CN107124251B

CN107124251B - Polarization code encoding method based on any kernel

Info

Publication number: CN107124251B
Application number: CN201710425690.9A
Authority: CN
Inventors: 武畅; 袁超; 杨路遥; 李豪; 闫康旭
Original assignee: University of Electronic Science and Technology of China
Current assignee: University of Electronic Science and Technology of China
Priority date: 2017-06-08
Filing date: 2017-06-08
Publication date: 2020-03-10
Anticipated expiration: 2037-06-08
Also published as: CN107124251A

Abstract

The invention relates to a polarization code coding method based on any kernel, which comprises the following steps: (1) factorization is carried out on N bits of input data to obtain a kernel factor, (2) a corresponding kernel matrix is obtained according to the kernel factor, (3) a polar code generating matrix is obtained by multiplying the kernel matrix through kronecker, and (4) the polar code generating matrix and a bit overturning matrix are subjected to kronecker multiplication to obtain a new generating matrix, and (5) the new generating matrix and an original bit sequence are calculated to obtain the polar code after bit overturning. The invention provides the construction of different kernels to generate a diversified coding matrix, and the method can generate a plurality of coding schemes, is necessary for the research of the polarization code, and has simple calculation and smaller calculation amount.

Description

Polarization code encoding method based on any kernel

Technical Field

The invention relates to the technical field of channel coding of digital communication, in particular to a polarization code coding method based on any kernel.

Background

Polar Codes (Polar Codes) are a constructive channel coding method which is strictly proved to reach channel capacity and proposed by e.arikan in 2009, and can generate an inner core with any size, wherein the inner core is a key for generating the Polar Codes, and a generating matrix of the Polar Codes can be easily generated by the inner core. The method can be used for generating matrixes with any size and is suitable for codes with any length. Meanwhile, the kernel generated by the method has better effect than the kernel with the size of 2 x 2 which is commonly used at present. By using the method, the complexity of generating the matrix can be reduced, and the realization is simpler.

At 18/11/2016, 3GPP determined Polar code scheme, pioneered by china corporation, hua, et al, as a control channel coding scheme for 5G eMBB (enhanced mobile broadband) scenarios, at RAN1#87 conference of 3GPP ending in lino, nevada, usa. So far, the technical scheme of channel coding for 5G eMBB (enhanced mobile broadband) scenarios is completely determined, wherein Polar codes are used as the coding scheme of the control channel. The more research on the polarization code, the more important it is for the generation of the polarization code.

In 2008, Erdal Arikan first proposed the concept of channel polarization (ChannelPolarization) at the international information theory ISIT conference; in 2009, a paper published in the journal "IEEE transmission on Information Theory" up to page 23 describes channel polarization in more detail, and a new coding method, named Polar Code, is given based on the channel polarization. The polar code has a deterministic construction method and is the only known channel coding method that can be strictly proven to "reach" the channel capacity.

Polar Code was constructed by introducing the concept of channel polarization. The channel polarization is divided into two phases, channel joint and channel split respectively. Through the combination and division of the channels, the symmetrical capacity of each sub-channel will show a tendency of two-stage differentiation: as the code length (i.e., the number of joint channels) N increases, the capacity of some of the subchannels tends to 1, while the capacity of the remaining subchannels tends to 0. Polar Code is exactly using this phenomenon of channel polarization to transmit message bits on K subchannels whose capacity tends to 1, and frozen bits (i.e. fixed bits known to both the transmitting and receiving parties, usually set to all zeros) on the remaining subchannels. The Code formed by the method is Polar Code, and the Code rate is K/N.

The polar code in the prior art has the basic coding elements of a general binary linear block code, so that the coding can be completed by writing out the generator matrix thereof in a display mode:

wherein the content of the first and second substances,

in order to be the original bit sequence,

for the coded bit sequence, G_NTo generate the matrix, the code length is N-2ⁿ。

The generator matrix is represented as:

wherein

Is to the matrix

The n-times kronecker product, we call F the kernel.

Having a recursive formula

By this method, the encoding matrix G can be generated.

FIG. 1 is a schematic diagram of a two-bit codeword encoding using a kernel

Similarly, if the code word is a four-bit code word, the code word is

The generating matrix is as follows:

the corresponding encoding scheme (after bit flipping) is shown in fig. 2:

the following is an example of 8 bits (after bit flipping):

in this way, a code length of N-2 can be constructedⁿGenerating a matrix. In a practical digital communication system, the code length is very flexible and does not necessarily satisfy the power of 2, and thus the code length is not N-2ⁿWhat to do with the existing method is as follows: if the code length is not the power of 2 when the polarization code is constructed, the number of channels is complemented to the power of 2 by a group of virtual channels with zero capacity, then each channel is interleaved and mapped according to the capacity equal division principle, the obtained channels are subjected to polarization transformation, in the transformed channels, the channel with larger channel capacity is selected according to the designed code rate to be used for transmitting the information bit sequence, and the rest channels are used for transmitting the fixed bit sequence known by a transmitting and receiving end.

Because the current research is common, the kernel is used

Few people use other kernels to research, for codes with any code length, the channel capacity after channel polarization is calculated, the number and the position of coded points are selected according to code rate, a lot of calculation is needed, and no corresponding generation method exists for matrixes with other sizes. Therefore, it is necessary to provide a method for generating kernels with any size, which has a small calculation amount and is simple to implement, so as to solve the need for different code lengths. At the same time, kernel performance of other dimension sizes and fixed dimension kernel

The resulting performance is not the same. The method can facilitate researchers to do better research and carry out deep research on the coming 5G.

Disclosure of Invention

The technical problem to be solved by the invention is to provide a polar code encoding method based on any kernel, so that the code length of the polar code encoding is allowed to be any kernel, the calculation amount is small, and the realization is simple.

The technical scheme for solving the technical problems is as follows:

a polar code coding method based on any kernel comprises the following steps:

(1) factorizing input N-bit data to obtain corresponding K kernel factors q_iAnd satisfies the following relationship:

wherein q is_i>1, taking an integer, wherein the value range of i is 1 to K;

(2) according to said kernel factor q_iComputing kernel matrices

Is a matrix of m x m dimensions, m q_i,

Column vector g expressed in m dimensions 1 x m_j：

The value range of j is 1 to m;

(3) the K kernel matrixes are processed

Obtaining a polarization code generation matrix G of N-bit data by Rogok multiplication_N：

Wherein the content of the first and second substances,

representing a kronecker multiplicative;

(4) generating the polarization code into a matrix G_NRatio of sumCarrying out kronecker multiplication on the special flip matrix to obtain a new generator matrix G_BN：

Wherein the content of the first and second substances,

is a bit flip matrix of dimension m x m, m q_i；

(5) Generating the new generator matrix G_BNAnd calculating with the original bit sequence to obtain the polarization code.

The invention has the beneficial effects that: the invention can simply generate kernel factors with any size, and the factors can be combined to generate a generating matrix with any size, thereby realizing simple and changeable polar code coding matrixes, performing bit reversal on each kernel factor to reduce the relevance of information, reducing the interference among information bits, improving the fault-tolerant performance of the whole code, deepening the polarization degree and realizing better error correction performance, and the method has simple calculation and smaller calculation amount.

On the basis of the technical scheme, the invention can be further improved as follows.

Further, the kernel matrix of the step (2)

Column vector g in (1)_jObtained by the following method:

g₁＝[1 1 1 1 1 … 1]，

g_m＝[0 0 0 0 0 … 1]，

wherein the content of the first and second substances,

representing any one value selected from the above, and represented as binary, m ═ q_i。

The further scheme has the advantages that the relevance degree of information can be reduced by using the bit flipping of each kernel factor, the interference among information bits is reduced, the fault-tolerant performance of the whole code can be improved, the polarization degree is deepened, and the better error-correcting performance is realized.

Further, the bit flipping matrix is:

wherein the content of the first and second substances,

is a permutation matrix of dimension 0m x m, m q_i；

The bit flipping matrix

The calculation process is as follows:

when m is an even number:

B_m/2is a matrix of (m/2) × (m/2) dimensions, B_m/2By the way,

iteration is carried out to obtain the product, and the iteration times are m/2 times;

then B is_m＝(b₁b₂... b_m)；

When m is an odd number, m1 is an even number if m1 is m-1;

then

B_m1/2Is a matrix of (m1/2) × (m1/2) dimensions, B_m1/2By the way,

iteration is carried out, and the iteration number is m 1/2;

let b_sIs a 1 x m dimensional matrix (000 … 1 … 0)^T，

Wherein 1 is in the matrix b_sThe subscript position of (b) is (m-1)/2_sInsert B_m1Is located in the middle position of (a),

then B is_m＝(b₁b₂… b_s… b_m1-1b_m1)。

The further scheme has the advantages that the bit flipping of each kernel factor can reduce the relevance of information, reduce the interference among information bits, improve the fault-tolerant performance of the whole code, deepen the polarization degree and realize better error-correcting performance.

Drawings

FIG. 1 is a schematic diagram of two-bit codeword encoding;

FIG. 2 is a diagram of four-bit codeword encoding;

FIG. 3 is a flow chart of the method of the present invention;

FIG. 4 is a graph of the simulation effect of the kernel 3 x 3 of the present invention;

FIG. 5 is a diagram of simulation results of different kernels of the present invention.

Detailed Description

The principles and features of this invention are described below in conjunction with the following drawings, which are set forth by way of illustration only and are not intended to limit the scope of the invention.

As shown in fig. 3, the present invention provides a polar code encoding method based on any kernel, which is characterized by comprising the following steps:

wherein q is_i>1, taking an integer, wherein the value range of i is 1 to K;

(2) according to said kernel factor q_iComputing kernel matrices

Is a matrix of m x m dimensions, m q_i,

Column vector g expressed in m dimensions 1 x m_j：

The value range of j is 1 to m;

(3) the K kernel matrixes are processed

Wherein the content of the first and second substances,

representing a kronecker multiplicative;

(4) generating the polarization code into a matrix G_NPerforming a kronecker product with the bit flip matrix to obtain a new generator matrix G_BN：

Wherein the content of the first and second substances,

is a bit flip matrix of dimension m x m, m q_i；

The bit flipping matrix is:

wherein the content of the first and second substances,

is a permutation matrix of m x m dimensions, m q_i；

The bit flipping matrix

The calculation process is as follows:

when m is an even number:

B_m/2is a matrix of (m/2) × (m/2) dimensions, B_m/2By the way,

then B is_m＝(b₁b₂... b_m)；

When m is an odd number, m1 is an even number if m1 is m-1;

then

B_m1/2Is a matrix of (m1/2) × (m1/2) dimensions, B_m1/2By the way,

iteration is carried out, and the iteration number is m 1/2;

let b_sIs a 1 x m dimensional matrix (000 … 1 … 0)^T，

Wherein 1 is in the matrix b_sThe subscript position of (b) is (m-1)/2_sInsert B_m1InAt the middle position of the upper plate, the lower plate,

then B is_m＝(b₁b₂… b_s… b_m1-1b_m1)。

The kernel q in the step (2)_iCorresponding kernel matrix

Column vector g in (1)_jObtained by the following method:

the general case is as follows:

g₁＝[1 1 1 1 1 … 1]，

g_m＝[0 0 0 0 0 … 1]，

wherein the content of the first and second substances,

In general, it is better to refer to the following cases, where N is a prime number, then the kernel factor q is taken as N directly, and if N can be logarithmic, then the small logarithmic factor m is taken, and the kernel factor q is taken as m; when generating a kernel matrix, we generally let the first column all be 1, g₁＝[11111…1]The values of the other column vectors are taken larger from the three, and generally do not result in column vectors that are all 0's.

The following is a specific method of generating kernels of arbitrary size:

(1) and carrying out logarithm taking and factorization on the input data with N bits to obtain the most basic kernel factor. For prime number M, factorization cannot be performed, and the value M is directly taken as a factor. For example, 51 is the kernel factor 51, and in other cases, for example, N is 1024, the logarithm is taken directly (generally, the minimum logarithm factor, but other values are also possible), and 1024 is 10242¹⁰The most basic kernel is used directly.

Performing 10 Crohn's products to generate a matrix

And (4) finishing.

For example, N36, to which the logarithm 36 is 6²Constructing cores of size 6 x 6 and 6 x 2 x 3, then constructing cores of

sizes

2 and 3.

(2) After obtaining the kernel factor, the kernel is constructed in the following way.

(a) Suppose the kernel factor is K, then the kernel is a matrix of dimensions K x K. The values of the elements of the matrix are only two: 1 and 0;

(b) in general: the value of the first column is 2^k-1, 2^k-1 is represented in binary, i.e. the elements of the first column, i.e. k 1, g₁＝[1 1 … 1]；

For the last column: g_k＝[0 0 ... 0 1],g_kIs a matrix of 1 x K.

The next second column up to the penultimate column having a value g_j＝2^j，g_j＝2^jOne of +/-1 is arbitrarily selected and expressed as binary, and the value is g_jThe elements of the column.

The general approach is that each column can be used, but there are generally no duplicate columns and the order of the columns can be swapped.

(c) There are a plurality of matrices generated in (b), the number of which is M > 3^k-1And (4) respectively. If the logarithm of the code length N is a single factor, the method in (b) is used. If it is multifactorial, e.g. 6-2 x 3, the corresponding generator matrix

The kronecker product is used to generate.

This method is different from that of any one produced by (b).

Because the generated kernel matrixes in the polarization processing are various, after bit inversion is added, the coding schemes are various, the performance of each coding scheme is different, the smaller kernel matrixes can be used for processing firstly, the performance of each small kernel matrix is calculated and verified to obtain the optimal performance, and then the optimal kernel matrix is used for performing the kronecker product, namely the optimal kernel matrix is used for performing the kronecker product, so that the performance is optimal.

As shown in fig. 4, the kernel is 3 × 3 production and effect:

production of G by this method₃(7,5,1)，7₂＝(1 1 1)，5₂＝(1 0 1)，1₂＝(0 0 1)

Then

In the same way

Not all of the above are listed.

The corresponding simulation results are as follows: with N equal to 3⁵243, the channel is gaussian white noise channel, the signal-to-noise ratio SNR is 0dB,

decoding adopts a Serial Cancellation List (SCL) decoding algorithm, and the parameter L is called a search width. Wherein L ═ 8.

Figure 5 shows a simulation diagram of a plurality of different cores,

are respectively provided with

The code rate is approximately 0.5, the added channel is a white Gaussian noise channel, the decoding adopts a Successive Cancellation List (SCL) decoding algorithm, and the parameter L is the search width。

It can be seen that different kernels have influence on the error rate, so that the establishment of different kernels is provided, and the research on the polarization code is necessary.

The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.