.
The main goal is to develop classification criteria, as well as, which is very important for the subsequent, to establish a certain terminology.
Description of signals by means of mathematical models.
Signals as physical processes can be studied using various instruments and devices - electronic oscilloscopes, voltmeters, receivers. This empirical method has a significant drawback. The phenomena observed by the experimenter always appear as particular, isolated manifestations, devoid of the degree of generalization that would make it possible to judge their fundamental properties, to predict the results under changed conditions.
In order to make signals objects of theoretical study and calculations, one should indicate the method of their mathematical description or, in the language of modern science, create a mathematical model of the signal under study.
A mathematical model of a signal can be, for example, a functional dependence, the argument of which is time. As a rule, in the future, such mathematical models of signals will be denoted by the symbols of the Latin alphabet s (t), u (t), f (t), etc.
The creation of a model (in this case, a physical signal) is the first essential step towards a systematic study of the properties of a phenomenon. First of all, the mathematical model allows one to abstract from the specific nature of the signal carrier. In radio engineering, the same mathematical model describes with equal success the current, voltage, strength of the electromagnetic field, etc.
The essential side of the abstract method, based on the concept of a mathematical model, lies in the fact that we get the opportunity to describe precisely those properties of signals that objectively appear as decisively important. At the same time, a large number of secondary signs are ignored. For example, in the overwhelming majority of cases it is extremely difficult to choose the exact functional dependences that would correspond to the electrical oscillations observed experimentally. Therefore, the researcher, guided by the entire set of information available to him, selects from the available arsenal of mathematical models of signals those that in a particular situation describe the physical process in the best and simplest way. So, choosing a model is pretty much a creative process.
Functions describing signals can take both real and complex values. Therefore, in what follows we will often talk about real and complex signals. The use of this or that principle is a matter of mathematical convenience.
Knowing the mathematical models of signals, one can compare these signals with each other, establish their identity and difference, and carry out a classification.
One-dimensional and multidimensional signals.
A typical signal for radio engineering is the voltage at the terminals of a circuit or the current in a branch.
Such a signal, described by one function of time, is usually called one-dimensional. In this book, one-dimensional signals will most often be studied. However, it is sometimes convenient to introduce into consideration multidimensional, or vector, signals of the form
formed by some set of one-dimensional signals. An integer N is called the dimension of such a signal (the terminology is borrowed from linear algebra).
A multidimensional signal is, for example, a system of voltages at the terminals of a multipole.
Note that a multidimensional signal is an ordered collection of one-dimensional signals. Therefore, in the general case, signals with different order of components are not equal to each other:
Multivariate signal models are especially useful in cases where the functioning of complex systems is analyzed using a computer.
Deterministic and random signals.
Another principle of classification of radio engineering signals is based on the possibility or impossibility of accurate prediction of their instantaneous values at any time.
If the mathematical model of the signal allows such a prediction, then the signal is called deterministic. The methods of its assignment can be varied - a mathematical formula, a computational algorithm, and finally, a verbal description.
Strictly speaking, deterministic signals, as well as deterministic processes corresponding to them, do not exist. The inevitable interaction of the system with the surrounding physical objects, the presence of chaotic thermal fluctuations and simply incomplete knowledge about the initial state of the system - all this forces us to consider real signals as random functions of time.
In radio engineering, random signals often manifest themselves as interference, preventing the extraction of information from the received waveform. The problem of countering interference, increasing the noise immunity of radio reception is one of the central problems of radio engineering.
It may seem that the concept of "random signal" is controversial. However, it is not. For example, the signal at the output of a radio telescope receiver directed at a source of cosmic radiation is chaotic oscillations, which, however, carry a variety of information about a natural object.
There is no insurmountable boundary between deterministic and random signals.
Very often, in conditions when the level of interference is much less than the level of a useful signal with a known shape, a simpler deterministic model turns out to be quite adequate to the task at hand.
The methods of statistical radio engineering, developed in recent decades for the analysis of the properties of random signals, have many specific features and are based on the mathematical apparatus of the theory of probability and the theory of random processes. A number of chapters of this book will be entirely devoted to this range of questions.
Pulse signals.
A very important class of signals for radio engineering are impulses, that is, oscillations that exist only within a finite period of time. At the same time, video pulses (Fig. 1.1, a) and radio pulses (Fig. 1.1, b) are distinguished. The difference between these two main types of impulses is as follows. If - video pulse, then the corresponding radio pulse (frequency and initial are arbitrary). In this case, the function is called the envelope of the radio pulse, and the function is called its filling.
Rice. 1.1. Pulse signals and their characteristics: a - video pulse, b - radio pulse; c - determination of the numerical parameters of the pulse
In technical calculations, instead of a complete mathematical model, which takes into account the details of the "fine structure" of the pulse, numerical parameters are often used, which give a simplified idea of its shape. So, for a video pulse that is close in shape to a trapezoid (Fig.1.1, c), it is customary to determine its amplitude (height) A. From the time parameters indicate the pulse duration, the front duration and the cutoff duration
In radio engineering, they deal with voltage pulses, the amplitudes of which range from fractions of a microvolt to several kilovolts, and the durations reach fractions of a nanosecond.
Analog, discrete and digital signals.
Concluding a brief overview of the principles of classification of radio engineering signals, we note the following. Often the physical process that generates a signal develops in time in such a way that the signal values can be measured in. any moments in time. Signals of this class are usually called analog (continuous).
The term "analog signal" emphasizes that such a signal is "analogous", completely similar to the physical process that generates it.
A one-dimensional analog signal is clearly represented by its graph (oscillogram), which can be either continuous or with break points.
Initially, signals of an exclusively analog type were used in radio engineering. Such signals made it possible to successfully solve relatively simple technical problems (radio communication, television, etc.). Analog signals were easy to generate, receive, and process using the means available at the time.
The increased requirements for radio engineering systems, a variety of applications forced the search for new principles for their construction. In some cases, analogue systems have been replaced by impulse systems, the operation of which is based on the use of discrete signals. The simplest mathematical model of a discrete signal is a countable set of points - an integer) on the time axis, at each of which the reference value of the signal is determined. Typically, the sampling rate for each signal is constant.
One of the advantages of discrete signals over analog signals is that there is no need to reproduce the signal continuously at all times. Due to this, it becomes possible to transmit messages from different sources over the same radio link, organizing multichannel communication with time division of channels.
Intuitively, fast time-varying analog signals require small steps to sample. In ch. 5 this fundamentally important issue will be explored in detail.
A special kind of discrete signals are digital signals. They are characterized by the fact that the reading values are presented in the form of numbers. For reasons of technical convenience of implementation and processing, binary numbers with a limited and usually not too large number of digits are usually used. Recently, there has been a trend towards widespread adoption of systems with digital signals. This is due to the significant advances made by microelectronics and integrated circuitry.
It should be borne in mind that in essence any discrete or digital signal (we are talking about a signal - a physical process, not a mathematical model) is an analog signal. So, a slowly changing analog signal can be compared with its discrete image, which has the form of a sequence of rectangular video pulses of the same duration (Fig. 1.2, a); the height of the ethnh impulses is proportional to the values at the reference points. However, you can act differently, keeping the height of the pulses constant, but changing their duration in accordance with the current reading values (Fig. 1.2, b).
Rice. 1.2. Discretization of the analog signal: a - at variable amplitude; b - with variable duration of the counting pulses
The two analog sampling methods presented here become equivalent if we assume that the values of the analog signal at the sampling points are proportional to the area of the individual video pulses.
The fixation of sample values in the form of numbers is also carried out by displaying the latter in the form of a sequence of video pulses. The binary number system is ideally suited for this procedure. You can, for example, associate a high level with one and a low potential level with zero, f Discrete signals and their properties will be studied in detail in Ch. 15.
