A polar code channel construction method based on frozen bit improvement

Abstract

The present invention specifically relates to a polar code channel construction method based on frozen bit improvement. The present invention belongs to the field of channel coding technology. The method first estimates the reliability measure of each polarization sub-channel by using the Gaussian approximation principle, selects the most reliable channel, then finds the row weight of the generator matrix corresponding to the selected channel, and selects the channel with smaller row weight again. Finally, according to the two proposed frozen channel setting principles, namely: the principle of setting frozen channels with appropriate bits for channel segments with different reliabilities and the principle of setting frozen channels at intervals of long information channel segments with smaller row weights and at the back, select channels with appropriate bits and set them as frozen channels, and the remaining channels are information channels, and SCL decoding is used for decoding. Simulation results show that the polar code channel construction method based on frozen bit improvement proposed by the present invention has obvious performance improvement, and the computational complexity is similar.

Description

Summary of the invention

[0001] In view of this, the purpose of the present invention is to provide a polar code channel construction method based on frozen bit improvement. By combining the construction of RM code, channels with high reliability but small row weight are selected, and the error correction and verification functions of frozen bits in SCL decoding are further analyzed, and two frozen channel setting principles are proposed: the principle of setting frozen channels with appropriate bits for channel segments with different reliability and the principle of setting frozen channels at intervals of long information channel segments with small row weights and at the back, setting the frozen channels in the selected channels, and making full use of the verification capability of frozen bits, thereby improving the decoding performance.

[0002] In order to achieve the above object, the present invention provides the following technical solutions:

[0003] First, if an all-zero sequence is input into the Additive White Gaussian Noise (AWGN) channel, since the logarithm likelihood ratio (LLR) corresponding to the source sequence is a Gaussian random variable with a variance twice the mean, the Gaussian approximation principle is used to estimate the LLR expectation of each polarization subchannel, and the larger the LLR expectation, the higher the corresponding channel reliability metric.

[0004] Then, list the row weights of the generator matrix corresponding to the K+f channels with the highest reliability (where K is the number of information sequences and f is the number of frozen channel settings), and select the channel set A with the smallest row weight again. m and the second smallest channel set A s , freeze the channels from set A m and A s When f is less than |A m |, the frozen channel is only set to A m In; when f is greater than |A m |, all minimum row weight channels are set as frozen channels, and in A s Select f-|A m |Set the channel to a frozen channel.

[0005] Finally, by analyzing the error correction and check function of frozen bits in SCL decoding, the frozen bits are regarded as an error correction code and a check code, and the number and position of frozen bits are studied and analyzed in depth, and two frozen channel setting principles are obtained, namely:

[0006] Principle 1: Set frozen channels with appropriate number of bits for channel segments with different reliability.

[0007] Principle 2: Set frozen channels at intervals in long information channel segments with smaller row weights and at the back.

[0008] According to the principle of setting two frozen channels, f-bit channels are selected from the K+f-bit channels with the highest reliability as frozen channels, and the remaining K-bit channels are information channels. SCL decoding is used for decoding.

[0009] The beneficial effect of the present invention is that the method of the present invention analyzes the error correction and verification functions of frozen bits in SCL decoding, and by changing the setting of the number and position of frozen channel bits, the error correction capability of the polar code itself in SCL decoding can be improved. Simulation results show that compared with the construction method of RM-Polar code in the literature [1] "W. Wu, Z. Zhai, PHSiegel. Improved Hybrid RM-Polar Codes and Decoding on Stable Permuted Factor Graphs [C]. 2021 11th International Symposium on Topics in Coding (ISTC), 2021: 1-5.", the FRM-Polar code construction method proposed in the present invention takes into account the distance and mutual connection between channels, as well as the verification function of frozen bits, and is more suitable for SCL decoding. Compared with the RM-Polar code construction method in reference [1] and the Gaussian approximation construction method in reference [2] "P. Trifonov. Efficient Design and Decoding of Polar Codes [J]. IEEE Transactions on Communications, 2012, 60 (11): 3221-3227.", the FRM-Polar construction method has better decoding performance. After cascading the cyclic redundancy check (CRC) code, a simulation comparison analysis was conducted with the RM-Polar code construction method and the Gaussian approximation construction method. The performance of the FRM-Polar construction method is still significantly improved at high signal-to-noise ratio. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] In order to make the purpose, technical solution and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0011] Figure 1 A technical roadmap for the method of the present invention;

[0012] Figure 2 Performance difference graph for setting channels with the same row weight and similar reliability as frozen channels;

[0013] Figure 3 Performance difference diagram for the same long information channel with the same row weights but different position channels set as frozen channels;

[0014] Figure 4 The block error rate curves of different channel construction methods;

[0015] Figure 5 The block error rate curves of different channel construction methods after cascading CRC codes; DETAILED DESCRIPTION

[0016] The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0017] (1) Combined with Figure 1 The specific implementation process of a polar code channel construction method based on frozen bit improvement is as follows:

[0018] Step 1: The polar code is subjected to an additive white Gaussian noise (AWGN) channel using GA to obtain the logarithm likelihood ratio (LLR) expectation and reliability metric of each polarization subchannel.

[0019] Consider using BPSK modulation in an AWGN channel with a polarization code length of N. The corresponding LLR is expressed as Assume the input sequence is an all-zero codeword, then Obey N(2 / σ 2 ,4 / σ 2 ), it can be seen that all LLRs are Gaussian random variables with variance twice the mean, so according to the Gaussian approximation principle, we can get equations (1) to (3).

[0020]

[0021]

[0022]

[0023] in

[0024]

[0025] By repeatedly using formula (1) and formula (2), the final calculation of each polarization subchannel is The LLR expectation is larger, and the corresponding channel After obtaining the reliability metrics of all channels, select the K+f channels with the highest reliability as information channels, where K is the number of information bits and f is the number of bits set for the frozen channel.

[0026] Step 2: Find the row weights of the generator matrix corresponding to the K+f channels with the highest reliability, and then select all the channels with the smallest and second smallest row weights.

[0027] For a code length of N = 2 n The polar code can be generated by the matrix G N Realization, G N The row weight of the i-th row (row numbers start at 1) is 2 wt(i) , where wt(i) represents the Hamming weight of the binary number i-1, and the minimum distance of the polar code is equal to G N The minimum row weight of the row corresponding to the information bit in the middle is selected, and the channel with smaller row weight is set as the frozen channel, which can reduce the low-weight codewords of the polar code and increase the MHW.

[0028] Before selecting the number and position of frozen channels, it is necessary to define formulas (5), (6) and (7):

[0029] f=log2N×(α-|K / N-1 / 2| 2 ) (12)

[0030] If f in formula (5) is not an integer, it is rounded to the nearest integer. α is a coefficient for adjusting the number of frozen bits, usually 0<α<1. According to formula (5), the number of frozen bits can be flexibly selected by adjusting the value of α.

[0031]

[0032]

[0033] In formula (6) and formula (7), d m and d s The MHW and sub-MHW are respectively the non-frozen bit sets, and the frozen bits are from set A. m and A s When f is less than |A m |, the frozen channel is only set in the channel with the minimum row weight; when f is greater than |A m |, all channels with the smallest row weights are set as frozen channels, and f-|A is selected from the channels with the second smallest row weights. m |Set the channel to a frozen channel.

[0034] Step 3: According to the two proposed frozen channel setting principles, namely, the principle of setting frozen channels with appropriate bits for channel segments with different reliabilities and the principle of setting frozen channels at intervals of long information channel segments with smaller and later row weights, select f-bit channels from the smallest and second smallest row weight channels and set them as frozen channels, and use SCL decoding for decoding.

[0035] In SCL decoding, when the frozen bit judgment result does not conform to the LLR judgment result, the path metric (PM) value will be added to the corresponding path. The path with a larger PM value is considered to have a higher probability of error and will be deleted. There are two situations that can cause frozen bit judgment errors: the first is the influence of noise, and the second is the error of the previous decoding bit. The noise is random and uncontrollable, so the position effect is not considered. For the second situation, the ideal state in the SCL decoding process includes 1 correct path and L-1 wrong paths. When the wrong path is translated to the frozen bit, the probability of LLR judgment error is greater. The increase in PM value will cause the wrong path to be pruned, which can increase the probability of the correct path being retained to the end. Therefore, the frozen bit also has some implicit verification capabilities. The frozen bit is regarded as an error correction code and a check code. The number of frozen bits and the placement position are deeply studied and analyzed, and two frozen channel setting principles are obtained, namely:

[0036] Principle 1: Set frozen channels with appropriate number of bits for channel segments with different reliability.

[0037] Principle 2: Set frozen channels at intervals at the back of long information channel segments with smaller row weights.

[0038] According to the principle of setting two frozen channels, f-bit channels are selected from the channels with smaller row weights as frozen channels, and the remaining K-bit channels are information channels. SCL decoding is used for decoding. This can not only reduce the low-weight codewords of the polarization code and optimize the distance spectrum of the polarization code, but also make full use of the verification capability of the frozen bits in SCL decoding to improve the error correction performance of the polarization code.

[0039] (2) Combined with attachment Figure 2 The first freezing channel setting principle is described as follows:

[0040] The simulation conditions are under AWGN channel, BPSK modulation, Gaussian approximation construction method with SNR=2.5dB, polar code length N=1024, code rate R=0.5, SCL decoding, and list size L=8. First, the distribution of (1024,512) polar code channels is listed as shown in Table 1.

[0041] Table 1: Number of segmented channels of (1024, 512) polar codes

[0042] Channel Segment Number of information channels Freeze Channel Number 1-128 1 127 129-256 18 110 257-384 28 100 385-512 88 40 513-640 42 86 641-768 100 28 768-896 108 20 897-1024 127 1

[0043] From Table 1, we can see that there are fewer frozen channels in the later channel segments. For example, in the 897-1024 channel segment, there is only one frozen channel among the 128 channels, which will result in extremely poor verification pruning capability of the information segment and reduced error correction performance. Next, we select four channels with similar reliability from the channels with the smallest row weights among all information channels: the 737th channel; the 837th channel; the 809th channel; and the 835th channel, whose reliability metrics are 26.5425; 26.5414; 24.1931; and 24.1908, respectively. We set them as frozen channels, and obtain the performance comparison chart as shown below: Figure 2 shown.

[0044] Depend on Figure 2 It can be seen that there is a significant performance gap at this time. When the two channels have the same weight and similar reliability, the performance is better when the position is later, and the performance gap is larger when the distance is larger. Combined with the distribution of channel segments, the 737th channel belongs to the sixth channel segment. The verification capability of the channel segment with more frozen channels is close to saturation. Adding a frozen channel has little impact on the performance. The 837th channel belongs to the seventh segment. The performance of the channel segment with fewer frozen channels is improved more by adding a frozen channel. In contrast, the 809th and 835th channels both belong to the seventh segment, and the performance gain is smaller. From the above analysis, we can get the first principle of frozen channel setting: set the appropriate number of frozen channels for channel segments with different reliability. Now analyze the phenomenon that the distance between the 835th and 837th channels is small, but the performance gap is large. Combined with the original frozen channel distribution, it is found that the 833th channel is a frozen channel. When the 835th channel is set as a frozen channel, only the 834th channel can be verified. The distance is too close, resulting in too few verification bits, and the verification capability of the frozen bit cannot be fully utilized. The reliability metric of the 837th channel is larger, but setting it as a frozen channel has better performance, which proves that the setting position of the frozen channel has a greater impact on the performance.

[0045] (3) Combined with attachment Figure 3 The second freezing channel setting principle is described as follows:

[0046] From the above analysis, we can know that when setting the frozen channel, the distance from the original frozen channel should be considered. Conventional construction sets the channel with the lowest reliability as the frozen channel, but most of these channels are located in the front, have fewer check bits, and have weak error correction capabilities. In order to balance the impact of these two factors, the following research was conducted: the two original frozen channels, 833 and 897, were found. Channels 834 to 896 are information channel segments of length 63, including channels 834, 835, 837, 841, 849, and 865, a total of six channels with a row weight of 16, and the distances from the 833 channel are 1, 2, 4, 8, 16, and 32, respectively. The corresponding position reliability metrics are 21.8897, 24.1908, 26.5414, 28.9317, 31.3541, and 33.8030. They are set as frozen channels respectively. The performance comparison chart is as follows Figure 3 shown.

[0047] Depend on Figure 3 It can be seen that although the reliability of the 865th channel is the highest, the performance is best when it is set as a frozen channel, which breaks the common sense that the frozen channel is set as the channel with the lowest reliability when constructing the polarization code. In addition, the distance between the 835th and 837th channels is small, but the performance gap is the most obvious. This is because the error probability of the first three positions of the R1 node is the largest, that is, the 834th; 835th; 836th channels. Setting the 837th channel as a frozen channel can verify these three positions with the highest error probability. The performance gap of setting the four channels, 837th; 841st; 849th; 865, as frozen channels is smaller because the error probability of the latter positions is smaller, so the performance improvement is not obvious, but the performance improvement at this time is still greater than the performance degradation caused by the decrease in reliability. From the above analysis, we can get the second principle of frozen channel setting: long information channel segments are set with frozen channels at intervals with smaller row weights and at the back.

[0048] In addition, due to the unique structure of polar codes: the reliability metric of channels at later positions is larger, so the proposed frozen channel setting principle is approximately equivalent to selecting the channel with the largest reliability metric among channels with smaller row weights. Therefore, the proposed polar code construction method and the RM-Polar code construction method are both channel reliability sorting problems with fixed row weights, so the complexity of these two methods is similar.

[0049] (4) Combined with attachment Figure 4 , 5 illustrates the superiority of the method of the present invention, the simulation is as follows:

[0050] The simulation conditions are as follows: under AWGN channel, BPSK modulation is adopted, Gaussian approximation construction method with signal-to-noise ratio SNR=2.5dB is adopted, long polar code length N=256, K=128, f=6, short polar code length N=128, K=64, f=4, list size L=8. Figure 4The SCL decoding performance of different construction methods is analyzed, and the block error rate of the FRM-Polar construction method proposed in this invention is compared with the RM-Polar code construction method in reference [1] and the Gaussian approximation construction method in reference [2].

[0051] from Figure 4 It can be seen that the FRM-Polar code construction method proposed in the present invention has the best performance compared with other construction methods. -3 When , the (256,128,6)FRM-Polar code proposed in this paper has about 0.35dB and 0.9dB gains compared with the (256,128,6)RM-Polar code and Gaussian approximation constructed Polar code, respectively. The (128,64,4)FRM-Polar code has about 0.3dB and 0.8dB gains compared with the (128,64,4)RM-Polar code and Gaussian approximation constructed Polar code, respectively. The proposed FRM-Polar code performs better than RM-Polar code and Polar code at different code lengths.

[0052] Since the concatenated CRC code can eliminate low-weight codewords, increase the MHW of the polar code, and improve the distance spectrum of the polar code, its performance under high signal-to-noise ratio is further improved. Therefore, in order to further verify the superiority of this construction method, a 4-bit CRC code is concatenated, and the corresponding generating polynomial is g 16 (x) = x 4 +x+1, and carried out simulation comparative analysis with RM-Polar code construction method and Gaussian approximation construction method.

[0053] Depend on Figure 5 It can be seen that the (128,64,4) CRC-FRM-Polar code proposed in this paper has a BLER of 10 -3 Compared with the (128,64,4)CRC-RM-Polar code and the traditional Gaussian approximation constructed CRC-Polar code, the (256,128,6)CRC-FRM-Polar code has a gain of about 0.4dB and 0.8dB respectively when BLER=10 -7 When compared with the (256,128,6) CRC-RM-Polar code and the traditional Gaussian approximation constructed CRC-Polar code, there are about 0.15dB and 0.3dB gains respectively. The new code performs slightly worse when the signal-to-noise ratio is low, because the channel with higher reliability is set as a frozen channel. However, as the signal-to-noise ratio increases, the performance gain of the new code will surpass the RM-Polar code, and this performance improvement will be more obvious when the signal-to-noise ratio is larger. In summary, the FRM-Polar construction method proposed in the present invention has a greater decoding performance improvement compared to the other two construction methods without much change in complexity.

Claims

1. A polar code channel construction method based on frozen bit improvement. Aiming at the problem that the minimum Hamming weight (MHW) of polar codes constructed by traditional Gaussian approximation (GA) is small and the error correction and verification functions of frozen bits in serial cancellation list (SCL) decoding are not fully utilized, a polar code channel construction method based on frozen bit improvement is proposed. The method first uses GA to estimate the reliability metric of the polar subchannel, selects the most reliable channel as in the traditional construction method, and then selects all channels with smaller row weights from the selected channels. These channels have higher reliability but smaller row weights. Finally, according to the two proposed frozen channel setting principles, namely, the principle of setting frozen channels with appropriate bits for channel segments with different reliability and the principle of setting frozen channels at intervals of long information channel segments with smaller row weights and at the back, channels with appropriate bits are selected and set as frozen channels, and the remaining channels are information channels. SCL decoding is used for decoding. The polar code channel construction method specifically includes the following steps: Step 1: The polar code is subjected to an additive white Gaussian noise (AWGN) channel using GA to obtain the expected logarithm likelihood ratio (LLR) and reliability metric of each polarization subchannel. Consider using BPSK modulation in an AWGN channel with a polarization code length of N. The corresponding LLR is expressed as Assume the input sequence is an all-zero codeword, then Obey N(2 / σ 2 ,4 / σ 2 ), it can be seen that all LLRs are Gaussian random variables with variance twice the mean, so according to the Gaussian approximation principle, we can get equations (1) to (3): in By repeatedly using formula (1) and formula (2), the final calculation of each polarization subchannel is The LLR expectation is larger, and the corresponding channel The higher the reliability, the higher the reliability. After obtaining the reliability measurement of all channels, the K+f channels with the highest reliability are selected as information channels, where K is the number of information bits and f is the number of bits set for the frozen channel; Step 2: Find the row weight of the generator matrix corresponding to the K+f-bit channel with the highest reliability, and select all the channels with the smallest and second smallest row weights; For a code length of N = 2 n The polar code can be generated by the matrix G N Realization, G N The row weight of the i-th row is 2 wt(i) , where i starts counting from 1, wt(i) represents the Hamming weight of the binary number i-1, and the minimum distance of the polar code is equal to G N The minimum row weight of the row corresponding to the information bit in the generator matrix is ​​the minimum Hamming weight (MHW) of the row corresponding to the information channel in the generator matrix. Selecting a channel with a smaller row weight as a frozen channel can reduce the low-weight codewords of the polar code and increase the MHW. Before selecting the number and position of frozen channels, it is necessary to define formulas (5), (6) and (7): f=log2N×(α-|K / N-1 / 2 2 ) (5) Where K is the number of information bits, f is the number of bits for setting the frozen channel. If f is not an integer, it is rounded to the nearest integer. α is the coefficient for adjusting the number of frozen bits. Usually, 0<α<1. According to formula (5), the number of frozen bits can be flexibly selected by adjusting the value of α. In formula (6) and formula (7), d m and d s are the MHW value in the non-frozen bit index set and the secondary MHW value just greater than the MHW value, A m and A s The Hamming weights are d m and d s The index value set of the row, the frozen bit from the set A m and A s Select from, when f is less than |A m |, the frozen channels are only set in the channels with the minimum row weights. When f is greater than |A m |, all channels with the smallest row weights are set as frozen channels, and f-|A is selected from the channels with the second smallest row weights. m |Set the bit channel to frozen channel; Step 3: According to the two proposed frozen channel setting principles, i.e., the principle of setting frozen channels with appropriate bits for channel segments with different reliability and the principle of setting frozen channels for long information channel segments at intervals with smaller and later positions with smaller row weights, select f-bit channels from the channels with the smallest and second smallest row weights and set them as frozen channels, and use SCL decoding during decoding; In SCL decoding, when the frozen bit judgment result does not conform to the LLR judgment result, the path metric (PM) value will be added to the corresponding path. The path with a larger PM value is considered to have a higher error probability and will be deleted. There are two situations that may lead to frozen bit judgment errors: the first is the influence of noise, and the second is the error of the previous decoding bit. Noise is random and uncontrollable, so the position influence is not considered. For the second situation, the ideal state in the SCL decoding process includes 1 correct path and L-1 wrong paths. When the wrong path is translated to the frozen bit, the LLR judgment error probability is greater. The increase in PM value will cause the wrong path to be pruned, which can increase the probability of the correct path being retained to the end. Therefore, the frozen bit also has a part of implicit verification capability. The frozen bit is regarded as an error correction code and a check code. The number of frozen bits and the placement position are deeply studied and analyzed, and two frozen channel setting principles are obtained, namely: Principle 1: Set frozen channels with appropriate number of bits for channel segments with different reliability. Principle 2: Set frozen channels at intervals at the back of long information channel segments with smaller row weights. According to the principle of setting two frozen channels, f-bit channels are selected from the channels with smaller row weights as frozen channels, and the remaining K-bit channels are information channels. SCL decoding is used for decoding. This can not only reduce the low-weight codewords of the polarization code and optimize the distance spectrum of the polarization code, but also make full use of the verification capability of the frozen bits in SCL decoding to improve the error correction performance of the polarization code.

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