Coursework: Analysis and optimization of a digital communication system. How the online service for optimizing mobile communications costs works. Methodology for optimizing mobile communications networks.
When designing cellular network mobile communications the following basic operations must be performed: assessing the cost of the designed network; network capacity assessment; assessment of radio coverage and location of cellular network elements; assessment of the maximum permissible density (degree of services); estimate the number of calls; assessment of future development of the cellular network. According to NOKIA specialists, the main stages of the network planning process are the following:
1.Collect information in the following sections:
Rules and laws;
Key information regarding demographics, income level, service area expansion forecast, service support, market research, etc.;
Availability of leased communication lines, availability of microwave frequencies, requirements for connections with other systems;
Number, address and routing principles;
Topographic maps;
Existing infrastructure, such as transmission networks and transmission media.
2. Determination of the required basic network parameters for radio coverage and capacity.
The main problem of this planning stage is network optimization according to the cost-effectiveness criterion. To implement this task in practice, detailed information about the cellular network is needed (increasing cost of the plan stage, protection available, required information infrastructure), as well as the formulation of the network’s objectives and requirements for its quality. The result of the second stage is a design of an integrated network topology, which should show the various services and the equipment required for its implementation. In addition, an initial detailed network implementation plan must be presented. The main purpose of this phase is to illustrate the comprehensive process of cellular network planning. Other types of planning should also be noted:
FTP (Fixed Transmission Planning) - fixed transmission planning;
NAP (Network Access Planning) - network access planning;
DCN (Data Communication Network Planning) - data network planning;
INP (Intelligent Network Planning) - intelligent network planning;
3G and IP Network Planning - planning for the development of the third generation (3G) cellular system, the use of IP network protocols, etc., which should be included in the complete cellular network design process.
3.Selection of MSC, BSC and locations base stations.
4.Surveying the location for given MSC, BSC and base stations, in other words, estimating the locations of MSC, BSC and BTS taking into account the environment surrounding these systems.
5.Detailed cellular network planning. This stage includes the following operations:
Computer-aided network design and tools to create the necessary radio coverage of the territory;
Analysis of interference (co-channel, external, noise);
Frequency planning;
Microwave channel planning;
Documentation, etc.
NOKIA has prepared a TOTEM set, including necessary tools for cellular network planning. Three areas in cellular network planning are discussed below:
SNP (Switching Network Planning) - network switching planning;
CTNP (Cellular Transmission Network Planning) - planning of a cellular transmission network;
RNP (Radio Network Planning) - radio network planning.
Features of planning a network switching system.
During this planning stage, the following tasks must be solved:
By measuring and taking into account the required network capacity (average call time, number of handovers, short message transmissions, etc.), the volume of switching is estimated;
The level of network execution is set in accordance with the specified switching capacity of the network;
The implementation of network switching and signaling systems is considered;
Rules for routing, protection, synchronization and switching control are developed;
Voice and signaling traffic matrices are determined;
The necessary equipment to implement the above tasks is assessed.
After the cellular network is marked (Fig. 7.9, 7.10), a detailed plan is executed with
Selected number of inputs (for example, network diagram, routing plan, digital analysis, control details, numbering plan, load plan, etc.). Additionally, the network switching system planner must, in addition to the above tasks, consider possible future expanded network plans.
Transmission network planning.
When planning a cellular transmission network main problem is to study the use of microwave communication lines (or fiber-optic communication lines) in the GSM network, providing, for example, interaction between the BTS and the BSC. Several planning options are possible:
Installation of own microwave relay communication lines (microwave radio relay communication lines);
Renting existing radio relay lines that fit into the cellular network being developed in terms of location and conditions of stable radio communication;
Laying fiber-optic communication lines.
When performing this network planning item, it is necessary to take into account the problem of connecting and coordinating large flows of various information. At this stage, it is necessary to develop a diagram of the main transmission network for BTS access and network nodes, which will provide a clear image network connections. This is also necessary to determine the required network capacity.
Both the synchronization principles and the gateway and switching connections must be identified. When planning microwave channels, it is necessary to select highly reliable broadband channels to ensure reliable communication between the BTS and the BSC. In addition, in a cellular network, instead of microwave channels, fiber-optic channels can be used.
Radio network planning.
The type and location of the BTS depend on the characteristics of the environment. In urban areas, cells are usually smaller in size than in rural areas. In addition, the volume of traffic also affects the number of radio channels in a typical cell. Since in the GSM standard the maximum theoretical distance from the BTS to the macrocell edge is 35 km, the ability of the MS to send packets that must arrive at the BTS in the correct slot is usually adapted to this.
Factors limiting cell sizes:
1) with an increase in the operating frequency, that is, with a decrease in the operating wavelength, the cell size decreases (the GSM 900 cell size is larger than the cell sizes for GSM 1800 and 1900);
2) external conditions: for open water spaces, the attenuation of radio signals is less than in forests or urban environments.
Thus, when planning a radio network in cellular system communication is necessary:
Implement the choice of radio channels by creating your own microwave radio relay lines, or by renting existing ones, or by laying fiber-optic communication lines;
Draw up a detailed network plan, including the results of the previous paragraph, as well as the results of measurements and testing of radio coverage of the territory.
Determination of traffic and number of channels in cells
The cell is the basic "building block" of a GSM network. A single cell is essentially a geographic area surrounding a single BTS, with cell sizes depending on the following factors:
From the environment;
From the number of users;
From the operating frequency range;
From the power of BTS transmitters, etc.
The cells are grouped around the base station controller BSC. Average cell sizes are determined by answering two fundamental questions: How large is the traffic channel (TCN - Traffic CHannel) that must be managed within the cell? How many channel traffic does a cell need? To answer these questions, you need to determine the amount of traffic in the cell.
where (k) (call/hour) is the average number of calls per hour; (t) — average talk time (hour). Quantitatively, traffic does not depend on the duration of observation. For example, if the study is carried out for 15 minutes, then in the formula for traffic A the denominator, instead of 3600 s, will be equal to 900 s.
Let's consider a numerical example. Let there be 540 calls per hour in a cell, and the average call duration is 100 s (100/60 = 1.66 min), then the traffic volume is: 
If you use the table. 7.1 (Erlang B model) of maximum traffic density, we obtain: number of channels Nk_c = 20, with failure probability Рв = 5%. Thus, in this case, the value of GOS = 5% (Grade Of Service), determined by the probability of failure, indicates that with an observation time of 1 hour, 5 out of 100 calls are refused due to a lack of cell resources , while the number of channels will be 20. Since in the GSM standard each radio channel supports 8 (voice) channels, we can make a rough assessment of the BTS equipment configuration: if we use three transceivers in the BTS, then Nk-c = 3x8 = will be provided 24 speech channels, which is more than the calculated value of 20 channels. This provides some margin in terms of traffic volume, since with Nk_r = 24 and Pb = 5% the traffic volume will be A = 19 Erl (from Table 7.1).
Frequency reuse.
Each BSS base station subsystem has a limited number of allocated frequencies. These frequencies must be allocated to each cell so that the required network capacity is satisfied by the different parts of the BSS.
Consider the following example. In Fig. Figure 7.11 shows a cellular network.
Rice. 7.11. Cellular network diagram.
Rice. 7.12. An example of uniform radio coverage of a territory with the selected frequency plan.
Let the network designer choose a cluster of order 9, that is, the number of allocated frequencies is 9 (for BSS). In Fig. Figure 7.12 shows a cluster distribution of frequencies using the frequency repetition principle. The next step is the assessment of LA (Local Area) - the local area of the network, which is performed in accordance with the traffic characteristics of each area. The final phase in planning a fixed network is to assess the required traffic and radio network.
Network optimization and development.
The network planning described above is only the first part of a long process of improving the cellular network that is being built. When further improving the designed cellular network, the following factors must be taken into account.
1. An increase in the number of subscribers requires expansion of the network in a given place and during a given period of time.
2. Taking into account the cost of the network for any operator is a competitive-forming parameter in the mobile communication services market.
3. Network capacity, on the one hand, must be minimized (provide the necessary traffic), and on the other, it should not be small, as this will worsen the quality of service of subscribers.
That is, there are conflicting requirements:
The network must have high quality and have wide radio coverage;
EDGE (Enhanced Date Rates for Global Evolution) - improved data transmission for the global evolution of communication systems (384 kbit/s);
SDH (Synchronous Digital Hierarchy) - synchronous digital hierarchy (using fiber-optic communication lines between nodes in a cellular network), etc.
The main reason for the introduction of high-speed data transmission systems is the growth in the number of users and the associated growth in traffic and the volume of various services in cellular mobile communication systems.
Therefore, to optimize and develop the cellular network, you should:
1) conduct field tests of the created network (quite expensive), which would make it possible to clarify not only the quality of information transmission, but also hardware problems, as well as the possibility of compressing (gathering) information, increasing the number of users with an unchanged structure and hardware networks, etc.
2) use the information received in the NMS (Fig. 7.13) to assess the conditions of geographic radio coverage (station tar), the power level of the BTS (serving BTS), and the emission levels of neighboring stations (neighbor stations) for three network messages.
This information allows you to implement network performance management from the NMS (network management system), obtain important information about the performance of various parts of the cellular network, which ultimately determines possible alternative solutions for the network operator.
Rice. 7.13. An example of assessing geographic radio coverage conditions, BTS power level and emissions from neighboring stations.
ANALYSIS AND OPTIMIZATION OF DIGITAL COMMUNICATION SYSTEM
1.3 Selecting the type of modulation and calculating transmission quality characteristics
Application
INTRODUCTION
Life modern society unthinkable without widely ramified information transmission systems. Without it, industry, agriculture, and transport would not be able to function.
Further development of all aspects of our society’s activities is unthinkable without the widest implementation automated systems control, the most important part of which is the communication system for exchanging information, as well as devices for storing and processing it.
Transfer, storage and processing of information take place not only when using technical devices. A normal conversation is an exchange of information. There are many different forms of presentation and storage of information, such as books, floppy disks, hard drives, etc.
Information transmission technology, perhaps more than any other technology, influences the formation of the structure of the world community. The past decade has seen revolutionary changes in the Internet and with it radical and often unpredictable changes in the way business is done on a global scale. From this follows a completely logical conclusion that without knowledge of the fundamentals of the theory of signal transmission, the creation of new advanced communication systems and their operation is impossible. Therefore, its study is an integral part of students’ theoretical training.
Transmitting a message from one point to another is the basis of the theory and technology of communication. The course “Telecommunication Theory” studies unified methods for solving various problems that arise when transmitting information from its source to the recipient.
1.1 Block diagram of a digital communication system
In a number of practical cases, the problem of transmitting continuous messages through a discrete communication channel arises. This problem is solved by using a digital communication system. One such system is a system for transmitting continuous messages using pulse code modulation (PCM) and harmonic carrier manipulation. The block diagram of such a system is shown in Fig. 1. It consists of a message source (IS), an analog-to-digital converter (ADC), a binary discrete communication channel (DCC), An integral part which is a continuous communication channel (NCC), a digital-to-analog converter (DAC) and a message recipient (MS). Each of the above parts of the system contains a number of other elements. Let's look at them in more detail.
A message source is some object or system, information about the state or behavior of which must be transmitted over a certain distance. The information that is transmitted from the IS is unexpected for the recipient. Therefore, its quantitative measure in telecommunication theory is expressed through the statistical (probabilistic) characteristics of messages (signals). A message is a physical form of representing information. Often messages are provided in the form of a time-varying current or voltage that represents the information being transmitted.
Figure 1.1 – Block diagram of a digital communication system
In the transmitter (IS), the message is first filtered in order to limit its spectrum to a certain upper frequency f B. This is necessary for the effective representation of the low-pass filter response x(t) as a sequence of samples x k = x(kT), k = 0, 1, 2, . .., which are observed at the output of the sampler. Note that filtering is associated with the introduction of an error e f (t), which reflects that part of the message that is attenuated by the low-pass filter. Further readings (x k) are quantized per level. The quantization process is associated with the nonlinear transformation of continuous-valued samples (x k) into discrete-valued ones (x k l), which also introduce an error, which is called the quantization error (noise) e sq (t). Quantum levels (y k = x k l ) are then encoded with a binary non-redundant (primitive) or noise-resistant code.
The sequence of code combinations (b k l) forms a PCM signal, which is sent to the modulator - a device that is designed to match the message source with the communication line. The modulator generates a linear signal S(t, b i), which is an electrical or electromagnetic oscillation that can propagate along a communication line and is uniquely associated with the message that is being transmitted (in this case, a PCM signal). The signal S(t, b i) is created as a result of discrete modulation (manipulation) - the process of changing one or more parameters of the carrier according to the PCM signal. When using a harmonic carrier U Н (t) = U m cos(2pf Н t+j 0), signals are distinguished: amplitude, frequency and phase manipulation (AM, FM and FM).
To prevent out-of-band emissions in single-channel communications or when organizing multi-channel communications, as well as to establish the desired signal-to-noise ratio at the receiver input, the linear signal is filtered and amplified in the output stage of the IC.
The signal S(t) from the output of the IC enters the communication line, where it is affected by noise n(t). At the receiver input (R) there is a mixture of z(t) = s(t) + n(t) of the transmitted signal and noise, which is filtered in the input stage of the R and fed to the demodulator (detector).
During demodulation, the law of change is extracted from the received signal information parameter, which in our case is proportional to the PCM signal. In this case, to recognize the transmitted binary signals, a decision device (DE) is connected to the output of the demodulator. When transmitting binary signals b i, i = 0, 1 via the DCS, the presence of interference in the NCS leads to ambiguous decisions (errors) of the switchgear, which in turn causes a discrepancy between the transmitted and received code combinations.
Finally, to restore the transmitted continuous message a(t), i.e. After receiving its estimate, the received code combinations are subjected to decoding, interpolation and low-pass filtering. In this case, in the decoder, the binary code combinations are used to restore L levels, m = 1 ... L-1.
The presence of errors in the binary DCS leads to transmission errors in the L-th DCS and the occurrence of transmission noise e P (t). The combined effect of filtering error, quantization and transmission noise leads to ambiguity between the transmitted and received messages.
1.2 Determination of ADC and DAC parameters
The time sampling interval T d is selected based on Kotelnikov’s theorem. The inverse quantity to Td - sampling frequency fd = 1/Td is selected from the condition
f d ≥ 2F m, (1.1)
where F m is the maximum frequency of the primary signal (message).
Increasing the sampling rate allows us to simplify the input low-pass filter (LPF) of the ADC, which limits the spectrum of the primary signal, and the output LPF of the DAC, which restores the continuous signal from the sample. But increasing the sampling rate leads to a decrease in the duration of the binary symbols at the ADC output, which requires an undesirable expansion of the communication channel bandwidth for transmitting these symbols. Typically, the parameters of the input low-pass filter of the ADC and the output low-pass filter of the DAC are chosen to be the same.
In Fig. 1.2 presents: S(f) - spectrum of readings, which are displayed by narrow pulses, S a (f) - spectrum of continuous message a(t), A(f) - operating attenuation of the low-pass filter.
In order for the low-pass filter to not introduce linear distortions into a continuous signal, the limiting frequencies of the low-pass filter passbands must satisfy the condition
f 1 ≥ F m (1.2)
In order to eliminate the overlap of the spectra of S a (f) and S a (f-f D), and also to ensure the attenuation of the composite S a (f-f D) by the restoring low-pass filter, the limiting frequencies of the low-pass filter stop bands must satisfy the condition
f 2 ≤ (f D - F m)(1.3)
Figure 1.2 - Spectrum of samples and frequency response of attenuation of ADC and DAC filters
So that the low-pass filters are not too complex, the ratio of the limiting frequencies is chosen from the condition
f 2 / f 1 = 1.3 ... 1.1.(1.4)
After substituting relations (1.2) and (1.3) into (1.4), you can select the sampling frequency f D.
In system digital transmission using the PCM method, the noise power at the DAC output is determined as
,(1.5)
where is the average quantization noise power;
Average noise power of measurement errors.
(1.6)
The quantization noise power is expressed in terms of the quantization step size Dx:
.(1.7)
The quantization step depends on the number of quantization levels N:
Dx = U max / (N-1)(1.8)
From expression (1.8) we determine the minimum possible number of quantization levels:
(1.9)
The length of the binary primitive code at the ADC output is an integer:
m = log 2 N .(1.10)
Therefore, the number of quantization levels N is chosen as an integer power of 2, at which
N ≥ N m i n .(1.11)
The duration of a binary symbol (bit) at the ADC output is determined as
T b = T D / m.(1.12)
The average amount of information transmitted over a communication channel per unit time - the information transmission speed H t - is determined by the formula
,(1.13)
where is the sample transmission rate;
– entropy.
, (1.14)
where is the signal level distribution law, is the number of quantization levels.
The sampling rate is equal to the sampling frequency:
.(1.15)
1.3 Modulation
We choose the type of modulation so that the speed of information transmission after modulation is not less than the productivity of the source, i.e.
,
where is the modulation rate,
Number of signal positions.
For AM, FM, OFM, KAM
Channel bandwidth.
,
where is the number of subchannels.
Then 
,
After determining the number of signal positions M, we calculate the error probabilities
Probability of error with AM-M:
,
Probability of error with FM-M:
Probability of error during OFM-M:
Probability of error with KAM-M:
M = 2 k, k is an even number.
Probability of error with OFDM:
where η is the number of amplitude levels;
M = 2 k, k is an even number.
The choice of modulation method is carried out in accordance with the criterion of minimum error probability.
1.4 Selecting the type of noise-resistant code and determining the length of the code combination
Noise-resistant, or redundant, coding is used to detect and/or correct errors that occur during transmission over a discrete channel. A distinctive property of error-correcting coding is that the redundancy of the source formed by the output of the encoder is greater than the redundancy of the source at the input of the encoder. Noise-resistant coding is used in various systems communications, when storing and transmitting data in computer networks, in household and professional audio and video equipment based on digital recording.
If economical coding reduces the redundancy of the message source, then error-correcting coding, on the contrary, consists in the purposeful introduction of redundancy in order to make it possible to detect and (or) correct errors that occur during transmission over the communication channel.
n=m+k – code combination length;
m – number of information symbols (bits);
k – number of check characters (digits);
Of particular importance for characterizing the correcting properties of the code is the minimum code distance d min, determined by pairwise comparison of all code combinations, which is called the Hamming distance.
In a non-redundant code, all combinations are allowed, and, therefore, its minimum code distance is equal to one - d min = 1. Therefore, it is enough for one symbol to be distorted for another allowed combination to be accepted instead of the transmitted combination. In order for the code to have corrective properties, it is necessary to introduce some redundancy into it, which would ensure a minimum distance between any two allowed combinations of at least two - d min > 2.
The minimum code distance is the most important characteristic noise-resistant codes, indicating the guaranteed number of errors detected or corrected by a given code.
When using binary codes, only discrete distortions are taken into account, in which one goes to zero (1 → 0) or zero goes to one (0 → 1). The transition 1 → 0 or 0 → 1 in only one element of the codeword is called a single error (single distortion). In general, the error multiplicity means the number of positions of the code combination at which, due to interference, some symbols were replaced by others. Double (t = 2) and multiple (t > 2) distortions of elements in the code combination within 0 are possible< t < n.
The minimum code distance is the main parameter characterizing the correcting capabilities of a given code. If the code is used only to detect errors of multiplicity t 0 , then it is necessary and sufficient that the minimum code distance be equal to
d min > t 0 + 1.(1.29)
In this case, no combination of t 0 errors can transform one allowed code combination into another allowed one. Thus, the condition for detecting all errors with multiplicity t 0 can be written as:
t 0 ≤ d min - 1.(1.30)
In order to be able to correct all errors with a factor of t or less, it is necessary to have a minimum distance that satisfies the condition:
In this case, any code combination with the number of errors t differs from each allowed combination in at least t and + 1 positions. If condition (1.31) is not met, it is possible that errors of multiplicity t will distort the transmitted combination so that it becomes closer to one of the allowed combinations than to the transmitted one or even turns into another allowed combination. In accordance with this, the condition for correcting all errors with a multiplicity of no more than t can be written as:
t and ≤ (d min - 1) / 2 .(1.32)
From (1.29) and (1.31) it follows that if the code corrects all errors with a multiplicity of t and, then the number of errors that it can detect is equal to t 0 = 2∙t and. It should be noted that relations (1.29) and (1.31) establish only the guaranteed minimum number of detected or corrected errors for a given d min and do not limit the possibility of detecting errors of higher multiplicity. For example, the simplest code with a parity check with d min = 2 allows you to detect not only single errors, but also any odd number of errors within t 0< n.
The length of the code combination n must be chosen in such a way as to provide the greatest throughput of the communication channel. When using a correcting code, the code combination contains n bits, of which m bits are information bits, and k bits are verification bits.
The redundancy of the correction code is the quantity
,(1.33)
whence follows
.(1.34)
This value shows what part of the total number of symbols of the code combination are information symbols. In coding theory, the value of B m is called the relative code rate. If the productivity of the information source is equal to H t symbols per second, then the transmission speed after encoding this information will be equal to
since in the encoded sequence, out of every n symbols, only m symbols are informational.
If the communication system uses binary signals (signals of type "1" and "0") and each unit element carries no more than one bit of information, then there is a relationship between the information transmission rate and the modulation rate
where V is the information transmission rate, bit/s; B - modulation speed, Baud.
Obviously, the smaller k, the more the m/n ratio approaches 1, the less V differs from B, i.e. the higher the throughput of the communication system.
It is also known that for cyclic codes with a minimum code distance d min = 3 the following relation holds true:
k³log 2 (n+1).(1.37)
It can be seen that the larger n, the closer the m/n ratio is to 1. So, for example, with n = 7, k = 3, m = 4, m/n = 0.571; with n = 255, k = 8, m = 247, m/n = 0.964; with n = 1023, k = 10, m = 1013, m/n = 0.990.
The above statement is also true for large d min, although there are no exact relationships for the connections between m and n. There are only upper and lower bounds that establish the relationship between the maximum possible minimum distance of the correction code and its redundancy.
Thus, the Plotkin bound gives the upper limit of the code distance d min for a given number of bits n in the code combination and the number of information bits m, and for binary codes:
(1.38)
At 
.(1.39)
The Hamming upper bound sets the maximum possible number of allowed code combinations (2 m) of any error-correcting code for given values of n and d min:
,(1.40)
where is the number of combinations of n elements based on i elements.
From here you can get an expression for estimating the number of check characters:
.(1.41)
For values (d min /n) ≤ 0.3, the difference between the Hamming limit and the Plotkin limit is relatively small.
The Varshamov-Hilbert bound for large values of n defines a lower bound on the number of check bits required to ensure a given code distance:
All of the above estimates give an idea of the upper bound of the number d min for fixed values of n and m or a lower estimate of the number of check symbols k for given m and d min .
From the foregoing, we can conclude that from the point of view of introducing constant redundancy into the code combination, it is advantageous to choose long code combinations, since with increasing n the relative throughput
R = V/B = m/n(1.43)
increases, tending to the limit equal to 1.
In real communication channels there is interference, leading to errors in code combinations. When an error is detected by the decoding device in systems with POS, a group of code combinations is asked again. During the questioning helpful information is not transmitted, so the speed of information transmission decreases.
It can be shown that in this case
,(1.44)
where P oo is the probability of detecting an error by the decoder (probability of asking again):
;(1.45)
P pp - probability of correct reception (error-free reception) of the code combination;
M - transmitter storage capacity in the number of code combinations
,(1.46)
where t p is the signal propagation time along the communication channel, s;
tk – time of transmission of a code combination of n bits, s.
Sign< >means that when calculating M, you should take the larger nearest integer value.
The signal propagation time over the communication channel and the code combination transmission time are calculated in accordance with the expressions
where L is the distance between terminal stations, km;
c is the speed of signal propagation along the communication channel, km/s (c = 3x10 5);
B - modulation speed, Baud.
If there are errors in the communication channel, the value of R is a function of P 0, n, k, B, L, s. Consequently, there is an optimal n (for given P 0, B, L, c), at which the relative throughput will be maximum.
To calculate the optimal values n, k, m, it is most convenient to use software package mathematical modeling, such as MathLab or MathCAD, by plotting the dependence R(n) in it. The optimal value will be when R(n) is maximum. When determining the values of n, k, m, it is also necessary to ensure that the following conditions are met:
where is the equivalent probability of a single bit reception error when using error-resistant coding with POC.
The value can be determined using the relation that when transmitting without the use of noise-resistant coding, the probability of erroneous registration of a code combination P 0kk of length n is equal to
.(1.48)
At the same time, when using noise-resistant coding
,(1.49)
where is the probability of undetected errors
;(1.50)
Probability of detected errors
.(1.51)
In addition to fulfilling condition (1.47), it is necessary to ensure
V ³ H t . (1.52)
From the above it follows that the process of searching for values of B, n, m, k is iterative and it is most convenient to arrange it in the form of a table, a sample of which is given in Table. 1.2
Table 1.2
| Ht = , Padd = . | ||||||
| to | n | m | K | IN | V | |
| 1 | ||||||
| 2 | ||||||
| 3 | ||||||
| … | ||||||
To detect errors, we select a cyclic code. Of all the known noise-resistant codes, cyclic codes are the simplest and most efficient. These codes can be used both to detect and correct independent errors and, in particular, to detect and correct serial errors. Their main property is that each code combination can be obtained by cyclically rearranging the symbols of combinations belonging to the same code.
Cyclic codes significantly simplify the description of a linear code, since for them, instead of specifying the elements of the binary matrix P, it is necessary to specify (n-k+1) binary coefficients of the polynomial g(D). They also simplify the encoding and decoding procedure for error detection. Indeed, to implement coding, it is enough to multiply polynomials, which is implemented using a linear register containing k memory cells and having feedback connections corresponding to the polynomial h(D).
The cyclic code is guaranteed to detect multiplicity errors and correct them. Therefore, in systems with a decisive feedback cyclic code coding is used.
When an error is detected on the receiving side, a request is sent via the reverse communication channel to the block in which it was detected, and then this block is retransmitted. This continues until the block is accepted without an error detected. Such a system is called a decision feedback system (DFS), since the decision to accept a block or to retransmit it is made at the receiving side. The system with POC are effective way increasing the noise immunity of information transmission.
When describing the encoding and decoding procedure with a cyclic code, it is convenient to use a mathematical apparatus based on comparing a set of code words with a set of power polynomials. This device allows you to identify more for a cyclic code simple operations encoding and decoding.
Among all the polynomials corresponding to the codewords of the cyclic code, there is a non-zero polynomial P(x) of the smallest degree. This polynomial completely determines the corresponding code and is therefore called generating.
The degree of the generating polynomial P(x) is equal to n - m, the free term is always equal to one.
The generating polynomial is the divisor of all polynomials corresponding to the codewords of the cyclic code.
The zero combination necessarily belongs to any linear cyclic code and can be written as (x n Å 1) mod (x n Å 1) = 0. Therefore, the generating polynomial P(x) must be a divisor of the binomial x n Å 1.
This gives constructive possibilities for constructing a cyclic code of a given length n: any polynomial that is a divisor of the binomial x n Å 1 can be used as a generator.
When constructing cyclic codes, they use tables of decomposition of binomials x n Å 1 into irreducible polynomials, i.e. polynomials that cannot be represented as a product of two other polynomials (see Appendix A).
Any irreducible polynomial included in the expansion of the binomial x n Å 1, as well as any product of irreducible polynomials, can be chosen as a generating polynomial, which gives the corresponding cyclic code.
To construct a systematic cyclic code, the following rule for constructing code words is used
where R(x) is the remainder of division m(x)×x n - m by P(x).
The degree of R(x) is obviously less than (n - m), and therefore in the code word the first m symbols will coincide with information ones, and the last n - m symbols will be verification ones.
The decoding procedure for cyclic codes can be based on the property of their divisibility without remainder by the generating polynomial P(x).
In error detection mode, if the received sequence is divided evenly by P(x), it is concluded that there is no error or it is not detected. Otherwise, the combination is rejected.
In error correction mode, the decoder calculates the remainder R(x) from dividing the received sequence F¢(x) by P(x). This remainder is called the syndrome. The received polynomial F¢(x) is the modulo two sum of the transmitted word F(x) and the error vector E osh (x):
Then syndrome S(x) = F¢(x) modP(x), since by definition of the cyclic code F(x) mod P(x) = 0. A certain syndrome S(x) can be associated with a certain error vector E osh(x). Then the transmitted word F(x) is found by adding .
However, the same syndrome can correspond to 2 m different error vectors. Let us assume that the syndrome S 1 (x) corresponds to the error vector E 1 (x). But all error vectors equal to the sum E 1 (x) Å F(x), where F(x) is any code word, will give the same syndrome. Therefore, by assigning the error vector E 1 (x) to the syndrome S 1 (x), we will carry out correct decoding in the case when the error vector is actually equal to E 1 (x), in all other 2 m - 1 cases, the decoding will be erroneous.
To reduce the probability of a decoding error, from all possible error vectors that give the same syndrome, the most probable one in a given channel should be selected as the one to be corrected.
For example, for a DSC, in which the probability P 0 of erroneous reception of a binary symbol is much less than the probability (1 - P 0) of correct reception, the probability of the appearance of error vectors decreases with increasing their weight i. In this case, the error vector with the smaller weight should be corrected first.
If only all error vectors of weight i and less can be corrected by the code, then any error vector of weight from i + 1 to n will lead to erroneous decoding.
The probability of erroneous decoding will be equal to the probability P n (>i) of the appearance of error vectors of weight i + 1 or more in a given channel. For DSC this probability will be equal to
.
The total number of different error vectors that a cyclic code can correct is equal to the number of non-zero syndromes – 2 n - m - 1.
In the course project, it is necessary, based on the value of k calculated in the previous paragraph, to select a generating polynomial according to the table given in Appendix A. Based on the selected generating polynomial, it is necessary to develop an encoder and decoder circuit for the case of an error detection.
1.5 Digital communication system performance indicators
Digital communication systems are characterized by quality indicators, one of which is the fidelity (correctness) of transmission.
To assess the efficiency of a communication system, the coefficient of utilization of the communication channel by power (energy efficiency) and the coefficient of utilization of the channel by frequency band (frequency efficiency) are introduced:
where V is the information transmission speed;
Signal-to-noise ratio at the demodulator input
; (1.55)
Frequency bandwidth occupied by the signal
, (1.56)
where M is the number of signal positions.
A generalized characteristic is the channel utilization rate in terms of throughput (information efficiency):
For a continuous communication channel taking into account Shannon’s formula
we get the following expression
. (1.58)
According to Shannon’s theorems for h=1, we can obtain the relationship between b and g:
b=g/(2 g - 1), (1.59)
which is called the Shannon boundary, which represents the best exchange between b and g in a continuous channel. It is convenient to depict this dependence as a curve on the b - g plane (Fig. 1.6).
Figure 1.6 - Shannon boundary
The efficiency of the system can be increased by increasing the speed of information transfer (increasing the entropy of messages). The entropy of messages depends on the probability distribution law. Therefore, to improve efficiency, it is necessary to redistribute the densities of message elements.
Eliminating or weakening the relationships between message elements can also improve the efficiency of systems.
Finally, improvements in system efficiency can be achieved through appropriate coding choices that save time during message transmission.
In the course project, it is necessary to mark the efficiency of the designed digital communication system with a dot on the constructed graph (Fig. 1.6).
1. Guidelines for course design in the discipline “Theory of Electrical Coupling” Bidny Yu.M., Zolotarev V.A., Omelchenko A.V. - Kharkov: KNURE, 2008.
2. Omelchenko V.A., Sannikov V.G. Theory of electrical communication. Parts 1, 2, 3. - K.: ISDO, 2001.
3. Theory of electrical communication: Textbook for universities / A.G. Zyuko. D.D. Klovsky, V.I. Korzhik, M.V. Nazarov; Ed. D.D. Klokovsky. – M.: Radio and communications. 1998.
4. Peterson W., Weldon E. Error Correcting Codes / Translation, from English. edited by R.L. Dobrushina and S.I. Samoilenko. - M-: Mir, 1999. - 596 p.
5. Andreev V.S. Theory of nonlinear electrical circuits. Textbook manual for universities. - M.: Radio and communication, 1999. - 280 p.
APPLICATION
Table of irreducible generating polynomials of degree m
Degree m = 7 x 7 + x 4 + x 3 + x 2 + 1 x 7 + x 3 +x 2 + x + 1 |
Degree m = 13 x 13 + x 4 + x 3 + x + 1 x 13 + x 12 + x 6 +x 5 + x 4 + x 3 + 1 x 13 + x 12 + x 8 + x 7 + x 6 + x 5 + 1 |
Degree m = 8 x 8 + x 4 + x 3 + x + 1 x 8 + x 5 + x 4 + x 3 + 1 x 8 + x 7 + x 5 + x +1 |
Degree m = 14 x 14 + x 8 + x 6 + x + 1 x 14 + x 10 + x 6 + 1 x 14 + x 12 + x 6 + x 5 + x 3 + x + 1 |
Degree m = 9 x 9 + x 4 +x 2 + x + 1 x 9 + x 5 + x 3 + x 2 + 1 x 9 + x 6 + x 3 + x + 1 |
Degree m = 15 x 15 + x 10 + x 5 + x + 1 x 15 + x 11 + x 7 + x 6 + x 2 + x + 1 x 15 + x 12 + x 3 + x + 1 |
Degree m = 10 x 10 + x 3 + 1 x 10 +x 4 +x 3 + x + 1 x 10 +x 8 +x з +x 2 + 1 |
Degree m = 16 x 16 + x 12 + x 3 + x + 1 x 16 + x 13 + x 12 + x 11 + x 7 + x 6 + x 3 + x + 1 x 16 + x 15 + x 11 + x 10 + x 9 + x 6 + x 2 + x + 1 |
Degree m = 11 x 11 + x 2 + 1 x 11 + x 7 + x 3 + x 2 + 1 x 11 + x 8 + x 5 + x 2 + 1 |
Degree m = 17 x 17 + x 3 + x 2 + x + 1 x 17 + x 8 + x 7 + x 6 + x 4 + x 3 + 1 x 17 + x 12 + x 6 + x 3 + x 2 + x + 1 |
Degree m = 12 x 12 + x 4 + x + 1 x 12 + x 9 + x 3 + x 2 + 1 x 12 + x 11 + x 6 + x 4 + x 2 + x+1 |
Fundamentals of optimization of information transmission systems, selection and principles of signal generation.
For radio channels with limited frequency and energy resources, the most important task is to use these resources efficiently. This means ensuring the maximum speed of information transfer from the message source with given resource parameters and reliability of message transmission.
In the modern theory of information transmission systems, it is customary to first optimize the communication system as a whole. Then the remaining elements of the system, in particular the receiver, are optimized, provided that the type of signals has already been selected.
When optimizing the system, we look for best view signal for a given radio channel and the corresponding optimal reception method.
“The founder of optimization of communication systems in general is K. Shannon, who proved the theorem:
“If a communication channel with a finite frequency response and additive white Gaussian noise (AWGN) has a capacity of “C”, and the source performance is equal to H′(A), then when H′(A) ≤ C, coding is possible that ensures the transmission of messages over this channel with arbitrarily small errors and at a speed arbitrarily close to the “C” value:
[bit/s], (3.1)
Where ∆f k– bandwidth of the rectangular frequency response of the communication channel;
R s- average signal power;
R w =N 0· ∆f k; (3.2)
N 0· - one-sided spectral density of AWGN.
For a discrete channel and random source encoding, this theorem can be written in a different form
where is the average probability of decoding error over a set of codes;
T- duration of the code block of the enlarged message source.
Since, [С−Н ′ (А) ≥ 0] according to the theorem, then with increasing T(by enlarging the source) and at H ′ (A)→C the value T→∞ and the decoding delay of the enlarged source code increases.
From (3.3) we can conclude conclusions:
- the longer the encoded message segment (T) and the less efficient
the channel capacity is used (the larger the difference [C-H ′ (A)]), the higher the reliability of the connection (1-);
- there is a possibility of exchange between the efficiency of use, the values of C, and T (decoding delay).
a) Let us analyze the capacity (3.1).
"C" can be increased by increasing ∆f k And R s. It must be taken into account that the power R w(3.2) also depends on ∆f k.
Based on the known relationship (at α=2, β = e) can be written down
Let's find the limit value depending on the band ∆f k and build a throughput graph.
At ∆f k→∞ . Then we expand the function log(1+x) in the Maclaurin series (i.e. at the point X=0) , which at x→0 equals ln(1+x)≈x. As a result we get
Let's plot function (3.4) depending on ∆f k with normalization on both axes N 0 /P c.
Fig.3.1. Graph of normalized channel capacity.
At R s / R w=1 in (3.1) → WITH= ∆f k. Taking into account the normalization along the axes of the graph, this equality corresponds to the point (C N 0 /P c =P w/P c=1) with coordinates (1,1).
Bandwidth increases noticeably with increasing ∆f k until P s /P w ≥1 and tends to the limit of 1.44 R s /N 0, i.e. the maximum value of parameter C occurs at h →0.
b) Let’s find Shannon’s boundary values for the specific bandwidth and energy costs at the information transmission rateR max = C .
By definition, the specific bandwidth costs in the communication channel are equal to
where R is the information transmission rate (bit/s) in the channel. Attempts to reduce these specific costs are associated with additional energy costs, characterized by the value of specific energy costs
Where E b- energy spent on transmitting 1 bit of information;
T 0- transmission time of 1 bit over the communication channel (duration of the channel symbol T ks);
Let's find the dependence of specific energy costs on specific strip costs. To do this, we express the quantities included in (3.1), assuming WITH=Rmax :
Substituting these values into (3.1) and dividing it by WITH we get
Based on the definition of logarithm log 2 N=a meaning N=2a can be written from where, taking the root of both sides, we get
As a result expression
determines the relationship between specific energy and bandwidth consumption in a channel with AWGN and finite frequency response. At the same time, because
then from (3.5) we obtain the dependence for the signal-to-noise ratio (SNR):
Thus, in a communication channel with a finite frequency response and AWGN, it is possible to implement an infinite number of different optimal systems. Spectrally efficient systems (baseband spectrum of the baseband signal) require a correspondingly increased SNR. Energy efficient systems require low SNR but must be wideband.
Real systems have values that lie on the graph in Fig. 3.2 above the Shannon limits. By comparing real systems with potential ones, it is possible to evaluate the reserve for improving the parameters of the communication system.