WO2019172856A1

WO2019172856A1 - Soft successive cancellation algorithm for polar codes

Info

Publication number: WO2019172856A1
Application number: PCT/TR2018/050083
Authority: WO
Inventors: Orhan GAZI; Alia ANDI; Ahmet Cagri ARLI
Original assignee: Cankaya Universitesi
Priority date: 2018-03-07
Filing date: 2018-03-07
Publication date: 2019-09-12

Abstract

The invention relates to a soft successive cancellation algorithm for polar codes that uses the soft likelihood ratios of the predecessor information bits for the determination of current bit being decoded. In classical successive cancellation decoding operation, for the decoding of current bit, the binary values of the previously decoded bits are used, i.e., hard decision values are used. In the proposed technique, we use the soft values of the previously decoded bits, i.e., their probabilities, to calculate the soft value, i.e., probability, of the current bit being decoded. After all the soft values of all the information bits are found, decisions are made for all the information bits, considering their soft values, at the same time.

Description

SOFT SUCCESSI VE CANCELLATI ON ALGORI THM FOR POLAR CODES

1 . The Technical Field of the I nvention

The invention relates to a soft successive cancellation algorithm for polar codes that uses the soft likelihood ratios of the predecessor information bit for the probability likelihood calculation of the current bit being decoded.

1 .1 Prior Art About the I nvention ( Previous Technique)

Polar codes are a class of channel codes designed in a nontrivial manner. Polar codes are the first mathematically provable channel codes available in channel coding world, and this can be considered as a major breakthrough in coding society. Polar codes suffer from low performance at moderate and low code-word lengths. To alleviate the low performance of polar codes, the list and stack decoding algorithms are proposed in [3] , [4] , [5] , [6] . Although, list and stack decoding algorithms show better performance than that of the classical successive cancelation method [ 1 ] , they involve much more computations regarding the classical successive cancelation algorithm. I n [7] , polar codes are concatenated with CRCs to prevent the performance degrading effect of error propagation. After the introduction of turbo codes, concatenated structures became popular to establish powerful code systems. The same idea was pursued for the construction of concatenated structures involving polar codes in [8] , [9] , [ 10] , [ 1 1 ] .

Although there are many polar decoders related patent applications, there are two patent applications that can thought to be familiar with the proposed technique. One of them has the title of "Polar code decoder and polar code decoding method based on probability calculation" [ 12] This application covers optimization of the decoder nodes to avoid access delays. Despite of mentioning soft information throughout the decoding stages, still hard decision is done at top node. The only similarity with the other patent application [ 13] about its name, "Soft decoding of polar codes". Despite word "soft" is used, the document covers a circuit that handles polar list decoder for a solid-state drive. As a result, there is no similar applications or documents about soft decision mechanism in successive cancellation algorithm and its derivative algorithms. They all inherit hard decisions. 1 .2 Polar Encoding and Decoding Unit and The Previously Derived Formulas

I n this section, brief information about successive cancelation algorithm and the formulas related to the successive cancelation algorithm will be provided. The main kernel unit used in polar encoder and decoder structures is depicted in Fig. 1 where the unit in Fig. 1 -A is used in encoder structures and the unit in Fig. 1 -B is used in decoder structures. From Fig. 1 B, it can be written that

from which, we can obtain

Equations (2) and (3) can be used for the likelihood calculation of â as in

The term LR(â) stands for the likelihood ratio (LR) of â. Using (2) and (3) in (4), we get

The value of â can be found using (5). After deciding on the value of d, the decoding operation for starts. I f the d is decided to be 0, i.e. , â = 0, then probabilities for

considering Figure 1 B, can be written as

from which, we can obtain

On the other hand, if the decoded â is decided to be 1 , i.e. , if â = 1, then probabilities for

can be calculated, considering Figure 1 B, as

from which, we get

The formulas in (8) and (11) can be combined as

The formulas in (8) and (1 1 ) are expressed in a recursive manner in [ 1 ] as

and

for the successive cancellation decoding algorithm.

2. Aims of The I nvention and a Brief Explanation

The successive cancelation decoding of polar codes are performed in a recursive manner using the formulas (13) and (14) introduced in [ 1 ] First (13) is calculated, and the value of is decided. Next, the decided value of is used in

for the calculation of (14). This means that hard decision is performed after calculation of (13), and the value of hard decision is used in the exponential part of (14). Since, hard decision results in loss of information, after calculation of (13), without making a hard decision for we want to use the result of (13) in (14) directly. For this purpose, we

modify the formula (14) such that we can use the likelihood calculation result of (13) directly in modified version of (14). I n this way, we do not lose information at each decoding stage. When all the likelihoods are calculated for all the information bits, decision is performed at the end of all likelihood calculations.

This invention relates to soft successive cancellation algorithm for polar codes. A new approach to the successive cancelation of polar codes is introduced. The proposed approach uses the soft likelihood ratios of the predecessor information bit rather than its exact values for the likelihood calculation of current information bit. The proposed method can be utilized for the construction of joint iterative communication systems exchanging soft likelihoods. Polar coding is the first kind of the capacity achieving codes which are defined for binary- input discrete memoryless channels. Polar coding is a channel coding procedure that makes the channels capacities approach to 0 or 1. By doing this, receiver knows whether the data can be decoded with high probability or not. Polar codes are utilized for both binary erasure and AWGN channels.

With this invention :

• The construction of soft decision based polar decoder is introduced.

• The equation (76) in [ 1 ] is modified such that it includes the probability value of the previous decoded bit rather than its decided hard value.

• Joint structures can be created with well-known coding schemes like turbo codes whose success comes from soft information change.

• For code-word lengths used in practical applications, the performance of the polar codes is poor compared to the well-defined and implemented coding schemes such as LDPC and turbo. Better results can be achieved by using soft polar decoders for shorter block sizes smaller than 1024.

• List and stack decoding algorithms introduced in [3-4] can utilize the proposed technique and can get much better results.

• Complexity remains still low. I n (25) and (27), low complexity logarithmic version of soft successive cancelation algorithm is depicted.

• Any joint communication structure involving iterative decoding operation can utilize the polar codes with the invented formulas in (17), (18) and (23).

• Any communication system involving polar codes can employ the invented formulas in (17), (18) and (23) and other iteratively decodable codes such as, turbo codes, LDPC codes, BCH codes, convolutional codes and any other block codes.

Another aspect of the invention is that the introduced technique can be used to construct joint code structures utilizing the exchange of soft information, i.e. , probabilities, in their decoding operations.

Another aspect of the invention is that with the proposed technique can be integrated with LDPC coding scheme.

Another aspect of the invention is that the proposed technique can be integrated with turbo and all the turbo-like codes employing belief , and all types of belief propagation methods. 3. The Descriptions of the Figures Explaining the I nvention

The figures used to better explain soft successive cancellation algorithm for polar codes developed with this invention and their descriptions are as follows:

Figure 1 A: Kernel Encoder structure Figure 1 B: Kernel Decoder structure

4. The Detailed Explanation of the I nvention

I n this part, the proposed soft successive cancelation algorithm is explained in details. We also present the log-domain implementation of the suggested technique. The log-domain implementation reduces the computational complexity and reduces the decoding latency. 4.1 Proposed Method and Soft Successive Cancellation Algorithm

I n this section, the proposed approach for the soft decoding of polar codes using the proposed method is going to be demonstrated.

4.1 .1 Using Soft Values for the Calculation of Likelihoods

I f the formulas (13) and (14) are inspected closely, it can be seen that the predecessor bit that is evaluated considering the likelihood ratio (13) is used in exponential term of (14), and this means that the previous hard decision is used for the decoding of current bit. I n our method, using (6) , (7) , (9) and (10) , the probabilities for b can be written as

from which, can be obtained as

Equation (17) shows that instead of the hard value of â as in (12) and (14) , soft value of â, i.e. , appears in the calculation of

Since employment of hard values results in the loss of information, employment of soft values results of a more reliable likelihood calculation. With the proposed approach, the recursive equation in (14) can be modified as in (18).

where symbol

denotes the likelihood calculation for the output of the XOR gate whose inputs includes likelihood ratios. Detailed explanation about this operation is going to be given in the next part. When (8), (11), (12) and (17) are inspected together, it can be seen that (8), (11) are nothing but limiting cases of (17), that is:

To the best of authors knowledge, the formulas given in (17), (18) and (23) and the log- domain formulas for soft successive cancelation algorithm given in (25) and (27) are new in the literature.

4.1 .2 Likelihood Combination for the Output of XOR Function

I n classical successive cancelation decoding of polar codes, the previously decoded bits, i.e. , hard values, are used for the decoding of successor bit. For this purpose, the previously decoded bits are used in the decoder structure of the successive cancelation algorithm . Logic XOR gates are used in both encoder and decoder structures. When some of the bits are decoded, these bits propagate through the decoder structure and some of the XOR gates have known input and output values. I f there are known bit values available at the XOR gate outputs, these XOR gates are called g nodes, otherwise, they are called / nodes. However, only soft values, i.e. , probabilities, for the decoded bits are used instead of hard values in our proposed technique, and for this reason, there are soft values at the inputs of the XOR gates. Since XOR gate can't handle soft values, an equivalent function is constructed below.

I f then the probabilities for x₁ can be written as

from which, we obtain

Equation (23) gives the likelihood value at XOR gate output if likelihood values instead of the bit values at the inputs of the XOR gates are considered. Symbol

is chosen to distinguish soft XOR operation from that of the logical XOR operation indicated by the symbol ®. The decision on the value of information bits, when the soft values are calculated for all the information bits, are done at the end of the decoding operation. That is, the likelihood values are calculated for all the information bits, then the decoding logic is performed according to

4.1 .3 Logarithmic Scale Likelihood Calculation for Soft Successive Cancellation Algorithm

To implement an invented method in hardware platform, it is vital to decrease the implementation complexity of the proposed approach. One technique is to implement the proposed approach in log domain. Leroux et. al. [2] offered logarithmic implementations of (5), (12) or (13), (14). The successive cancelation decoding formulas and their log equivalents are depicted in Table-1 .

Table-1 Successive cancellation decoding formulas and their Log domain equivalents

We derived the log-domain equivalent form of the proposed Soft Successive Cancellation formulas. The log-domain equivalent of (17) is derived as in

where LLR(â ) is found using

And the log version of the soft XOR combining in (23) is derived as in

The method of invention is used to create joint structures involving other coding schemes which are iteratively decodable employing belief propagation algorithms.

References

[ 1 ] E. Arikan, "Channel polarization : A method for constructing capacity achieving codes for symmetric binary-input memoryless channels," I EEE Trans on I nf . Theory, vol. 55, no. 7, pp. 30513073, July 2009.

[2] Camille Leroux, Alexandre J. Raymond, Gabi Sarkis, I do Tal, Alexander Vardy, Warren J. Gross, "Hardware I mplementation of Successive Cancellation Decoders for Polar Codes", J

Sign Process Syst 69:305-315, July 2012.

[3] I . Tal and A. Vardy, "List decoding of polar codes," in Proc. of I EEE I nt. Symp. on I nf . Theory, 201 1 , pp. 15. [4] K. Niu and K. Chen, "Stack decoding of polar codes," Electronics Letters, vol. 48, no. 12, pp. 695697, June 2012.

[5] K. Chen, K. Niu, and J. Lin, "I mproved successive cancellation decoding of polar codes," l EEE Trans. on Communications, vol. 61 , no. 8, pp. 31003107, August 2013. [6] B. Li, H. Shen, and D. Tse, "An adaptive successive cancellation list decoder for polar codes with cyclic redundancy check," I EEE Communications Letters, vol. 16, no. 12, December 2012.

[7] K. Niu and K. Chen, "CRC-aided decoding of polar codes," I EEE Communications Letters, vol. 16, no. 10, pp. 16681671 , October 2012. [8] P. Trifonov and P. Semenov, "Generalized concatenated codes based on polar codes," in

Proc. of l EEE I nt. Symp. on Wireless Communication Systems, 201 1 , pp. 442446.

[9] J. Guo, M. Qin, A. G. Fabregas, and P. Siegel, " Enhanced belief propagation decoding of polar codes through concatenation," in Proc. of l EEE I nt. Symp. on I nf . Theory, 2014.

[ 10] H. Mahdavifar, M. El-Khamy, J. Lee, and I . Kang, "On the construction and decoding of concatenated polar codes," in Proc. of I EEE I nt. Symp. on I nf. Theory, 2013.

[ 1 1 ] M. Bakshi, S. Jaggi, and M. Effros, "Concatenated polar codes," in Proc. of I EEE I nt. Symp. on I nf. Theory, 2010.

[ 12] Beijing University of Posts and Telecommunications, "Polar code decoder and polar code decoding method based on probability calculation ", CN104079382 (A) -2014-10-01 . [ 13] Seagate Technology Lie, "Soft decoding of polar codes", US 9317365 B2 -2014-19-03.

Claims

CLAI MS

1 . The method of soft successive cancellation decoding algorithm for polar codes at the receiver side comprising the steps of;

a) calculating of likelihood ratio of the predecessor bit b) calculating likelihood ratio of the current bit using below formula ( 17)

or using recursive form of formula ( 17) which is below formula ( 18)

where denotes the operation for the calculation of the soft output of XOR gate whose inputs are likelihoods, and its operation is given as below formula (23)

2. The method according to claim 1 where likelihood ratio of the predecessor bit in step (a) is calculated using below formula (5)

or in a recursive manner using below formula (13)

3. The method of soft successive cancellation decoding algorithm for polar codes according to claim 1 in log domain at the receiver side comprising the steps of ;

• Calculating the likelihood ratio of the current bit in log domain using formulas (25) as

and the operation for the calculation of the soft output of XOR gate whose inputs are likelihoods in log domain is performed using formula (27) as in

and the LLR(â) in (25) is calculated using (26) as in

4. The method of soft successive cancellation algorithm for polar codes according to claim 1 ; wherein the method is adapted to create joint structures involving other coding schemes which are iteratively decodable employing belief propagation algorithms.