CN108259128B - Method for constructing system Raptor code based on non-random generator matrix - Google Patents
Method for constructing system Raptor code based on non-random generator matrix Download PDFInfo
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Abstract
The invention relates to a method for constructing a systematic Raptor code based on a non-random generator matrix. The Raptor code in the invention is an inner code and an outer code, wherein the inner code is a non-rate LT code, and the outer code is a high-rate LDPC code. Both the LDPC code and the LT code are systematic codes. The Raptor encoder encodes the source bits into Raptor code words, and a first part of the generated Raptor code words is uncoded systematic code words and a second part is check code words. The sending end firstly sends uncoded systematic code words and then sends check code words in sequence according to the size sequence of the illumination. And after receiving a certain number of code words, the receiving end uses the global BP algorithm to carry out iterative decoding. Compared with the prior art, the invention has the advantages of higher throughput rate, smaller BER, simple structure and the like.
Description
Technical Field
The invention relates to physical layer coding in a wireless communication system, in particular to a method for constructing a system Raptor code based on a non-random generator matrix.
Background
In a wireless communication system, channel coding is one of the most central techniques for improving reliability of data transmission. The code rate of the traditional channel coding (such as LDPC, Turbo, etc.) only has a plurality of stepwise choices, and the switching of any code rate can not be realized. More importantly, a transmitting end using conventional channel coding predicts the quality of a channel according to the feedback condition of the channel, and then selects a certain code rate according to the quality of the channel. Since the wireless channel is time-varying, the prediction of the channel quality tends to be very delayed. This will result in the code rate and channel quality selected by the transmitting end not matching, resulting in a low throughput of the entire wireless communication system. The rateless code (or fountain code) is a code with an unfixed code rate, the encoder can continuously generate code words with any code rate, and a sending end does not need to predict the channel quality and only needs to continuously send the code words until an ACK signal fed back by a receiving end is received. LT codes are the first practical rateless codes, and Raptor codes have been proposed shortly. The Raptor is a cascade rateless code and is composed of an LDPC outer code with high code rate and an LT inner code, so that the performance of the Raptor code is better than that of a pure LT code. The earliest LT codes and Raptor codes were both application layer codes proposed for erasure channels. In order to ensure that the receiving end can recover the original information source by collecting any enough code words. The Raptor coder generates the code word in a random mode, which mainly includes two aspects: (1) raptor when generating a codeword, the degree of each codeword is randomly selected. (2) After the degree of each code word is determined, a plurality of LDPC middle bits are randomly selected to carry out modulo two addition. The wireless channel mainly comprises a white gaussian noise channel and a fading channel, and the two channels do not lose any code word, but only generate additive and multiplicative noise to the code word, so that the value of the code word is distorted. The conventional random Raptor code design is not well suited for wireless channels. Although many researchers have applied Raptor codes to the physical layer of wireless channels for research and improvement, the disadvantage of Raptor codes that use random generation matrices for codeword construction is not overcome. Therefore, the performance of the Raptor code still has a great space for improvement at present.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a method for constructing a systematic Raptor code based on a non-random generator matrix, which has high throughput rate and low decoding error rate.
The purpose of the invention can be realized by the following technical scheme:
a construction method of a systematic Raptor code based on a non-random generator matrix is characterized in that the Raptor code is divided into an inner code and an outer code, the inner code is a non-rate LT code, and the outer code is a high-code-rate LDPC code. The Raptor encoder encodes the source bits into Raptor codewords. The scheme comprises the following steps:
(1) let the one-dimensional vector composed of K source bits be s ═ siK-1, s passes through the LDPC encoder to generate N intermediate codewords b ═ biI ═ 0,1,. ang, N-1 }. The intermediate code words b further pass through an LT encoder to generate variable quantityRaptor codewords r. Let the non-random generator matrix of the LT encoder be G, then r ═ b · G.
(2) Since the LT code is a systematic code, the Raptor codeword r can be divided into two parts, which can be represented as r ═ r ', r ″, where r' is N uncoded systematic codewords, the same as the LDPC intermediate bit b. r "is L check codewords (the value of L is variable due to the nature of rateless codes and can theoretically be infinite). G can be written as G ═ I, P ], where I is a unit matrix of N × N and P is a sparse matrix of N × L.
(3) The uncoded codeword is solely one data packet, and the check codeword is divided into a plurality of data packets.
(4) The degree of the Raptor codeword is the number of LDPC intermediate bits selected per codeword.
(5) The code words in the check code word r' are grouped together, and the degrees of the code words are arranged in descending order from big to small. I.e. the number of non-zero elements in each column of G is fixed, rather than random. P can therefore be rewritten as: p ═ P1,P2,…,Pi,…,PI]In which P isiIs N × LiSubmatrix of degree D for productioniThe check codeword of (1).
(6) And sequentially selecting LDPC middle bits according to the sequence of each Raptor code word (including a system code word without codes). I.e. the positions of each column of non-zero elements in G are not randomly placed but are collectively placed together. By using
To represent PiL of (1)iRow, then
Including (N-D)i) Element with value 0 and DiAn element having a value of 1. In order to avoid the generation of rings in the generated matrix, the source bits need to be interleaved and rearranged once each time a certain number of check code words are generated.
In the above technical method, the non-random generator matrix G obtained in step (1) is the key technology of the present invention. Existing Raptor codes use a randomly generated matrix G. The reason for this is that Raptor codes are proposed for BEC erasure channels. A part of Raptor code words can be randomly lost in an erasure channel, so that the original source bits can be recovered by any enough Raptor code words collected by a receiving end. The existing Raptor encoder adopts a random generation matrix G to generate a code word, which is mainly embodied in two aspects: (1) when the Raptor generates the code words, the degree of each code word is randomly selected, namely the number of non-zero elements of each column in G (namely the degree of each Raptor code word) is random. (2) After the degree of each Raptor code word is determined, the code word randomly selects a plurality of LDPC middle bits to perform modulo two addition. I.e. the position of each column of non-zero elements in G is randomly placed. The wireless channel mainly includes a white gaussian noise channel and a fading channel. Unlike erasure channels, wireless channels do not lose any codewords, but only generate additive and multiplicative noise on Raptor codewords, distorting the codeword values. Therefore, the existing design scheme of the random Raptor code is not very suitable for a wireless channel. In order to adapt to a wireless channel, the invention innovatively provides a Raptor code construction method based on a non-random generator matrix G.
When the system is applied,
1. the sending end sends out the Raptor code words generated by the method of the invention in sequence according to a certain sequence. Specifically speaking: all the uncoded systematic code words r' have the degree of 1, and the systematic code words are gathered together and are the first code word, and the transmitting end firstly transmits all the uncoded code words. The uncoded systematic codeword is followed by the set of the most highly checked codewords, after which the degrees of the checked codewords decrease sequentially. The sending end sends the check code word with larger degree first and then sends the check code word with smaller degree until receiving the ACK signal fed back by the receiving end.
2. And the receiving end carries out iterative decoding by using a global BP algorithm. The method specifically comprises the following steps: and after the receiving end collects all the N uncoded code words, decoding by using a global BP iterative decoder. And if the decoding cannot be successfully decoded, continuously waiting for the transmitting end to transmit more check code words. When a certain number of check code words are collected, decoding is tried together with the previous uncoded code words. And continuing until decoding is successful, and sending an ACK signal to the sending end after the decoding is successful. At the receiving end, the implementing step may include
(1) In each iteration of the BP algorithm, LLR (log-likelihood ratio) completes updating of the V node and the C node in the LT decoder once, and then completes updating of the V node and the C node in the LDPC decoder once.
(2) And outputting decoding bits through hard decision after multiple global iterations.
The innovation point of the invention is embodied in two aspects:
(1) the present invention relates to the position of the code word of the rateless code to the channel condition for the first time. When the channel condition is good, the transmitting end only needs to transmit a small number of code words, and the receiving end can decode correctly. When the channel condition is poor, the transmitting end needs to transmit more code words and the receiving end can decode correctly. Therefore, when the channel condition is better, the code word sent by the sending end has a larger degree, so that each code word can cover more information source bits, thereby saving the number of the code words needing to be sent and improving the throughput rate of the whole system. When the channel condition is poor, the code word sent by the sending end needs to have a smaller degree, so that the code word polluted by the channel can be decoded successfully more easily. Based on the premise, in the Raptor code words generated by the method, the check code words with the same degree are grouped together, and the distribution of the degree is arranged in a descending order from big to small. That is, after the sending end sends the systematic code word without codes, the check code word sent first has a larger degree, and the check code word sent later has a smaller degree.
(2) In the conventional Raptor code, several LDPC intermediate bits are randomly selected per Raptor codeword. When the codeword length is short, some LDPC intermediate bits may be selected multiple times, while other LDPC intermediate bits may not be selected, which may result in the receiver not being able to fully recover the source, resulting in a lower throughput. In the invention, each code word (including a system code word without code) sequentially selects LDPC middle bits according to the sequence.
Compared with the prior art, the system Raptor code designed based on the non-random generator matrix provided by the invention can obviously improve the throughput rate of the system and the BER after decoding.
Drawings
Fig. 1 is a schematic flow chart of a transmitting end and a receiving end in the application of the present invention.
Fig. 2 is a schematic diagram of an LT encoder in the present invention generating Raptor codewords from LDPC intermediate bits.
Fig. 3 is a detailed partial view of fig. 2, which shows a method for selecting LDPC intermediate bits by a Raptor check codeword.
Fig. 4 is a comparison of throughput (goodput) performance of the present invention when combined with 16QAM versus the conventional scheme.
Fig. 5 shows the comparison of the throughput (goodput) performance of the present invention when combined with 64QAM versus the conventional scheme.
Fig. 6 is a comparison of Bit Error Rate (BER) performance of the present invention when combined with 16QAM versus the conventional scheme.
Fig. 7 is a comparison of Bit Error Rate (BER) performance of the present invention when combined with 64QAM versus the conventional scheme.
Detailed Description
The invention is described in detail below with reference to the figures and specific embodiments.
Examples
The invention provides a method for constructing a system Raptor code based on a non-random generator matrix. The Raptor code of the invention is a cascade code and consists of an LDPC outer code with high code rate and an LT inner code without rate. The method comprises the steps that an information source bit firstly enters an LDPC coder to generate an LDPC middle bit, and the LDPC middle bit then enters an LT coder to generate a Raptor code word. The first part of the Raptor code word is an uncoded systematic code word and the second part is a check code word. The check code words with the same degree are gathered together, and the degrees are arranged in descending order from big to small. And sequentially selecting LDPC middle bits according to the sequence of each Raptor code word (comprising a system code word and a check code word). However, in order to avoid the generation of loops in the generator matrix, the source bits need to be interleaved and rearranged every time a certain number of check code words are generated. The sending end firstly sends all system code words and then sends check code word data packets in sequence until receiving the ACK signal fed back by the receiving end. And after collecting a certain number of Raptor code words, the receiving end decodes the code words by using a global BP iterative algorithm. The block diagram of the transmitting end and the receiving end of the whole system is shown in fig. 1.
The specific example steps are as follows:
first, fig. 2 is a bipartite graph of an LT encoder encoding process. The circles in the figure represent the N LDPC intermediate bits and the squares represent Raptor codewords. From the figure we can see that the first N Raptor codewords are degree 1 uncoded systematic codewords, which are identical to the LDPC intermediate bits. The systematic codeword is followed by a check codeword. The set of degrees of these check codewords is { D1,D2,…,Di,…,DI}(Di>Di+1) The corresponding code word length is { L }1,L2,…,Li,…,LI}. Wherein L isiThe relationship with N is: l isi＝N×xi(1. ltoreq. I. ltoreq.I), where xiCan be adjusted according to actual conditions. In a particular design application, the source bit length is 4096. The code rate of the LDPC code is taken to be 0.95. The set of degrees of the check code word designed by enumeration is {12,6,4,2}, xiIs taken as x1≈0.27,x2≈0.146,x3≈0.27,x4The value of (A) is not fixed and can be infinite.
Step two, fig. 3 is a partial detailed view of the LT encoder in fig. 2. Wherein
For successive 4 check code words, code words
Degree of (D)jCode word rI,0,rI,1,rI,2The degrees of all the points are 2. { bn-2,bn-1,bn,bn+1,…,bn+5Is the consecutive 8 LDPC middle bits. In the context of figure 3, it is shown,
the last intermediate bit selected is bn-1Then r isI,0Selection bn-1The next two consecutive intermediate bits bn,bn+1R in the same wayI,1Selection bn+1The next two consecutive intermediate bits bn+2,bn+3}，rI,2Selection bn+3The next two consecutive intermediate bits bn+4,bn+5}。
Step three, according to the design in step one and step two, the generator matrix is represented by G, and G can be written as G ═ I, P]Where I is a unit matrix of N x N and P is a sparse matrix of N x L (where L is variable in length and can theoretically be infinitely long), P can be rewritten as P ═ P1,P2,…,Pi,…,PI]In which P isiIs N × LiA sub-matrix of
To represent PiL of (1)iRow, then
Including (N-D)i) Element with value 0 and DiAn element having a value of 1. And the positions of these elements with a value of 1 are grouped together, assuming their positions from k' to k ". Then the following formula is given:
and fourthly, after the receiving end collects all the N uncoded code words, decoding by using a global BP iterative decoder. And if the decoding cannot be successfully decoded, continuously waiting for the transmitting end to transmit more check code words. When a certain number of check code words are collected, decoding is tried together with the previous uncoded code words. And continuing until decoding is successful, and sending an ACK signal to the sending end after the decoding is successful.
The BP iterative decoding algorithm is specifically as follows:
(41) in each iteration of the BP algorithm, LLR (log-likelihood ratio) completes updating of the V node and the C node in the LT decoder once, and then completes updating of the V node and the C node in the LDPC decoder once.
(42) And outputting decoding bits through hard decision after multiple global iterations.
Fig. 4 is a comparison of throughput (goodput) performance of the present invention when combined with 16QAM versus the conventional scheme. Fig. 5 shows the comparison of the throughput (goodput) performance of the present invention when combined with 64QAM versus the conventional scheme. In fig. 4 and 5, the horizontal axis represents the signal-to-noise ratio (SNR) of the channel, and the vertical axis represents the throughput curve. Fig. 6 shows the comparison of the Bit Error Rate (BER) performance of the present invention in combination with 16QAM (10 dB channel signal to noise ratio) versus the conventional scheme. Fig. 7 shows the comparison of the Bit Error Rate (BER) performance of the present invention in combination with 64QAM (10 dB channel signal to noise ratio) versus the conventional scheme. In fig. 6 and 7, the horizontal axis represents the reciprocal (1/R) of the code rate of the Raptor code, and the vertical axis represents the Bit Error Rate (BER) curve. The curves indicated by circles ' o ' in the four graphs of fig. 4 to fig. 7 are performance graphs of the Raptor code proposed by the present invention, and the curves indicated by asterisks ' are performance graphs of the Raptor code of the existing scheme, and it can be seen from fig. 4 and fig. 5 that the performance of the present invention is much better than that of the existing Raptor code under the condition of medium and high signal-to-noise ratio, and is slightly better than that of the existing Raptor code under the condition of low signal-to-noise ratio. It can be seen from fig. 6 and 7 that the performance of the present invention in terms of BER is much better than that of the existing Raptor code.
The above description is only a preferred example of the present invention, and the scope of the claims of the present invention is not limited thereto. The present invention is also directed to various embodiments, and various changes and modifications may be made by one skilled in the art without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (1)
1. A construction method of a systematic Raptor code based on a non-random generator matrix is characterized in that the Raptor code is divided into an inner code and an outer code, wherein the inner code is a rate-free LT code, and the outer code is a high-code-rate LDPC code; the Raptor coder codes the source bit into Raptor code word; the method comprises the following steps:
(1) let the one-dimensional vector composed of K source bits be s ═ siK-1, s passes through the LDPC encoder to generate N intermediate codewords b ═ biI ═ 0,1, ·, N-1 }; the intermediate code word b further generates Raptor code words r with variable quantity after passing through an LT encoder; let the non-random generating matrix of the LT encoder be G, then r equals b.G;
(2) since the LT code is a systematic code, the Raptor codeword r is divided into two parts, denoted as r ═ r ', r ″, where r' is N uncoded systematic codewords, the same as the LDPC intermediate bit b; r' is L check code words; g is written as G ═ I, P ], where I is a unit matrix of N × N and P is a sparse matrix of N × L;
(3) the uncoded systematic code words are independently a data packet, and the check code words are divided into a plurality of data packets;
(4) the degree of the Raptor code word is the number of LDPC middle bits selected by each code word;
(5) the moderate same code words in the check code words r' are gathered together, and the degrees of the code words are arranged in a descending order from big to small, namely the number of the nonzero elements in each column in G is fixed and is not random; thus, P is rewritten as: p ═ P1，P2，…，Pi，…，PI]In which P isiIs N × LiSubmatrix of degree D for productioniThe check code word of (1);
(6) sequentially selecting LDPC middle bits according to the sequence of each Raptor code word; that is, the positions of each column of non-zero elements in G are not randomly placed, but are collectively placed together; by using
To represent PiL of (1)iRow, then
Including N-DiElement with value 0 and DiAn element having a value of 1.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN102820952A (en) * | 2012-08-02 | 2012-12-12 | 林子怀 | Method for optimizing throughput of wireless network system based on no-rate encoding |
| CN106685597A (en) * | 2017-02-14 | 2017-05-17 | 华南理工大学 | Raptor code part ring elimination coding method based on degree optimization |
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Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN102244922A (en) * | 2011-08-08 | 2011-11-16 | 浙江大学 | Power control method applicable to Raptor Codes under additive white Gaussian noise channel |
| CN102820952A (en) * | 2012-08-02 | 2012-12-12 | 林子怀 | Method for optimizing throughput of wireless network system based on no-rate encoding |
| CN106685597A (en) * | 2017-02-14 | 2017-05-17 | 华南理工大学 | Raptor code part ring elimination coding method based on degree optimization |
Non-Patent Citations (1)
| Title |
|---|
| Work in progress: A new algorithm to improve the decoding success probability of Raptor code;Huihui Li;《9th International Conference on Communications and Networking in China》;20140816;全文 * |
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