Before embarking on the study of any new phenomena, processes or objects, science always strives to classify them according to the greatest possible features. To consider and analyze signals, we single out their main classes. This is necessary for two reasons. First, checking whether a signal belongs to a particular class is an analysis procedure. Secondly, to represent and analyze signals of different classes, it is often necessary to use different tools and approaches. The basic concepts, terms and definitions in the field of radio signals are established by the national (formerly state) standard “Radio signals. Terms and Definitions". Radio signals are extremely diverse. A part of a brief classification of signals according to a number of features is shown in Fig. 1. More details about a number of concepts are given below. 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
(bivariate signal; n = 2) or more (multivariate signal n > 2) independent variables. One-dimensional signals are only functions of time, while multi-dimensional ones, in addition, reflect the position in n-dimensional space.
Fig.1. Classification of radio signals
For definiteness and simplification, we will mainly consider one-dimensional time-dependent signals, however, the material of the tutorial allows generalization to the multidimensional case, when the signal is represented as a finite or infinite set of points, for example, in space, the position of which depends on time. In television systems, the black-and-white image signal can be viewed as a function f(x, y, f) of two spatial coordinates and time, representing the radiation intensity at point (x, y) at 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 (we can also consider these three functions as components of a three-dimensional vector fields). Besides, different kinds television signals may occur when transmitting a television image together with sound.
A multidimensional signal is an ordered set of one-dimensional signals. A multidimensional signal is created, for example, by a system of voltages at the terminals of a multipole (Fig. 2). 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 is an integer, the dimension of the signal.
R 
is. 2. Multipole voltage system
According to the features of the structure of the temporal representation (Fig. 3), all radio signals are divided into analog (analog), discrete (discrete-time; from Latin discretus - divided, intermittent) and digital (digital).
If the physical process that generates a one-dimensional signal can be represented as a continuous function of time u(t) (Fig. 3, a), then such a signal is called analog (continuous), or, more generally, continual (continuos - multistage), if the latter has jumps , discontinuities along the amplitude axis. Note that traditionally the term "analog" is used to describe signals that are continuous in time. A continuous signal can be treated as a real or complex oscillation in time u(t), which is a function of a continuous real time variable. The concept of "analogue" signal is due to the fact that any instantaneous value of it is similar to the law of change of the corresponding physical quantity in time. An example of an analog signal is a voltage that is applied to the input of an oscilloscope, resulting in a continuous waveform as a function of time on the screen. Since modern continuous signal processing using resistors, capacitors, operational amplifiers, and the like has little to do with analog computers, the term "analog" today seems to be not entirely unfortunate. It would be more correct to call continuous signal processing what is commonly referred to today as analog signal processing.
In radio electronics and communication technology, impulse systems, devices and circuits are widely used, the operation of which is based on the use of discrete signals. For example, an electrical signal that reflects speech is continuous both in level and time, and a temperature sensor that outputs its values every 10 minutes serves as a source of signals that are continuous in value but discrete in time.
A discrete signal is obtained from an analog signal by a special conversion. The process of converting an analog signal into a sequence of samples is called sampling (sampling), and the result of such a conversion is a discrete signal or a discrete series (discrete series).
The simplest mathematical model of a discrete signal 
- a sequence of points on the time axis, taken, as a rule, at regular intervals 
, called the sampling period (or interval, sampling step; sample time), and in each of which the values of the corresponding continuous signal are given (Fig. 3, b). The reciprocal of the sampling period is called the sampling frequency: 
(other designation 
). The corresponding angular (circular) frequency is determined as follows: 
.
Discrete signals can be created directly by the source of information (in particular, discrete readings of sensor signals in control systems). The simplest example of discrete signals is temperature information transmitted in radio and television news programs, but there is usually no weather information in the pauses between such transmissions. It should not be thought that discrete messages are necessarily converted into discrete signals, and continuous messages into continuous signals. Most often, it is continuous signals that are used to transmit discrete messages (as their carriers, that is, carriers). Discrete signals can be used to transmit continuous messages.
Obviously, in the general case, the representation of a continuous signal by a set of discrete samples leads to a certain loss of useful information, since we do not know anything about the behavior of the signal in the intervals between samples. However, there is a class of analog signals for which such information loss practically does not occur, and therefore they can be reconstructed with a high degree of accuracy from the values of their discrete samples.
A variety of discrete signals is a digital signal (digital signal), In the process of converting discrete signal samples into digital form (usually into binary numbers), it is quantized by the voltage level (quantization) 
. In this case, the values of the signal levels can be numbered by binary numbers with a finite, required number of digits. A signal discrete in time and quantized in level is called a digital signal. By the way, signals quantized in level but continuous in time are rare in practice. In a digital signal, discrete signal values 
first, they are quantized by the level (Fig. 3, c) and then the quantized samples of the discrete signal are replaced by numbers 
most often implemented in a binary code, which is represented by high (one) and low (zero) levels of voltage potentials - short pulses with a duration 
(Fig. 3, d). Such a code is called unipolar. Since the readings can take a finite set of values of voltage levels (see, for example, the second reading in Fig. 3, d, which in digital form can be written almost equally likely as the number 5 - 0101, and the number 4 - 0100), then when presenting the signal, it is inevitable it is rounded off. The resulting rounding errors are called quantization errors (or noise) (quantization error, quantization noise).
The sequence of numbers representing the signal during digital processing is a discrete series. The numbers that make up the sequence are the values of the signal at separate (discrete) points in time and are called digital signal samples (samples). Further, the quantized value of the signal is represented as a set of pulses characterizing zeros ("0") and ones ("1") when representing this value in binary system calculus (Fig. 3, d). A set of pulses is used to amplitude modulate the carrier wave and obtain a pulse code radio signal.
As a result of digital processing, nothing “physical” is obtained, only numbers. And numbers are an abstraction, a way of describing the information contained in a message. Therefore, we need to have something physical that will represent the numbers or "carry" the numbers. So, the essence of digital processing is that a physical signal (voltage, current, etc.) is converted into a sequence of numbers, which is then subjected to mathematical transformations in a computing device.
Transformed digital signal(sequence of numbers) can be converted back to voltage or current if necessary.
Digital signal processing provides ample opportunities for transmitting, receiving and converting information, including those that cannot be implemented using analog technology. In practice, when analyzing and processing signals, digital signals are most often replaced by discrete ones, and their difference from digital ones is interpreted as quantization noise. In this regard, the effects associated with level quantization and digitization of signals, in most cases, will not be taken into account. It can be said that both in discrete and digital circuits (in particular, in digital filters) discrete signals are processed, only inside the structure of digital circuits these signals are represented by numbers.
Computing devices designed for signal processing can operate with digital signals. There are also devices built mainly on the basis of analog circuitry that work with discrete signals, presented in the form of pulses of various amplitudes, durations or repetition rates.
One of the main features by which signals differ is the predictability of the signal (its values) in time.
R 
is. 3. Radio signals:
a - analog; b - discrete; c - quantized; g - digital
According to the mathematical representation (according to the degree of a priori presence, from Latin a priori - from the previous, i.e. pre-experimental information), all radio engineering signals are usually divided into two main groups: deterministic (regular; determined) and random (casual) signals (Fig. 4).
Radio engineering signals are called deterministic, the instantaneous values of which are reliably known at any time, i.e., predictable with a probability equal to one. Deterministic signals are described by predetermined time functions. Incidentally, the instantaneous value of a signal is a measure of how much and in which direction a variable deviates from zero; thus, the instantaneous values of the signal can be both positive and negative (Fig. 4, a). The simplest examples of a deterministic signal are a harmonic oscillation with a known initial phase, high-frequency oscillations modulated according to a known law, a sequence or burst of pulses, the shape, amplitude and time position of which are known in advance.
If the message transmitted over the communication channels were deterministic, that is, known in advance with complete certainty, then its transmission would be meaningless. Such a deterministic message does not, in fact, contain any new information. Therefore, messages should be considered as random events (or random functions, random variables). In other words, there must be some set of message options (for example, a set of different pressure values given by the sensor), of which one is realized with a certain probability. In this regard, the signal is also a random function. A deterministic signal cannot be a carrier of information. It can only be used for testing a radio engineering information transmission system or testing its individual devices. The random nature of messages, as well as interference, determined the importance of the theory of probability in constructing the theory of information transmission.
Rice. 4. Signals:
a - deterministic; b - random 
Deterministic signals are divided into periodic and non-periodic (pulse). Final energy signal significantly different from zero for a limited time interval commensurate with the completion time transition process in the system for which it is intended to act, is called a pulse signal.
Random signals are signals whose instantaneous values at any time are not known and cannot be predicted with a probability equal to one. In fact, for random signals, you can only know the probability that it will take on any value.
It may seem that the concept of "random signal" is not entirely correct.
But it's not. For example, the voltage at the output of a thermal imager receiver directed at an IR radiation source represents chaotic oscillations that carry various information about the analyzed object. Strictly speaking, all signals encountered in practice are random and most of them represent chaotic functions of time (Fig. 4b). Paradoxical as it may seem at first glance, but a signal carrying useful information can only be a random signal. The information in such a signal is embedded in a set of amplitude, frequency (phase) or code changes of the transmitted signal. Communication signals change instantaneous values in time, and these changes can only be predicted with a certain probability less than one. Thus, communication signals are in some way random processes, and therefore their description is carried out using methods similar to methods for describing random processes.
In the process of transmitting useful information, radio signals can be subjected to one or another transformation. This is usually reflected in their name: signals are modulated, demodulated (detected), coded (decoded), amplified, delayed, sampled, quantized, etc.
According to the purpose that the signals have in the modulation process, they can be divided into modulating (primary signal that modulates the carrier wave) or modulated (carrier wave).
By belonging to one or another type of radio engineering systems, and in particular information transmission systems, there are “communication”, telephone, telegraph, broadcasting, television, radar, radio navigation, measuring, control, service (including pilot signals) and other signals .
This brief classification of radio signals does not fully cover all their diversity.
Questions for the state exam
on the course "Digital Signal Processing and Signal Processors"
(Korneev D.A.)
distance learning
Classification of signals, energy and power of signals. Fourier series. Sine-cosine form, real form, complex form.
CLASSIFICATION OF SIGNALS USED IN RADIO ENGINEERING
From an informational point of view, signals can be divided into deterministic And random.
deterministic any signal is called, the instantaneous value of which at any time can be predicted with a probability of one. Examples of deterministic signals are pulses or bursts of pulses whose shape, amplitude and position in time are known, as well as a continuous signal with given amplitude and phase relationships within its spectrum.
TO random refer to signals whose instantaneous values are not known in advance and can only be predicted with a certain probability less than one. Such signals are, for example, electrical voltage corresponding to speech, music, sequences of characters of the telegraph code when transmitting a non-repeating text. Random signals also include a sequence of radio pulses at the input of the radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, the position of the target, and some other reasons. Many other examples of random signals can be given. Essentially, any signal that carries information should be considered random.
listed above deterministic signals, "fully known", no longer contain information. In what follows, such signals will often be referred to as oscillations.
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.
Analog signal Discrete signal
Quantized signal Digital signal
Rice. 1.2. Signals 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)
Meanwhile, the signals from the message source 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 magnitude and continuous in time (Fig. 1.2, a);
arbitrary in magnitude 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).
First class signals (Fig. 1.2, a) are sometimes called analog, since they can be interpreted as electrical models physical quantities, or continuous, since they are given along the time axis on an uncountable set of points. Such sets are called continuous. In this case, along the ordinate axis, the signals can take on any value in a certain interval. Since these signals may have discontinuities, as in Fig. 1.2, a, then, in order to avoid incorrectness in the description, it is better to denote such signals by the term continual.
So, the continuum 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 created directly by the source of information (for example, discrete sensors in control systems or telemetry) or formed as a result of discretization of continuous signals.
On fig. 1.2, b shows a signal given for discrete values of time t (on a countable set of points); the magnitude of the signal at these points can take on any value in a certain interval along the ordinate axis (as in Fig. 1.2, a). Thus, the term discrete characterizes not the signal itself, but the way it is specified on the time axis.
The signal in fig. 1.2, in is given on the entire time axis, however, its value can only take on discrete values. In such cases, one speaks of a signal quantized by level.
In what follows, the term discrete will be used only in relation to discretization in time; 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 levels can be numbered with numbers with a finite number of digits. Therefore, a signal discrete in time and quantized in terms of level (Fig. 1.2, d) will be called digital in the future.
Thus, one can distinguish between continual (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 signal type and circuit type is shown in functional diagram(Fig. 1.3).
When processing a continuous signal using an analog circuit, no additional signal conversions are required. When processing a continuum signal using a discrete circuit, two transformations are required: signal sampling in time at the input of the discrete circuit and the inverse transformation, i.e., restoration of the continuum structure of the signal at the output of the discrete circuit.
For an arbitrary signal s(t) = a(t)+jb(t), where a(t) and b(t) are real functions, the instantaneous signal power (energy distribution density) is determined by the expression:
w(t) = s(t)s*(t) = a 2 (t)+b 2 (t) = |s(t)| 2.
The signal energy is equal to the integral of the power over the entire interval of signal existence. In the limit:
E s = w(t)dt = |s(t)| 2dt.
Essentially, instantaneous power is the power density of the signal, since power measurements are only possible through the energy released at certain intervals of non-zero length:
w(t) = (1/Dt) |s(t)| 2dt.
The signal s(t) is studied, as a rule, at a certain interval T (for periodic signals - within one period T), while average power signal:
W T (t) = (1/T) w(t) dt = (1/T) |s(t)| 2dt.
The concept of average power can also be extended to undamped signals, the energy of which is infinitely high. In the case of an unlimited interval T, a strictly correct determination of the average signal power is made by the formula:
Ws = w(t)dt.
The idea that any periodic function can be represented as a series of harmonically related sines and cosines was proposed by Baron Jean Baptiste Joseph Fourier (1768−1830).
Fourier series function f(x) is represented as
.
Fundamentals of digital signal processing (DSP).
Lecturer: Kuznetsov Vadim Vadimovich
https://github.com/ra3xdh/DSP-RPD
https://github.com/ra3xdh/RTUiS-labs
Question. Radio signals. Classification.
Signals can be voltage, current, field strength. In most cases, carriers of radio signals are electromagnetic oscillations. The mathematical model of the signal is usually a functional dependence whose argument is time (the dependence of the voltage in the circuit on time). For deterministic signals based on mathematical model you can find out the instantaneous value of the signal at any time. An example of a deterministic signal is a sinusoidal voltage, f=50Hz w=314s^-1.
Pulse signals exist only within a finite time interval. Examples of pulse signals: video pulse (Fig. 2a) and radio pulse (Fig. 2b).
If the physical process generating the signal develops in time in such a way that it can be measured at any time, then the signals of this class are called analog. An analog signal can be represented by a graph of its change in time, that is, an oscillogram.
Discrete signals are described by a set of samples at regular intervals. An example of a discrete signal is shown in Figure 3.
Digital signals are a special kind of discrete ones. Reference values are presented as numbers. Usually binary numbers with some dimension are used. An example of a digital signal is shown in Table 1.
analog signals.
A periodic signal S(t), period T has the following property: S(t)=S(t±nT) n=1,2,.. An example of a periodic signal is shown in Figure 4.
The period of the signal is related to the frequency f and the circular frequency w as follows: f=1/T=w/2π. Other examples of periodic signals are shown in Figure 5.
Question. modulated signal. Basics of modulation.
In a modulated signal, the amplitude, frequency, and phase of a sinusoidal RF signal changes in time with the LF. The low frequency signal is superimposed on the carrier.
1. Amplitude modulation (AM).
S(t) - sound signal, - RF signal, carrier, M - modulation factor.
An example of a modulated signal is shown in Figure 6.
2. Frequency modulation (FM:FM). The carrier amplitude remains unchanged, and the carrier frequency changes in time with the modulated signal.
The oscillogram of the frequency modulated signal is shown in Figure 7.
3. Phase modulation (FM:PM). . the oscillogram of the PM signal is shown in Figure 8.
During the positive half-cycle, the phase of the modulated oscillation leads the phase of the carrier frequency oscillation, while the oscillation period decreases and the frequency increases. During the negative period of the modulating voltage, the phase of the modulated waveform lags in phase with the carrier frequency waveform. Thus, FM is at the same time FM. For FM, the opposite is true: frequency modulation is also phase modulation. FM is used in professional radio communications.
Sigma and delta functions.
The sigma function is given by the following expression:
Delta function is an impulse of infinitely large amplitude and infinitely short duration. (Fig. 10).
The delta function is the derivative of the sigma function.
If the signal given by the continuous function is multiplied by the delta functions and integrated over time, the result will be the instantaneous value of the signal at the point where the delta pulse is concentrated.
From the filtering properties of the delta function follows the scheme of the instantaneous signal value meter.
Sigma and delta functions are used to analyze the passage of analog and digital signals through linear systems. The response of the system, if a delta pulse is applied to it, is called impulse response system H(t).
Question. Signal power and energy.
If not a constant voltage is applied to the resistor, but an alternating signal s(t), then the power will also be variable (instantaneous power).
In signal theory, it is usually assumed that R=1. w=s(t)^2. To find the signal energy it is necessary to integrate the power over the entire range;
For signals infinite in time, the average power can be determined as follows:
W=[W], E=[(V^2)*s]
It is this energy that is released on a 1 ohm resistor if a voltage s (t) is applied to it.
If the signal is emitted over a certain interval T, then the average signal power is considered.
Spectral analysis of signals.
Question. Decomposition of an analog signal in a Fourier series.
An example of the representation of a sawtooth signal as a sum of sinusoidal signals with different amplitude and phase is shown in fig. 12.
Let's introduce the fundamental frequency of a periodic signal with period T: w_1=2pi/T. When expanded in a Fourier series, a periodic signal is represented as a sum of sinusoidal signals or harmonics, with frequencies that are multiples of the fundamental frequency: 2w_1, 3w_1... The amplitudes of these signals are called expansion coefficients. The Fourier series is written as a sum of harmonics:
The real form of the Fourier series:
Using the well-known notation from the course of electrical engineering in the form of a complex number, the Fourier series is represented as:
This expression includes harmonics with negative frequencies. Negative frequency is not a physical concept, it has to do with the way complex numbers are represented. Since the sum of the harmonics must be a real number, each harmonic has a complex conjugate with –ω. By absolute value, the amplitudes of harmonics with positive and negative frequencies are equal.
Question. Spectral diagrams.
There are amplitude and phase spectral diagrams. By horizontal axis lay off the frequencies of harmonics, along the vertical - amplitudes (phases). If the modulus of the Fourier series is shown in complex form, then the positive and negative circular frequencies ω are plotted along the X axis.
An example of the spectrum of an analog periodic signal. (PWM)
Consider a sequence of rectangular pulses with period T, duration τ, and amplitude A.
Duty cycle.
The oscillogram of such a signal is shown in Figure 13.
The DC component of a square wave.
bn = 0.
The spectral diagram for a sequence of rectangular pulses is shown in fig. 14.
It can be seen from the spectrum of the diagram that with an increase in the duty cycle, the pulse duration decreases. The sequence of rectangular pulses has a richer spectral content, more harmonics and more amplitudes are present in the spectrum. Thus, a shortening of the pulse duration leads to a broadening of the spectrum. Broad spectrum signals can cause interference.
The Fourier series is calculated using mathematical packages.
Fourier transform.
It is used to expand the range of valid signals.
Distinguish between direct and inverse transformation.
Question. direct conversion(transition from signal to spectrum).
Let s(t) be a single pulse signal of finite duration. Let's supplement it with the same, periodically following signal, with period T. We get a sequence of pulses (Fig. 15).
To pass to the Fourier transform and find the spectrum of a single pulse, it is necessary to find the limiting form of the Fourier series in complex form at
Spectrum calculation:
The physical meaning of the spectral density is that it is a proportionality factor between the length of a small frequency interval Δf near the frequency f 0 and the amplitude of a harmonic signal with a frequency f 0 . The signal s(t) is, as it were, made up of many different sinusoidal signals of small amplitude. The density spectrum shows the contribution to the signal of elementary sinusoidal signals of each frequency.
The probability density spectrum is a complex number and is plotted as a curve on the complex plane.
Real number - amplitude spectrum
power spectrum
Phase spectrum
Properties of the Fourier Transform
Linearity - the spectrum of the sum of several signals multiplied by constant coefficients is equal to the sum of these signals. If the signal amplitude changes by a factor of A, then its spectral density also changes by a factor of A.
Property of the real and imaginary parts of the spectrum. The real part of the spectrum, that is, the amplitude spectrum, is an even function of frequency. The amplitude spectrum is symmetrical with respect to the zero frequency. The imaginary part of the spectrum is an odd function of frequency. The phase spectrum is antisymmetric with respect to zero frequency.
Time shift of the signal. When the signal is shifted in time, the amplitude spectrum does not change, but the phase spectrum shifts in phase.
The spectrum of the product of the signals is equal to the convolution of the spectra and vice versa.
The property is used to find the signal at the output if the frequency response is known.
The linear system and the signals at its input and output are shown in Figure 20.
The spectrum of the delta function.
The delta pulse spectrum contains all frequencies from 0 to .
Spectrum of derivative and integral.
Connection with Fourier series.
Knowing the transformation for one period of a periodic signal, you can calculate its expansion in a Fourier series.
An example of calculating the spectrum of an impulse signal.
Let us calculate the spectrum of a rectangular video pulse with amplitude and duration . The pulse is located symmetrically with respect to the origin (Fig. 22).
We pass from the circular frequency to the frequency f.
The amplitude spectrum is shown in (Figure 23).
The phase spectrum is shown in (Figure 24).
The power spectrum is shown in (Figure 25).
Question. Inverse Fourier Transform.
The condition for the existence of the spectral density of the signal.
Spectral analysis of integrable signals.
A signal can be mapped to spectral density if the signal is absolutely integrated.
Doesn't apply to a perfectly integrated signal. harmonic vibrations and direct current.
Examples of absolutely integrable and non-integrable signals in (Fig. 16).
The spectra of such signals are represented in terms of delta functions.
The spectrum of the constant level signal A is a delta pulse located at zero frequency ().
The physical meaning of this expression is a signal that is constant in absolute value and has a constant component in time only at zero frequency.
The spectrum of a sinusoidal signal.
Any periodic signal can be represented by a Fourier series in a complex form, that is, as a sum of sinusoidal signals.
Spectra direct current, sinusoidal and periodic signal are shown in (Fig. 17).
On a spectrum analyzer, the spectrum of a periodic signal will be observed as a train of spiked pulses. The amplitudes of these pulses are proportional to the amplitudes of the harmonics. A typical view of the spectrum is shown in (Fig. 18).
Spectral analysis can also be applied to random signals. For them, the power spectrum is considered. For example, consider white noise (Fig. 1).
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 time can be predicted with a probability of one. Examples of deterministic signals are pulses or bursts of pulses whose shape, amplitude and position in time are known, as well as a continuous signal with given amplitude and phase relationships within its spectrum.
Random signals include signals whose instantaneous values are not known in advance and can only be predicted with a certain probability less than one. Such signals are, for example, electrical 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 the radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, the position of the target, and some other reasons. Many other examples of random signals can be given. Essentially, any signal that carries information should be considered random.
The deterministic signals listed above, "fully known", no longer contain information. In what follows, such signals will often be referred to as oscillations.
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 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 interference, are often combined by the term random fluctuations or random processes.
Further subdivision of signals can be related to their nature: one can speak of a signal as a physical process or as encoded, for example, in a binary code, numbers.
In the first case, a signal is understood as some time-varying electrical quantity (voltage, current, charge, etc.) associated in a certain way 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 transmitters and radiated 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 channel are often divided into control signals and radio signals; the former are 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.).
The processing of digitally encoded signals is carried out with the help of computer technology.
Shown in Fig. 1.1 and described in § 1.2 structural scheme the communication channel does not contain indications of the type of signal used to transmit the message and the structure of individual devices.
Meanwhile, the signals from the message source, 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 magnitude and continuous in time (Fig. 1.2, a);
arbitrary in magnitude 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 continuum. In this case, along the ordinate axis, the signals can take on any value in a certain interval. Since these signals may have discontinuities, as in Fig. 1.2, a, then, in order to avoid incorrectness in the description, it is better to denote such signals by the term continual.
So, the continuum 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 created directly by the source of information (for example, discrete sensors in control systems or telemetry) or formed as a result of discretization of continuous signals.
On fig. 1.2, b shows a signal given for discrete values of time t (on a countable set of points); the magnitude of the signal at these points can take on any value in a certain interval along the ordinate axis (as in Fig. 1.2, a). Thus, the term discrete characterizes not the signal itself, but the way it is specified on the time axis.
The signal in fig. 1.2, in is given on the entire time axis, however, its value can only take on discrete values. In such cases, one speaks of a signal quantized by level.
In what follows, the term discrete will be used only in relation to discretization in time; 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 levels can be numbered with numbers with a finite number of digits. Therefore, a signal discrete in time and quantized in terms of level (Fig. 1.2, d) will be called digital in the future.
Thus, one can distinguish between continual (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 continuum signal using a discrete circuit, two transformations are required: signal sampling in time at the input of the discrete circuit and the inverse transformation, i.e., restoration of the continuum structure of the signal at the output of the discrete circuit.
Rice. 1.3. Types of signal and their corresponding circuits
Finally, when digitally processing a continuous signal, two more additional conversions are required: analog-to-digit, i.e., quantization and digital coding at the input of the digital circuit, and inverse digital-to-analogue conversion, i.e., decoding at the output of the 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 continuum signal. Therefore, digital technology has become most widespread in the processing of signals at relatively low frequencies (sound and video frequencies). However, advances in microelectronics contribute to a rapid increase in the upper limit of the processed frequencies.
General information about radio signals
When transmitting information over a distance with the help of radio engineering systems, various types of radio engineering (electrical) signals are used. Traditionally radio engineering signals are considered to be any electrical signals related to the radio range. From a mathematical point of view, any radio signal can be represented by some function of time u(t ), which characterizes the change in its instantaneous values of voltage (most often), current or power. According to the mathematical representation, the whole variety of radio engineering signals is usually divided into two main groups: deterministic (regular) and random signals.
deterministic called radio signals, the instantaneous values of which are reliably known at any time, i.e., predictable with a probability equal to one /1/. An example of a deterministic radio engineering signal is a harmonic oscillation. It should be noted that, in fact, a deterministic signal does not carry any information and almost all of its parameters can be transmitted over a radio channel with one or more code values. In other words, deterministic signals (messages) essentially contain no information, and there is no point in transmitting them.
random signals are signals, the instantaneous values of which are not known at any time and cannot be predicted with a probability equal to one /1/. Almost all real random signals, or most of them, are chaotic functions of time.
According to the features of the structure of the temporal representation, all radio signals are divided into continuous and discrete.and by the type of transmitted information: analog and digital.In radio engineering, pulse systems are widely used, the operation of which is based on the use of discrete signals. One of the varieties of discrete signals is digital signal /1/. In it, the discrete values of the signal are replaced by numbers, most often implemented in binary code, which represent high (unit) And low (zero) voltage potential levels.
Functions describing signals can take both real and complex values. Therefore, in radio engineering they talk about real and complex signals. The use of one form or another of the signal description was a matter of mathematical convenience.
Spectrum concept
Direct analysis of the impact of complex waveforms on radio circuits very difficult and not always possible. Therefore, it makes sense to represent complex signals as the sum of some simple elementary signals. The principle of superposition justifies the possibility of such a representation, stating that in linear circuits the effect of the total signal is equivalent to the sum of the effects of the corresponding signals separately.
Harmonics are often used as elementary signals. This choice has a number of advantages:
a) The expansion into harmonics is implemented quite easily by using the Fourier transform.
b) When a harmonic signal is applied to any linear circuit, its shape does not change (remains harmonic). The frequency of the signal is also stored. Amplitude and phase change, of course; they can be calculated relatively simply using the method of complex amplitudes.
c) In engineering, resonant systems are widely used, which make it possible to experimentally isolate one harmonic from a complex signal.
Representing a signal as a sum of harmonics given by frequency, amplitude and phase is called signal decomposition into a spectrum.
The harmonics included in the signal are specified in trigonometric or imaginary exponential form.
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