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Dear readers! From November 18, 2019 to December 17, 2019, our university was provided with free test access to a new unique collection in the Lan EBS: “Military Affairs”.
A key feature of this collection is educational material from several publishers, selected specifically on military topics. The collection includes books from such publishing houses as: "Lan", "Infra-Engineering", "New Knowledge", Russian State University Justice, MSTU im. N. E. Bauman, and some others.

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“Maps and diagrams in the collections of the Presidential Library”

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Dear readers! On November 13 at 10:00, the LETI library, within the framework of a cooperation agreement with the B.N. Yeltsin Presidential Library, invites employees and students of the University to take part in the conference-webinar “Maps and diagrams in the collections of the Presidential Library.” The event will be held in a broadcast format in the reading room of the socio-economic literature department of the LETI library (5 building room 5512).

graduate work

2.1 Mathematical model of the radar environment

The radar situation is characterized by the location and nature of radar objects (targets) in the radar coverage area, as well as the conditions environment, affecting the propagation of radar signals.

When propagating radio waves, the phenomenon of wave dispersion should be taken into account, i.e. dependence of phase velocity on signal frequency. The phenomenon of dispersion is observed due to the fact that the refractive index of the atmosphere differs from unity, i.e. the speed of electromagnetic waves in this case is slightly less than the speed of light.

Another significant effect of radio wave propagation in a real environment is the bending of the direction of propagation, or wave refraction. This phenomenon can occur in a heterogeneous environment, i.e. environment with the refractive index varying from point to point /4/.

Since all these effects weakly change the characteristics of the radar signal, they can be neglected.

Any radar target or object is characterized by its location in space, motion parameters, effective reflecting surface (RCS), as well as the function of ESR distribution over the surface of the object (for distributed objects).

The location of an object (target) is characterized by the position of the center of mass of this object (target) in some reference coordinate system /2/. In radar, the local spherical coordinate system is most often used, the origin of which is located at the location of the radar antenna.

In a ground-based radar, one of the axes of the coordinate system usually coincides with the northern direction of the meridian passing through the position of the radar antenna, and the location of the target C is found based on the results of measuring the slant range D, azimuth b and elevation angle c (Figure 2.1). In this case, the system is motionless relative to the earth's surface.

Figure 2.1 - Local spherical coordinates

Measuring the range to a target using radio engineering methods is based on the constancy of the speed and straightness of the propagation of radio waves, which are maintained in real conditions with fairly high accuracy. Range measurement comes down to recording the moments of emission of the probing signal and reception of the reflected signal and measuring the time interval between these two moments. Reflected pulse delay time:

where D is the distance between the radar and the target (Figure 2.1), m;

c is the speed of propagation of radio waves, m/s.

To determine the radial speed of a moving object, the Doppler effect /3/ is used, which consists in changing the frequency of observed oscillations if the source and observer move relative to each other. Therefore, the task of determining the radial velocity comes down to determining the frequency of reflected oscillations in comparison with emitted ones. The simplest and most convenient derivation of quantitative relationships for the Doppler effect for radar is based on considering the “transmission - reflection - reception” process as a single one. Let vibrations enter the antenna:

The signal reflected from a stationary target and delayed by time t3 at the receiver input will have the form:

There is a phase shift here:

as well as a constant phase shift μ μ that occurs during reflection. When moving away from the radar with a constant radial speed, the range.

where V P is the radial speed of the target (Figure 2.2), m/s.

Figure 2.2 - Radial speed of the target relative to the radar

Substituting the corresponding value from (1) into (4), we get:

The frequency of reflected oscillations, determined by the derivative of the oscillation phase μ C with respect to time, is equal to:

From here (8)

those. When the target moves away from the radar, the frequency of reflected oscillations is lower than that of emitted ones.

Magnitude

called Doppler frequency.

The power of the reflected signal at the input of the radar receiver depends on a number of factors /4/ and, above all, on the reflective properties of the target. The primary (incident) radio wave induces conduction currents (for conductors) or displacement currents (for dielectrics) on the target surface. These currents are a source of secondary radiation in different directions.

The reflective properties of targets in a radar are usually assessed by the effective scattering area (RCS) of the target S 0:

where o is the depolarization coefficient of the secondary field (0 ? o ? 1);

P OTR = S·D 0 ·П 1 - reflected signal power, W;

P 1 is the power flux density of the radar signal on a sphere of radius R in the vicinity of the point where the target is located, W/m 2 ;

D 0 - the value of the backscatter diagram (BSD) in the direction to the radar;

S - total scattering area of ​​the target, m 2.

The EPR of the target is expressed in square meters a coefficient that takes into account the reflective properties of the target and depends on the configuration of the target, the electrical properties of its material and the ratio of the target size to the wavelength.

This value can be considered as a certain target area equivalent to a normal radio beam with area S0, which, isotropically dissipating all the wave power incident on it from the radar, creates at the receiving point the same power flux density as the real target. The effective scattering area does not depend on either the intensity of the emitted wave or the distance between the station and the target.

Since measuring the EPR of real objects is difficult in practice due to the complex shape of the latter, sometimes in calculations they operate with the amount of energy reflected from a radar object or the ratio of reflected energy to emitted energy.

If the radar object is distributed, i.e. consists of many independent emitters, then to find the EPR, one of two reflection models is used. In both models, the target is represented as a set of n point elements, among which there is no dominant reflector (first model), or there is one dominant reflector (second model), which gives a stable reflected signal.

In the technical radar literature /2, 4/ on radar, a generalized Swerling model is used with a distribution of the form:

where is the average EPR value, m 2.

This expression corresponds to a 2 distribution with 2k degrees of freedom, where k determines the complexity of the target reflection model. For k = 1, we obtain a model with an exponential EPR distribution, and for k = 2, we obtain a model of a target in the form of a large reflector that changes orientation in space within small limits, or a set of equal reflectors plus the largest one.

The law of distribution of amplitudes of the reflected signal is reduced to the generalized Rayleigh law /4/:

where E is the amplitude of the reflected signal, V;

E 0 - amplitude of the reflected signal from the dominant emitter, V;

y 2 - dispersion of orthogonal amplitude components, V 2;

I 0 - modified Bessel function of the first kind of zero order:

In the case of a group emitter consisting of n point emitters, the EPR distribution diagram along azimuths has a very complex lobe structure, depending on the relative position of the reflecting elements and the relative distances between them. Therefore, group targets, depending on their angular position relative to the line of sight, can give significant fluctuations in the power of the reflected signals. These oscillations occur relative to an average level proportional to the average EPR value for incoherent addition. Simultaneously with fluctuations in the power of the reflected signal, random changes in its delay time and angle of arrival are observed.

For moving distributed targets, the phenomenon of interference of secondary radiation oscillations from different points arises, which is based on a change in the relative position of the target’s point reflectors. The Doppler effect is a consequence of this effect. To describe the phenomenon, a backscatter diagram (BSD) is used, which characterizes the dependence of the amplitude of the reflected signal on the direction /2/.

In addition, when targets are irradiated, the phenomenon of depolarization of the probing signal occurs, i.e. the polarization of the reflected and incident waves do not coincide. For real purposes, fluctuating polarization takes place, i.e. all elements of the polarization matrix /1/ are random and it is necessary to use the matrix of numerical characteristics of these random variables.

In a statistical approach to the analysis of radar objects, a correlation function or a correlation matrix /8/ is used to describe the functions of the latter, which characterize the change in the parameters of the object over time. The disadvantage of this model is the complexity of calculations due to the need to use statistical methods and the complexity of organizing the input of initial parameters.

Based on the above, to describe a radar object, it is necessary to know its position in space, its extent in range and azimuth (for distributed objects), the EPR and its distribution model, the model of object motion or the law of change in the Doppler frequency increment of the reflected signal, the number of point emitters (for group emitters).

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Designing modern military radar systems is no easy task. But the use of the latest modeling tools and techniques allows us to resolve many of the difficulties of the development process.

HONGLEI CHEN, SOFTWARE ENGINEER, RICK GENTILE, PRODUCT MANAGER MATHWORKS

The development of radar systems is a complex, multi-domain task. With the rise of phased array antenna (PAA) technology, engineers have access to new capabilities such as electronic beam steering and spatial signal processing. But new opportunities have led to the complication of systems as a whole. In addition, an increase in the number of interference sources, “filling” the radio frequency spectrum with their radiation, coupled with an ever-decreasing effective scattering surface (RCS) of targets, creates new difficulties in achieving the required performance indicators of radar systems.

A convenient dynamic simulation environment can become a decisive factor in optimizing the radar development process and help reduce the risks that inevitably arise when designing complex systems operating in difficult conditions. Simulation of multi-domain radar systems will help to adopt right decisions during the development process, and will also allow you to detect design errors at the most early stages. For example, using the model, you can evaluate the radar’s ability to detect targets with small RCS, or test signal processing algorithms in conditions of noise and interference. At later stages, the same models can be used to demonstrate the need for modification to an existing system and demonstrate the benefit of such modification before purchasing or manufacturing any additional components. In addition, the model can be used to predict the behavior of the system in the event of failure of one or more components.

From probe pulses to detections

Let's try to highlight several aspects of how the model can help with estimating system parameters. Figure 1 shows a multi-domain system model created in Simulink. The model contains blocks of the radar system responsible for the generation, reception, transmission and spatial processing of signals. Mathematical descriptions of targets and propagation environments are also included in the system model.

Figure 1. Multi-domain radar model.

This is a model of an X-band radar that allows you to detect targets with low RCS values ​​(<0.5 м 2). Требуемая дальность в данном примере – 35 км с разрешением по дальности 5 метров. Каждый из блоков, показанных на Рис. 1, может быть с лёгкостью описан на языке MATLAB или настроен в соответствии с выбранной конфигурацией системы. Например, такие параметры, как тип сигнала, требуемая мощность передатчика или коэффициент усиления антенны могут быть явно установлены в каждом из блоков.

Development of probing pulses

Once we have determined the range and velocity resolution parameters, as well as the minimum and maximum coverage range of our radar, we can interactively select the modulating pulse parameters to suit the system requirements. Figure 2 shows the configuration of the probe pulse parameters that are set interactively. The resulting “signal characteristics” are highlighted with a frame, and we can verify that they satisfy the system requirements. Figure 3 shows the response of the corresponding matched filter.

Figure 2. Modulating pulse.

Figure 3. Corresponding matched filter.

For such radar systems, we try to minimize the transmitter power, and therefore reduce the cost. Despite the power limitation, we are faced with the task of detecting targets with small RCS. This can be achieved by using antenna arrays with high gain in the system.

Development of antenna arrays

We can interactively design and analyze lattice parameters, including geometry, spacing of elements, relative positions of elements, and weighting functions. An example is shown in Figure 4 - a rectangular lattice of 36x36 equally spaced elements. The beam generated by such gratings can be deflected both in azimuth and elevation. Figure 5 shows the radiation pattern of the designed antenna. An array of this size for X-band radars can be easily installed on many platforms, including mobile ones.

The P-15 (P-15MN) radar station of the decimeter wave range was intended to detect targets flying at medium, low and extremely low altitudes. Entered service in 1955. It was used as part of radar posts of radio engineering units and as a reconnaissance and target designation station for anti-aircraft missile units.

The P-15 station was mounted on one vehicle along with the antenna system and was deployed into a combat position in 10 minutes. The power supply unit was transported in a trailer.

Model from ZZ MODELL, the base vehicle ZIL-157 was supplied (most likely) from ICM and is made of plastic, in my opinion, not bad at all. There was no particular hassle during assembly. Kung resin station. During the assembly process it was necessary to tinker with the fit of the rear wall (where the double doors are). The jacks are also made of resin and are quite fragile; one broke. The antenna-feeder system is made of photo-etched material.

The model was painted with Tamia Color acrylic paints, and the whole thing was blown over with Humbrol matte varnish.

From the modifications to the model presented to you, I decided to do the following:

• tool boxes located under the rear wall of the kung on both sides;
• the second fuel tank of the car (there is only one included with the model for some reason unknown to me);
• rear license plate mount;
• waveguide on the upper antenna feed;
• the bottom step to the ladder on the rear side wall of the kung.

I didn’t lift it high on jacks, because... According to the instructions - still Soviet - it is enough only for the wheels of the suspended equipment to turn if it is located on a hard surface. There is also such a thing as to preserve rubber in the summer, the wheels are painted white. Although in my practice I have seen painted wheels a couple of times.

Of the shortcomings I noticed in the assembly diagram, I noticed one little thing. In the circuit, the feed holders of the upper and lower antennas are attached in the same way - with tubes to which the radio frequency cable is attached downwards. Although in a real station, on the lower antenna, it is mounted in reverse (see photo). I noticed this thing by accident when trying to imitate a radio frequency cable, when everything was already assembled. The lower waveguide part of the lower photo-etched antenna is also not made accurately - it does not correspond to the original, it had to be corrected.

As for the degree of correspondence of the entire model to the original, I was quite satisfied with it. Although there is some work to be done.

2.2 Mathematical model of radar

As already noted in paragraph 1.1, the main radar modules are the antenna unit, together with the antenna switch, transmitter and receiver. A large class of various devices can be used as a terminal device, differing in the way they display information and not affecting the received radar signals, so this class of devices is not considered.

2.2.1 Mathematical model of the antenna

One of the main characteristics of the antenna is its directional pattern (DDP) /5/, which characterizes the dependence of the radiated power on the direction (Figure 2.3).

Figure 2.3 – Antenna power pattern

The antenna radiation pattern in the azimuth-range plane at a constant elevation angle with a uniform field distribution across the aperture is expressed by the function:

(14)

The angle β for uniform motion of the antenna in a circle can be found using the formula:

(15)

where ω is the angular speed of rotation of the antenna, rad/s.

Let's consider the shape of the reflected signal in a 360-degree radar. As the antenna rotates, the amplitude of the probing pulses irradiating the target changes in accordance with the radiation pattern. Thus, the probing signal irradiating the target turns out to be modulated and described by a function of time

where s P (t) – radio pulses of the transmitter.

Let us assume that the target practically does not change the duration of the reflected pulses, and that the movement of the target during the irradiation time can be neglected. Then the reflected signal is characterized by the function:

where k is a constant coefficient.

For a single-antenna radar, in which the antenna radiation pattern during reception is described by the same function F E (t) as during transmission, the signal at the receiver input is written in the form:

Because the antenna rotation speed is relatively low and the beam displacement during the delay time is much less than the width of the radiation pattern, then F E (t)≈F E (t – t W). In addition, a function characterizing the power radiation pattern:

(19)

where β is the angle measured in one direction from the maximum to the target azimuth, degrees;

Θ 0.5 – width of the radiation pattern at half power, measured in both directions from the maximum (Figure 2.3), degrees.

Taking into account the above, (17) can be represented as:

those. The pulses at the receiver input are modulated in amplitude in accordance with the power directional pattern of the antenna.

The target azimuth is determined by the parameters of the angle-code converter sensor (Figure 2.4).

Figure 2.4 – Scheme for connecting the angle-code converter sensor

When the antenna rotates, the signals from the photo emitter are recorded by the photo receiver after the signals pass through holes in the plate located on the axis of the antenna. Signals from the photodetector are transmitted to the counter, which generates pulses called MAI pulses (short azimuth intervals). The angle of rotation of the antenna, and, consequently, the azimuth of the received radar signal is determined by the MAI pulses. The number of MAI coincides with the meter's conversion factor and determines the accuracy with which azimuth is measured.

Based on the above, the antenna module is characterized by the following parameters: the shape of the radiation pattern and its width, the antenna gain, the number of MAIs.

2.2.2 Mathematical model of the transmitting device

The transmitting device can be characterized by the radiation power, the number and type of probing signals and the law of their arrangement.

The radar range in the case of optimal signal processing and a given spectral noise density depends on the energy of the probing signal, regardless of its shape /5/. Considering that the maximum power of electronic devices and antenna-feeder devices is limited, an increase in range is inevitably associated with an increase in pulse duration, i.e. with a decrease in potential range resolution.

Complex or power-intensive signals resolve conflicting demands for increased detection range and resolution. Detection range increases when using high energy signals. An increase in energy is possible by increasing either the power or the duration of the signal. The power in a radar is limited from above by the capabilities of the radio frequency generator and especially by the electrical strength of the feed lines connecting this generator to the antenna. Therefore, it is easier to increase the signal energy by increasing the signal duration. However, long duration signals do not have good range resolution. Complex signals with a large base can resolve these contradictions /7/. Currently, frequency modulated (FM) signals are widely used as one of the types of complex signals.

The entire set of FM signals can be described using the formula:

(21)

where T is the pulse duration, s;

t – time, function argument, varies within , c;

b k – coefficients of signal phase series expansion;

f 0 – signal carrier frequency, Hz.

Indeed, with n = 1 we obtain a linearly frequency-modulated (chirp) signal, whose coefficient b 0 - the signal base - can be found as:

(22)

where Δf is the frequency deviation of the chirp signal, Hz.

If we take n = 1 and frequency deviation Δf = 0 Hz, we obtain a MONO signal or video pulse with a rectangular envelope, which is also widely used in radar for detecting targets at short distances.

Another way to increase the signal energy while maintaining a short pulse duration is to use bursts of pulses, i.e. a series of pulses separated by interpulse intervals is considered as a single signal. In this case, the signal energy is calculated as the sum of the energies of all pulses /7/.