CN103986498A - Pseudo-random code optimization method based on graph theory - Google Patents

Abstract

The invention relates to a pseudo-random code optimization method based on a graph theory, and belongs to the technical field of code design and optimization in digital communications. According to the method, pseudo-random codes are mapped as nodes in the graph theory, the cross-correlation measure is mapped as the length of the edge of the graph, a cross-correlation measure matrix is converted to be an adjacent matrix in the graph theory through the threshold value judgment, the cross-correlation optimization problem of the pseudo-random codes is converted to be the maximum clique problem in the graph theory, and the codes with the good cross-correlation performance are optimized through the maximum clique search. Compared with a conventional traversal search optimization method, the calculation complexity is decreased from the index complexity to the linear complexity, and the cross-correlation performance of the pseudo-random codes can be rapidly optimized.

Description

A kind of pseudo noise code optimization method based on graph theory

Technical field

The present invention relates to a kind of pseudo noise code optimization method based on graph theory, belong to code Design and optimisation technique field in digital communication.

Background technology

Pseudo noise code is to realize the necessary condition of spread spectrum communication, and the pseudo noise code that 01 balance is good, self correlated peak sharp-pointed, cross-correlation performance is good all requiring in as satellite navigation system, mobile communication system in many important spread spectrum communications and measuring system is as spreading code.The design of pseudo noise code is divided into two stages: construction phase and optimizing phase, construction phase refers to that employing is someway as the process of the structure such as linear shift register, number theory method candidate codewords set, candidate codewords set is also referred to as code pond (CodePool), and the optimizing phase refers to from candidate codewords set according to the preferred part code word of certain performance criteria as practical code word.Optimizing phase comprises two heavy elements: performance measure and optimized algorithm, and performance measure has defined the index request that " good code " should be satisfied, and optimized algorithm is the method for finding " good code " from candidate codewords set.For different concrete application, the standard of " good code " is not identical, and same optimized algorithm is also varied, uses different performance measure and optimized algorithm can produce different design results in identical candidate collection.

Although the standard of " good code " is not unique, relatively consistent for the requirement of code correlation.Desirable pseudo noise code comprises the requirement of correlation: be that 0 auto-correlation function disturbs anti-multipath beyond main lobe, be that 0 cross-correlation function antagonism multiple access disturbs everywhere everywhere.In the design of modern pseudo noise code, except traditional periodic correlation function, also need the impact of considering that data bit is overturn on correlation, therefore need to consider strange correlation function, two kinds of different situations of even correlation function.Be the pseudorandom code word a={a of L bit for length ₀, a ₁... a _l-1, b={b ₀, b ₁... b _l-1, wherein a _i, b _i∈ 1, and+1}, 4 kinds of correlation function definition are as shown in table 1.

Table 1 Pseudo-code Correlation function definition

Taking European Galileo satellite navigation system and GPS of America satellite navigation system as example, the normalization correlated performance of typical navigation signal is as shown in table 2.

The typical satellite navigation signals pseudo noise code of table 2 normalization correlation (dB)

In the optimizing process of pseudo noise code, the computation complexity that different performance is estimated is widely different, can be divided into two classes according to the relation of performance measure and code word: independent performance measure and composite behaviour are estimated, the calculating of independent performance measure only needs an independent code word as input, and composite behaviour is estimated needs 2 or more code word as input.Auto-correlation function belongs to independent performance measure, and belonging to composite behaviour, estimates cross-correlation function, pseudo noise code preferably in, the optimization of independent performance measure can realize with threshold value judgement, complexity is linear complexity, but more complicated of the optimization that composite behaviour is estimated, exceeds linear complexity conventionally, the optimization that therefore composite behaviour is estimated is the difficult point that pseudo noise code is optimized.For the code set C={c that comprises N pseudorandom code word _i, i=1...N, adopts the performance measure of maximum strange, even cross-correlation peak value as whole set, conventionally

R _xy(c _i,c _j)＝max(|EvenCCF(c _i,c _j,τ)|,|OddCCF(c _i,c _j,τ)|)

Target function is

CCF (C)=max (R _xy(c _i, c _j)), i, j=1...N, and i ≠ j

The target of optimizing is that CCF (C) reaches minimum value, and CCF (C) is all relevant to N code word.Be the cross-correlation optimization of M, the target code word number of sets pseudo noise code that is N for candidate codewords number of sets, obtain best cross-correlation optimization and need to travel through the combination of whole N code word, total plant combination, according to the Stirling formula in Combinational Mathematics, the computation complexity of cross-correlation performance optimum is exponential complexity, almost cannot realize for the problem of medium-scale (more than 100 element).

Graph theory (GraphTheory) is the mathematical theory of description node and correlation thereof, in a simple graph G=(V, E), figure G is made up of node set V and limit set E, V has comprised the node in figure, and E represents internodal annexation and distance.The theory of graph theory can be for describing the cross correlation of pseudo noise code, for a pseudo noise code set C who comprises N code word, code word c _i, i=1...N represents with the node in figure, cross-correlation performance is estimated R _xy(c _i, c _j) can represent the length (distance) on the limit between node, pseudo noise code set that comprises 4 code words with illustration and cross-correlation performance thereof are as shown in Figure 1.For setting up the graph theory model of cross-correlation function, cross-correlation performance is estimated by N × N matrix notation be

R=[r _ij] _{n × N}, wherein $r_{\mathrm{ij}}=\left\{\begin{array}{cc}{R}_{\mathrm{xy}}(c_i,c_j),& i\≠j\\ 0,& i=j\end{array}\right.$

When using a thresholding Th to carry out accessibility judgement to R matrix, R matrix becomes R' matrix

R'=[rr _ij] _{n × N}, wherein ${\mathrm{rr}}_{\mathrm{ij}}=\left[\begin{array}{c}1,r_{\mathrm{ij}}\≤\mathrm{Th}\\ 0,r_{\mathrm{ij}}>\mathrm{Th}\end{array}\right]$

Its physical significance is after exceeding threshold value Th, to think that when i, j spacing between two nodes the two is unreachable.In Graph Theory, matrix R' is exactly adjacency matrix (AdjacencyMatrix) A=[a of figure _ij] _{n × N}, work as a _ij=1 represents that between node, existence connects (limit), otherwise represents not connect, and after adopting thresholding Th=7 to adjudicate as shown in Figure 2, corresponding matrix R, R', A are Fig. 1:

$R=\left[\begin{array}{c}0467\\ 4085\\ 6809\\ 7590\end{array}\right],R^{\′}=\left[\begin{array}{c}1111\\ 1101\\ 1010\\ 1101\end{array}\right],A=\left[\begin{array}{c}1111\\ 1101\\ 1010\\ 1101\end{array}\right]$

Can see between egress 1,2,4 and connect for complete, be to have limit between any two nodes, in the correlation matrix of pseudo noise code, its implication is that the maximum of the co-related measure of code word 1,2,4 is less than thresholding Th=7, has 1,2,4 node groups of composition (Clique) of full annexation in Graph Theory.Therefore, the cross-correlation optimization problem of pseudo noise code is equivalent to Clique problem (the Maximum Clique Problem in graph theory, MCP), MCP problem is a classical np problem in graph theory, the verified polynomial complexity solution that do not exist, this has proved that pseudo noise code optimization is NP complexity issue from another point of view.Although there is not the algorithm of multinomial complexity for MCP problem, but have a lot of sub-optimal algorithm, comprise and determine algorithm and heuritic approach, different implementation methods and software kit exceed 20 kinds, solving of MCP is a kind of mathematical tool that the present invention realizes, and its principle and realization are considered to known technology.Solving of MCP will obtain nodes and the node set of Clique, and the two,, corresponding to pseudo noise code number and the codeword set optimized under thresholding Th, has optimized in the time that number of codewords reaches number required N, the principle that the pseudo noise code that Here it is based on graph theory is optimized.

Summary of the invention

The object of the invention is to optimize the poor problem of preferred properties in nonpolynomial complexity, finite time in order to solve pseudo noise code, propose a kind of pseudo noise code optimization method based on graph theory, new method is linear complexity, can realize the rapid Optimum of pseudo noise code.

The present invention is achieved through the following technical solutions:

A pseudo noise code optimization method based on graph theory, in candidate codewords set, number of codewords is M, the number of codewords that requires preferred result is N, N < M, pseudo noise code length is L bit, comprises the steps:

Step 1, structure co-related measure matrix.

C in calculated candidate codeword set _i, c _jeven cross-correlation function EvenCCF (c _i, c _j) and strange cross-correlation function OddCCF (c _i, c _j), with the maximum R of strange, even cross-correlation function absolute value _xy(c _i, c _j) estimate as cross-correlation performance.The building method that cross-correlation performance is estimated matrix R is

$R(i,j)=\left\{\begin{array}{cc}0,& i=j\\ R_{\mathrm{xy}}(c_i,c_j),& i\&NotEqual;j\end{array}\right.$

Wherein, c _iand c _jrepresent different code words, R (i, j) represents that cross-correlation performance estimates that the i of matrix R is capable, j column element, i=1...M, j=1...M, R _xy(c _i, c _j) expression spread spectrum code word c _iand c _jcross-correlation performance estimate.

Step 2, initialization decision threshold.

Search cross-correlation performance is estimated the non-zero value MinR of minimum in matrix R, sets it as the initial value of decision threshold Th, i.e. Th=MinR.

Step 3, adjacency matrix and node set in structure graph theory model.

Adopt Graph Theory, M candidate codewords is mapped in graph theory model, a corresponding node of candidate codewords.Use decision threshold Th judgement cross-correlation performance to estimate matrix R, the adjacency matrix A in structure graph theory model, building method is

$A(i,j)=\left[\begin{array}{c}1,R(i,j)\&le;\mathrm{Th}\\ 0,R(i,j)>\mathrm{Th}\end{array}\right]$

I that wherein A (i, j) is adjacency matrix is capable, j column element.In the time of A (i, j)=1, represent code word c _iand c _jcorresponding graph theory model node is connected; In the time of A (i, j)=0, represent code word c _iand c _jbetween corresponding graph theory model node, do not connect.

With the node set V in the sequence number structure graph theory model of M pseudo noise code, element V (i)=i wherein, i=1...M, preserve the sequence number of M pseudo noise code.

Step 4, the Clique in search graph theory model.

The input of searching for as Clique using adjacency matrix A and the node set V of step 3, Clique search Output rusults is the node set V' of Clique nodes M' and Clique.

In the time of M'< N, represent preferred pseudo noise code lazy weight, decision threshold Th is added to 1, return to step 3; Finish if M' >=N represents to optimize, proceed to step 5.

Step 5, output optimum results.

According to the code word sequence number in the node set V' of Clique, concentrate the code word that extracts corresponding sequence number to export as optimum results from candidate codewords.

So far, the pseudo noise code optimizing process based on graph theory finishes.

Beneficial effect

A kind of pseudo noise code optimization method based on graph theory that the present invention proposes, pseudorandom code word is mapped as the node in graph theory, co-related measure is mapped as the length on the limit in figure, the judgement of co-related measure matrix passing threshold is converted into the adjacency matrix in graph theory, the cross-correlation optimization problem of pseudo noise code is converted to the Clique problem in graph theory, searches for the code word preferably with good cross-correlation performance by Clique.The optimization method that contrasts conventional traversal search, computation complexity drops to linear complexity from exponential complexity, can realize the rapid Optimum of pseudorandom code cross-correlation performance.

Brief description of the drawings

Fig. 1 is the cross-correlation performance diagram of 4 pseudo noise codes in background technology;

Fig. 2 is the diagram that adopts the pseudo noise code after threshold judgement in background technology;

Fig. 3 is that the pseudo noise code based on graph theory of the present invention is optimized realization flow;

Fig. 4 is judging threshold and Clique interstitial content relation in embodiment.

Embodiment

Below in conjunction with drawings and Examples, technical scheme of the present invention is described further objects and advantages of the present invention in order better to illustrate.In this embodiment, will use american global positioning system gps system L5 navigation signal to describe for example.

Gps system L5 signal is the navigation signal towards life security service of a new generation, L5 specification of signals is defined in 2011 to be announced in IS-GPS-705B file, pseudo noise code length L=10230 bit, be limited to the initial state that generates register, 8191 kinds of combinations in candidate codewords set, after deleting choosing based on balance, autocorrelation and spectral characteristic, remain 107 of candidate's code characters, be M=107, consider that 37 satellites of GPS basic configuration constellation need 37 × 2=74 code word altogether, be N=74, cross-correlation optimization will make 74 pseudo noise codes have good correlation.

In order to describe the needs of algorithm, further dwindle the scale of problem, select M=6, N=3, as initial conditions, has 6 code characters, i.e. C={c in the set of candidate's code character ₁, c ₂..., c ₆, retain 3 pseudo noise codes as optimum results, adopt traversal search to need altogether inferior search, the demonstration program that does not affect this example that problem scale dwindles.

A pseudo noise code optimization method based on graph theory, concrete steps are:

Step 1, structure co-related measure matrix.

The cross-correlation function of 6 code characters in calculated candidate codeword set, using the maximum of strange, even cross-correlation function absolute value as performance measure, the cross-correlation performance of 6 code characters is estimated 6 × 6 cross-correlation performances of composition and is estimated matrix R.

$R=\left[\begin{array}{cccccc}0& 414& 436& 468& 406& 418\\ 414& 0& 420& 434& 394& 494\\ 436& 420& 0& 404& 440& 386\\ 468& 434& 404& 0& 392& 402\\ 406& 394& 440& 392& 0& 430\\ 418& 494& 386& 402& 430& 0\end{array}\right]$

Step 2, initialization decision threshold.

Traversal search cross-correlation performance is estimated the non-zero value MinR=386 of minimum in matrix R, sets it as the initial value of decision threshold, i.e. Th=386.

Step 3, adjacency matrix and node set in structure graph theory model.

Use decision threshold Th judgement cross-correlation performance to estimate matrix R, the adjacency matrix A in structure graph theory model.

$A=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 1\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 1\end{array}\right]$

Node set V in structure graph theory model, V=[1 2345 6].

Step 4, the Clique in search graph theory model.

Using adjacency matrix A and node set V as input, the Clique in search graph, Output rusults is the node set V'=[36 of Clique nodes M'=2 and Clique].

When M'=2 < N=3 represents preferred pseudo noise code lazy weight, decision threshold is added to 1, Th=387, return to step 3;

Decision threshold and Clique nodes are as shown in table 3.

Table 3 decision threshold and Clique nodes

Decision threshold	Clique nodes
		386	2
387	2
		…	…
402	2
		403	2
404	3

Table 3 shows, along with the increase of decision threshold, Clique nodes increases, and in the time of Th=404, Clique comprises 3 nodes, V'=[346], stop search.

Step 5, output optimum results.

According to the node set V'=[346 of Clique] in code word sequence number, from candidate codewords concentrate extract the 3rd, 4,6 code words export as optimum results.

So far, the pseudo noise code optimizing process based on graph theory finishes.

For the advantage of the pseudo noise code optimization method of concrete analysis based on graph theory, with the generation method construct candidate code character set of GPSL5 signal pseudo noise code, choose the pseudo noise code of quantity and GPSL5 signal ICD file equal number, be N=74, carry out performance of PN code and estimate comparison, result is as shown in table 4, the relation of new method decision threshold and Clique nodes as shown in Figure 4, along with the increase Clique interstitial content of decision threshold progressively increases.74 selected code words of new method are better than the design of GPS of America L5 on cross-correlation performance is estimated, optimizing process running time on PC is 2 seconds, PC is configured to: IntelCore2Q6600 processor 2.4GHz, 4GB internal memory, Matlab software platform, realizes pseudo noise code optimizing process and has good time complexity.

The comparison of table 4 performance of PN code

The inventive method is not limited to the optimization of specific satellite navigation system pseudo noise code, but is applicable to use arbitrarily radionavigation, the communication system of binary sequence as pseudo noise code.

Although described by reference to the accompanying drawings embodiments of the present invention, to those skilled in the art, under the premise without departing from the principles of the invention, can also make some distortion and improvement, these also should be considered as belonging to protection scope of the present invention.

Claims (1)

1. the pseudo noise code optimization method based on graph theory, is characterized in that: in candidate codewords set, number of codewords is M, the number of codewords that requires preferred result is N, and N < M, specifically comprises the steps:

Step 1, structure co-related measure matrix;

C in calculated candidate codeword set _i, c _jeven cross-correlation function EvenCCF (c _i, c _j) and strange cross-correlation function OddCCF (c _i, c _j), with the maximum R of strange, even cross-correlation function absolute value _xy(c _i, c _j) estimate as cross-correlation performance; The building method that cross-correlation performance is estimated matrix R is

$R(i,j)=\left\{\begin{array}{cc}0,& i=j\\ R_{\mathrm{xy}}(c_i,c_j),& i\&NotEqual;j\end{array}\right.$

Wherein, c _iand c _jrepresent different code words, R (i, j) represents that cross-correlation performance estimates that the i of matrix R is capable, j column element, i=1...M, j=1...M, R _xy(c _i, c _j) expression spread spectrum code word c _iand c _jcross-correlation performance estimate;

Step 2, initialization decision threshold;

Search cross-correlation performance is estimated the non-zero value MinR of minimum in matrix R, sets it as the initial value of decision threshold Th, i.e. Th=MinR;

Step 3, adjacency matrix and node set in structure graph theory model;

Adopt Graph Theory, M candidate codewords is mapped in graph theory model, a corresponding node of candidate codewords; Use decision threshold Th judgement cross-correlation performance to estimate matrix R, the adjacency matrix A in structure graph theory model, building method is

$A(i,j)=\left[\begin{array}{c}1,R(i,j)\&le;\mathrm{Th}\\ 0,R(i,j)>\mathrm{Th}\end{array}\right]$

I that wherein A (i, j) is adjacency matrix is capable, j column element; In the time of A (i, j)=1, represent code word c _iand c _jcorresponding graph theory model node is connected; In the time of A (i, j)=0, represent code word c _iand c _jbetween corresponding graph theory model node, do not connect;

With the node set V in the sequence number structure graph theory model of M pseudo noise code, element V (i)=i wherein, i=1...M, preserve the sequence number of M pseudo noise code;

Step 4, the Clique in search graph theory model;

The input of searching for as Clique using adjacency matrix A and the node set V of step 3, Clique search Output rusults is the node set V' of Clique nodes M' and Clique;

In the time of M'< N, represent preferred pseudo noise code lazy weight, decision threshold Th is added to 1, return to step 3; Finish if M' >=N represents to optimize, proceed to step 5;

Step 5, output optimum results;

According to the code word sequence number in the node set V' of Clique, concentrate the code word that extracts corresponding sequence number to export as optimum results from candidate codewords;

So far, the pseudo noise code optimizing process based on graph theory finishes.