KR100748501B1 - Graph decoding method using partial correction - Google Patents

Abstract

,

) 중 작은 값을 갖는 사이트 조합을 선택하는 단계와; 다수의 사이트 조합 중 가장 큰 비중을 차지하는 K개의 사이트 조합만이 보정 팩터로 선정되도록

를 업데이트 처리하는 단계와; 상기 선택 결과에 상기 업데이트 결과를 더하여 디코딩 과정에서 발생되는 손실을 보정하는 단계에 의해 달성된다.The present invention relates to a technique that adds some correction factor to the Min-Sum algorithm for decoding an LDPC code so that decoding performance close to the Sum-Product algorithm can be exhibited. The present invention relates to a method for graph decoding a received LDPC code with a Min-Sum algorithm.

,

Selecting a site combination having a small value; Only the K site combinations, which make up the largest proportion of multiple site combinations, are selected as the correction factor.

Updating the process; By adding the update result to the selection result to correct a loss generated in the decoding process.

Description

Graph decoding method using partial correction {GRAPH DECODING METHOD USING PARTIAL CORRECTION}

1 is a Tanner graph of a Sum-Product algorithm according to the prior art.

2 is a Tanner graph of the Min-Sum algorithm according to the prior art.

Figure 3 is a Tanner graph of the Min-Sum algorithm represented by the difference in cost function in the prior art.

4 is a block diagram of a correction module according to the prior art.

5 is an explanatory diagram of a structure for correction of Min-Sum in the prior art.

Figure 6 is an explanatory diagram of a structure for the correction of Min-Sum in the present invention.

7 is an explanatory diagram of a forward-backward recursion structure for efficient correction in the present invention.

8 is a graph showing the simulation results for the first embodiment of the present invention.

9 is a graph showing the simulation results for the second embodiment of the present invention.

10 is a table showing the complexity of each algorithm for the first embodiment of the present invention.

Fig. 11 is a table of the complexity and complexity of each algorithm of the second embodiment of the present invention.

*** Description of the symbols for the main parts of the drawings ***

41: comparison and selection 42: subtractor

43: absolute value calculator 44: ROM                 

45: summer

The present invention relates to a technique for decoding a block code, and in particular, by adding a small correction factor to the Min-Sum algorithm for decoding an LDPC code, using partial correction to achieve decoding performance close to that of the Sum-Product algorithm. It relates to a graph decoding method.

As a conventional technique for decoding a block code, there is an algebraic decoding using a hard decision value of a received signal. However, recently, studies on graph decoding methods that enable iterative decoding using soft decision values of received signals have been actively conducted.

In order to perform graph decoding, a graph using a parity check matrix of a block code is prepared. The graph corresponds to the sites and rows corresponding to the columns of the parity check matrix. Tanner graphs with checks are used a lot. In the parity check matrix, the checks for the site and the row corresponding to the column of the component having a value of 1 are connected to each other and send and receive probability values.

As a code suitable for graph decoding, there is a Low Density Parity Check (LDPC) code, and a decoding algorithm proposed to decode the code includes a Sum-Product algorithm and a Min-Sum algorithm.

Sum-Product algorithm for decoding LDPC code shows the best performance when the Tanner graph does not have cycles, and Min-Sum algorithm is generally less complicated than Sum-Product algorithm. There is this.

Figure 1 shows the probability transfer process in the Sum-Product algorithm previously proposed on the Tanner graph. This algorithm is performed according to the relation shown in Equation 1 to be described later. To implement this, a nonlinear function and a multiplier are required. In Fig. 1, "s" is a site, "E" is a Check, and each variable is defined as follows.

Sum-Product algorithm is executed by the following process.

Recursive Iteration: Performs the following update a fixed number of times.

이면 사이트 s에 해당하는 수신 신호는 1로 디코딩되고,

이면 0으로 디코딩된다. At this time,

, The received signal corresponding to site s is decoded to 1,

If it is, it is decoded to 0.

Figure 2 shows the metric (Metric) transfer process in the proposed Min-Sum algorithm on the Tanner graph. This algorithm is performed according to the relation shown in Equation 6 to be described later. Unlike the Sum-Product algorithm, a nonlinear function and a multiplier are not required, but the performance is inferior to that of the Sum-Product algorithm. The Min-Sum algorithm is derived through approximation in the Sum-Product algorithm, which facilitates the hardware implementation by eliminating the nonlinear functions and multiplication operations required by the Sum-Product algorithm.

Taking the natural log on both sides in Equation 1 and multiplying by -1, Equation 3 below is obtained.

If an approximation equation such as [Equation 4] is applied to the second equation of [Equation 3], it is simply approximated as in [Equation 5].

와 같이 각각의 log 함수를 변수로 치환하면 상기 [수학식 3],[수학식 5]를 아래의 [수학식 6]과 같이 정리할 수 있으며, 이를 Min-Sum 알고리즘이라 한다.

By substituting each log function as a variable, Equation 3 and Equation 5 can be summarized as Equation 6 below, which is called a Min-Sum algorithm.

In order to reduce the memory capacity required in actual hardware implementation, the cost function of Equation 6 may be improved as shown in Equation 7 below with a difference between a = 1 and a = 0. FIG. 3 illustrates a structure in which a cost function to be transferred is changed into a difference between two cost functions in order to efficiently implement the Min-Sum algorithm shown in FIG. 2, which is performed according to the relation shown in Equation 7 Its performance is the same as the Min-Sum algorithm of FIG.

In the case of the Min-Sum algorithm, however, the complexity is lower than that of the Sum-Product algorithm, but the performance is poor.

The loss generated in the approximation process of Equation 4 may be recovered by the following method. First, an approximation formula consisting of two terms can be expressed as Equation 8 below.

를 보정 팩터(Correction factor)라 하고, 실제 하드웨어 구현 시 메모리 혹은 롬에 함수 값을 저장하여 사용하게 된다. 즉, 사이트 조합

과

의 차이를 근거로 메모리 혹은 롬에 저장된 보정값을 읽어서 더하게 된다. 일반적인 n개의 항으로 이루어진 경우 상기 [수학식 8]의 과정을 아래의 [수학식 9]와 같이 재귀적으로 수행하여 보정 팩터를 추가할 수 있다.here,

Is called a correction factor, and the actual hardware implementation stores the function value in memory or ROM. That is, a site combination

and

The correction value stored in the memory or ROM is added based on the difference of. In the case of general n terms, a correction factor may be added by performing the process of Equation 8 recursively as shown in Equation 9 below.

이다. 즉, 상기 [수학식 8]과 같이 먼저 두 개의 항으로 이루어진 보정 팩터를 계산한 후 그 결과와 세 번째 항을 이용하여 같은 과정을 수행한다. n개의 항일 경우 n-1회의 보정 팩터가 재귀적으로 더해진다.here,

to be. In other words, as shown in Equation 8, first, a correction factor consisting of two terms is calculated, and the same process is performed using the result and the third term. For n terms, n-1 correction factors are added recursively.

가 되므로, a=1인 경우와 a=0인 경우 각각에 대하여

회의 보정이 수행된다. 이러한 구조를 도식화 한 것이 도 5이며, 이는 다음의 [수학식 10]으로 표현된다.In case of LDPC code, if K sites are connected to one check, Equation 9

Therefore, for a = 1 and a = 0, respectively

Conference correction is performed. 5 is a diagram illustrating this structure, which is represented by Equation 10 below.

는 사이트 s가 a일 때 패리티 체크 E를 만족하는 나머지 사이트의 조합(Site combination)이다. 보정 팩터

은 상기 [수학식 8]과 같다. here,

Is the site combination of the remaining sites that satisfy parity check E when site s is a. Correction factor

Is the same as [Equation 8].

FIG. 4 is a structure in which a correction factor is added to improve the Min-Sum algorithm shown in FIG. 2, and shows a structure for correcting all losses generated while approximating by Equation 4.

,

가 입력될 때 비교 및 선택부(41)는 그 중에서 크기가 작은 사이트 조합

를 선택하여 출력한다. 또한, 감산기(42)에서는 상기 사이트 조합

,

의 차를 구하고, 절대값 연산기(43)에서는 그 차값의 절대값

를 구하여 롬(44)으로 출력한다. 상기 롬(44)에는 보정 테이블이 구비되어 있어 상기 절대값

에 상응되는 보정값

을 출력하게 되며, 이는 합산기(45)에서 상기 비교 및 선택부(41)로부터 출력되는

에 더해지며, 그 결과치가 최종 출력

가 된다.That is, FIG. 4 is an exemplary view of implementing the correction module shown in Equation 8, and combining two sites.

,

Is selected, the comparison and selection section 41 selects a site combination having a smaller size.

Select to print. In the subtractor 42, the site combination is used.

,

Is obtained, and the absolute value calculator 43 calculates the absolute value of the difference value.

Is obtained and output to the ROM 44. The ROM 44 is provided with a correction table so that the absolute value is

Correction value corresponding to

Is output from the comparison and selection unit 41 in the summer 45

Is added to the final output

Becomes

By the way, in the case of the calibration module as shown in Figure 4, the performance is close to the normal Sum-Product, but there is a disadvantage that the complexity increases by adding a correction factor.

의 계산 과정을 나타낸 것으로, 첫 번째 비트 조합

에서부터 마지막 비트 조합

까지 모든 비트 조합을 더하는 구조로 되어 있다. 여기서, M으로 표시된 블록은 도 4에서 합산기(45)에 해당되는 것이며, E,s,k는 도 1에서와 같은 의미를 갖는다. 각각의 입력신호

는 상기 [수학식 10]과 같이 계산된다. 보 정을 위해 필요한 M 블록은 하나의 a값에 대하여 모두

개이다.On the other hand, Figure 5 is a check-to-site cost function for improving the performance of the Min-Sum algorithm

Shows the process of calculating the first bit combination

Last bit combination from

It is a structure that adds all bit combinations up to. Here, the block denoted by M corresponds to the summer 45 in FIG. 4, and E, s, and k have the same meanings as in FIG. 1. Each input signal

Is calculated as shown in Equation 10 above. The M blocks needed for correction are all for a value of a

Dog.

As described above, in the decoding method according to the related art, the Min-Sum algorithm is implemented to be less complicated than the Sum-Product algorithm, but has a disadvantage of poor performance. In the case of hardware or software, there is a disadvantage in that it has a complicated structure.

Therefore, an object of the present invention is to partially compensate for adding only a correction factor considering only the K site combinations that occupy the largest weight, so that the performance of the Sum-Product algorithm can be achieved without increasing the complexity of the Min-Sum algorithm. It provides a graph decoding method using.

,

) 중 작은 값을 갖는 사이트 조합을 선택하는 단계와; 다수의 사이트 조합 중 가장 큰 비중을 차지하는 K개의 사이트 조합이 보정 팩터로 선정되도록

를 업데이트 처리하여 각 사이트에 대한 코스트 펑션을 구하는 단계와; 상기 선택 결과에 상기 업데이트 결과를 더하여 디코딩 과정에서 발생되는 손실을 보정하는 단계로 이루어지는 것으로, 이와 같은 본 발명의 디코딩 방법을 첨부한 도 4, 도 6 내지 도 도11을 참조하여 상세히 설명하면 다음과 같다.Graph decoding method using the partial correction according to the present invention, in the correction module for decoding the LDPC code received through the white Gaussian noise channel by the Min-Sum algorithm, the received site combination (

,

Selecting a site combination having a small value; The K site combinations, which account for the largest proportion of the multiple site combinations, are selected as the correction factor.

Obtaining a cost function for each site by updating processing; Comprising a step of correcting the loss generated in the decoding process by adding the update result to the selection result, will be described in detail with reference to Figures 4, 6 to 11 attached to the decoding method of the present invention as follows. same.

According to a first aspect of the present invention, the decoding algorithm according to the present invention is performed by the following process when one site is connected to J checks on a Tanner graph and K sites are connected to one check. do.

Recursive Iteration: Performs the following update a fixed number of times.

는 비선형 함수인

를 양자화 하여 롬(44)의 룩업테이블에 저장해 둔 값이다.

를 업데이트 하기 위한 구조는 도 6과 같은 형식을 따른다.here,

Is a nonlinear function

Is quantized and stored in the lookup table of the ROM 44.

The structure for updating follows the format as shown in FIG.

이면 사이트에 해당하는 수신 신호는 1로 디코딩되고,

이면 0으로 디코딩된다. At this time,

, The received signal corresponding to the site is decoded to 1,

If it is, it is decoded to 0.

를 보정함에 있어서 가장 큰 비중을 차지하는 K개의 사이트 조합만을 고려함으로써

개의 사이트 조합을 고려해야 하는 기존의 방식에 비하여 훨씬 낮을 복잡도로 구현할 수 있게 된다.According to a second aspect of the present invention, the decoding algorithm in the first aspect may exhibit a performance close to that of the Sum-Product algorithm.

By considering only the K site combinations that account for the largest

This can be implemented with much lower complexity than the traditional approach of considering two site combinations.

According to the third aspect of the present invention, the graph decoding algorithm in the first aspect indicates the cost function to be transmitted as the difference between the two cost functions, thereby reducing the number of memories required.

는 두 번의 덧셈과 한 번의 룩업 테이블 억세스에 의해 획득되며, 그 룩업 테이블이 롬에 존재하는 경우 도 4에서와 같이 구현된다.According to a fourth aspect of the invention, a function in the first aspect

Is obtained by two additions and one lookup table access, and is implemented as shown in FIG. 4 when the lookup table exists in the ROM.

According to the fifth aspect of the present invention, the decoding algorithm in the first aspect exhibits performance that is close to the decoding performance of the Sum-Product algorithm, while having a much lower complexity than the conventional scheme.

는

를 업데이트함에 있어서, 포워드-백워드 재귀(forward-backward recursion) 형태의 구조로 구현되므로 복잡도를 최적으로 줄일 수 있고, 그 구조는 도 7과 같다. According to a sixth aspect of the invention, a function is provided in the first and fourth

Is

In updating, the complexity can be optimally reduced because it is implemented in a forward-backward recursion type structure, and the structure is shown in FIG. 7.

According to the seventh aspect of the present invention, the decoding structure in the first to sixth features can be applied to all types of algorithms approximated using a Log function, such as the Max-Log-MAP algorithm in turbo code.

Hereinafter, the decoding method will be described in more detail with reference to an embodiment of the present invention.

504 패리티 체크 행렬로 나타낼 수 있는데, 이와 같은 경우 각각의 열에는 3개의 1이 존재하고, 각각의 행에는 6개의 1이 존재한다.(P1) First, an LDPC parity check code having a code rate of 1/2, a number of information bits of 252, and a codeword length of 504 bits (504, 3, 6) is 242.

504 parity check matrix, in which case there are three 1s in each column and six 1s in each row (P1).

The LDPC code of the first process (P1) is transmitted from the transmitter and delivered to the receiver through a white Gaussian noise (AWGN) channel (P2).

When the Tanner graph is created to decode the LDPC code received in the second process (P2), all of the checkers consist of 252 checks and 504 sites, where each check is connected to 6 sites, and each site is 3 Is connected to two checks. In such a case, the variable J = 3 and K = 6, which are commonly shown in Figs. 1 to 7, will be (P3).

으로 초기화 되며, 도 3과 도 6의 알고리즘으로 디코딩되는 경우 모든 체크와 사이트에 대하여

으로 초기화 된다.(P4) When decoding with the Sum-Product algorithm of FIG. 1 on the Tanner graph created by the third process P3, mu_E, s (1) = mu_E, s (0) = mu _s, E ( 1) = mu _s, E (0) = 1 is initialized. Also, all the checks and sites are decoded by the algorithm of FIGS. 2 and 5.

It is initialized to, and when it is decoded by the algorithm of Fig. 3 and Fig. 6 for all checks and sites

Initialized to (P4).

When decoding to Sum-Product by the first to fourth processes (P1-P4), the update is performed using Equation 1, and in the case of the Min-Sum algorithm of FIG. ] To update. In addition, in the case of adding the correction factor, the correction algorithm of FIG. 5 is updated by using Equation 10, and the correction algorithm of FIG. 6 is updated by using Equation 12 (P5).

The number of update operations may be arbitrarily determined in the first to fifth processes P1 to P5. For example, 200 updates may be used. (P6)

The number of operations required to update once for each algorithm according to the first to sixth processes P1-P6 is as shown in the table of FIG. 10. It can be seen that the correction algorithm according to the present invention has much less complexity than the conventional correction algorithm. (P7)

As another application example of the present invention, the above algorithm can reduce the loss caused by applying the Sum-Product type algorithm consisting of a nonlinear function and a multiplier to approximate the log function, specifically, a block code It can be applied to a graph decoding algorithm for decoding. In addition, this algorithm can be extended to the Log-MAP algorithm used for decoding convolutional codes. This can be implemented with much lower complexity than the method of adding a correction factor while reducing the performance degradation caused by a structure without a correction factor.

 As described in detail above, the present invention adopts the Min-Sum algorithm, but corrects the method by adding only a correction factor considering only the K site combinations that occupy the largest weight, thereby approaching the Sum-Product without increasing the complexity. This has the effect of achieving decoding performance.

Claims (4)

In the method for graph decoding the received LDPC code with a Min-Sum algorithm, Site combinations (

,

Selecting a site combination having a small value; The equation below is used so that only the K site combinations, which account for the largest proportion of the multiple site combinations, are selected as the correction factors.

here,

,

Is a nonlinear function

Is quantized to the value stored in the lookup table.

Updating the process; And correcting the loss generated in the decoding process by adding the update result to the selection result. The method of claim 1, wherein selecting a site combination comprises: A method of decoding a graph using partial correction, characterized in that J checks are connected to one site on a Tanner graph and K sites are connected to one check. The method of claim 1, wherein

Is, A method of decoding a graph using partial correction, characterized in that it is obtained by two additions and one lookup table access. delete

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