Analog and discrete signals. Analog, discrete, digital signals
Signals are information codes that people use to convey messages in an information system. The signal can be given, but it is not necessary to receive it. Whereas a message can only be considered a signal (or a set of signals) that was received and decoded by the recipient (analog and digital signal).
One of the first methods of transmitting information without the participation of people or other living beings were signal fires. When danger arose, fires were lit sequentially from one post to another. Next, we will consider the method of transmitting information using electromagnetic signals and will dwell in detail on the topic analog and digital signal.
Any signal can be represented as a function that describes changes in its characteristics. This representation is convenient for studying radio engineering devices and systems. In addition to the signal in radio engineering, there is also noise, which is its alternative. Noise does not carry useful information and distorts the signal by interacting with it.
The concept itself makes it possible to abstract from specific physical quantities when considering phenomena related to the encoding and decoding of information. The mathematical model of the signal in research allows one to rely on the parameters of the time function.
Signal types
Signals based on the physical environment of the information carrier are divided into electrical, optical, acoustic and electromagnetic.
According to the setting method, the signal can be regular or irregular. A regular signal is represented as a deterministic function of time. An irregular signal in radio engineering is represented by a chaotic function of time and is analyzed by a probabilistic approach.
Signals, depending on the function that describes their parameters, can be analog or discrete. A discrete signal that has been quantized is called a digital signal.
Signal Processing
Analog and digital signals are processed and directed to transmit and receive information encoded in the signal. Once information is extracted, it can be used for various purposes. In special cases, information is formatted.
Analog signals are amplified, filtered, modulated, and demodulated. Digital data can also be subject to compression, detection, etc.
Analog signal
Our senses perceive all information entering them in analog form. For example, if we see a car passing by, we see its movement continuously. If our brain could receive information about its position once every 10 seconds, people would constantly get run over. But we can estimate distance much faster and this distance is clearly defined at each moment of time.
Absolutely the same thing happens with other information, we can evaluate the volume at any moment, feel the pressure our fingers exert on objects, etc. In other words, almost all information that can arise in nature is analogue. The easiest way to transmit such information is through analog signals, which are continuous and defined at any time.
To understand what an analog electrical signal looks like, you can imagine a graph that shows amplitude on the vertical axis and time on the horizontal axis. If we, for example, measure the change in temperature, then a continuous line will appear on the graph, displaying its value at each moment in time. To transmit such a signal using electric current, we need to compare the temperature value with the voltage value. So, for example, 35.342 degrees Celsius can be encoded as a voltage of 3.5342 V.
Analog signals used to be used in all types of communications. To avoid interference, such a signal must be amplified. The higher the noise level, that is, interference, the more the signal must be amplified so that it can be received without distortion. This method of signal processing spends a lot of energy generating heat. In this case, the amplified signal may itself cause interference for other communication channels.
Nowadays, analog signals are still used in television and radio, to convert the input signal in microphones. But in general, this type of signal is being replaced or replaced by digital signals everywhere.
Digital signal
A digital signal is represented by a sequence of digital values. The most commonly used signals today are binary digital signals, as they are used in binary electronics and are easier to encode.
Unlike the previous signal type, a digital signal has two values “1” and “0”. If we remember our example with temperature measurement, then the signal will be generated differently. If the voltage supplied by the analog signal corresponds to the value of the measured temperature, then a certain number of voltage pulses will be supplied in the digital signal for each temperature value. The voltage pulse itself will be equal to “1”, and the absence of voltage will be “0”. The receiving equipment will decode the pulses and restore the original data.
Having imagined what a digital signal will look like on a graph, we will see that the transition from zero to maximum is abrupt. It is this feature that allows the receiving equipment to “see” the signal more clearly. If any interference occurs, it is easier for the receiver to decode the signal than with analog transmission.
However, it is impossible to restore a digital signal with a very high noise level, while it is still possible to “extract” information from an analog type with large distortion. This is due to the cliff effect. The essence of the effect is that digital signals can be transmitted over certain distances, and then simply stop. This effect occurs everywhere and is solved by simply regenerating the signal. Where the signal breaks, you need to insert a repeater or reduce the length of the communication line. The repeater does not amplify the signal, but recognizes its original form and produces an exact copy of it and can be used in any way in the circuit. Such signal repetition methods are actively used in network technologies.
Among other things, analog and digital signals also differ in the ability to encode and encrypt information. This is one of the reasons for the transition of mobile communications to digital.
Analog and digital signal and digital-to-analog conversion
We need to talk a little more about how analog information is transmitted over digital communication channels. Let's use examples again. As already mentioned, sound is an analog signal.
What happens in mobile phones that transmit information via digital channels
Sound entering the microphone undergoes analog-to-digital conversion (ADC). This process consists of 3 steps. Individual signal values are taken at equal intervals of time, a process called sampling. According to Kotelnikov’s theorem on channel capacity, the frequency of taking these values should be twice as high as the highest signal frequency. That is, if our channel has a frequency limit of 4 kHz, then the sampling frequency will be 8 kHz. Next, all selected signal values are rounded or, in other words, quantized. The more levels created, the higher the accuracy of the reconstructed signal at the receiver. All values are then converted into binary code, which is transmitted to the base station and then reaches the other party, which is the receiver. A digital-to-analog conversion (DAC) procedure takes place in the receiver's phone. This is a reverse procedure, the goal of which is to obtain a signal at the output that is as identical as possible to the original one. Next, the analog signal comes out in the form of sound from the phone speaker.
Inputs and outputs are the basic concept of any smart home controller, be it an industrial controller (Beckhoff, Aries, Siemens, ABB - any) or a distributed KNX or HDL system. Any system has elements of the type “Binary input module” or “Analog output block”.
Since in order to calculate the system and generally understand where its cost comes from, it is very important to know the difference between inputs and outputs, I will tell you more about them.
Controller input
An input is a terminal for connecting any device that transmits information to the controller. Signal sources are connected to the controller inputs.
Switch is the source of the signal. The signal can be either “pressed” or “not pressed”. That is, either a logical zero or a logical one. The switch is connected to a terminal on the controller that sees whether it is pressed or not.
Here we move on to the concept that the input and output can be discrete (binary or digital it can be called) or analog. Discrete means perceiving either one or zero. The switch is connected to a discrete input, since it is either pressed or not pressed, there are no other options.
A discrete input can either wait for some voltage to appear or for the input to be shorted to ground. For example, the ARIES PLC controller perceives as a logical unit the appearance of a voltage from +15 to +30 volts at the input. And the WirenBoard controller expects ground (GND) to appear at the input. In the first case, you need to supply +24V to the switch, so that when you press the button, +24 volts will come to the controller input, in the second, we supply a common negative (ground) to the switch, and when you press it, it will come to the controller.
Motion Sensor also connects to the controller's discrete input. The sensor either gives a signal that there is movement, or that there is no movement. Here is the connection diagram for the Colt XS sensor:
The two left contacts are the sensor supply voltage, +12 volts. The two middle contacts are an alarm contact, they are normally closed. That is, if there is no movement, then N and C are closed, if movement appears, then N and C are open. This is done so that if an attacker cuts the sensor wire or damages the sensor, the circuit will break, which will lead to the alarm going off.
In the case of the Aries controller (as well as Beckhoff and most other controllers), we need to apply +24 volts to N, and connect C to the controller input. If the controller sees +24V at the input, that is, a logical one, then everything is in order, there is no movement. As soon as the signal disappears, it means the sensor has worked. In the case of a controller that detects ground rather than voltage, we connect N to the common minus of the controller, C also to its input.
The T sensor contacts are a tamper. They are also normally closed and open when the sensor housing is opened. Many elements of security systems have such contacts.
Water leakage sensor - also connects to a discrete input. The principle is the same, but it is usually normally open. That is, if there is no leakage, there is no signal.
Analog controller input He doesn’t just see whether there is a signal or not, he sees the magnitude of the signal. A universal analog signal is from 0 to 10 volts DC; this signal is given by many different sensors. Or from 1 to 10 volts. There is also a current signal - from 4 to 30 milliamps. Why not from zero, but from 1 volt or 4 milliamps? To understand whether the sensor is working at all. If a sensor with an output signal of 1-10 volts produces 1 volt, then this corresponds to the minimum level of the measured value. If 0 volts it means it is off or broken or the wire is broken.
Temperature sensors can output between 0 and 10 volts. If, according to the passport, the sensor measures temperature in the range from 0 to +50 degrees, it means that a 0 volt signal corresponds to 0 degrees, a 5 volt signal corresponds to +25 degrees, and a 10 volt signal corresponds to +50 degrees. If the sensor measures temperature in the range from -50 to +50 degrees, then 5 volts from the sensor corresponds to 0 degrees, and, say, 8 volts from the sensor corresponds to +30 degrees.
The same with a humidity or light sensor. We look at the measurement range of the parameter, look at the output signal and we can get the exact measured value.
That is, the analog input measures the magnitude of the signal: current or voltage. Or, for example, resistance, if we talk about resistive sensors. Many sensors are available in different versions: with current or voltage output. If we need to find some rare sensor for the system, for example, the level of a certain gas in the air, then most likely it will have an output of either 0-10V or 4-20mA. The more advanced ones have an RS485 interface, more on that later.
Carbon monoxide, natural gas (methane) and propane sensors usually have a discrete output, that is, they connect to a discrete input of the controller and give a signal when the measured gas concentration becomes dangerous. Carbon dioxide or oxygen level sensors provide an analog value corresponding to the level of gas in the air so that the controller itself can decide on some action.
Controller outputs
Outputs are terminals to which the controller itself can send a signal. The controller sends a signal to control something.
Discrete output - This is the output to which the controller can supply either a logical zero or a logical one. That is, either turn it on or turn it off.
Light without brightness adjustment is connected to a discrete output.
Electric underfloor heating - also to a discrete output.
A water shut-off valve or an electrical outlet, or an exhaust fan or a radiator drive - they are connected to the discrete outputs of the controller.
Depending on the specific discrete output module, the output can be either transistor, that is, requiring a relay to control some powerful device, or relay, that is, you can immediately connect something to it. It is necessary to look at the output characteristics - switched voltage and current. It is important to understand that if it is written that the output switches a 230 volt 5 ampere resistive load, then this only applies to an incandescent light bulb. LED lamp - you need to divide the current by ten. Power supplies and electric motors are also far from resistive loads.
Analog output - a terminal to which the controller can send a signal not only on/off, but also a certain control value. These are the same 0-10 (or 1-10) volts or 4-20 milliamps. Next, we connect either a lighting dimmer, a fan speed controller, or something else that has a corresponding input to this control signal.
The lighting control is a power dimmer that, depending on the 0-10 volt signal from the controller, produces an output of 0 to 230 volts AC to power incandescent or dimmable LED lamps.
For LED strips, a PWM dimmer (or PWM driver or dimming power supply) is used; using a 0-10 or 1-10 volt signal from the controller, it supplies a pulse-width modulated signal to the strip for dimming.
For fans, a thyristor regulator is used.
Controller Interfaces
Any controller also has different communication interfaces that determine what other devices it can communicate with. Communication interfaces are usually two-way, that is, the controller can transmit information to them and receive status information.
The Ethernet interface is a connection to a computer network and the Internet for control from a mobile application or communication with other controllers.
The RS-485 ModBus interface is the most common for communication with various equipment. These are air conditioners, ventilation machines, various sensors and actuators, expansion modules and much more.
RS-232 is an interface with a short line range. Usually these are, for example, GSM modems.
KNX is a communication interface with the KNX bus, on which many devices of all types can be located.
We get the following summary picture of the controller inputs and outputs:
Example
Let's take ARIES PLC160 as an example.
It has 16 discrete inputs, 4 of which are fast-acting, that is, suitable for connecting rapidly changing signals, for example, pulse counters. The input voltage must be from 15 to 30 volts for the controller to consider it as one.
12 discrete outputs with switching up to 250 volts 3 amps. That is, it is 690 watts at a voltage of 230V. Suitable for dozens of incandescent or LED lamps. For warm floors or sockets, you need to install an additional relay with a higher switching current.
8 analog inputs. The inputs can be configured to receive standardized signals 0-10V, 0-5mA, 0-20mA, 4-20mA.
4 analog outputs. Depending on the modification of the controller, the output signal will be either voltage (0-10), or current (4-20), or variable.
It has many communication interfaces: Ethernet, RS-485, RS-232, USB (for firmware).
At a cost of 32 thousand, this is an excellent controller on which you can implement a lot of things even without additional blocks. And this is an industrial-grade reliability controller.
You can read about what a smart home on an industrial controller is, as well as more details about inputs and outputs here:
How is a measuring signal different from a signal? Give examples of measuring signals used in various fields of science and technology
A measuring signal is a material carrier of information containing quantitative information about the physical quantity being measured and representing a certain physical process, one of the parameters of which is functionally related to the physical quantity being measured. This parameter is called informative. And the signal carries quantitative information only about the informative parameter, and not about the physical quantity being measured.
Examples of measurement signals could be
Output signals of various generators (magnetohydrodynamic, lasers, masers, etc.), transformers (differential, current, voltage)
Various electromagnetic waves (radio waves, optical radiation, etc.)
List the characteristics by which measuring signals are classified
Based on the nature of measuring informative and time parameters, measuring signals are divided into analog, discrete and digital. According to the nature of changes over time, signals are divided into constant and variable. According to the degree of availability of a priori information, variable measuring signals are divided into deterministic, quasi-deterministic and random.
How are analog, discrete and digital signals different from each other?
An analog signal is a signal described by a continuous or piecewise continuous function Y a (t), and both this function itself and its argument t can take on any values at given intervals (Y min ; Y max) and (t min ; t max).
A discrete signal is a signal that varies discretely in time or in level. In the first case, it can take nT at discrete moments in time, where T = const is the sampling interval (period), n = 0; 1; 2; ... - an integer, any values in the interval (Y min ; Y max) called samples, or samples. Such signals are described by lattice functions. In the second case, the values of the signal Yd(t) exist at any time t in the interval (t min ; t max), however they can take on a limited range of values h j = nq, multiples of the quantum q.
Digital signals are level-quantized and time-discrete signals Y c (nT), which are described by quantized lattice functions (quantized sequences) that at discrete times nT take only a finite series of discrete values - quantization levels h 1 h 2, ..., hn.
Tell us about the characteristics and parameters of random signals
A random signal is a time-varying physical quantity, the instantaneous value of which is a random variable.
The family of realizations of a random process is the main experimental material on the basis of which its characteristics and parameters can be obtained.
Each realization is a non-random function of time. The family of implementations for any fixed time t o is a random variable called the cross section of the random function corresponding to the time t o . Consequently, a random function combines the characteristic features of a random variable and a deterministic function. With a fixed value of the argument, it turns into a random variable, and as a result of each individual experiment it becomes a deterministic function.
Random processes are most fully described by distribution laws: one-dimensional, two-dimensional, etc. However, it is very difficult to operate with such generally multidimensional functions, therefore in engineering applications, such as metrology, they try to make do with the characteristics and parameters of these laws, which describe random processes not completely, but partially. Characteristics of random processes, in contrast to the characteristics of random variables, which are discussed in detail in Chap. 6 are not numbers, but functions. The most important of them are mathematical expectation and variance.
The mathematical expectation of a random function X(t) is a non-random function
mx(t) = M = xp(x, t)dx,
which for each value of the argument t is equal to the mathematical expectation of the corresponding section. Here p(x, t) is the one-dimensional distribution density of the random variable x in the corresponding section of the random process X(t). Thus, the mathematical expectation in this case is the average function around which specific implementations are grouped.
The variance of a random function X(t) is a non-random function
Dx(t) = D = 2 p(x, t)dx,
the value of which for each moment of time is equal to the dispersion of the corresponding section, i.e. dispersion characterizes the spread of realizations relative to mx(t).
The mathematical expectation of a random process and its dispersion are very important, but not exhaustive characteristics, since they are determined only by a one-dimensional distribution law. They cannot characterize the relationship between different sections of a random process for different values of time t and t". For this, a correlation function is used - a non-random function R(t, t") of two arguments t and t", which for each pair of argument values is equal to the covariance of the corresponding cross sections of a random process:
The correlation function, sometimes called autocorrelation, describes the statistical relationship between the instantaneous values of a random function separated by a given time value φ = t"-t. If the arguments are equal, the correlation function is equal to the variance of the random process. It is always non-negative.
In practice, the normalized correlation function is often used
It has the following properties: 1) if the arguments t and t" are equal, r(t, t") = 1; 2) symmetric with respect to its arguments: r(t,t") = r(t",t); 3) its possible values lie in the range [-1;1], i.e. |r(t,t")| ? 1. The normalized correlation function is similar in meaning to the correlation coefficient between random variables, but depends on two arguments and is not a constant value.
Random processes that occur uniformly in time, and whose partial implementations oscillate around the average function with a constant amplitude, are called stationary. :Quantitatively, the properties of stationary processes are characterized by the following conditions.
* The mathematical expectation of a stationary process is constant, i.e. m x (t) = m x = const. However, this requirement is not essential, since it is always possible to move from a random function X(t) to a centered function for which the mathematical expectation is zero. It follows from this that if a random process is nonstationary only due to a time-varying (over sections) mathematical expectation, then by the operation of centering it can always be reduced to a stationary one.
* For a stationary random process, the cross-sectional dispersion is a constant value, i.e. Dx(t) = Dx = const.
*: The correlation function of a stationary process does not depend on the values of the arguments t and t", but only on the interval φ = t"-t, i.e. R(t,t") = R(φ). The previous condition is a special case of this condition, i.e. Dx(t) = R(t, t) = R(φ = O) = const. Thus, the dependence autocorrelation function only from the interval "t" is the only essential condition for the stationarity of a random process.
An important characteristic of a stationary random process is its spectral density S(w), which describes the frequency composition of the random process at w?0 and expresses the average power of the random process per unit frequency band:
The spectral density of a stationary random process is a non-negative function of frequency S(n)?0. The area contained under the curve S(u) is proportional to the dispersion of the process. The correlation function can be expressed in terms of spectral density
R(φ) = S(φ)cosφdφ.
Stationary random processes may or may not have the property of ergodicity. A stationary random process is called ergodic if any of its implementations of sufficient duration is, as it were, an “authorized representative” of the entire set of implementations of the process. In such processes, any implementation will sooner or later pass through any state, regardless of what state this process was in at the initial moment of time.
Probability theory and mathematical statistics are used to describe errors. However, first it is necessary to make a number of significant reservations:
* the application of methods of mathematical statistics to the processing of measurement results is valid only under the assumption that the individual readings obtained are independent of each other;
* most of the probability theory formulas used in metrology are valid only for continuous distributions, while error distributions due to the inevitable quantization of samples, strictly speaking, are always discrete, i.e. the error can only take a countably many values.
Thus, the conditions of continuity and independence for measurement results and their errors are observed approximately, and sometimes are not observed. In mathematics, the term “continuous random variable” is understood as a significantly narrower concept, limited by a number of conditions, than “random error” in metrology.
Given these limitations, the process of occurrence of random errors in measurement results minus systematic and progressive errors can usually be considered as a centered stationary random process. Its description is possible on the basis of the theory of statistically independent random variables and stationary random processes.
When performing measurements, it is necessary to quantify the error. For such an assessment, it is necessary to know certain characteristics and parameters of the error model. Their nomenclature depends on the type of model and the requirements for the estimated error. In metrology, it is customary to distinguish three groups of characteristics and error parameters. The first group is measurement errors (standards of errors) specified as required or acceptable standards for characteristics. The second group of characteristics are errors attributed to the totality of measurements performed according to a certain technique. The characteristics of these two groups are used mainly in mass technical measurements and represent probabilistic characteristics of the measurement error. The third group of characteristics - statistical estimates of measurement errors reflect the proximity of a separate, experimentally obtained measurement result to the true value of the measured value. They are used in the case of measurements carried out during scientific research and metrological work.
As characteristics of the random error, the standard deviation of the random component of the measurement error and, if necessary, its normalized autocorrelation function are used.
The systematic component of the measurement error is characterized by:
* RMS deviation of the non-excluded systematic component of the measurement error;
* boundaries within which the non-excluded systematic component of the measurement error is located with a given probability (in particular, with a probability equal to unity).
Requirements for error characteristics and recommendations for their selection are given in the regulatory document MI 1317-86 "GSI. Results and characteristics of measurement error. Forms of presentation. Methods of use when testing product samples and monitoring their parameters."
We looked at various definitions of the concept "information" and came to the conclusion that information can be defined in many different ways, depending on the chosen approach. But we can speak clearly about one thing: information - knowledge, data, information, characteristics, reflections, etc. - category intangible . But we live in a material world. Therefore, in order to exist and spread in our world, information must be associated with some kind of material basis. Without it, information cannot be transmitted and stored.
Then the material object (or environment) with the help of which this or that information is presented will be information carrier
, and we will call a change in any characteristic of the carrier signal
.
For example, imagine a uniformly burning light bulb; it does not convey any information. But, if we turn the light bulb on and off (that is, change its brightness), then with the help of alternating flashes and pauses we can convey some message (for example, through Morse code). Likewise, a uniform hum does not convey any information, but if we change the pitch and volume of the sound, we can form some kind of message (which is what we do with spoken language).
In this case, signals can be of two types: continuous
(or analog
) And discrete
.
The textbook gives the following definitions.
Continuous the signal takes on many values from a certain range. There are no breaks between the values it takes.
Discrete the signal takes on a finite number of values. All values of a discrete signal can be numbered with integers.
Let us clarify these definitions a little.
The signal is called continuous(or analog) if its parameter can accept any value within a certain interval.
The signal is called discrete, if its parameter can take final the number of values within a certain interval.
The graphs of these signals look like this:
Examples continuous signals can be music, speech, images, thermometer readings (the height of the mercury column can be any and represents a series of continuous values).
Examples discrete signals can be readings from mechanical or electronic watches, texts in books, readings from digital measuring instruments, etc.
Let's return to the examples discussed at the beginning of the message - a flashing light bulb and human speech. Which of these signals is continuous and which is discrete? Answer in the comments and justify your answer. Can continuous information be converted into discrete information? If yes, please provide examples.
In technical branches of knowledge, the term signal is
1) a technical means for transmitting circulation and using information.
2) the physical process of displaying an information message (changing any parameter of the information carrier)
3) the semantic content of a certain physical state or process.
Signal – information/messages/information about any processes/states or physical quantities of objects of the material world, expressed in a form convenient for transmission, processing, storage and use of this information.
From a mathematical point of view, a signal is a function, that is, the dependence of one quantity on another.
Purpose of Signal Processing
The purpose of signal processing is considered to be the study of certain information information that is displayed in the form of target information and the transformation of this information into a form convenient for further use.
Purpose of Signal Analysis
By “analysis” of signals we mean not only their purely mathematical transformations, but also drawing conclusions about the specific features of the corresponding processes and objects based on these transformations. The goals of signal analysis are usually: - Determination or evaluation of numerical parameters of signals (energy, average power, root mean square value, etc.). - Decomposition of signals into elementary components to compare the properties of different signals. - Comparison of the degree of proximity, “similarity”, “relatedness” of various signals, including with certain quantitative estimates.
Signal registration
The concept of signal is inextricably linked with the term signal registration, the use of which is as broad and ambiguous as the term signal itself. In the most general sense, this term can be understood as the operation of isolating a signal and converting it into a form convenient for further use, processing and perception.. Thus, when receiving information about the physical properties of any objects, signal registration is understood as the process of measuring the physical properties of an object and transferring the measurement results to the material carrier of the signal or the direct energy transformation of any properties of the object into information parameters of the material carrier of the signal (usually electrical ). But the term signal recording is also widely used for the processes of separating already formed signals carrying certain information from the sum of other signals (radio communications, telemetry, etc.), and for the processes of recording signals on long-term memory media, and for many other processes related with signal processing.
Internal and external noise sources
Noises, as a rule, are stochastic (random) in nature. Interference includes distortion of useful signals under the influence of various destabilizing factors (electrical interference, vibration, types of noise and interference are distinguished by their sources, energy spectrum). Depending on the nature of the impact on the signal, sources of noise and interference can be internal or external.
Internal interference is inherent in the physical nature of signal sources and detectors, as well as material media. External sources of interference can be of artificial or natural origin. Artificial noise includes industrial noise and interference from operating equipment.
What does the mathematical model of the signal provide?
The theory of analysis and processing of physical data is based on mathematical models of the corresponding physical fields and physical processes on the basis of which mathematical models of signals are created; they make it possible to generally abstract from the physical nature to judge the properties of signals, predict changes in signals under various conditions, in addition, it becomes possible to ignore a large number secondary signs. Knowledge of mathematical models makes it possible to classify signals according to various criteria (for example, signals are divided into deterministic and stochastic).
Signal classification
Signal classification carried out on the basis of essential features of the corresponding mathematical models of signals . All signals are divided into two large groups: deterministic and random.
Harmonic signals
Harmonic signals (sinusoidal), are described by the following formulas:
s(t) = A×sin (2f o t+f) = A×sin ( o t+f), s(t) = A×cos( o t+), (1.1.1)
Rice. 5. Harmonic signal and spectrum of its amplitudes
where A, f o , o , f are constant values that can act as information parameters of the signal: A is the signal amplitude, f o is the cyclic frequency in hertz, o = 2f o is the angular frequency in radians , and f are the initial phase angles in radians. The period of one oscillation is T = 1/f o = 2/ o. When j = f-p/2, sine and cosine functions describe the same signal. The frequency spectrum of the signal is represented by the amplitude and initial phase value of the frequency f o (at t = 0).Polyharmonic signals
Polyharmonic signals constitute the most widespread group of periodic signals and are described by the sum of harmonic oscillations:
s(t) =A n sin (2f n t+ n) ≡ A n sin (2B n f p t+ n), B n ∈ I, (1.1.2)
or directly by the function s(t) = y(t ± kT p), k = 1,2,3,..., where T p is the period of one complete oscillation of the signal y(t), specified over one period. The value f p =1/T p is called the fundamental oscillation frequency.
Rice. 6. Signal model Fig. 7. Signal spectrum
Polyharmonic signals are the sum of a certain constant component (f o =0) and an arbitrary (in the limit - infinite) number of harmonic components with arbitrary values of amplitudes A n and phases j n , with frequencies that are multiples of the fundamental frequency f p . In other words, on the period of the fundamental frequency f p , which is equal to or a multiple of the minimum harmonic frequency, a multiple number of periods of all harmonics fits, which creates the periodicity of the signal repetition. The frequency spectrum of polyharmonic signals is discrete, and therefore the second common mathematical representation of signals is in the form of spectra (Fourier series).Almostperiodic signal
Almost periodic signals are close in their form to polyharmonic. They also represent the sum of two or more harmonic signals (in the limit - to infinity), but not with multiples, but with arbitrary frequencies, the ratios of which (at least two frequencies minimum) do not relate to rational numbers, as a result of which the fundamental period of the total oscillations is infinite big rice 9.
Rice. 9. Almost periodic signal and spectrum of its amplitudes
Analog signals
Analog signal (analog signal) is a continuous or piecewise continuous function y=x(t) of a continuous argument, i.e. both the function itself and its argument can take any value within a certain interval y 1 £y £ y 2 , t 1 £t £ t 2 . If the intervals of signal values or its independent variables are not limited, then by default they are assumed to be equal to -¥ to +¥.
The set of possible signal values forms a continuum - a continuous space in which any signal point can be determined with infinity accuracy.
Sources of analog signals are physical processes and phenomena; examples of analog signals are most often given by changes in the strength of the electric, magnetic and electromagnetic fields over time.
Discrete signals
Discrete signal
Rice. 13. Discrete signal (discrete signal) – fig. 13 in its values is also a continuous function, but defined only by discrete values of the argument. According to the set of its values, it is finite (countable) and is described by a discrete sequence of samples (samples) y(nt), where y 1 £y £ y 2, t is the interval between samples (interval or sampling step, sample time) , n = 0, 1, 2,...,N. The reciprocal of the sampling step: f = 1/t is called the sampling frequency. If a discrete signal is obtained by sampling an analog signal, then it represents a sequence of samples whose values are exactly equal to the values of the original signal.Digital signal
Digital signal (digital signal) is quantized in its values and discrete in its argument. It is described by a quantized lattice function y n = Q k, where Q k is a quantization function with the number of quantization levels k, and the quantization intervals can be either uniform or uneven, for example, logarithmic. A digital signal is specified, as a rule, in the form of a discrete series of numerical data - a numerical array of successive values of the argument at t = const, but in the general case the signal can also be specified in the form of a table for arbitrary values of the argument.
Rice. 14. Digital signal
Essentially, a digital signal in its values (counts) is a formalized version of a discrete signal when the latter’s counts are rounded to a certain number of digits, as shown in Fig. 14. A digital signal is finite in its many values. The process of converting analog samples with infinite values into a finite number of digital values is called level quantization, and the rounding errors of samples (discarded values) that arise during quantization are called noise or quantization errors.Kotelnikov-Shannon theorem
Physical meaning of the theorem Kotelnikov-Shannon: if the maximum frequency in the signal is f, then it is enough to have at least 2 samples with known values of t 1 and t 2 on one period of this harmonic, and it becomes possible to write a system of two equations (y 1 =a cos 2ft 1 and y 2 =a cos 2ft 2) and solve the system with respect to 2 unknowns - amplitude a and frequency f of this harmonic. Therefore, the sampling frequency must be 2 times the maximum frequency f in the signal. For lower frequencies this condition will be satisfied automatically.
In practice, this theorem is widely used, for example, in converting audio recordings. The range of frequencies perceived by humans is from 20 Hz to 20 kHz; therefore, for lossless conversion it is necessary to perform sampling at a frequency of more than 40 kHz; therefore, cd dvd mp3 is digitized at a frequency of 44.1 kHz. The quantization operation (analog-to-digital conversion of the ADC ADC) consists of converting a discrete signal into a digital signal encoded in a binary system. dead reckoning
System concept
A system for any purpose always has an input to which an input signal or input action (in the general case multidimensional) is applied and an output from which the processed output signal is removed. If the design of the system and internal transformation operations are not of fundamental importance, then the system as a whole can be perceived as a black box in a formalized form.
A formalized system represents a specific system operator(algorithm) for converting the input signal – impact s(t), into the signal at the system output y(t) – response or output reaction systems. Symbolic designation of the transformation operation:
For deterministic input signals, the relationship between input and output signals is uniquely specified by the system operator.
System Operatort
System operator T is a rule (set of rules, algorithm) for converting signal s(t) into signal y(t). For well-known signal conversion operations, extended symbols of transformation operators are also used, where the second symbol and special indices indicate a specific type of operation (for example, TF - Fourier transform, TF -1 - inverse Fourier transform).
Linear and non-linear systems
In the case of implementing a random input signal at the input of the system, there is also a one-to-one correspondence between the processes at the input and output, but in this case the statistical characteristics of the output signal change. Any signal transformations are accompanied by changes in their spectrum and, according to the nature of these changes, they are divided into 2 types: linear and nonlinear
Nonlinear is when new harmonic components appear in the signal spectrum, and when the signals change linearly, the amplitudes of the component spectrum change. Both types of changes can occur with the preservation and distortion of useful information. Linear systems constitute the main class of signal processing systems.
The term linearity means that the signal conversion system must have an arbitrary, but necessarily linear relationship between the input and output signals.
A system is considered linear if, within a specified area of input and output signals, its response to input signals is additive (the principle of superposition of signals is fulfilled) and homogeneous (the principle of proportional similarity is fulfilled).
Additivity principle
Principle additivity requires that the reaction to the sum of two input signals be equal to the sum of the reactions to each signal separately:
T = T+T.
Principle of homogeneity
Principle uniformity or proportional similarity requires maintaining the unambiguity of the transformation scale for any amplitude of the input signal:
T= c T.
Basic system operations
The basic linear operations from which any linear transformation operators can be formed include the operations of scalar multiplication, shift and addition of signals:
y(t) = b x(t), y(t) = x(t-t), y(t) = a(t)+b(t).
Rice. 11.1.1. System Operation Graphics
Addition and multiplication operations are linear only for discrete and analog signals.
For systems with a dimension of 2 or more, there is also another basic operation called the operation spatial masking, which can be considered as a generalization of scalar multiplication. So, for two-dimensional systems:
z(x,y) = c(x,y)u(x,y),
where u(x,y) is a two-dimensional input signal, c(x,y) is a spatial mask of constant (weighting) coefficients. Spatial masking is the element-wise product of the signal values with the mask coefficients.
Differential equations as a universal tool for studying signals
Differential equations are a universal tool for specifying a specific relationship between input and output signals, both in one-dimensional and multidimensional systems, and can describe the system both in real time and a posteriori. Thus, in an analog one-dimensional linear system, such a relationship is usually expressed by a linear differential equation
a m = b n .
(11.1.1)
When normalized to a o = 1, it follows
Essentially, the right side of this expression in the most general mathematical form displays the content of the input signal conversion operation, i.e. the operator for transforming the input signal into the output signal is specified. To uniquely solve equations (11.1.1), in addition to the input signal s(t), certain initial conditions must be specified, for example, the values of the solution y(0) and its time derivative y"(0) at the initial time.
A similar connection in a digital system is described by difference equations
a m y((k-m)t) =b n s((k-n)t).
(11.1.2)
y(kt) =b n s((k-n)t) –a m y((k-m)t).< 0. Интервал дискретизации в цифровых последовательностях отсчетов обычно принимается равным 1, т.к. выполняет только роль масштабного множителя.
(11.1.2")
The last equation can be considered as an algorithm for sequentially calculating the values y(kt), k = 0, 1, 2, …, from the values of the input signal s(kt) and the previous calculated values y(kt) with known values of the coefficients a m , b n and taking into account the setting of initial conditions - values s(kt) and y(kt) at k
Recursive systems
Backbone wide-area networks are used to form peer-to-peer connections between large local networks belonging to large departments of an enterprise. Backbone territorial networks must provide high throughput, since the backbone combines the flows of a large number of subnets. In addition, backbone networks must be constantly available, that is, provide a very high availability factor, since they carry the traffic of many business-critical applications. Due to the special importance of highways, they can be forgiven for their high cost. Since an enterprise usually does not have many large networks, backbone networks are not required to maintain an extensive access infrastructure.
Access networks are understood as territorial networks necessary for connecting small local networks and individual remote computers with the central local network of an enterprise. If great attention has always been paid to the organization of backbone connections when creating a corporate network, then the organization of remote access for enterprise employees has become a strategically important issue only recently. For many types of enterprise activities, quick access to corporate information from any geographical location determines the quality of decision-making by its employees. The importance of this factor is growing with the increase in the number of employees working at home (telecommuters) who are often on business trips, and with the increase in the number of small branches of enterprises located in different cities and, perhaps, different countries.
Multiplexing
Multiplexing is the use of one communication channel to transmit data to several subscribers. Communication lines (channel) consist of a physical medium through which information signals of data transmission equipment are transmitted.
Types of communication channels
simplex - when the receiver communicates with the transmitter over one channel, with unidirectional transmission of information (for example, in television and radio broadcasting networks);
half-duplex - when two communication nodes are connected by one channel, through which information is transmitted alternately in one direction, then in the opposite direction (in information-reference and request-response systems);
duplex - allows you to transmit data simultaneously in two directions through the use of a four-wire communication line (two wires for transmitting, the other two for receiving data), or two frequency bands.
Characteristics of communication lines
The main characteristics of the communication channel - throughput and reliability of data transmission
Channel capacity (the amount of information transmitted per unit of time) is estimated by the number of bits of data transmitted over the channel per second BIT/sec
The reliability of data transmission is assessed by the bit error rate (BER), which is determined by the probability of distortion of the transmitted data bit. The bit error rate for communication channels without additional error protection is 10 -4 to 10 -6
Main characteristics of cables
Computer networks use cables that comply with international standards ISO 11801. These standards regulate the following basic characteristics of cables:
– attenuation (dB/m);
– resistance of the cable to internal sources of interference (if there is more than one pair of wires in the cable);
Impedance (characteristic impedance) - the effective input resistance of the cable for alternating current;
The level of external EM radiation in the conductor characterizes the cable's noise immunity.
The degree of attenuation of external interference from various sources. The most widely used types of cables are unshielded twisted pair / shielded twisted pair / coaxial cable / fiber optic.
Unshielded-
Shielded is better than unshielded
Cable (RG8 and RG11 - thick coaxial cable has a characteristic impedance of 8 Ohms and an outer diameter of 2.5 cm)
RG58 & RG59 cables – thin coaxial cables with a characteristic impedance of 75 Ohms
Data transmission media (wired and wireless)
Depending on the physical medium of data transmission, communication lines can be divided:
wired communication lines without insulating and shielding braids;
cable, where communication lines such as twisted pair cables, coaxial cables or fiber optic cables are used to transmit signals;
wireless (radio channels of terrestrial and satellite communications), using electromagnetic waves that propagate over the air to transmit signals.
