CN115333676B - Code word construction method of convolutional coding suitable for polarization adjustment - Google Patents

Abstract

The invention belongs to the technical field of communication, in particular to a code word construction method of convolutional coding suitable for polarization adjustment. In the traditional convolution coding code word construction method of polarization adjustment, the best effect is the Reed Mu Legou construction method, but only code words with specific code rate can be constructed, and the Reed Muller assisted joint boundary construction method provided by the invention selects each information bit according to the joint boundary determined by the Hamming distance spectrum on the basis of the Reed Mu Legou construction method, and can be applied to code words with any code rate. Compared with the traditional code word construction scheme, the code word construction scheme provided by the invention avoids the process of reselecting the optimal channel under different signal-to-noise ratio channels, and can well make up the non-universality of the convolution coding code word construction of polarization adjustment. Meanwhile, simulation results prove that under the scene of short codes, the invention constructs the convolution code with optimal performance for polarization adjustment at present.

Description

Code word construction method of convolutional coding suitable for polarization adjustment

Technical Field

The invention belongs to the technical field of communication, and particularly relates to a code word construction method of convolutional coding suitable for polarization adjustment. The invention relates to Polarization-adjusted convolutional coding (PAC), serial cancellation list decoding (Successive Cancellation List, SCL) and other technologies.

Background

Polarization codes are the only channel codes that have been shown to achieve shannon channel capacity in the present theory, but provided that the code length goes to infinity. For this reason, the polarization-adjusted convolutional code is proposed by the polarization code presenter e.arikan to compensate for the problem of channel capacity loss of the polarization code due to insufficient channel polarization in a short code scenario. Experimental results prove that the PAC code with the code length of 128 and the code rate of 1/2 can be close to the dispersion approximate boundary of the limited code length. However, the code word construction method of the PAC code (128, 64), namely the reed Mu Legou method, can only construct PAC codes with specific code rate, and the code word construction algorithm of the traditional polarization code is applicable to PAC codes, but cannot exert good performance.

Disclosure of Invention

Aiming at the problems, the invention provides a novel codeword construction method based on Reed-Muller (RM) auxiliary joint bound (Union Upper Bound, UUB) for PAC codes. In the short code stage, the minimum Hamming distance of the code word has important influence on the coding performance, and on the basis, the invention uses the characteristic that the minimum Hamming distance of the code word is limited by the Reed-Mueller construction method to provide a Reed-Mueller aided joint boundary construction method based on the Hamming distance spectrum, which is used for constructing the code word with any code rate, and improves the performance of the PAC code while improving the universality of the original algorithm.

For easy understanding, the Reed Muller assisted joint boundary construction method proposed by the present invention is described as follows:

a system block diagram of PAC codes is shown in fig. 1.

PAC codes are different from other error correction codes in that their generator matrix is fixed. For a PAC codeword with a code length of N and a code rate of (K/N), the codeword construction scheme, i.e., rate analysis, refers to selecting K channels from the N channels for transmitting information bits, and the remaining N-K channels for transmitting frozen bits. The method for selecting the information bit channel is different, and the coding result is greatly different, so the selection of the information bit is a very critical step in the PAC coding process. Conventional methods for selecting information bits are Gaussian Approximation (GA), papanicolaou parametric (BHA), polarization Weighting (PW) and reed Mu Lefa (RM). The gaussian approximation and the papanicolaou parameter method differ in the channel chosen for transmitting the information bits at different signal to noise ratios. While reed Mu Lefa can only construct codewords of a specific code rate. Although scholars have tried to combine the gaussian approximation and the pasteurization parameter method with the reed Mu Legou method to obtain the reed muller-assisted gaussian approximation (RM-GA) and the reed muller-assisted pasteurization parameter method (RM-BHA), respectively, the actual effect is not ideal. Therefore, there is currently a lack of a fast and efficient codeword construction method for PAC codes.

For ease of understanding, a description will be given of the reed muller construction method. For any integer m and r, 0.ltoreq.r.ltoreq.m, there is an r-order RM code, denoted RM (r, m), whose parameters are as follows: code length: m=2 ^m Dimension number: minimum hamming distance: d, d _min ＝2 ^m-r . One commonality of RM code and polarization code is that they have the same generation matrix, but the information bit channels are selected differently, the generation matrix is +.>Wherein->and Represents the m-time kronecker product of F. The RM code selects an information bit channel according to the row weights of the generator matrix. According to F _m The minimum Hamming distance of the RM codeword is 2 if the order of the codeword is r ^m-r . According to the design rule of the RM code, the code rate can only be specific for a given code length, for example, an RM code with a code length of 128, and the number of information bits can only be 1,8, 29, 64, 99, 128.

The information vector d of length K is mapped to a length N (n=2 by rate analysis ⁿ ) Before the polarization code encoding, a pre-convolution process with a code rate of 1 is performed. Assume that the convolution polynomial of the convolutional code is: g= [ g ] ₀ ,g ₁ ,...,g _t ]The convolutional code coding can be expressed as:

wherein i is more than or equal to 1 and less than or equal to N, and t represents the convolution constraint length.

The polarization transformation process can be represented by a polarization matrix G:

x＝uG (2)

wherein the method comprises the steps of Representing the kronecker product of n times.

x is subjected to BPSK modulation and then transmitted through an AWGN channel to obtain a received signal y:

y _i ＝x _i +n _i , (3)

wherein, the liquid crystal display device comprises a liquid crystal display device,is Gaussian additive white noise, sigma ² Representing the noise variance.

Serial cancellation list decoding (Successive Cancellation List, SCL) is a classical decoding algorithm for polar codes. The SCL decoding algorithm comes from the SC decoding algorithm. When the SC decoder decodes the ith (i is not less than 1 and not more than N) code word, if the ith code word is a frozen bit, the decoding is 0. If the ith codeword is an information bit, the decoding result is determined by the log likelihood ratio of the ith codeword. Only one path is decoding from beginning to end of SC decoding, and in order to improve the decoding performance of SC decoder, SCL decoding algorithm is proposed. SCL decoding is a parallel irregular tree search algorithm, and when decoding the ith bit, L paths are decoded simultaneously, each path has a corresponding path metric value, and the larger the path metric value is, the greater the probability that the codeword corresponding to the path is correct is indicated. If the ith bit is an information bit, the value of each bit has two possibilities of 0 and 1, 2L paths are provided when the decoding of the ith code word is finished, and the L paths with the largest path metric value are selected from the 2L paths to decode the next bit. If the ith bit is a frozen bit, the path metric value of the L paths is directly updated, and then the decoding of the next bit is performed.

The SCL decoding algorithm is equally applicable to PAC codes, except that one more deconvolution is performed in the decoding process of each bit. When decoding i bits, if the i bit is a frozen bit0, and then go throughThe deconvolution transformation to the corresponding +.>Updating the path metric value of L paths, and if the ith bit is an information bit, respectively making +.>0 and 1, and then convolutionally transformed into the corresponding +.>And updating the path metric values of 2L paths, selecting L paths with larger path metric values from the 2L paths to decode the next bit, and selecting a codeword corresponding to the path with the largest path metric value as a final decoding result when decoding is finished.

The technical scheme of the invention is as follows:

a polarization-adjusted convolutional encoded codeword construction scheme comprising a code length N (n=2 ⁿ ) Number of information bits K. Let the number of information bits K satisfy:

K ₁ <K<K ₂ , (4)

wherein, the liquid crystal display device comprises a liquid crystal display device,is expressed as satisfying K ₁ K is as follows ₂ An integer of these two equations.

Aggregation of reamsWherein G is _q Represents the q-th row of the generator matrix G, wt (G _q ) Row weight representing the q-th row of the generator matrix G, i.e. the number of 1's in the q-th row,/->Representing that the weight of the row in the generation matrix G is greater than or equal to 2 ^n-p Index set of subscripts of->Representing that the weight of the row in the generation matrix G is greater than or equal to 2 ^n-p-1 And less than 2 ^n-p Is described. For a PAC code of (N, K), the problem to be solved can be converted into, firstly, from +.>Select->Channels, again from->Is selected from the group consisting of |K-K ₁ The invention mainly aims at providing a method for generating a channel in the set +.>Is selected from the group consisting of |K-K ₁ Criteria for i channels. The method comprises the following specific steps:

s1, initializing a setGiving a very high signal-to-noise ratio SNR and a very large serial cancellation list number L;

s2, assembling the commandWherein->

S3, constructing onePAC codeword of (C) then set->A set of indices of the information bit channel index. The invention adopts 'B.Li, H.shen, and D.Tse,' An adaptive successive cancellation list decoder for polar codes with cyclic redundaThe hamming distance spectrum of PAC codewords is obtained by the method in ncy check, "IEEE Communications Letters, vol.16, no.12, pp.2044-2047,2012. In the coding process, the information bit is made to be all-zero bit, the transmitting code word is also all-zero code word, the transmission channel is Gaussian additive white noise channel (AWGN), serial offset list coding is used at the decoding end, when the number of the list is large, besides the coding result of the first path is correct, the rest (L-1) code words are the code words which are the most easily misplaced by all-zero code words, and the rule of the linear block code can know that the hamming weight of the (L-1) code words can be approximately understood as the hamming distance spectrum of the PAC code. Recording the hamming distance spectrum A of the PAC codeword _i ；

S4, repeating the steps S2-S3, and selecting only the set each timeOne element of (a) until the set is traversed>All elements in (1) according to the formula

To calculate the selectionAs a joint boundary for each element of the information bits. Wherein, array A _i Representation ofHamming distance spectrum obtained with each element of (a) as an information bit channel,/d->Representing code rate, E _b /N ₀ Represents the signal-to-noise ratio, Q represents the right-tail function of the standard normal distribution, for E _b /N ₀ Without any provision forA definite fixed value is generally satisfied if the calculated joint bound is less than 1 e-6;

s5, selecting the index of the subscript corresponding to the smallest joint boundary according to the calculation result in the step S4, and gatheringThe index-th element of (a) is added to the set +.>From the collection->Delete the element;

s6, repeating the steps S2-S5 until all the required information bit channels are selected, namely

Compared with the traditional construction method, the method has the advantages that the method can construct code words with any code rate and improve the performance, and simultaneously, the selected channels are the same under AWGN (Additive White Gaussian Noise) channels with different signal to noise ratios, so that the universality of the method is improved.

Drawings

FIG. 1 is a coding and decoding flow chart of polarization-adjusted convolutional encoding;

fig. 2 is a schematic diagram showing the bit error rate performance of the proposed codeword construction algorithm RM-UUB compared with the conventional algorithm in the case of the code length n=128, k=40;

fig. 3 is a schematic diagram showing the bit error rate performance of the proposed codeword construction algorithm RM-UUB compared with the conventional algorithm in the case of the code length n=128, k=72.

Detailed Description

The technical method of the present invention is described in detail below with reference to the accompanying drawings and examples:

examples

The present example constructs a (128, 40) PAC code, let snr=100db, l=1e5, convolution polynomial g= [1,0,1,1,0,1,1], specifically including:

s1, selecting a travel weight from a generation matrix G of PAC codes to be greater than or equal to 2 ⁵ Index subscripts of =32 constitute a setThen->Selecting index subscripts with travel weight of 16 or more and less than 32 from the generated G to form a set +.>Then->

S2, assembling the commandWherein->

S3, constructing onePAC codeword of (C) then set->A set of indices of the information bit channel index. In the encoding process, when the information bits are all zero bits, the transmitting code words are all zero code words, the transmission channel is a Gaussian additive white noise channel (AWGN), serial offset list decoding is used at the decoding end, when the number of the list is large, the decoding result of the first path is correct, the remaining 99999 decoding errors are the 99999 code words which are the easiest to be mistaken by all zero code words, and the hamming weight of the 99999 code words can be approximately understood as the hamming distance spectrum of the PAC code. Recording the hamming distance spectrum A of the PAC codeword _i ；

S4, repeating the steps S2-S3, and selecting only the set each timeOne element of (a) until the set is traversed>All elements in (1) according to the formula

Calculating a joint boundary corresponding to each element;

s5, selecting the index of the subscript corresponding to the smallest joint boundary according to the calculation result in the step S4, and gatheringThe index-th element of (a) is added to the set +.>From the collection->The element is deleted, and the matrix A is emptied;

s6, repeating the steps S2-S5 until all the required information bit channels are selected, namely

Fig. 2 shows a comparison of the bit error rate performance at n=128 and k=40, and it can be seen that the RM-UUB construction method is significantly better than the conventional codeword construction scheme, and especially at snr=1.5 dB, the RM-UUB has about 0.3dB more coding gain than the current best codeword construction scheme. In addition, the frame error rate of the RM-UUB can approach the dispersion approximation boundary of the limited code length, which fully explains the superiority of the RM-UUB construction method. Even in the high code rate scenario, the RM-UUB can still maintain its advantages, and fig. 3 shows a comparison of the error rate performance of n=128 and k=72, and it can be seen that, even in the high code rate scenario, the RM-UUB is still significantly better than the conventional PAC codeword construction scheme. This fully illustrates the meaning and value of the proposed codeword construction scheme of the present invention.

Claims (1)

1. A code word construction method of convolution coding suitable for polarization adjustment defines the code length of PAC code as N, N=2 ⁿ The number of information bits is K, and the number of information bits K satisfies:

K ₁ <K<K ₂

wherein, the liquid crystal display device comprises a liquid crystal display device,p is an integer and 1.ltoreq.p.ltoreq.n;

definition setWherein G is _q Represents the q-th row, wt (G _q ) Row weight representing the q-th row of the generator matrix G, i.e. the number of 1's in the q-th row,/->Representing that the weight of the row in the generation matrix G is greater than or equal to 2 ^n-p Index of subscript->Representing that the weight of the row in the generation matrix G is greater than or equal to 2 ^n-p-1 And less than 2 ^n-p Is a subscript index of (2); the method for constructing the code word is characterized by comprising the following steps:

s1, initializing a setGiving a signal-to-noise ratio SNR and a serial cancellation list path number L;

s2, assembling the commandWherein->

S3, constructing onePAC codeword of (C) then set->The index is the index set of information bit channel, and the method of transmitting all-zero code word under SNR and counting error code word weight at decoding end is adopted to estimate +.>Hamming distance spectrum a of PAC code _i The method specifically comprises the following steps: in the encoding process, the information bit is made to be all-zero bit, then the transmitting codeword is also made to be all-zero codeword, the serial offset list is used for decoding at the decoding end, the number L of the serial offset list paths is made to meet the requirement that the decoding result except the first path is correct, the rest (L-1) decoding errors are made, the hamming weight of the (L-1) codewords is approximate to the hamming distance spectrum of the PAC code, and the hamming distance spectrum A is obtained _i ；

S4, repeating the steps S2-S3, and selecting only the set each timeUntil traversing the set +.>All elements in (1) according to the formula

Calculating a joint boundary corresponding to each element; wherein, the arrayA _i Representation ofHamming distance spectrum obtained with each element of (a) as an information bit channel,/d->Representing code rate, E _b /N ₀ Representing the signal-to-noise ratio, Q representing the right tail function of the standard normal distribution;

s5, selecting the index of the subscript corresponding to the smallest joint boundary according to the calculation result in the step S4, and gatheringThe index-th element of (a) is added to the set +.>From the collection->Delete the element;

s6, repeating the steps S2-S5 until all the required information bit channels are selected, namely