CN111884662A - Method and system for identifying polarization code parameters under error code condition - Google Patents

Method and system for identifying polarization code parameters under error code condition Download PDF

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CN111884662A
CN111884662A CN202010896852.9A CN202010896852A CN111884662A CN 111884662 A CN111884662 A CN 111884662A CN 202010896852 A CN202010896852 A CN 202010896852A CN 111884662 A CN111884662 A CN 111884662A
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吴昭军
张立民
钟兆根
但波
金堃
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Naval Aeronautical University
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    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
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Abstract

The invention relates to a method and a system for identifying parameters of a polarization code under an error code condition. Constructing a polar code word matrix by using the intercepted bit sequence and the set traversed code length range; determining a kronecker product power matrix according to the polarization code codeword matrix; sequentially rejecting line data in the kronecker product power matrix, and determining a corresponding dual matrix by using the kronecker product power matrix after the line data are rejected each time; judging whether the currently rejected row data is an information bit position or a frozen bit position according to the dual matrix and the polarization code codeword matrix; counting the currently traversed code length and all the currently determined information bit positions to determine the code rate corresponding to the currently traversed code length; identifying the parameters of the polarization code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the invention realizes the correct identification of the parameters of the polarization code.

Description

Method and system for identifying polarization code parameters under error code condition
Technical Field
The invention relates to the field of channel coding identification, in particular to a method and a system for identifying parameters of a polarization code under an error code condition.
Background
To combat the interference of channel noise, channel coding techniques are widely used in modern digital communication systems, and with the rapid development of channel coding theory, a series of coding schemes with performance approaching to shannon limit, such as Turbo code, LDPC code, and polar code, are found, and these coding schemes are widely used in 5G mobile communication or satellite communication. For non-cooperative communication parties, blind identification of channel coding is always a research hotspot, at present, documents for coding identification mainly focus on algebraic structure coding, convolutional codes, Turbo codes and LDPC codes, and documents for polar code identification alone are not reported yet, so that the problem of polar code parameter identification under a high error rate condition is a problem to be solved urgently for the field of coding identification.
Disclosure of Invention
The invention aims to provide a method and a system for identifying parameters of a polarization code under an error code condition, so as to realize correct identification of the parameters of the polarization code.
In order to achieve the purpose, the invention provides the following scheme:
a method for identifying polarization code parameters under error code conditions comprises the following steps:
constructing a polarization code codeword matrix by using the intercepted bit sequence and the set traversed code length range;
determining a kronecker product power matrix according to the polarization code codeword matrix;
sequentially rejecting line data in the kronecker product power matrix, and determining a corresponding dual matrix by using the kronecker product power matrix after the line data are rejected each time;
judging whether the currently rejected row data is an information bit position or a frozen bit position according to the dual matrix and the polarization code codeword matrix;
counting the currently traversed code length and all the currently determined information bit positions to determine the code rate corresponding to the currently traversed code length until the code rates corresponding to all the traversed code lengths in the set traversed code length range are determined;
identifying the parameters of the polarization code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the parameters of the polarization code include a code length, an information bit position, and a frozen bit position.
Optionally, the sequentially rejecting line data in the kronecker power matrix, and determining a corresponding dual matrix by using the kronecker power matrix after the line data rejected each time specifically include:
using formulas
Figure BDA0002658716940000021
Determining a corresponding dual matrix of the Croncke product power matrix after the row data eliminated each time; wherein HiDetermining a corresponding dual matrix for the Croncke product power matrix after the ith eliminated row of data,
Figure BDA0002658716940000022
is HiIs transposed, SiFor a column transformation matrix, P(n-1)×1The conditions are satisfied as follows: ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1]Wherein R isiFor a row transformation matrix, QiTo remove the Kroecker power matrix after the ith row, I(n-1)×(n-1)Is (n-1) × (n-1) identity matrix.
Optionally, the determining, according to the dual matrix and the polar code codeword matrix, whether the currently rejected row of data is an information bit position or a frozen bit position specifically includes:
determining statistics and a decision threshold according to the dual matrix and the polarization code codeword matrix;
judging whether the statistic is smaller than the judgment threshold;
if the statistic is smaller than the judgment threshold, determining that the currently-rejected line data is an information bit position, and judging the next-rejected line data;
and if the statistic is not smaller than the judgment threshold, determining that the currently removed row data is a frozen bit position, and judging the next removed row data.
Optionally, the counting the currently traversed code length and the currently determined information bit positions to determine the code rate corresponding to the currently traversed code length until determining the code rates corresponding to all traversed code lengths within the set traversed code length range includes:
and determining the code rate corresponding to the currently traversed code length according to the ratio of all the currently determined information bit positions to the currently traversed code length.
A system for identifying polarization code parameters under error conditions, comprising:
the polar code codeword matrix constructing module is used for constructing a polar code codeword matrix by utilizing the intercepted bit sequence and the set traversed code length range;
the kronecker product power matrix determining module is used for determining a kronecker product power matrix according to the polarization code codeword matrix;
the dual matrix determining module is used for sequentially rejecting line data in the kronecker product power matrix and determining a corresponding dual matrix by using the kronecker product power matrix after the line data are rejected each time;
the bit position determining module is used for judging whether the currently rejected row data is an information bit position or a frozen bit position according to the dual matrix and the polar code codeword matrix;
a code rate determining module, configured to count a currently traversed code length and all currently determined information bit positions to determine a code rate corresponding to the currently traversed code length until determining code rates corresponding to all traversed code lengths within the set traversed code length range;
the parameter identification module of the polarization code is used for carrying out parameter identification on the polarization code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the parameters of the polarization code include a code length, an information bit position, and a frozen bit position.
Optionally, the dual matrix determining module specifically includes:
a dual matrix determination unit for using a formula
Figure BDA0002658716940000031
Determining a corresponding dual matrix of the Croncke product power matrix after the row data eliminated each time; wherein HiDetermining a corresponding dual matrix for the Croncke product power matrix after the ith eliminated row of data,
Figure BDA0002658716940000032
is HiIs transposed, SiFor a column transformation matrix, P(n-1)X 1 satisfies the condition: ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1]Wherein R isiFor a row transformation matrix, QiTo remove the Kroecker power matrix after the ith row, I(n-1)×(n-1)Is (n-1) × (n-1) identity matrix.
Optionally, the bit position determining module specifically includes:
a statistic and decision threshold determining unit, configured to determine a statistic and a decision threshold according to the dual matrix and the polar code codeword matrix;
the first judging unit is used for judging whether the statistic is smaller than the judgment threshold or not;
an information bit position determining unit, configured to determine that the currently rejected row data is an information bit position if the statistic is smaller than the decision threshold, and perform a judgment on a next rejected row data;
and the frozen bit position determining unit is used for determining that the currently removed line data is the frozen bit position and judging the next removed line data if the statistic is not smaller than the judgment threshold.
Optionally, the code rate determining module specifically includes:
and the code rate determining unit is used for determining the code rate corresponding to the currently traversed code length according to the ratio of all the currently determined information bit positions to the currently traversed code length.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
according to the parameter identification method and system of the polarization code under the error code condition, the code word matrix and the kronecker product matrix are constructed by traversing possible code length values, then the suspected information bit position is traversed, the vector row corresponding to the kronecker product matrix is removed, and the check vector is solved; secondly, solving a judgment threshold of the information bit position based on a minimum error judgment criterion so as to estimate the code rate and the information bit position under the traversal code length; and the identified code length is the minimum code length corresponding to the minimum code rate, and the position of the information bit under the code length is identified at the same time, so that the identification of the parameters is finally completed. Under the condition of error codes, the method can finish correct identification of polarization code parameters, has stronger fault tolerance and has the error rate of 10-3In order of magnitude, under the condition that the code length is 1024, the parameter identification rate can reach more than 95%, and the calculation complexity of the method only has a linear relation with the maximum code length of traversal and the intercepted data volume.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
Fig. 1 is a schematic flow chart of a method for identifying polarization code parameters under error code conditions according to the present invention;
FIG. 2 is a schematic diagram illustrating a principle of a method for identifying polarization code parameters under error conditions according to the present invention;
FIG. 3 is a schematic diagram of a system for identifying polarization code parameters under error conditions according to the present invention;
FIG. 4 is a diagram illustrating a result of identifying a polarization code parameter in a noise-free environment;
FIG. 5 is a diagram illustrating a polar code parameter identification result in a noisy environment;
FIG. 6 is a diagram illustrating the effect of code length on algorithm performance;
FIG. 7 is a diagram illustrating the effect of different numbers of intercepted codewords on the algorithm;
fig. 8 is a schematic diagram of the influence of the polar code rate on the algorithm performance.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a method and a system for identifying parameters of a polarization code under an error code condition, so as to realize correct identification of the parameters of the polarization code.
The basic principle of the polarization code is as follows:
repeatedly using any binary symmetric memoryless channel W for n times to obtain n mutually independent channels W, and then converting the n mutually independent channels into a group of n mutually associated channels
Figure BDA0002658716940000051
Defining channels therein
Figure BDA0002658716940000052
X→Yn×Xi-1Wherein X is a channel
Figure BDA0002658716940000053
Input symbol set of YnFor channel n as the output symbol set, Xi-1Is composed ofi-1-dimensional input symbol set, "×" is cartesian product. When n is large enough, the channel
Figure BDA0002658716940000054
A part of the capacity of (a) approaches 1 while the remaining part approaches 0. Transmitting information bits in a channel with a channel capacity of 1, and in a channel with a channel capacity of 0The frozen bits are transmitted in the channel, thereby achieving reliable transmission of information. For the polarization code, the coding process is divided into two parts of channel combination and channel splitting, wherein the channel combination is to combine n independent channels W into WnAnd channel splitting is to split WnDecomposed into n channels with certain relevance
Figure BDA0002658716940000055
The polarization code principle is briefly explained below in terms of channel merging and channel splitting.
The channel combination is to combine n independent channels W to generate Wn:Xn→Yn. Wherein n is 2m(m is a positive integer). GnFor generating a polynomial matrix of a polarization code, u1,u2,…,unFor the sequence of input channels, x1,x2,…,xnFor input sequence through GnThe latter coding sequence, y1,y2,…,ynA sequence is output for the channel. Namely, it is
Figure BDA0002658716940000056
Figure BDA0002658716940000057
Then
Figure BDA0002658716940000058
And
Figure BDA0002658716940000059
satisfies the following conditions:
Figure BDA00026587169400000510
the channel transition probability can be expressed as:
Figure BDA0002658716940000061
for any code length n (n-2)mM is a positive integer) of a polar code, which generates a polynomial GnSatisfy the formula(3) The relationship, namely:
Figure BDA0002658716940000062
wherein,
Figure BDA0002658716940000063
m-times of the Crohn's product power of the matrix F
Figure BDA0002658716940000064
Cr×s＝(bi,j) Then, then
Figure BDA0002658716940000065
And Br×sThe kronecker product of (a) is defined as:
Figure BDA0002658716940000066
according to the definition of the formula (4),
Figure BDA0002658716940000067
satisfy the requirement of
Figure BDA0002658716940000068
And is
Figure BDA0002658716940000069
In the formula (3), BnIs a bit-reversal permutation matrix for
Figure BDA00026587169400000610
The row order of (2) permutes. When in use
Figure BDA00026587169400000611
When the row number of (1, 0,2, 3) is (1, 1), the binary number is (00,01,10,11), and the bit is inverted to obtain (00,10,01,11), which corresponds to the decimal number of (0,2,1,3), that is, the binary number is (00,01,10,11)
Figure BDA00026587169400000612
The order of the row permutation. Under normal conditions
Figure BDA00026587169400000613
Therefore, it is
Figure BDA00026587169400000614
Can be further expressed in a recursive form:
Figure BDA00026587169400000615
for n mutually independent channels to be combined into WnThen splitting the channel into n polarized channels with correlation
Figure BDA00026587169400000616
X→Yn×Xi-1I is more than or equal to 1 and less than or equal to n, and the transition probability of the ith polarized channel is defined as:
Figure BDA00026587169400000617
in the formula (6)
Figure BDA00026587169400000618
For the calculation of the channel transition probability in equation (6), the channel combining structure of the polarization code is generally utilized, and the calculation is performed in a recursive manner, that is:
Figure BDA00026587169400000619
Figure BDA00026587169400000620
in the formulae (7) and (8),
Figure BDA00026587169400000621
represents a sequence of pairs (u)1,u2,…,u2i-2) The estimation of the odd-numbered sequence in (c),
Figure BDA00026587169400000622
represents a sequence of pairs (u)1,u2,…,u2i-2) And (4) estimation of medium even sequences. When n is sufficiently large, a channel polarization phenomenon occurs, i.e., a part of the polarized channel capacity approaches 1, and the remaining polarized channel capacity approaches 0. For polarization channel reliability, it is mainly measured by the babbitt parameter, which is defined as:
Figure BDA0002658716940000071
wherein, W is X → Y, X ═ 0,1, and Y is the channel output symbol set.
From the definition of the babbitt parameter, when z (w) is ≈ 1, the channel reliability is worst; conversely, when z (w) is ≈ 0, the channel reliability is best. Binary memoryless channel parameter recursion calculation expression:
Figure BDA0002658716940000072
Figure BDA0002658716940000073
the equation (10) is true if and only if the channel is a binary erasure channel.
The polar code mainly utilizes a polar channel with high reliability to transmit information bits, and simultaneously transmits the residual polar channel with poor reliability to freeze bits. Frozen bit positions are usually taken to be 0. Combining the channel merging, channel splitting and babbit parameter calculation processes, the polarization code encoding process is as follows:
step 1: constructing a polarization code generation matrix Gn；
Step 2: computing split polarization channel transition probability
Figure BDA0002658716940000074
And step 3: computing a polarized channel
Figure BDA0002658716940000075
(ii) a barpanica parameter of (d);
and 4, step 4: sorting the Pasteur parameters from small to large, saving the labels corresponding to the first k parameters in a set, and then, for an information sequence usetAnd a freezing sequence
Figure BDA0002658716940000076
The code of (2) is:
Figure BDA0002658716940000077
wherein c is a polar code encoding codeword, Gn(set) denotes by GnThe middle row being a matrix formed by rows of elements in the set, setcIs a complement of set, symbol
Figure BDA0002658716940000078
Representing addition in the binary domain. In general, the transmitted symbol at the frozen bit position is 0, so equation (12) can be further written as:
c＝uset·Gn(set) (13)
for the identification of the polarization code, the parameters to be identified include the code length of the polarization code, the position of the information bit and the position of the frozen bit, and the position of the frozen bit is determined after the position of the information bit is identified because the position of the information bit and the position of the frozen bit are in a complementary relationship. For a communication system, each frame of data is preceded by a synchronization code sequence (typically 1 to 2 bytes in length), which can be used to easily implement the polarization code codeword synchronization, and under the condition of error code, the polarization code length, information bits and frozen bit parameters are identified.
The traversed code length corresponding to the minimum code rate is the code length of the polarization code, and the specific determination process is as follows:
as known from the coding structure of the polar code, the length n of the polar code satisfies n-2m(m is a positive integer) relationship, whereby interception can be utilizedAnd constructing polarization code words under different m conditions by the data, analyzing dual space characteristics of the polarization code words, and searching for a rule.
And dividing the polarization code with the actual code length of n and the code rate of k/n into a code word sequence with the code length of n/2, wherein the divided code words are equivalent to a new polarization code, and the code rate of the new polarization code is greater than k/n.
And (3) proving that: let n be 2mConstructing a m-degree Crohn's product power matrix Dn×nComprises the following steps:
Figure BDA0002658716940000081
wherein D is(n/2)×(n/2)Is a matrix of m-1 times the Crohn's product power, 0(n/2)×(n/2)Is an all-zero matrix of (n/2) × (n/2). After bit reversal permutation, extract Dn×nThe line number of the middle k line is not recorded as: π (1), π (2), …, π (k). Assuming that the set { π (1), π (2), …, π (i) } is a subset of the set {1,2, …, n/2}, and the set { π (i +1), π (i +2), …, π (k) } is a subset of the set { n/2+1, n/2+2, …, n }, when the polarization code encoding matrix G is presentn(set) can be written as:
Figure BDA0002658716940000082
wherein, the element dπ(i),jIs D(n/2)×(n/2)Medium matrix elements.
When the polarization code is divided into code words with the code length of n/2, a new code word coding matrix obtained by division becomes:
Figure BDA0002658716940000083
in the formula (16), the obvious repeated line after the double folding in the formula (15) is removed.
By observing formula (16), each row can be at D(n/2)×(n/2)Is present in (1), so G'n/2Can be used as a polarization code with the code length of n/2, so that the segmented code words are equivalent to a new polarization code. Matrix G 'is discussed further below'n/2Rank case.
Case 1: no repeating line in formula (16), i.e. G'n/2Every two rows in the row are not repeated. Due to G'n/2Each row in D(n/2)×(n/2)Is found in, and due to D(n/2)×(n/2)Is a full rank matrix, so G'n/2One is determined to be a row full rank matrix, so from G'n/2The code rate of the generated polarization code is 2k/n, and the code rate after the division is more than k/n.
Case 2: in equation (16), there are repeated matrix rows. Let the number of repeated rows be t and the set of repeated row elements be set 0. Due to D(n/2)×(n/2)Is a full rank matrix, simultaneously composed of Gn(set) Structure No., G'n/2No duplicate lines occur in lines 1 to ith, and no duplicate lines occur in lines i +1 to kth, so set0 is equal to G'n/2The intersection of the middle, front i line and the back k-i line, namely:
set0＝{qπ(1),qπ(2),…,qπ(i)}∩{qπ(i+1),qπ(i+2),…,qπ(k)} (17)
wherein q isπ(j)Is matrix G'n/2Row j of (2).
Since the split channel corresponding to the transmitted information bit is selected according to the reliability of the channel, i is less than or equal to k/2, if the equal sign is established when the code rate of the polar code is equal to 1, and k is equal to n, t is less than or equal to i < k/2 because the code rate of the actually transmitted polar code is less than 1. At this time, G 'is removed from the repeating line'n/2The rank of (2) is k-t, which corresponds to a code rate of 2(k-t)/n, and since t is less than k/2, 2(k-t)/n is greater than k/n, thus obtaining the evidence.
Theorem 1: the properties of the given polar code are recursive, that is, after the polar code with the actual code length of n is divided into n/2, n/4, …,4,2, each divided code word is equivalent to a new polar code, and the code rate of the new polar code increases in sequence with the reduction of the division length until the code rate increases to 1.
Theorem 2: and expanding the polarization code with the code length of n and the code rate of k/n into a code word with the code length of 2 · n, wherein the polarization code with the code length of 2 · n exists, the dual space of the polarization code is the same as the dual space of the expanded code word, and the code rate of the expanded code word is equal to k/n.
And (3) proving that: g 'is not assumed to be an unexpanded polar code encoding matrix'nIn the matrix, the line numbers of each line in the formula (14) are pi '(1), pi' (2), …, pi '(k), so G'nCan be expressed as:
Figure BDA0002658716940000091
in the formula, d ″)π′(i),j(i is more than or equal to 1 and less than or equal to k, j is more than or equal to 1 and less than or equal to n) is Dn×nAnd (5) medium element.
When the code length of the polarization code is extended to be 2 · n, the front n bit sequence and the rear n bit sequence of the code word after new extension are G'nThe generated code word, so the extended code word generation matrix can be expressed as:
Figure BDA0002658716940000101
because of G'nIs a row full rank matrix, so G'2nIs also determined to be a row full rank matrix, so G'2nThe code rate of the corresponding code word is 2k/(2n), which is equal to the code rate of the original polarization code.
Since the row transform does not affect the dual space of the matrix, G'2nAnd superposing the middle front n rows and the back n rows to obtain a transformed matrix:
Figure BDA0002658716940000102
when the formula (20) is observed, G ″)2nIs exactly D(2·n)×(2·n)In some cases, therefore G ″)2nCan construct a polar code coding matrix with the code length of 2n, the dual space of which is just equal to G2′nThe corresponding dual spaces are the same, thus obtaining the evidence.
Theorem 2 also has recurrences, namely: after the polarization code with the code length n is expanded into 2 · n,4 · n,8 · n, …, the code rate of the expanded code word is still equal to k/n.
As can be seen from theorems 1 and 2, when the traversed code length is smaller than the actual code length of the polarization code, the code rate of the constructed code word is greater than the actual code rate of the polarization code; on the contrary, when the traversed code length is greater than the actual code length, the constructed code word code rate is equal to the actual code rate of the polarization code, so that the code length of the polarization code is equal to the minimum code length corresponding to the minimum code rate. When the code rate is obtained for the constructed code word, a gaussian elimination method can be adopted, but the method has poor recognition performance under the condition of error codes, and the calculation amount and the required data amount are also increased sharply as the traversal code length is increased. In order to overcome the problems of the elimination method, the information bit positions corresponding to the code words are solved from the dual space for constructing the code words, and because the ratio of the number of the information bit positions to the traversed code length is exactly equal to the code rate, the code rate is indirectly solved, and meanwhile, the information bit positions can be identified.
The information bit position identification specifically includes:
according to the construction principle of the polar code encoding matrix, the code word actually participating in encoding generates the matrix GnEach row in (set) corresponds exactly to GnThe line in which the information bit position is located, and identifying the information bit position is equivalent to identifying Gn(set) in each row GnThe middle position. Due to Gn(set) and GnThe number of lines is different, the space of the code word generated by the two is also different from the dual space, and theorem 3 gives the relationship between the two coding spaces and the dual space.
Theorem 3: setting a space V1Is set by the vector set1＝{a1,a2,…,alStretched out and does not mark V1＝L(set1) Set is set1A subset of2＝{aπ(1),aπ(2),…,aπ(i)Where pi (j) (1. ltoreq. j. ltoreq.i) corresponds to a set1Middle mark, marked by set2The space formed by stretching is V2＝L(set2) Then, then
Figure BDA0002658716940000111
And dual space
Figure BDA0002658716940000112
And (3) proving that: when in use
Figure BDA0002658716940000113
Time, equivalent to dual space
Figure BDA0002658716940000114
Is established, theorem 3 proves
Figure BDA0002658716940000115
And (4) finishing. Do not set the vector V as the space V2Any element in (1), then V can be represented by V2Medium baseline sexual representation, namely:
Figure BDA0002658716940000116
wherein alpha isj(j is more than or equal to 1 and less than or equal to i) epsilon GF (2). Let set2At set1The complement of (C) is set3＝{aπ(i+1),aπ(i+2),…,aπ(l)V can be expressed as
Figure BDA0002658716940000117
So v can also be set1The middle element is linearly expressed, so V is equal to V1Thus having
Figure BDA0002658716940000118
Obtaining the syndrome.
From theorem 3, the space of the polar code word is equal to GnThe space spanned by the row vectors in (set), and the codeword space must be GnThe middle row vector spans a subspace of space. If G is to benRemoving a certain line, wherein when the position of the removed line is not the position of the information bit, the dual space corresponding to the space formed by the removed residual line is necessarily orthogonal to the code word space; conversely, when the position of the culled row is exactly the information bit position, the orthogonal relationship no longer exists. Therefore, G can be eliminated through sequential traversalnIn a certain row, then obtaining its correspondent dual space, when the dual matrix and code word are formed into check relationship, said position can be judged as frozenA junction bit position; otherwise, the position is the information bit position. Let GnThe row in the row is eliminated, and the matrix formed by the residual row vectors is QiDue to GnIs a row full rank matrix, so QiA full-rank matrix of rows with dimension (n-1) x n, obtained by row transformation and column transformation:
Ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1](21)
wherein R isiFor a row transformation matrix, SiFor a column transformation matrix, I(n-1)×(n-1)Is an identity matrix. Q can be obtained quickly from the formula (21)iDual matrix of (H)iComprises the following steps:
Figure BDA0002658716940000119
namely, it is
Figure BDA00026587169400001110
Wherein
Figure BDA00026587169400001111
Represents HiThe transposing of (1).
After the dual vector is solved, whether the dual vector and a code word space form an orthogonal relation needs to be judged, and under the condition of error codes, the judgment is mainly based on the fact that the dual vector and the non-dual vector enable the code word checking relation to be established as the basis. First, two types of hypothesis testing are given:
Figure BDA00026587169400001112
Hithe dual vectors of the code words, namely the eliminated row positions are frozen bit positions;
Figure BDA0002658716940000121
Hithe dual vectors which are not codewords, namely the row positions of the culling are information bit positions.
Under the assumption of conditions
Figure BDA0002658716940000122
If the error code exists, the check relation is still true, and the position of the error code in the code word corresponds to HiThe number of error codes occurring at the same time must be even number at the position of the middle element 1, and at this time, the check relation is still established through modulo 2 operation. Without setting the channel error rate to pe，HiCode weight of wiIf the check relation establishment probability is:
Figure BDA0002658716940000123
in the formula,
Figure BDA0002658716940000124
and C represents the operation of taking the combination number.
Taking the average T of the difference between the number of established code word check relations and the number of unsatisfied code words as statistic, and setting the number of constructed code words as N, then under the assumed condition
Figure BDA0002658716940000125
The mean of the statistics T is: 2P-1, variance 4P (1-P)/N, when N is sufficiently large, T approximately follows a gaussian distribution:
Figure BDA0002658716940000126
under the assumption of conditions
Figure BDA0002658716940000127
Under the condition of HiNot dual vector, the check relation is random at this time, the probability of establishment is 0.5, so the average value of the statistic T is 0, the variance is 1/N, when the number N of code words is sufficiently large, the code word is processed
Figure BDA0002658716940000128
Subject to a gaussian distribution:
Figure BDA0002658716940000129
for convenience of explanation, let us note μ1＝2P-1，
Figure BDA00026587169400001210
μ0＝0，
Figure BDA00026587169400001211
And if the decision threshold under the two assumed conditions is Λ, the false-alarm probability is:
Figure BDA00026587169400001212
the false alarm probability is:
Figure BDA00026587169400001213
under the condition of no prior knowledge, the probability of the occurrence of the two types of hypothesis conditions is the same, so the average error decision probability PtComprises the following steps:
Figure BDA00026587169400001214
will PtTaking the derivative of Λ and making it equal to 0 yields:
Figure BDA0002658716940000131
taking logarithms on two sides, converting the logarithms into a unitary quadratic equation, and solving a minimum error judgment threshold as follows:
Figure BDA0002658716940000132
in order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Fig. 1 is a schematic flow chart of a method for identifying parameters of a polarization code under an error code condition, as shown in fig. 1, the method for identifying parameters of a polarization code under an error code condition includes:
s101, constructing a polar code word matrix by using the intercepted bit sequence and the set traversed code length range.
S102, determining a Kernel product power matrix according to the polarization code codeword matrix.
S103, sequentially eliminating the row data in the kronecker product power matrix, and determining the corresponding dual matrix by using the kronecker product power matrix after the row data are eliminated each time.
Using formulas
Figure BDA0002658716940000133
Determining a corresponding dual matrix of the Croncke product power matrix after the row data eliminated each time; wherein HiDetermining a corresponding dual matrix for the Croncke product power matrix after the ith eliminated row of data,
Figure BDA0002658716940000134
is HiIs transposed, SiFor a column transformation matrix, P(n-1)X 1 satisfies the condition: ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1]Wherein R isiFor a row transformation matrix, QiTo remove the Kroecker power matrix after the ith row, I(n-1)×(n-1)Is (n-1) × (n-1) identity matrix.
And S104, judging whether the currently rejected row data is an information bit position or a frozen bit position according to the dual matrix and the polar code codeword matrix.
S104 specifically comprises the following steps:
and determining statistics and a decision threshold according to the dual matrix and the polarization code codeword matrix.
And judging whether the statistic is smaller than the judgment threshold.
And if the statistic is smaller than the judgment threshold, determining that the currently-rejected line data is an information bit position, and judging the next-rejected line data.
And if the statistic is not smaller than the judgment threshold, determining that the currently removed row data is a frozen bit position, and judging the next removed row data.
S105, counting the currently traversed code length and all the currently determined information bit positions to determine the code rate corresponding to the currently traversed code length until the code rates corresponding to all the traversed code lengths in the set traversed code length range are determined.
And determining the code rate corresponding to the currently traversed code length according to the ratio of all the currently determined information bit positions to the currently traversed code length.
S106, identifying the parameters of the polarization code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the parameters of the polarization code include a code length, an information bit position, and a frozen bit position.
Fig. 2 is a schematic diagram of a principle of a method for identifying polarization code parameters under error code conditions, as shown in fig. 2, identifying a true polarization code length, and simultaneously obtaining an information bit position under the code length, wherein the specific algorithm steps are as follows:
step 1: setting the code length range n of traversal as nminTo nmaxWherein n is 2m(m is a positive integer);
step 2: constructing a data matrix with the length of n of code words by using intercepted data, and then constructing a matrix construction method by using a polarization code to generate a matrix D of m-degree Crohn's product powersn×n；
And step 3: i is sequentially valued from 1 to n, and D is removed each timen×nObtaining a dual vector by using the ith row of data and the formula (21) and the formula (22);
and 4, step 4: the data matrix and the dual vector are utilized to obtain statistic T, and a decision threshold lambda is calculated at the same timeoptIf T is less than ΛoptIf yes, judging that the position i is an information bit position, and returning to the step 3 until i is equal to n, and if the position i is equal to i + 1;
and 5: counting the number of information bit positions under the condition that the traversal code length is nCalculating the corresponding code rate, traversing the next code length at the same time, and returning to the step 2 until n is nmax；
Step 6: and outputting the minimum code length and the information bit position corresponding to the minimum code rate to finish the identification of the polarization code parameters.
Because the structure of the polarization code is relatively fixed, before identification, the dual vector under each traversal code length and information bit position can be obtained and stored, and the dual vector can be directly loaded when calculating the statistic, so that a large amount of repeated calculation can be reduced.
To further determine the usability of the present invention and to perform computational complexity analysis, the analysis process is as follows:
the intercepted data volume is L, and the traversed code length range is
Figure BDA0002658716940000151
To
Figure BDA0002658716940000152
When the code length of traversal is 2m(mmin≤m≤mmax) When necessary, pair 2mTraversing each information bit position, wherein traversing each information bit position requires performing
Figure BDA0002658716940000153
A sub-vector multiplication operation,
Figure BDA0002658716940000154
A sub-addition operation and a 1-time threshold solution. For convenience of description, 1-time threshold computation is approximately equivalent to 5-time vector multiplication operation, so that the pair code length is 2mFor the course of the pass, 5.2 is requiredm+ L vector multiplications and L additions. The calculated quantities under different traversal code lengths are summed to obtain the total multiplication complexity
Figure BDA0002658716940000155
The addition complexity is (m)max-mmin+1) L. It can be seen that the method herein calculates a quantitative approximationIncreasing linearly with the length of the intercepted data and the maximum code length traversed.
Fig. 3 is a schematic structural diagram of a system for identifying parameters of a polar code under an error code condition, as shown in fig. 3, the system for identifying parameters of a polar code under an error code condition, provided by the present invention, is characterized by comprising: a polar code codeword matrix construction module 301, a kronecker power matrix determination module 302, a dual matrix determination module 303, a bit position determination module 304, a code rate determination module 305 and a polar code parameter identification module 306.
The polar code codeword matrix constructing module 301 is configured to construct a polar code codeword matrix using the intercepted bit sequence and the set traversed code length range. .
The kirenk product power matrix determining module 302 is configured to determine a kirenk product power matrix according to the polar code codeword matrix.
The dual matrix determining module 303 is configured to sequentially remove row data in the kronecker power matrix, and determine a corresponding dual matrix by using the kronecker power matrix after the row data is removed each time.
The bit position determining module 304 is configured to determine whether the currently rejected row of data is an information bit position or a frozen bit position according to the dual matrix and the polar code codeword matrix.
The code rate determining module 305 is configured to count the currently traversed code length and all currently determined information bit positions to determine a code rate corresponding to the currently traversed code length until determining code rates corresponding to all traversed code lengths within the set traversed code length range.
The polarized code parameter identification module 306 is configured to perform parameter identification on the polarized code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the parameters of the polarization code include a code length, an information bit position, and a frozen bit position.
The dual matrix determination module 303 specifically includes: and a dual matrix determination unit.
A dual matrix determination unit for utilizing the formula
Figure BDA0002658716940000161
Determining a corresponding dual matrix of the Croncke product power matrix after the row data eliminated each time; wherein HiDetermining a corresponding dual matrix for the Croncke product power matrix after the ith eliminated row of data,
Figure BDA0002658716940000162
is HiIs transposed, SiFor a column transformation matrix, P(n-1) × 1 satisfies the condition: ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1]Wherein R isiFor a row transformation matrix, QiTo remove the Kroecker power matrix after the ith row, I(n-1)×(n-1)Is (n-1) × (n-1) identity matrix.
The bit position determining module 304 specifically includes: a statistic and decision threshold determining unit, a first judging unit, an information bit position determining unit and a frozen bit position determining unit.
And the statistic and decision threshold determining unit is used for determining statistic and decision threshold according to the dual matrix and the polarization code codeword matrix.
The first judging unit is used for judging whether the statistic is smaller than the judgment threshold.
And the information bit position determining unit is used for determining that the currently removed row data is the information bit position and judging the next removed row data if the statistic is smaller than the judgment threshold.
And the frozen bit position determining unit is used for determining that the currently removed line data is the frozen bit position and judging the next removed line data if the statistic is not smaller than the judgment threshold.
The code rate determining module 304 specifically includes:
and the code rate determining unit is used for determining the code rate corresponding to the currently traversed code length according to the ratio of all the currently determined information bit positions to the currently traversed code length.
In order to further illustrate the effectiveness of the method and system for identifying the polarization code parameters under the error code condition, the following provides the relevant verification:
the code length of the polarization code selected by simulation is 256, the code rate is 1/2, the Babbitt parameters are calculated according to the formula (10) and the formula (11), the calculated Babbitt parameters are sorted from small to large, the position of the front 1/2 is selected to transmit information bits, and the position of the rear 1/2 is selected to transmit frozen bits. The algorithm verifies the correctness of the algorithm under two environments of no error code and error code. Under the condition of code error, setting the channel error rate to be 0.01, the number of intercepted code words to be 10000 and the traversed code length to be 2m(2 m 10), i.e. the code length ranges from 4 to 1024. According to the algorithm steps, firstly, the code rate value under the traversal code length is obtained, then the information bit position corresponding to the identified code length is output, and the identification results under two noise environments are shown in fig. 4 and fig. 5.
First, as seen from the result of fig. 4(a), under the condition of no error code, when the estimated code length is less than 256, the solved code rate is significantly greater than 0.5, and when the code length is greater than 256, the code rate is exactly equal to 0.5, which is consistent with the conclusions of theorem 1 and theorem 2, and at this time, it is recognized that the code length of the polarization code is 256; traversing the information bit position under the condition that the code length is 256, wherein the error code is 0, so that the threshold of the solution is equal to 1, and when the traversed position is the frozen bit position, the statistic T is equal to 1 because the check relation is established; otherwise, the verification is not true, and the statistic T can only linger around 0, which is consistent with the derived statistical property, which indicates that the conclusion of theorem 3 is correct.
Secondly, traversing the code length of the polarization code under the condition that the error rate is 0.01, wherein when m is 8, the code rate has the minimum value of 0.5, identifying that the code length of the polarization code is 256, simultaneously traversing the position of the information bit under the condition of the error rate, and when the traversed position is just the position of the information bit, the statistic is smaller than the threshold value, and the analyzed statistical characteristics and the solved minimum error judgment threshold are reasonable. In conclusion, the method provided by the invention can complete the correct identification of the polarization code parameters under the condition of error codes.
Simulation sets 5 types of polarization codes, the code length n of each polarization code is 64, 128, 256, 512, 1024, the code rate R is 1/2, the number of intercepted code words is 5000, the channel error rate range is 0 to 0.15, the interval is 0.005, 1000 monte carlo tests are performed at each error rate, and the results of counting the identification rates of the polarization code length and the information bit parameter are shown in fig. 6.
From the results of fig. 6, the code length has a large influence on the algorithm, and the recognition performance of the algorithm gradually decreases as the code length increases. The main reason is that when the code length is increased, the information bit position to be identified is inevitably increased under the condition of the same code rate, and the misjudgment probability of the information bit position is also increased at the moment, which causes the performance of the algorithm to be reduced; meanwhile, as can be seen from the comparison of the results of fig. 6(a) and fig. 6(b), the code length recognition performance is significantly better than the recognition of the information bit position, and the main reason is that the recognition of the information bit position requires not only the number of information bits but also the specific position to be recognized, and the code length recognition only requires that the code rate under the actual code length is the minimum, so the requirement of the information bit position recognition on the algorithm is more severe.
The simulation sets the length of the polar code to be 128, the code rate R to be 1/2, and the number of the intercepted code blocks is 1000, 5000, 10000, 15000, 20000, 5. The range of the channel error rate is 0 to 0.14, the interval is 0.005, the accuracy of parameter identification is counted under different numbers of intercepted code words and channel error rates, the number of Monte Carlo tests in simulation is 1000, and the obtained result is shown in figure 7.
From the results of fig. 7, the code rate has a large influence on code length identification, and as the code rate increases, the code length identification performance tends to decrease; for information bit identification, the code rate has little effect on the code length after the code length is correctly identified. The main reason is that the larger the actual code rate is, the closer the estimated code rate between the traversed adjacent code lengths will be, and once the error code is increased, the larger the error recognition probability will be for code length recognition; for the information bit positions, after the code length is correctly identified, the identification of the parameters depends on whether the judgment of each information bit position is accurately finished, and under the actual code length and code rate, the number of the information bit positions is not greatly different, so the code rate has little influence on the identification of the information bit positions.
In the simulation, the code length of the polarization code is set to be 128, the 4 coding code rates R are set to be 1/3, 1/2, 2/3 and 3/4, the number of the intercepted code words is 5000, the error rate range of the channel is set to be 0 to 0.15, the interval is 0.005, the Monte Carlo simulation times are 1000, the correct rate of parameter identification under different code rates and error rates is counted, and the result is shown in FIG. 8.
From the results of fig. 8, the code rate has a large influence on code length identification, and as the code rate increases, the code length identification performance tends to decrease; for information bit identification, the code rate has little effect on the code length after the code length is correctly identified. The main reason is that the larger the actual code rate is, the closer the estimated code rate between the traversed adjacent code lengths will be, and once the error code is increased, the larger the error recognition probability will be for code length recognition; for the information bit positions, after the code length is correctly identified, the identification of the parameters depends on whether the judgment of each information bit position is accurately finished, and under the actual code length and code rate, the number of the information bit positions is not greatly different, so the code rate has little influence on the identification of the information bit positions.
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.

Claims (8)

1. A method for identifying polarization code parameters under error code conditions is characterized by comprising the following steps:
constructing a polarization code codeword matrix by using the intercepted bit sequence and the set traversed code length range;
determining a kronecker product power matrix according to the polarization code codeword matrix;
sequentially rejecting line data in the kronecker product power matrix, and determining a corresponding dual matrix by using the kronecker product power matrix after the line data are rejected each time;
judging whether the currently rejected row data is an information bit position or a frozen bit position according to the dual matrix and the polarization code codeword matrix;
counting the currently traversed code length and all the currently determined information bit positions to determine the code rate corresponding to the currently traversed code length until the code rates corresponding to all the traversed code lengths in the set traversed code length range are determined;
identifying the parameters of the polarization code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the parameters of the polarization code include a code length, an information bit position, and a frozen bit position.
2. The method according to claim 1, wherein the sequentially removing row data in the kronecker power matrix and determining a corresponding dual matrix by using the kronecker power matrix after each removed row data specifically comprises:
using formulas
Figure FDA0002658716930000011
Determining a corresponding dual matrix of the Croncke product power matrix after the row data eliminated each time; wherein HiDetermining a corresponding dual matrix for the Croncke product power matrix after the ith eliminated row of data,
Figure FDA0002658716930000012
is HiIs transposed, SiFor a column transformation matrix, P(n-1)×1The conditions are satisfied as follows: ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1]Wherein R isiFor a row transformation matrix, QiTo remove the Kroecker power matrix after the ith row, I(n-1)×(n-1)Is (n-1) × (n-1) identity matrix.
3. The method according to claim 1, wherein said determining whether the currently rejected row of data is an information bit position or a frozen bit position according to the dual matrix and the polar code codeword matrix specifically comprises:
determining statistics and a decision threshold according to the dual matrix and the polarization code codeword matrix;
judging whether the statistic is smaller than the judgment threshold;
if the statistic is smaller than the judgment threshold, determining that the currently-rejected line data is an information bit position, and judging the next-rejected line data;
and if the statistic is not smaller than the judgment threshold, determining that the currently removed row data is a frozen bit position, and judging the next removed row data.
4. The method according to claim 1, wherein the counting of the currently traversed code length and all currently determined information bit positions determines the code rate corresponding to the currently traversed code length until the code rates corresponding to all traversed code lengths within the set traversed code length range are determined, specifically includes:
and determining the code rate corresponding to the currently traversed code length according to the ratio of all the currently determined information bit positions to the currently traversed code length.
5. A system for identifying polarization code parameters under error conditions, comprising:
the polar code codeword matrix constructing module is used for constructing a polar code codeword matrix by utilizing the intercepted bit sequence and the set traversed code length range;
the kronecker product power matrix determining module is used for determining a kronecker product power matrix according to the polarization code codeword matrix;
the dual matrix determining module is used for sequentially rejecting line data in the kronecker product power matrix and determining a corresponding dual matrix by using the kronecker product power matrix after the line data are rejected each time;
the bit position determining module is used for judging whether the currently rejected row data is an information bit position or a frozen bit position according to the dual matrix and the polar code codeword matrix;
a code rate determining module, configured to count a currently traversed code length and all currently determined information bit positions to determine a code rate corresponding to the currently traversed code length until determining code rates corresponding to all traversed code lengths within the set traversed code length range;
the parameter identification module of the polarization code is used for carrying out parameter identification on the polarization code according to the traversed code length corresponding to the minimum code rate and the information bit position corresponding to the minimum code rate; the parameters of the polarization code include a code length, an information bit position, and a frozen bit position.
6. The system of claim 5, wherein the dual matrix determination module specifically comprises:
a dual matrix determination unit for using a formula
Figure FDA0002658716930000031
Determining a corresponding dual matrix of the Croncke product power matrix after the row data eliminated each time; wherein HiDetermining a corresponding dual matrix for the Croncke product power matrix after the ith eliminated row of data,
Figure FDA0002658716930000032
is HiIs transposed, SiFor a column transformation matrix, P(n-1)×1The conditions are satisfied as follows: ri·Qi·Si＝[I(n-1)×(n-1)|P(n-1)×1]Wherein R isiFor a row transformation matrix, QiTo remove the Kroecker power matrix after the ith row, I(n-1)×(n-1)Is (n-1) × (n-1) identity matrix.
7. The system of claim 5, wherein the bit position determining module specifically comprises:
a statistic and decision threshold determining unit, configured to determine a statistic and a decision threshold according to the dual matrix and the polar code codeword matrix;
the first judging unit is used for judging whether the statistic is smaller than the judgment threshold or not;
an information bit position determining unit, configured to determine that the currently rejected row data is an information bit position if the statistic is smaller than the decision threshold, and perform a judgment on a next rejected row data;
and the frozen bit position determining unit is used for determining that the currently removed line data is the frozen bit position and judging the next removed line data if the statistic is not smaller than the judgment threshold.
8. The system of claim 5, wherein the code rate determining module specifically comprises:
and the code rate determining unit is used for determining the code rate corresponding to the currently traversed code length according to the ratio of all the currently determined information bit positions to the currently traversed code length.
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