RU2522300C1 - Manipulation coding method - Google Patents

Abstract

FIELD: information technology.

SUBSTANCE: manipulation coding method includes first establishing a number of signal constellation points and ordering said points for Gray coding; establishing a number of points in the signal constellation equal to any natural number; determining the Euclidean distance between signal constellation points and generating a vector of Euclidean distances between signal constellation points; setting a graph incidence matrix, a source and drain vertex and a plurality of vertices that are compulsory for passage; determining the initial vector of graph edge allocations using a Hungarian method of solving the allocation problem; calculating the gradient vector for the initial vector; finding the minimum in the direction of the gradient vector using a one-dimensional search method; recalculating the gradient vector at the minimum point; determining the minimum in a new direction using the one-dimensional search method; obtaining an improved approach of the allocation vector, and using the Newton-Raphson method to find the optimum routing solution in Euclidean space; generating a graph vertex bypass sequence and determining code combinations corresponding to signal constellation points in accordance with the Gray coding rule.

EFFECT: high noise-immunity of transmitting discrete information.

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Description

The invention relates to the field of encoding information for transmitting discrete messages over communication channels.

A known method of manipulation coding [Nechaev Yu.B. Manipulation codes for systems with iterative processing of the received signal. / Nechaev Yu.B., Malyutin A.A. Infocommunication technologies. - 2009. No. 2. - p. 70-74], based on computer enumeration of all possible options for displaying a block of code symbols in a transmitted signal. The disadvantage of this method is the low noise immunity of the transmission of discrete information.

A known method of matching modulation and coding [Vasiliev K.K. Theory of electrical communication: textbook / Vasiliev K.K., Glushkov V.A., Dormidontov A.V., Nesterenko A.G .; under the general. ed. Vasilyeva K.K. - Ulyanovsk: UlSTU, 2008. - p. 387-388] based on the partition of the ensemble of signals into nested sub-ensembles. The method of matching modulation and coding based on dividing the ensemble of signals into nested sub-ensembles reduces the dimension of the exhaustive task of synthesizing the signal-code structure, but does not provide guaranteed construction of the signal-code structure with maximum frequency-energy characteristics and does not always allow matching Euclidean and Hamming distances, which determines low noise immunity of this method of matching modulation and coding. Another significant limitation of the application of this method is the requirement of parity of the total number of points in the ensemble of signals and the parity of the number of points in each of the subassemblies obtained by splitting, which determines the possibility of its use only for signal-code constructions with the number of points multiple of the power of 2.

The closest in technical essence (prototype) to the claimed method is a modulation and demodulation method, a modulation device and a demodulation device (RU 2384960 C2 03/20/2010), in the claims of which a modulation and demodulation method is presented in which data is transmitted (2n + 1) bits (where "n" is an integer greater than 1), and the multilevel value is set to 2 ^{(2n + 1)} , the method comprising: dividing the signal points ordered in each of the four quadrants into 8 subgroups corresponding to 3 bits from the data ( 2n + 1) bits, with four the quadrants are separated by a common-mode axis and an orthogonal axis perpendicular to each other; encoding 3 bits so that the average Hamming distance between adjacent signal points in 8 subgroups becomes minimal; performing Gray coding for 2 bits from the data of (2n + 1) bits, as a signal enabling the identification of four quadrants.

A disadvantage of the known prototype is that it is applicable only to signals with multi-level quadrature amplitude modulation, and the total number of points should be determined by the formula 2 ^{(2n + 1)} , while the modulation and demodulation method described in the prototype has a low noise immunity for transmitting discrete information.

The objective of the invention is to provide a method of manipulation coding, which improves the noise immunity of the transmission of discrete information through manipulation coding for signal constellations with an arbitrary number of points and an arbitrary arrangement of signal points on the signal diagram.

This problem is solved in that the method of manipulation coding, which consists in setting the number of points of the signal constellation and arranging them for coding according to Gray, according to the invention is supplemented by setting the number of points in the signal constellation equal to any natural number, determining the Euclidean distance between the points of the signal constellation, form a vector of Euclidean distances between the points of the signal constellation, specify the incidence matrix of the graph, the source and drain vertices, the set of vertices required for the passage, determine the initial vector of assignments of the graph arcs using the Hungarian method of solving the assignment problem, calculate the gradient vector for the initial vector, find the minimum in the direction of the gradient vector by the one-dimensional search method, recalculate the gradient vector at the minimum point, determine the minimum in the new one by the one-dimensional search method direction, get an improved approximation of the destination vector, using the Newton-Raphson method find the optimal solution to the routing problem in Euclidean space, form Coziness The sequence of traversal of graph vertices, in accordance with the Gray coding rule, determines the code combinations corresponding to the points of the signal constellation.

The listed new set of essential features provides the possibility of increasing the noise immunity of the transmission of discrete information by obtaining the possibility of manipulation coding for signal constellations with an arbitrary number of points and their arbitrary location on the signal diagram.

The analysis of the prior art made it possible to establish that analogues that are characterized by a combination of features that are identical to all the features of the claimed solution are absent, which indicates the compliance of the claimed method with the condition of patentability "novelty".

Search results for known solutions in this and related fields of technology in order to identify features that match the distinctive features of the claimed object from the prototype showed that they do not follow explicitly from the prior art. The prior art also did not reveal the popularity of the influence provided by the essential features of the claimed invention of the transformations to achieve the specified technical result. Therefore, the claimed invention meets the condition of patentability "inventive step".

The drawing shows a block diagram of a sequence of actions that implement the claimed method of manipulation coding.

The implementation of the invention is achieved as follows.

1. Set the number of points of the signal constellation equal to V, and arrange them, numbering from 1 to V, thereby setting the V vertices of the graph. The Euclidean distance between the points of the signal constellation is determined, the vector of Euclidean distances (the weights of the arcs) between the points of the signal constellation is formed, and the incident matrix of the fully connected graph I is set. The source and sink vertices are specified, and the vertex of the source and sink can be any vertex of the graph, for example, the first. The set of graph vertices that are required to go through a closed path will be all vertices of the graph, since it is necessary to determine the manipulation coding method as a correspondence between the points of the signal constellation and the transmitted code combinations for all points of the signal constellation.

The Euclidean distance between the i-th and j-th points of the signal constellation is calculated by the formula:

, $$T_{ij}=\sqrt{{\displaystyle \sum _{ij}{\displaystyle \sum _{k=\mathrm{one}}^P{\left(a_i{}_k-a_j{}_k\right)}^2}}}$$

,

,Where $$i,j=\overline{\mathrm{one,}V}$$

,

, $$a_j{}_k$$

- k-я координата i-ой (j-ой) точки в сигнальном пространстве, $$a_i{}_k$$

, $$a_j{}_k$$

- k-th coordinate of the i-th (j-th) point in the signal space,

P is the dimension of the signal space (for example, for quadrature amplitude modulation P = 2).

The incident matrix I consists of the incident matrices of the outgoing arcs of the graph I _ex and the incident matrices of the incoming arcs of the graph I _in ,

I = I _out + I _in .

Initial data:

- incidence matrix of outgoing arcs of graph I _ref ;

- incidence matrix of incoming arcs of graph I _in ;

, где D - число дуг;is the vector of the magnitude of the Euclidean distances (weights of the arcs of the graph) $$\overrightarrow{T}={\left[t_{\mathrm{one}},...,t_D\right]}^T$$

where D is the number of arcs;

, $$b_{вх}{}_j=-1$$

;- the number of vertices of the source and drain of the selected path $$b_{\mathrm{and}\mathrm{from}x}{}_i=\mathrm{one}$$

, $$b_{\mathrm{at}x}{}_j=-\mathrm{one}$$

;

- the set of vertices required for the passage {V _pr };

- the set of free vertices {V _sw }.

The task of routing along the vertices of the graph is presented in the following formulation:

$${\displaystyle \sum _{k=\mathrm{one}}^D{T}_k{ƒ}_k\Rightarrow \frac{\min }{{\overrightarrow{ƒ}}^T=\left({ƒ}_{\mathrm{one}},...,{ƒ}_k,...,{ƒ}_D\right)}}\left(\text{one}\right)$$

where T _k is the Euclidean distance between the signal points,

- вектор назначений дуг графа, $$k=\overline{\mathrm{1,}D}$$

. $$\overrightarrow{ƒ}$$

is the vector of assignments of the arcs of the graph, $$k=\overline{\mathrm{one,}D}$$

.

The objective function (1) reflects the need to minimize the total Euclidean distance in the route formed by the successive passage of arcs in the vector ƒ _k .

свяжем с декомпозированной матрицей инциденций для V-узлов и D-дуг, вектором истоков $${\overrightarrow{b}}_{исх}^{\left(i,j\right)}$$

и вектором стоков $${\overrightarrow{b}}_{вх}^{\left(i,j\right)}$$

с помощью соотношений:Auxiliary structural variables $$\overrightarrow{x}$$

we associate with the decomposed incident matrix for V-nodes and D-arcs, the source vector $${\overrightarrow{b}}_{\mathrm{and}\mathrm{from}x}^{\left(i,j\right)}$$

and stock vector $${\overrightarrow{b}}_{\mathrm{at}x}^{\left(i,j\right)}$$

using the ratios:

$$I_{\mathrm{and}\mathrm{from}x}\overrightarrow{ƒ}-{\overrightarrow{x}}_{\mathrm{and}\mathrm{from}x}={\overrightarrow{b}}_{\mathrm{and}\mathrm{from}x};\left(\text{2}\right)$$

$$I_{\mathrm{at}x}\overrightarrow{ƒ}+{\overrightarrow{x}}_{\mathrm{and}\mathrm{from}x}={\overrightarrow{b}}_{\mathrm{at}x}.\left(\text{3}\right)$$

, $$b_{в{х}_j}=-1$$

.When specifying the source and sink vectors, it is necessary to put $$b_{\mathrm{and}\mathrm{from}x}{}_i=\mathrm{one}$$

, $$b_{\mathrm{at}{x}_j}=-\mathrm{one}$$

.

Let for the formation of options for the selection of routes given sets:

{V _pr } - the set of vertices required to pass along the route i → j;

{V _sw } is the set of vertices that are free to pass along the route i → j (that is, the route may or may not pass through a vertex belonging to this set).

Obviously: {V _pr } ∪ {V _sw } = {V}.

, ∀k∈{V_pr}; $$b_{вх}{}_l=-1$$

, ∀l∈{V_pr}.Then: $$b_{\mathrm{and}\mathrm{from}x}{}_k=\mathrm{one}$$

, ∀k∈ {V _pr }; $$b_{\mathrm{at}x}{}_l=-\mathrm{one}$$

, ∀l∈ {V _pr }.

In the process of solving the routing problem, it is necessary to find auxiliary structural variables that reflect the use of free vertices to form routes:

$$x_{\mathrm{and}\mathrm{from}x}{}_n=\{\begin{array}{l}\mathrm{one,}\text{if the path passes through the nth free vertex;}\hfill \\ \text{0}\text{if the path does not pass through the nth free vertex; n}\in \left\{{\text{V}}_{\text{sw}}\right\}.\hfill \end{array}$$

The restriction on the absence of cycles in the graph paths is written as:

$$E_d^T{\left(-I_{\mathrm{and}\mathrm{from}x}diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^T{I}_{\mathrm{at}x}^T\right)}^{V+\mathrm{one}}{\text{E}}_{\text{d}}=0\left(\text{four}\right)$$

- вектор размерности V×1.Where $$E_d^T{=}_{\mathrm{one}}{\left(\mathrm{one,}...\mathrm{,one,}...\mathrm{,one}\right)}_V$$

is a vector of dimension V × 1.

We introduce routing fixity using Boolean variables

$${ƒ}_k\in \left\{0.1\right\},\text{k}=\overline{\text{one}\text{, D}};\left(\text{5}\right)$$

$$x_n\in \left\{0.1\right\},\text{n}\in \left\{{\text{V}}_{\text{sw}}\right\}.\left(\text{6}\right)$$

, $$\overrightarrow{x}$$

записываются в видеWe use the property of idempotency of the constraint: x ² = x, which allows only the Booleanness of the variable x (x = 0 or 1). Then the constraints (5, 6) on the continuum sets of variables $$\overrightarrow{ƒ}$$

, $$\overrightarrow{x}$$

are written as

$${ƒ}_k^2={ƒ}_k,\text{k}=\overrightarrow{\mathrm{one,}D};\left(\text{7}\right)$$

$$x_n^2=x_n,\text{n}\in \left\{{\text{V}}_{\text{sw}}\right\},\left(\text{8}\right)$$

and thus the integer problem (1), (2-6) is transformed (immersed) into the general nonlinear programming problem (1), (2-4, 7, 8).

This problem is an optimization problem with constraints in the form of equalities. Therefore, we apply the method of Lagrange multipliers to it. Then we write the Lagrange function in the form: lambda - Lagrange multipliers

$$\begin{array}{l}F\left(\overrightarrow{ƒ},\overrightarrow{x},\overrightarrow{\lambda ƒ},\overrightarrow{\lambda x},\overrightarrow{\lambda p},\overrightarrow{\lambda ƒc}\right)={\overrightarrow{T}}^T\overrightarrow{ƒ}+{\overrightarrow{\lambda ƒ}}^T\left[{}_{2V}\left[\begin{array}{l}{I}_{\mathrm{and}\mathrm{from}x}\hfill \\ ...\hfill \\ I_{\mathrm{at}x}\hfill \end{array}\right]{}^D\overrightarrow{ƒ}-{}_{2V}\left[\begin{array}{l}{\overrightarrow{x}}_{\mathrm{and}\mathrm{from}x}\hfill \\ ...\hfill \\ -{\overrightarrow{x}}_{\mathrm{and}\mathrm{from}x}\hfill \end{array}\right]{}^{\mathrm{one}}-{}_{2V}\left[\begin{array}{l}{\overrightarrow{b}}_{\mathrm{and}\mathrm{from}x}\hfill \\ ...\hfill \\ {\overrightarrow{b}}_{\mathrm{at}x}\hfill \end{array}\right]{}^{\mathrm{one}}\right]+\\ +{\displaystyle \sum _{i=\mathrm{one}}^n\lambda x_i\left(x_i^2-x_i\right)+\lambda p\left[E_d^T{\left(-I_{\mathrm{and}\mathrm{from}x}diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^T{I}_{\mathrm{at}x}^T\right)}^{V+\mathrm{one}}Ed\right]+{\displaystyle \sum _{j=\mathrm{one}}^D\lambda ƒc_j\left({ƒ}_j^2-{ƒ}_j\right)}}\left(9\right)\end{array}$$

размерностью D+n+2V+n+1+D.We introduce the general vector of the desired variables in the form of a composite vector $${\overrightarrow{X}}^T={\overrightarrow{ƒ}}^T,{\overrightarrow{x}}^T,{\overrightarrow{\lambda ƒ}}^T,{\overrightarrow{\lambda x}}^T,\lambda p,\overrightarrow{\lambda ƒc}$$

dimension D + n + 2V + n + 1 + D.

We write the necessary conditions for the extremum in the form of a system of equations:

; $$\begin{array}{l}\frac{\partial F}{\partial \overrightarrow{ƒ}}=2\times \lambda p\times dg\left[B\left[{\displaystyle \sum _{j=0}^V\left\{{\left[A\times diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^{T}B\right]}^{V-j}{E}_d{E}_d^T{\left[A\times diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^{T}B\right]}^j\right\}}\right]A\right]\overrightarrow{ƒ}+\\ +\overrightarrow{T}+{}_{2V}\left[\begin{array}{l}{I}_{\mathrm{and}\mathrm{from}x}\hfill \\ ...\hfill \\ I_{\text{in}}\hfill \end{array}\right]{}^D\overrightarrow{\lambda ƒ}+\left[\begin{array}{l}\lambda ƒc_{\mathrm{one}}\left(2{ƒ}_{\mathrm{one}}-\mathrm{one}\right)\hfill \\ ..................\hfill \\ \lambda ƒc_D\left(2{ƒ}_D-\mathrm{one}\right)\hfill \end{array}\right]=\overrightarrow{0}\end{array}$$

;

; $$\frac{\partial F}{\partial \overrightarrow{x}}=\left[\begin{array}{l}..................\hfill \\ -\lambda {ƒ}_n+\lambda {ƒ}_{n+V}\hfill \\ ..................\hfill \end{array}\right]+\left[\begin{array}{l}..................\hfill \\ \lambda x_n\left(2x_n-\mathrm{one}\right)\hfill \\ ..................\hfill \end{array}\right]=\overrightarrow{0},\text{n}\in \left\{{\text{V}}_{\text{sw}}\right\}$$

;

$$\frac{\partial F}{\partial \overrightarrow{\lambda ƒ}}={}_{2V}\left[\begin{array}{l}{I}_{\mathrm{and}\mathrm{from}x}\hfill \\ ...\hfill \\ I_{\mathrm{at}x}\hfill \end{array}\right]{}^D\overrightarrow{ƒ}-{}_{2V}{}^{\mathrm{one}}\left[\begin{array}{l}...\hfill \\ x_n\hfill \\ ...\hfill \\ x_{n+V}=-x_n\hfill \\ ...\hfill \end{array}\right]{}^{\mathrm{one}}-{}_{2V}{}^{\mathrm{one}}\left[\begin{array}{l}{\overrightarrow{b}}_{\mathrm{and}\mathrm{from}x}\hfill \\ ⋮\hfill \\ {\overrightarrow{b}}_{\mathrm{at}x}\hfill \end{array}\right]{}^{\mathrm{one}}=\overrightarrow{0},\text{n}\in \left\{{\text{V}}_{\text{sw}}\right\};\left(10\right)$$

; $$\frac{\partial F}{\partial \overrightarrow{\lambda x}}=\left[\begin{array}{l}.........\hfill \\ x_n^2-x_n\hfill \\ .........\hfill \end{array}\right]=\overrightarrow{0},\text{n}\in \left\{{\text{V}}_{\text{sw}}\right\}$$

;

; $$\frac{\partial F}{\partial \lambda p}=E_d^T{\left(-I_{\mathrm{and}\mathrm{from}x}diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^T{I}_{\mathrm{at}x}^T\right)}^{W+\mathrm{one}}{\text{E}}_{\text{d}}=\overrightarrow{0}$$

;

$$\frac{\partial F}{\partial \overrightarrow{\lambda ƒc}}=\left[\begin{array}{l}{ƒ}_{\mathrm{one}}^2-{ƒ}_{\mathrm{one}}\hfill \\ .........\hfill \\ {ƒ}_D^2-{ƒ}_D\hfill \end{array}\right]=\overrightarrow{0}.$$

Thus, a system is formed of m = D + n + 2V + n + 1 + D equations with m = D + n + 2V + n + 1 + D unknowns.

We choose the Newton-Raphson method [Hamdy A. Taha. Introduction to Operations Research, 6th edition: Per. from English - M .: Publishing house "Williams", 2001. - S.744-745] (second order) for solving a system of nonlinear equations. Moreover, as the objective function of the equivalent minimization problem, we put the minimum discrepancy between the left and right sides of the system of equations (10):

$${\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)\Rightarrow \underset{\overrightarrow{X}}{\min }.\left(\text{eleven}\right)$$

Algorithms based on the Newton-Raphson method converge with a good initial approximation to the exact solution. Therefore, the developed algorithm should contain operations for determining the initial approximation.

In the general case, using the necessary extremum condition (10) to find the stationary points of the Lagrange function (9) is fraught with difficulties arising in the numerical solution of the corresponding system of equations. The Newton-Raphson method offers an iterative procedure for solving a system of nonlinear equations. It belongs to the number of gradient methods for the numerical search for the extremum of a function in the absence of restrictions.

, i=1, 2, …, m, метод использует итерационную процедуру $${\overrightarrow{X}}_{k+1}={\overrightarrow{X}}_k-J{\left({\overrightarrow{X}}_k\right)}^{-1}\frac{\partial F}{\partial {\overrightarrow{X}}_k}$$

, где: $$J\left({\overrightarrow{X}}_k\right)=\frac{{\partial }^{2}F\left({\overrightarrow{X}}_k\right)}{\partial {\overrightarrow{X}}_k\partial X_k^T}$$

- матрица якобиана.So, for a system of equations $${ƒ}_i=\left(\overrightarrow{X}\right)=0$$

, i = 1, 2, ..., m, the method uses an iterative procedure $${\overrightarrow{X}}_{k+\mathrm{one}}={\overrightarrow{X}}_k-J{\left({\overrightarrow{X}}_k\right)}^{-\mathrm{one}}\frac{\partial F}{\partial {\overrightarrow{X}}_k}$$

where: $$J\left({\overrightarrow{X}}_k\right)=\frac{{\partial }^{2}F\left({\overrightarrow{X}}_k\right)}{\partial {\overrightarrow{X}}_k\partial X_k^T}$$

- Jacobian matrix.

Algorithms constructed by the Newton-Raphson method have a small radius of convergence [Hamdy A. Taha. Introduction to Operations Research, 6th edition: Per. from English - M .: Publishing house "Williams", 2001. - S.745]. Therefore, we will use the two-stage procedure for finding the initial approximation. At the first stage, we apply the polynomial algorithm of the Hungarian method [Papadimitriou X. Combinatorial optimization. Algorithms and complexity / Papadimitriou X., Steiglitz K. - M .: Mir, 1985. - p.255-262] solving the assignment problem, which does not contain restrictions on the cyclicity of the path, formulated as:

, $${\displaystyle \sum _{i=\mathrm{one}}^V{\displaystyle \sum _{j=\mathrm{one}}^V{c}_{ij}{x}_{ij}\Rightarrow \underset{x_{ij}}{\min }}}$$

,

$${\displaystyle \sum _{i=\mathrm{one}}^V{x}_{ij}}=\mathrm{one,}\forall \text{j}=\overline{\text{one}\text{, V}\text{,}}\left(\text{12}\right)$$

. $${\displaystyle \sum _{j=\mathrm{one}}^V{x}_{ij}=\mathrm{one,}\forall }\text{i}=\overline{\text{one}\text{, V}}$$

.

с помощью венгерского метода решения задачи о назначениях на основе матрицы евклидовых расстояний между вершинами графа. Таким образом, получают начальное приближение общего вектора искомых переменных $${\overrightarrow{X}}_1^T=\left({\overrightarrow{ƒ}}_1^T,{\overrightarrow{0}}^T,{\overrightarrow{0}}^T,{\overrightarrow{0}}^T\mathrm{,0,}{\overrightarrow{0}}^T\right)$$

.2. Determine the initial vector of assignments of the arcs of the graph $${\overrightarrow{ƒ}}_{\mathrm{one}}$$

using the Hungarian method of solving the assignment problem based on the matrix of Euclidean distances between the vertices of the graph. Thus, an initial approximation of the general vector of the desired variables is obtained $${\overrightarrow{X}}_{\mathrm{one}}^T=\left({\overrightarrow{ƒ}}_{\mathrm{one}}^T,{\overrightarrow{0}}^T,{\overrightarrow{0}}^T,{\overrightarrow{0}}^T\mathrm{,\; 0,}{\overrightarrow{0}}^T\right)$$

.

3. Calculate the gradient vector for the initial vector.

Gradient correction vector $$\overrightarrow{gradSTRAF}=\overrightarrow{T}$$

At the second stage, we use the fastest gradient descent of the first order, while the algorithm of the second stage is included in the algorithm of the Newton-Raphson method.

4. Find by the method of one-dimensional search a minimum in the direction of the gradient vector. For this:

- put U = ∞;

- set the step number i = 1, the accuracy of the algorithm: Tosp = 10 ^-3 ;

;- calculate the total time delay for the first stage of the initial approximation $$Ce{l}_i={\overrightarrow{T}}^T{\overrightarrow{ƒ}}_i$$

;

;- calculate the gradient vector $$J\left(\overrightarrow{X}\right)\frac{\partial F}{\partial X}+\overrightarrow{gradSTRAF}$$

;

- check the condition | Cel _i -U |> Tsp, if it is satisfied, then go to the next step, and if not, then determine the value of the auxiliary variable i1 = 1 and determine the minimum in a new direction by the one-dimensional search method;

- put U = Cel _i ;

- increase the value of i by 1;

модифицированной функции невязки $${\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)+{\overrightarrow{T}}^T\times \overrightarrow{ƒ}$$

в направлении градиента $$\left(J\left(\overrightarrow{X}\right)\frac{\partial F}{\partial X}+\overrightarrow{gradSTRAF}\right)$$

системы нелинейных уравнений $$\frac{\partial F}{\partial \overrightarrow{X}}=0$$

.- conduct a one-dimensional search for the minimum point $$\left(\overrightarrow{X}\right)$$

modified residual function $${\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)+{\overrightarrow{T}}^T\times \overrightarrow{ƒ}$$

in the direction of the gradient $$\left(J\left(\overrightarrow{X}\right)\frac{\partial F}{\partial X}+\overrightarrow{gradSTRAF}\right)$$

systems of nonlinear equations $$\frac{\partial F}{\partial \overrightarrow{X}}=0$$

.

Input data for one-dimensional minimum point search:

- общий вектор исходных переменных;- $$\overrightarrow{X}$$

- the general vector of the source variables;

;- grad - gradient $$\left(J\left(\overrightarrow{X}\right)\frac{\partial F}{\partial X}+\overrightarrow{gradSTRAF}\right)$$

;

- вектор евклидовых расстояний (весов дуг графа). - $$\overrightarrow{T}$$

is the vector of Euclidean distances (weights of the arcs of the graph).

Minimum one-dimensional search order

Step 1. Assign a step number: i = 1.

Step 2. Set the threshold value: Porog = 10 ¹³ .

; min=W_x.Step 3. Assign $$W_x={\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)+{\overrightarrow{T}}^T\overrightarrow{ƒ}$$

; min = W _x .

.Step 4. Set the value of the gradient step: Δ = -1 × 10 ^-0 , $$\overrightarrow{teka}=0$$

.

Step 5. Check the condition | W _x -Porog}> 10 ^-3 , if so, go to step 6, if not, then go to step 14.

Step 6. Assign i = i + 1.

Step 7. Threshold value Porog = W _x .

, если да, то к шагу 9, если нет, то к шагу 10.Step 8. Verify the condition $${\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)+{\overrightarrow{T}}^T\overrightarrow{ƒ}\ge \min$$

if yes, then to step 9, if not, then to step 10.

Step 9. Calculate the value of the step: Δ = Δ × 10 ^-3 .

.Step 10. Calculate the value of the vector: $$\overrightarrow{tekap}=\overrightarrow{teka}+\Delta grad$$

.

.Step 11. Calculate: $$W_x={\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)+{\overrightarrow{T}}^T\overrightarrow{ƒ}$$

.

Step 12. Checking the condition W _x <min.

и перейти к шагу 5.Step 13. Assign values: min = W _x , $$\overrightarrow{teka}=\overrightarrow{teka}+\Delta grad$$

and go to step 5.

.Step 14. Generation of output data: $$\overrightarrow{X}+\overrightarrow{teka}$$

.

5. Recalculate the gradient vector at the minimum point.

.Calculate new gradient value $$J\left(\overrightarrow{X}\right)\frac{\partial F}{\partial \overrightarrow{X}}+\overrightarrow{gradSTRAF}$$

.

.The value of the objective function is calculated. $$Ce{l}_i={\overrightarrow{T}}^T\overrightarrow{ƒ}$$

.

The value of the auxiliary variable i1 = 1 is determined.

6. The one-dimensional search method determines the minimum in a new direction.

7. Get improved destination vector approximation.

$$NewLag{r}_{i\mathrm{one}}={\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)$$

8. Find the optimal solution to the routing problem in Euclidean space using the Newton-Raphson method.

$${\overrightarrow{X}}_k^T={\overrightarrow{ƒ}}^T,{\overrightarrow{x}}^T,{\overrightarrow{\lambda ƒ}}^T,{\overrightarrow{\lambda x}}^T,\lambda p,{\overrightarrow{\lambda ƒc}}^T$$

Next, we introduce the notation of matrices with an indication of their dimension, as follows:

- матрица размера m×n. $${}_m{}^{\mathrm{one}}A{}^n$$

- a matrix of size m × n.

, $$J\left({\overrightarrow{X}}_k\right)={}_m{}^{\mathrm{one}}\left[\begin{array}{llllll}{}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{ƒ}}^T}\right]{}^D\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{x}}^T}\right]{}^n\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^{2V}\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^n\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial \lambda p}\right]{}^{\mathrm{one}}\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda ƒc}}^T}\right]{}^D\hfill \\ {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{ƒ}}^T}\right]{}^D\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{x}}^T}\right]{}^n\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^{2V}\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^n\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial \lambda p}\right]{}^{\mathrm{one}}\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda ƒc}}^T}\right]{}^D\hfill \\ {}_{2V}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{ƒ}}^T}\right]{}^D\hfill & {}_{2V}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{x}}^T}\right]{}^n\hfill & {}_{2V}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^{2V}\hfill & {}_{2V}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^n\hfill & {}_{2V}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial \lambda p}\right]{}^{\mathrm{one}}\hfill & {}_{2V}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒc}}^T}\right]{}^D\hfill \\ {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{ƒ}}^T}\right]{}^D\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{x}}^T}\right]{}^n\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^{2V}\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^n\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial \lambda p}\right]{}^{\mathrm{one}}\hfill & {}_n{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{\lambda ƒc}}^T}\right]{}^D\hfill \\ {}_{\mathrm{one}}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{ƒ}}^T}\right]{}^D\hfill & {}_{\mathrm{one}}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{x}}^T}\right]{}^n\hfill & {}_{\mathrm{one}}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^{2V}\hfill & {}_{\mathrm{one}}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda x}}^T}\right]{}^n\hfill & {}_{\mathrm{one}}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \lambda p\partial \lambda p}\right]{}^{\mathrm{one}}\hfill & {}_{\mathrm{one}}{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda ƒc}}^T}\right]{}^D\hfill \\ {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial {\overrightarrow{ƒ}}^T}\right]{}^n\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial {\overrightarrow{x}}^T}\right]{}^n\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial {\overrightarrow{\lambda ƒ}}^T}\right]{}^{2V}\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial {\overrightarrow{\lambda x}}^T}\right]{}^n\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial \lambda p}\right]{}^{\mathrm{one}}\hfill & {}_D{}^{\mathrm{one}}\left[\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial {\overrightarrow{\lambda ƒc}}^T}\right]{}^D\hfill \end{array}\right]{}^m$$

,

Where

, $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{ƒ}}^T}=2\times \lambda p{\overrightarrow{ƒ}}^T\otimes I_D\times dg\left\{\left(A^T\otimes B{\displaystyle \sum _{j=0}^W\left(k_2+k_{\mathrm{one}}\right)+AAA\left(ƒ\right)}\right)\right\}+2\left[\begin{array}{cc}\lambda ƒc_{\mathrm{one}}& 0\\ 0& \lambda ƒc_D\end{array}\right]$$

,

here the symbol ⊗ denotes the Kronecker product of the matrices A and B:

, $${}_m{}^{\mathrm{one}}A{}^n\otimes {}_p{}^{\mathrm{one}}B{}^q={}_{m\times p}{}^{\mathrm{one}}\left(\begin{array}{ccc}{a}_{\mathrm{eleven}}& B\cdots a_{\mathrm{one}n}& B\\ ⋮& & ⋮\\ a_{m\mathrm{one}}& B\cdots a_{mn}& B\end{array}\right){}^{n\times q}$$

,

$$AAA\left(\overrightarrow{ƒ}\right)=dg\left[B\left[{\displaystyle \sum _{j=0}^W\left\{{\left[A\times diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^{T}B\right]}^{W-j}{E}_d{E}_d^T{\left[A\times diag\left(\overrightarrow{ƒ}\right)diag{\left(\overrightarrow{ƒ}\right)}^{T}B\right]}^j\right\}}\right]A\right]\overrightarrow{ƒ}$$

dg () is the operator of forming a diagonal matrix with elements equal to the diagonal elements of the matrix ().

A = -I _ref ;

B = I _in .

; $$\begin{array}{l}{k}_{\mathrm{one}}=\left[I_V\otimes \left\{W{\left(\overrightarrow{ƒ}\right)}^{W-j}\times E_V{E}_V^T\right\}\right]\times \left\{{\displaystyle \sum _{i=\mathrm{one}}^j{\left(W{\left(\overrightarrow{ƒ}\right)}^T\right)}^{j-i}\otimes W{\left(\overrightarrow{ƒ}\right)}^{i-\mathrm{one}}}\right\}\times \left(B^T\otimes A\right)\times \\ \times \left\{2I_D\otimes diag\left(\overrightarrow{ƒ}\right)\right\}\times diag\left(vec{I}_D\right)\times \left(I_D\otimes E_D\right)\end{array}$$

;

; $$\begin{array}{l}{k}_2=\left\{{\left[E_D{E}_D^T\times W{\left(\overrightarrow{ƒ}\right)}^j\right]}^T\otimes I_V\right\}\times \left\{{\displaystyle \sum _{i=\mathrm{one}}^{W-j}{\left(W{\left(\overrightarrow{ƒ}\right)}^T\right)}^{W-j-i}\otimes W{\left(\overrightarrow{ƒ}\right)}^{i-\mathrm{one}}}\right\}\times \\ \times \left(B^T\otimes A\right)\left\{2I_D\otimes diag\left(\overrightarrow{ƒ}\right)\right\}\times diag\left(vec{I}_D\right)\times \left(I_D\otimes E_D\right)\end{array}$$

;

I _L is the identity matrix of size L × L;

E _L is a unit vector of size L × 1;

; $$W\left(\overrightarrow{ƒ}\right)=A\times diag\left(\overrightarrow{ƒ}\right)\otimes diag{\left(\overrightarrow{ƒ}\right)}^T\times B$$

;

- диагональная матрица с вектором $$\overrightarrow{ƒ}$$

на главной диагонали; $$diag\left(\overrightarrow{ƒ}\right)$$

- diagonal matrix with vector $$\overrightarrow{ƒ}$$

on the main diagonal;

vecI _D is the vectorization operator of the matrix I _D.

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[\begin{array}{c}{I}_{исх}\\ \dots \dots \\ I_{вх}\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial \lambda p}=2\times AAA\times \overrightarrow{ƒ}$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[\begin{array}{c}{I}_{\mathrm{and}\mathrm{from}x}\\ ......\\ I_{\mathrm{at}x}\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial \lambda p}=2\times AAA\times \overrightarrow{ƒ}$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{ƒ}\partial {\overrightarrow{\lambda ƒc}}^T}=\left[\begin{array}{ccc}2& {ƒ}_{\mathrm{one}}-\mathrm{one}& 0\\ 0& 2{ƒ}_D& -\mathrm{one}\end{array}\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{x}}^T}2\left[\begin{array}{ccc}\ddots & & 0\\ & \lambda x_n& \\ 0& & \ddots \end{array}\right]$$

, n∈{V_sw}; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}{\overrightarrow{\partial ƒ}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{x}}^T}2\left[\begin{array}{ccc}\ddots & & 0\\ & \lambda x_n& \\ 0& & \ddots \end{array}\right]$$

, n∈ {V _sw };

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda x}}^T}=\left[\begin{array}{ccc}\ddots & & 0\\ & 2x_n-1& \\ 0& & \ddots \end{array}\right]$$

, n∈{V_sw}; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[\begin{array}{cccc}0...& -{\mathrm{one}}_{n,n}& ...0& ...0\\ ......& ......& ......& ......\\ 0...& 0...& {\mathrm{one}}_{n+V,n+V}& ...0\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda x}}^T}=\left[\begin{array}{ccc}\ddots & & 0\\ & 2x_n-\mathrm{one}& \\ 0& & \ddots \end{array}\right]$$

, n∈ {V _sw };

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda ƒc}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial \lambda p}=\left[\overrightarrow{0}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{x}\partial {\overrightarrow{\lambda ƒc}}^T}=\left[0\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{x}}^T}=\left[\begin{array}{cccc}0\dots & -1_{n,n}& \dots 0& \dots 0\\ \dots \dots & \dots \dots & \dots \dots & \dots \dots \\ 0\dots & 0\dots & 1_{n+V,n+V}& \dots 0\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{ƒ}}^T}=\left[\begin{array}{c}{I}_{\mathrm{and}\mathrm{from}x}\\ ......\\ I_{\mathrm{at}x}\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{x}}^T}=\left[\begin{array}{cccc}0...& -{\mathrm{one}}_{n,n}& ...0& ...0\\ ......& ......& ......& ......\\ 0...& 0...& {\mathrm{one}}_{n+V,n+V}& ...0\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[0\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial \lambda p}=\left[\overrightarrow{0}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒc}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial \lambda p}=\left[\overrightarrow{0}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒc}}^T}=\left[0\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{x}}^T}=\left[\begin{array}{ccc}\ddots & & 0\\ & 2x_n-1& \\ 0& & \ddots \end{array}\right]$$

; n∈{V_sw}; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{ƒ}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{x}}^T}=\left[\begin{array}{ccc}\ddots & & 0\\ & 2x_n-\mathrm{one}& \\ 0& & \ddots \end{array}\right]$$

; n∈ {V _sw }; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[0\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}{\partial }_{\lambda p}}=\left[\overrightarrow{0}\right]$$

; $$\frac{{\partial }^{{}_2}F}{\partial \overrightarrow{\lambda x}\partial \overrightarrow{\lambda ƒc^T}}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}\partial {\overrightarrow{\lambda x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda x}{\partial }_{\lambda p}}=\left[\overrightarrow{0}\right]$$

; $$\frac{{\partial }^{{}_2}F}{\partial \overrightarrow{\lambda x}\partial \overrightarrow{\lambda ƒc^T}}=\left[0\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{x}}^T}=\left[{\overrightarrow{0}}^T\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda ƒ}}^T}=\left[{\overrightarrow{0}}^T\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial \overrightarrow{\lambda x}}=\left[{\overrightarrow{0}}^T\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{ƒ}}^T}2{\overrightarrow{ƒ}}^T{\left[AAA\left(\overrightarrow{ƒ}\right)\right]}^T$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{x}}^T}=\left[{\overrightarrow{0}}^T\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda ƒ}}^T}=\left[{\overrightarrow{0}}^T\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial \overrightarrow{\lambda x}}=\left[{\overrightarrow{0}}^T\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda ƒc}}^T}=\left[{\overrightarrow{0}}^T\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial \lambda p}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \lambda p\partial {\overrightarrow{\lambda ƒc}}^T}=\left[{\overrightarrow{0}}^T\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒc}\partial {\overrightarrow{ƒ}}^T}=\left[\begin{array}{ccc}2{ƒ}_{\mathrm{one}}& -\mathrm{one}& 0\\ 0& 2{ƒ}_D& -\mathrm{one}\end{array}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{x}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda ƒ}}^T}=\left[0\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial {\overrightarrow{\lambda x}}^T}=\left[0\right]$$

;

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial \overrightarrow{\lambda ƒc}}=\left[0\right]$$

. $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial \lambda p}=\left[\overrightarrow{0}\right]$$

; $$\frac{{\partial }^{2}F}{\partial \overrightarrow{\lambda ƒ}\partial \overrightarrow{\lambda ƒc}}=\left[0\right]$$

.

U = ∞

Check condition: | NewLagr _i1 -U |>Tosp; if not, then proceed to the formation of a sequence of traversal of the vertices of the graph.

i1 = i1 + 1

$$\Delta \overrightarrow{X}=J{\left(\overrightarrow{X}\right)}^{-\mathrm{one}}\times \left(-\frac{\partial F}{\partial \overrightarrow{X}}\right)$$

$$\overrightarrow{X}=\overrightarrow{X}+\Delta \overrightarrow{X}$$

$$NewLag{r}_{i\mathrm{one}}={\left(\frac{\partial F}{\partial \overrightarrow{X}}\right)}^T\times \left(\frac{\partial F}{\partial \overrightarrow{X}}\right)$$

return to check conditions | NewLagr _i1 - U |> Tosp.

9. Form a sequence of traversal of the vertices of the graph

. $${\overrightarrow{X}}^{*T}=\left({\overrightarrow{ƒ}}^{*T},{\overrightarrow{x}}^{*T},{\overrightarrow{\lambda ƒ}}^{*T},{\overrightarrow{\lambda x}}^{*T},\lambda p^{*},{\overrightarrow{\lambda ƒc}}^{*T}\right)$$

.

10. In accordance with the Gray coding rule, code combinations corresponding to the points of the signal constellation are determined.

In the source [Dayan Adionel Guimaraes. Digital Transmission: A Simulation-Aided Introduction with VisSim / Comm. Springer-Verlag Berlin Heidelberg, 2009. - P.397] an estimate of the boundary of the probability of error in the case of using a manipulation code and in the alternative case is proposed:

, $$\frac{P_e}{{\log }_{2}M}\le P_b\le \frac{M\cdot P_e}{2M-2}$$

,

where P _b is the probability of erroneous reception of a bit (error in one binary bit of the code combination),

P _e is the probability of erroneous reception of a symbol,

M - the number of points in the signal constellation (positioning system

modulation).

We define the gain in noise immunity as the ratio of the probability of erroneous reception of a bit using the worst method of manipulation coding to the probability of erroneous reception of a bit using the proposed method of manipulation encoding

$$\frac{\frac{M\cdot P_e}{2M-2}}{\frac{P_e}{{\log }_{2}M}}=\frac{M\cdot {\log }_{2}M}{2M-2}$$

So, for example, when using the described method of manipulation coding for M = 256, the probability of erroneous reception of a bit will decrease by about 4 times.

Thus, with such a combination of essential features during manipulation coding, an increase in the noise immunity of the transmission of discrete information is achieved by obtaining the possibility of manipulation coding for signal constellations with an arbitrary number of points and an arbitrary arrangement of signal points on the signal diagram.

Claims (1)

The method of manipulation coding, which consists in setting the number of points of the signal constellation and arranging them for gray coding, characterized in that setting the number of points in the signal constellation equal to any natural number, determines the Euclidean distance between the points of the signal constellation, forms a vector of Euclidean distances between points of the signal constellation, specify the incidence matrix of the graph, the source and drain vertices, the set of vertices required for passage, determine the initial destination vector using the Hungarian method of solving the assignment problem, calculate the gradient vector for the initial vector, find the minimum in the direction of the gradient vector by the one-dimensional search method, recalculate the gradient vector at the minimum point, determine the minimum in the new direction by the one-dimensional search method, get an improved vector approximation destinations, using the Newton-Raphson method, find the optimal solution to the routing problem in Euclidean space, form a sequence of traversal of graph vertices, in In accordance with the Gray coding rule, code combinations corresponding to the points of the signal constellation are determined.