JP2008154238A - Method and apparatus for calculating multiplicity in rs decoding, and decoder and decoding method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method and apparatus for calculating multiplicity in RS decoding in which complexity in multiplicity calculation is reduced, and an RS decoding method and RS decoder with which complexity in algebraic soft decision decoding of RS codes is reduced. <P>SOLUTION: A calculation method according to the present invention includes the steps of: inputting a reliability matrix and the target value of multiplicity calculation and representing the target value with a monotonically increasing function or monotonically decreasing function of one independent parameter; calculating an independent parameter value corresponding to the target value by applying high-speed iteration to the independent parameter while utilizing the characteristic that the target value is the monotonically increasing function or monotonically decreasing function of the independent parameter, and calculating the multiplicity matrix on the basis of the reliability matrix and the calculated independent parameter value. Furthermore, an apparatus for calculating multiplicity in RS decoding, an RS decoder and a decoding method thereof are disclosed. The application of the present invention enables complexity in multiplicity calculation to be obviously reduced and complexity in algebraic soft decision decoding of RS codes to be reduced. <P>COPYRIGHT: (C)2008,JPO&INPIT

Description

 The present invention relates to the field of RS decoding technology, and more particularly, to a method for calculating the degree of redundancy in RS decoding, an apparatus, a decoder, and a decoding method.

 Reed-Solomon (RS) code was already submitted in the 1960s of the 20th century, and is now a digital communication system such as a wired, wireless and deep space communication system, a system such as data storage and code compression Has been widely applied to. The RS code is a kind of linear block codes, and its codeword has a maximum distance separation (MDS) characteristic, and has good resistance to burst error and channel fading. Have.

 For many years, the error correction potential of the RS code has not been sufficiently exhibited by people decoding the RS code by a hard decision or non-algebraic soft decision method. Therefore, the decoding performance of the RS code cannot be compared with a turbo code close to the Shannon limit or a low density parity check (LDPC) code for a long time.

 Recently, the algebraic soft decision decoding of RS codes has made some breakthroughs, so that the code has many applications including channel fading countermeasures, short codes, information source compression, etc. In the field, it can be seen that there are further performance advantages over Turbo codes and LDPC codes. The work of algebraic soft-decision decoding of RS codes started from the work of list decoding of RS codes of 1997 Sudan. The error correction capability of the RS code decoder has been greatly enhanced by Sudan's list decoding work. Guruswami et al. Improved the work of Sudan and further enhanced the error correction performance of the RS code decoder. Over the work of Guruswami and Sudan, Koetter and Vardy have submitted algebraic soft-decision decoding, so that the decoding performance of the code can be comparable to Turbo code and LDPC code, and may even exceed these codes. .

 Next, algebraic soft decision decoding of the RS code will be described. RS code algebraic soft-decision decoding includes three key steps: redundancy calculation, bivariate polynomial interpolation calculation, and bivariate polynomial decomposition. For the redundancy calculation, in most cases, the information input to the RS decoder is the probability reliability information of the received symbol. Generally, this information is called “soft information”. In order to be able to perform Guruswami-Sudan (GS) list decoding, the probability soft information needs to be converted into the redundancy of interpolation points of GS list decoding.

 In contrast to hard-decision decoding and non-algebraic soft-decision decoding, an algebraic soft-decision decoder of an RS code has better decoding performance, but the decoding complexity is also higher. To reduce the complexity of RS code algebraic soft-decision decoding, extensive research has been done on bivariate polynomial interpolation and bivariate polynomial decomposition, including a variety of divide-and-conquer methods. The method was studied.

 However, even if some improvements are made, the complexity of the algebraic soft decision decoding of the RS code is still very high because the complexity of the redundancy calculation step in the prior art is still high. In addition, in the prior art, even though the complexity of the redundancy calculation step can be controlled through a certain method, the target value cannot be controlled. As a result, the complexity of the interpolation and decomposition steps is increased as well as decoding. May lead to deterioration of vessel performance. Both of these cases cannot be assumed in a practical system.

 The present invention has been made in view of the above circumstances. The main object of the present invention is to provide a method for calculating the degree of redundancy in RS decoding so as to reduce the complexity of calculating the degree of redundancy.

 Another object of the present invention is to provide an apparatus for calculating redundancy in RS decoding so as to reduce the complexity of calculating redundancy.

 Another object of the present invention is to provide an RS decoder so as to reduce the complexity of algebraic soft decision decoding of RS codes.

 Another object of the present invention is to provide an RS decoding method so as to reduce the complexity of RS code algebraic soft decision decoding.

 In order to achieve the above object, the solution of the present invention is realized as follows.

A method for calculating the degree of duplication in RS decoding,
Inputting a reliability matrix and a target value for calculating the degree of overlap, and expressing the target value as a monotonically increasing function or a monotonically decreasing function of one independent variable;
Using the characteristic that the target value is a monotonically increasing function or a monotonically decreasing function of the independent variable, the independent variable corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the trust Calculating a redundancy matrix based on a degree matrix and the calculated independent variable value.

  Performing fast iteration on the independent variable includes performing fast iteration on the independent variable using a bisection algorithm or golden section algorithm.

The target value is the number of interpolation points of the multiplicity calculation, and the fast iteration is specifically:
Step A1 that takes an upper limit value that is larger than the independent variable value corresponding to the target value and a lower limit value that is smaller than the independent variable value corresponding to the target value for the independent variable;
Calculates one objective function value between the upper limit value and the lower limit value, and determines whether the objective function value at that point is smaller than the number of interpolation points. If the value is updated and not smaller, step B1 for updating the upper limit value with the independent variable value of the point;
It is determined whether or not the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the high-speed iteration is terminated. If not, step C1 is returned to step B1.

The lower limit is

And the upper limit is

And

Here, N is the code length of the RS code, s is the number of interpolation points,

Is the modulation order.

The target value is a cost value, and the fast iteration is specifically:
Step A2 for taking an upper limit value larger than the independent variable value corresponding to the target value and a lower limit value smaller than the independent variable value corresponding to the target value for the independent variable;
The objective function value at one point between the upper limit value and the lower limit value is calculated, and it is determined whether the objective function value at the point is smaller than the cost value. Updating, if not small, step B2 for updating the upper limit value with the independent variable value of the point;
It is determined whether or not the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the high-speed iteration is terminated. If not, step C2 is returned to step B2.

The lower limit is

And the upper limit is

And

Here, N is the code length of the RS code, C is the cost value,

Is the modulation order,

  Is an element of the i-th row and j-th column of the reliability matrix.

  The reliability matrix is a reliability full matrix or a reliability sparse matrix.

An apparatus for calculating the degree of redundancy in RS decoding,
An input module for receiving a reliability matrix and a target value for calculating the degree of overlap;
A target value display module for expressing the target value by a monotone function of one independent variable;
Using the characteristic that the target value is a monotonic function of an independent variable, the independent variable value corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the independent variable value calculated with the reliability matrix And a multiplicity matrix calculation module for calculating a multiplicity matrix based on.

  The target value display module represents the target value as a monotonically increasing function of one independent variable, or represents the target value as a monotone decreasing function of one independent variable.

  The input module receives a confidence complete matrix or a confidence sparse matrix.

  The multiplicity matrix calculation module performs high-speed iteration on the independent variable by a dichotomy algorithm or a golden section algorithm.

An RS code decoder comprising:
Calculate the multiplicity matrix based on the input reliability matrix and the target value of the multiplicity calculation, where the target value is represented by a monotone function of one independent variable, and the target value is a monotone function of the independent variable Using the characteristics, the independent variable corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the multiplicity matrix is calculated based on the calculated independent variable value and the reliability matrix. A multiplicity calculator,
A bivariate polynomial interpolator for obtaining a bivariate polynomial by calculating a coefficient of the bivariate polynomial based on the calculated multiplicity matrix;
A bivariate polynomial decomposition apparatus that decomposes the obtained bivariate polynomial to obtain a candidate codeword for list decoding.

An RS code decoding method comprising:
Input the reliability matrix and the target value of the multiplicity calculation, represent the target value by a monotone function of one independent variable, and use the characteristic that the target value is a monotone function of the independent variable to Calculating an independent variable value corresponding to a target value by performing high-speed iteration on, and calculating a multiplicity matrix based on the reliability matrix and the calculated independent variable value;
Calculating a coefficient of a bivariate polynomial based on the calculated multiplicity matrix to obtain a bivariate polynomial, and decomposing the obtained bivariate polynomial to obtain a candidate codeword for list decoding.

The target value is the number of interpolation points of the multiplicity calculation, and the fast iteration is specifically:
Step A3 for taking an upper limit value larger than the independent variable value corresponding to the target value and a lower limit value smaller than the independent variable value corresponding to the target value for the independent variable;
Calculates one objective function value between the upper limit value and the lower limit value, and determines whether the objective function value at that point is smaller than the number of interpolation points. If the value is updated and not smaller, step B3 for updating the upper limit value with the independent variable value of the point;
It is determined whether or not the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the high-speed iteration is terminated. If not, step C3 is returned to step B3.

The target value is a cost value, and the fast iteration is specifically:
Step A4 for taking an upper limit value larger than the independent variable value corresponding to the target value and a lower limit value smaller than the independent variable value corresponding to the target value for the independent variable;
The objective function value at one point between the upper limit value and the lower limit value is calculated, and it is determined whether the objective function value at the point is smaller than the cost value. Updating, if not smaller, step B4 for updating the upper limit value with the independent variable value of the point;
It is determined whether or not the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the high-speed iteration is terminated. If not, step C4 is returned to step B4.

 As can be seen from the above solution, in the present invention, the target value is represented by a monotone function of one independent variable, and the target value is a monotonic function of the independent variable, and a fast iteration is performed on the independent variable. To calculate an independent variable value corresponding to the target value, and calculate a redundancy matrix based on the reliability matrix and the calculated independent variable value. As can be seen from the above, in the present invention, a monotonic function characteristic is introduced into the algebraic soft decision decoding redundancy calculation of the RS code, an appropriate initial value is found, and the redundancy matrix is fastened by various fast iterative algorithms. Can be calculated. Therefore, the complexity of the redundancy calculation is significantly reduced, thereby reducing the complexity of decoding the RS code.

 In the present invention, one point is selected between the lower limit value and the upper limit value of the set independent variable, the magnitude of the function value of the point and the target value is compared, and based on the comparison result A high-speed search is realized by updating the lower limit value or the upper limit value.

  In the present invention, the iterative algorithm employed may be various fast iterative algorithms including algorithms such as the golden section method and the bisection method, and is very convenient for various specific applications.

 In order to further clarify the objects, solutions and merits of the present invention, the present invention will be described in more detail below with reference to the drawings and specific examples.

  RS code algebraic soft-decision decoding has better decoding performance but higher complexity than traditional hard-decision decoding and non-algebraic soft-decision decoding. The amount of computation in the redundancy calculation step occupies a considerable specific gravity in the entire algebraic soft decision decoder.

In a normal case, the amount of calculation of the degree of redundancy for realizing the algebraic soft decision decoding of the RS code is

It is. here,

Is the code length. When the modulation order q is relatively large, the amount of calculation required for the redundancy calculation is large. For a q value that is always selectable, for example, 256, the number of required arithmetic operations (for example, memory access, comparison, etc.) for realizing the redundancy calculation is

  There is a possibility of reaching.

  In the present invention, the complexity of the RS code algebraic soft decision decoder can be clearly reduced by reducing the complexity of the redundancy calculation step.

  First, the algorithm principle of the present invention will be described in detail.

Code length

Information length

For the RS code, the information polynomial is

And where the information symbol

Are elements defined in a finite field GF (q).

In the finite field GF (q),

Select different non-zero elements and set

Get. set

  Can be used for RS encoding.

The set of RS codewords may be described as follows:

RS code sequence to be transmitted

Suppose that

The sequence is sent to the modulated channel, affected by channel noise and interference, and arrives at the receiving side.

  It becomes.

RS code algebraic soft-decision decoder

  Based on the above, we reconstruct some maximum transmittable information polynomial on the transmitting side. This process is called list decoding.

  In order to realize RS code list decoding, it is necessary to extract interpolation point information and redundancy information corresponding to the interpolation point from the received input information.

  In a practical system, the information obtained by the receiver is always probability reliability information. In order to realize algebraic soft decision decoding of the RS code, it is necessary to convert these pieces of probability information into interpolation points and corresponding redundancy information. The conversion is a redundancy calculation.

Decoder is polynomial

The received vector

The probability reliability matrix corresponding to

  Suppose that

Where the matrix

Is the received series

The transmitted information polynomial is

  The probability that the jth encoded symbol is i−1.

is there

Then interpolation point

Get. The interpolation point is a non-negative integer called multiplicity

  Corresponding to

Interpolation point

The degree of overlap corresponding to

Then, for the point, in the interpolation step of the algebraic soft decision decoding of the RS code,

  A constraint equation is generated.

line; queue; procession; parade

The multiplicity matrix corresponding to

If so, the number of constraint equations generated in the interpolation step is

  And is called cost.

  The higher the cost value, the higher the complexity required for the interpolation and decomposition steps.

On the other hand, each element of the multiplicity matrix

The value multiplied by the probability is

It is. Multiplicity matrix

In response to

The number of such products, and the value obtained by adding these products

  Is defined as a score. As can be seen from a study by Koeter et al., The higher the score, the better the output performance of the algebraic soft decision decoder.

  An actual decoder can only accept a finite complexity. . The redundancy calculation needs to maximize the score when the interpolation and decomposition step complexity is constant, or minimize the interpolation and decomposition step complexity when the score is constant.

In other words, the redundancy calculation needs to maximize the score when the cost is constant, or minimize the cost when the score is constant. The former is a multiplicity matrix set.

Is actually a matrix

  Is to look for.

  In order to more clearly explain the distinction between the prior art and the present invention, first, the conventional Kotter algorithm will be schematically described. In order to calculate the optimal redundancy matrix, the following algorithm is given by Kotter et al.

Input is a confidence matrix

, And the number of interpolation points s to be calculated based on the degree of overlap (first case):
Input is a confidence matrix

, And the number of interpolation points s to calculate with overlap, the output is

Optimal multiplicity matrix satisfying

It is. In step 1, settings are made as follows.

In step 2, the matrix

Maximum value position in

The position corresponding to

Find and position

Values and counters corresponding to

Is updated as follows.

In step 3,

If so, go back to the matrix

Output

  If so, return to Step 2.

Input is a confidence matrix

And cost value

If (the second case):
Input is confidence matrix and cost value

And the output is

Optimal multiplicity matrix satisfying

  It is.

In step 1, settings are made as follows.

In step 2, the matrix

Maximum value position in

The position corresponding to

Find and position

The value corresponding to is updated as follows.

In step 3,

If,

And return to step 2,

If so, go back to the matrix

  Is output.

In the first case, first, the number of interpolation points s and the reliability matrix

And then

And counter

Is given an initial value, then the matrix

Maximum value position in

The position corresponding to

Looking for and procession

And multiplicity matrix

Position of

Update the element corresponding to. Next, the counter is incremented by 1 and it is determined whether or not the number of iterations has reached the specified number of interpolation points s. If not, the iteration is continued. line; queue; procession; parade

  Is output.

  Regarding the process in the second case, since only the stop control parameter is different from that in the first case, detailed description of the algorithm steps is omitted.

  Next, consider the computational complexity of algorithms such as Koetter. Since the first case and the second case correspond to the same algorithm and are distinguished only by their stop control parameters, only the complexity of the first case is considered.

In the first case, size

The maximum value search is performed s times within the matrix. Here for RS code

Is established. Therefore, the complexity of the process is

  It becomes.

Consider sparse matrix, ie, for each received symbol, closest to the perimeter of the received point

If only the corresponding probabilities of the parameter points are put in the confidence matrix, each column of the confidence matrix

There are only elements, the computational complexity of the first case

It becomes.

Is a multiplicity matrix generated in the s-th case of the first case of an algorithm such as Koetter,

Defined in the execution column of

Elements in

Meet.

  Koeter et al. Proved the following theorem.

Each positive number

Against

There is a non-negative integer that satisfies
Conversely, for each non-negative integer s, a positive real number that holds the above equation

  Exists.

  Based on the above theorem, we analyze as follows.

  According to the theorem, the applicant can obtain the following inference (inference 1 and inference 2).

(Inference 1) Two positive real numbers

Meets non-negative integers

Each is a multiplicity matrix

If it corresponds to

Is always true.

  And

In the first case of an algorithm such as Koetter, each iteration step has its corresponding multiplicity

Only one position where 1 is added

Because there is

Is established.

If you pay attention to

It becomes. Then

  become.

(Inference 2) Iterative step in the first case of an algorithm such as Koetter

Multiplicity matrix generated by

Corresponding to

If is a positive number, the cost value

Is a parameter

  The increase is monotonous.

  Since the inference 2 can be obtained directly from the inference 1, its proof is omitted here.

Therefore, the target value for calculating the degree of duplication in algebraic soft decision decoding of RS code (number of interpolation points s or cost value to be calculated by degree of duplication)

) Is a parameter

  The conclusion is that it can be expressed as a monotonically increasing function. That is, the redundancy calculation can be converted into a problem of finding a solution of the target value of the monotonically increasing function. In order to find the target value of the monotonically increasing function or monotonically decreasing function, it is possible to adopt a method such as the bisection method or the golden section method so as to obtain a solution at high speed without adopting a highly complex iterative method. it can. The bisection method and the golden section method have a low convergence rate because of their fast convergence.

If the target value of the algebraic soft decision decoding of RS code can be expressed by a monotonically increasing function of an independent variable, the monotonically increasing function

Against the target value

Are given, respectively

Two initial values satisfying

If you can find the equation, you can use various methods such as dichotomy or golden section method.

  And the target value can be calculated at high speed.

  In the present invention, the monotone function characteristic is introduced into the algebraic soft decision decoding redundancy calculation of the RS code, and by finding an appropriate initial value, the redundancy matrix can be obtained by a fast iterative method such as the bisection method or the golden division method. It can be calculated at high speed.

  Since the monotone function characteristics are introduced, the required result can be obtained with only a small number of iteration steps by using the dichotomy method or the golden section method with a fast convergence speed. The present invention avoids the disadvantage that the amount of calculation by the sequential search of the method such as Koeter is large, and extremely reduces the complexity of the redundancy calculation.

Specifically, for the completeness matrix of reliability, the complexity required for calculating the degree of redundancy is the same as that of a method such as Koetter.

And the required complexity for a sparse reliability matrix is that of a method such as Koetter

  Reduce to linear complexity.

The amount of calculation required by the present invention is

  Since the present invention has the same input / output as the Koetter method, the decoding performance of the entire decoder can be reduced without increasing the complexity of the interpolation and decomposition steps. Absent.

  Next, a fast search for a target value of a monotonically increasing function will be described by taking a bisection method as an example.

function

Is monotonically increasing, and

And the target value

Solution corresponding to

Is the value to be obtained. First,

The function value corresponding to is calculated.

Because

Is the new limit,

The function value corresponding to is calculated.

Because

  Is set as a new lower limit, and the target value can be obtained at high speed by repeating the process.

In order to calculate the multiplicity matrix using the bisection method or the golden section method, it is also necessary to set an initial value. function

Is an increase function of the target value

And given the equation

Meet

Ask for. Respectively

Two initial values satisfying

Can be given in advance, the target value can be quickly searched using the bisection method or the golden section method immediately. Of course, if the initial value is not set, the target value can be searched for the monotonically increasing function or the monotonically decreasing function using the bisection method or the golden section method, but the initial value is set in the first step of the algorithm. Since it needs to be determined, its complexity is slightly higher than the method with the initial value set.

  There are the following inferences (inference 3 and inference 4) for setting the initial value of the multiplicity calculation.

(Inference 3) Reliability matrix

And a positive integer s,

And are set. Non-negative integer

Each is a multiplicity matrix

If you respond to

It becomes.

  It becomes.

(Inference 4) Reliability matrix

And positive real numbers

There is

And are set.

Defined as
Inequality

Is established.

Can easily verify that it is one of the roots of

  That is, the left half inequality holds.

  Next, we prove the inequality of the right half.

Each positive number

Against

There is a non-negative integer s that satisfies, and conversely, for each non-negative integer s, a positive real number that holds the above expression

From the theorem that there exists

  I understand.

here,

If you write

It becomes. Then

Can be obtained.

The value of the first term of the last equation of the equation is exactly

  And the third term is a non-negative number.

If we write the second term alone,

In conjunction with the previous equation

  Get.

In inference 3 and inference 4, the target value is the number of interpolation points s and the cost value, respectively.

An initial value is given for. Based on these two inferences, if the initial value is set according to the above formula,

Be sure

I explained that it was between.

  From the above detailed analysis, the following can be understood.

Target value for calculating the degree of duplication of RS code algebraic soft decision decoding (number of interpolation points s or cost value to be calculated based on degree of duplication)

) Is a parameter

It can be seen that it can be expressed by a monotonically increasing function or a monotonically decreasing function. Also, set the upper and lower limits of the initial value, the target value s or

Corresponding to

May be between the lower and upper limits. Then, for example, high-speed calculation of the redundancy matrix can be realized by using the bisection method or the golden section method.

  As described above, FIG. 1 is a flowchart illustrating an example of a method for calculating the degree of duplication in RS decoding according to the present invention. As shown in FIG. 1, the method includes the following steps.

  In step 101, a reliability matrix and a target value for multiple calculation are input, and the target value is represented by a monotone function of one independent variable.

  Here, the monotone function may be a monotone increase function or a monotone decrease function. The reliability matrix may be a complete reliability matrix or a sparse reliability matrix.

  In step 102, using the characteristic that the target value is a monotonic function of the independent variable, the independent variable value corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the reliability matrix and the calculation are performed. A multiplicity matrix is calculated based on the independent variable values.

  Here, it is possible to perform high-speed iteration on independent variables using various high-speed iteration algorithms such as a bisection algorithm and a golden section algorithm.

  Specifically, when the target value is the number of interpolation points for the redundancy calculation, the fast iteration specifically corresponds to the target value with an upper limit value that is larger than the independent variable value corresponding to the target value for the independent variable first. Taking a lower limit value that is smaller than the independent variable value, then calculating an objective function value at one point between the upper limit value and the lower limit value, and determining whether the objective function value at that point is smaller than the number of interpolation points. If it is smaller, the lower limit value is updated with the independent variable value of the point. If not, the upper limit value is updated with the independent variable value of the point. Then, the difference between the upper limit value and the lower limit value is preset. Determining whether it is less than the threshold value, and if it is smaller, terminate the fast iteration, and if not smaller, subsequently performing the iteration.

Preferably, the lower limit is

And the upper limit is

And

Here, N is the code length of the RS code, s is the number of interpolation points,

  Is the modulation order.

  When the target value is a cost value, the fast iteration specifically specifies, for the independent variable, an upper limit value that is larger than the independent variable value corresponding to the target value and a lower limit value that is smaller than the independent variable value corresponding to the target value. Then, the objective function value of one point between the upper limit value and the lower limit value is calculated, and it is determined whether the objective function value at the point is smaller than the cost value. The lower limit value is updated with the independent variable value, and if it is not smaller, the upper limit value is updated with the independent variable value at the point, and then whether or not the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. Determining, if small, ending the fast iteration; if not, returning and continuing to repeat.

Preferably, the lower limit is

And the upper limit is

Here, N is the code length of the RS code, C is the cost value,

Is the modulation order,

Is an element of the i-th row and j-th column of the reliability matrix.

In the following, the computational complexity of the algorithm of the present invention will be considered. Each iteration step of the present invention is

Addition operation,

Multiplication operations and

Rounds of integer rounding operations are required. The dichotomy and the golden section method can complete the computation with fewer iteration steps (usually the requirement is already met with less than 20 iterations. More finite word length effects due to the computer's finite word length effect) The number of iterations only increases the amount of computation unnecessarily and has no real value.) Thus, the computational complexity of the algorithm of the present invention is

  It becomes.

Consider sparse matrix, ie, for each received symbol, closest to the perimeter of the received point

If only the corresponding probabilities of the parameter points are put in the confidence matrix, each column of the confidence matrix

There are only elements, and the computational complexity of our algorithm is

  It becomes.

  The above describes the algorithm of the present invention in detail. Next, a preferred embodiment of the present invention will be described in detail with reference to the above algorithm.

Input is a confidence matrix

  For the first case where the number of interpolation points s is calculated based on the degree of overlap, the first embodiment of the present invention will be specifically described below.

here,
Input is a confidence matrix

  , And the number s of interpolation points to be calculated based on the degree of overlap.

The output is

Optimal multiplicity matrix satisfying

  It is.

In step 1, settings are made as follows.

In step 2, set as follows,

In step 3,

  FIG. 2 is a flowchart of a redundancy calculation method in RS decoding according to the first embodiment of the present invention. As shown in FIG. 2, the method includes the following steps.

In step 201, the number of interpolation points s and the reliability matrix

, And preset parameters

  Enter.

here,

Are two positive numbers set in advance. Parameters

And may be set to 0.5 or 0.618, for example, corresponding to the bisection method or the golden section method.

Is a small positive number, for example, 0.0001. In iteration

The length of the line segment is

  In the following cases, it is assumed that the line segment has already converged to one point. Thus, the iteration may be stopped.

In step 202, initial iteration values are set as follows.

In step 203,

Calculate

In step 204,

Objective function value corresponding to

  Calculate

In step 205, the objective function value is compared with the input target value s. If the function value is smaller than the target value, in step 206,

If the function value is not smaller than the target value in step 207,

  Update the value of.

In step 208,

Is a preset threshold value

Check if it is less.

If it is smaller, it means that the condition for stopping the iteration is satisfied.

Is output.

  If not, return to step 203 and then repeat.

Input is a confidence matrix

And cost value

  Hereinafter, the second embodiment of the present invention will be described in detail.

Input is a confidence matrix

And cost value

It is.

The output is

Optimal multiplicity matrix satisfying

  It is.

In step 1, settings are made as follows.

In step 2, set as follows,

In step 3,

  If yes, go back to step 2.

  FIG. 3 is a flowchart of a redundancy calculation method in RS decoding according to the second embodiment of the present invention. As shown in FIG. 3, the method includes the following steps.

In step 301, the cost value

, Confidence matrix

, And preset parameters

  Enter.

here,

Are two positive numbers set in advance. Parameters

Thus, for example, 0.5 or 0.618 may be set in accordance with the bisection method or the golden section method.

Is a small positive number, for example, 0.0001. In iteration

The length of the line segment is

  In the following cases, it is assumed that the line segment converges to one point. Thus, the iteration may be stopped.

In step 302, the initial iteration value is set as follows.

In step 303,

And then in step 304:

Objective function value corresponding to

  Calculate

In step 305, the objective function value and the input target value

If the function value is smaller than the target value, in step 306,

If the function value is not smaller than the target value in step 307,

Update the value of.

In step 308,

Is a preset threshold value

Check if it is less.

If it is smaller, it indicates that the condition for stopping the iteration is satisfied.

Is output.

  If not, return to step 303 and repeat.

In inference 3 and inference 4, the target values are s and

An initial value is given for. Based on these two inferences, the desired goal

Explained that there is. FIG. 4 is a diagram showing an example of selection of the upper limit value and the lower limit value in the first embodiment of the present invention. FIG. 5 is a diagram showing an example of selection of the upper limit value and the lower limit value in the second embodiment of the present invention.

Moreover, based on the said algorithm and process flow, this invention provides the calculation apparatus of the duplication degree in RS decoding. FIG. 6 is a block diagram showing an example of a duplication degree calculation apparatus in RS decoding according to the present invention. As shown in FIG.
An input module 601 for receiving a reliability matrix and a target value for calculating the degree of overlap;
A target value display module 602 that represents the target value as a monotonic function of one independent variable;
Using the characteristic that the target value is a monotonic function of an independent variable, the independent variable value corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the independent variable value calculated with the reliability matrix And a redundancy matrix calculation module 603 that calculates a redundancy matrix based on

  Here, the target value display module 602 may be configured to represent the target value as a monotonically increasing function of one independent variable, or to represent the target value as a monotone decreasing function of one independent variable. The confidence matrix received by the input module 601 may be a complete confidence matrix or a sparse confidence matrix.

  Preferably, the multiplicity matrix calculation module 603 is configured to perform high-speed iteration on the independent variable by a dichotomy algorithm or a golden section algorithm.

Further, the present invention provides an RS code decoder based on the above-described redundancy calculation apparatus for RS decoding. FIG. 7 is a block diagram showing an example of an RS decoder according to the present invention. As shown in FIG.
Calculate the multiplicity matrix based on the input reliability matrix and the target value of the multiplicity calculation, where the target value is represented by a monotone function of one independent variable, and the target value is a monotone function of the independent variable Using the characteristics, the independent variable corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the multiplicity matrix is calculated based on the calculated independent variable value and the reliability matrix. A multiplicity calculator 701;
A bivariate polynomial interpolator 702 for obtaining a bivariate polynomial by calculating a coefficient of the bivariate polynomial based on the calculated multiplicity matrix;
A bivariate polynomial decomposition apparatus 703 that decomposes the obtained bivariate polynomial to obtain a candidate codeword for list decoding.

Specifically, considering the decoding of an RS code with a code length of N and an information length of K,
The bivariate polynomial interpolator 702 arranges some linear equations whose coefficients are defined in the finite field GF (q) after the interpolation points and the degree of overlap corresponding to each interpolation point are obtained. Since the unknowns of the obtained simultaneous linear equations correspond to the coefficients of the bivariate polynomial Q (x, y), it is possible to obtain the coefficients of the binary polynomial Q (x, y) by solving the simultaneous linear equations. The operation for obtaining the coefficient of Q (x, y) is called Q (x, y) interpolation.

  The bivariate polynomial decomposition apparatus 703 obtains a factor yf (x) of Q (x, y) after obtaining the bivariate polynomial Q (x, y). Here, f (x) is a K-1 or lower order polynomial defined in GF (q). The decomposed polynomials of the order of K-1 or lower are candidate codeword polynomials for list decoding, and an optimum decoded word can be obtained by comparing every arranged codeword with the received sequence.

The present invention also provides an RS code decoding method. FIG. 8 is a flowchart showing an example of the RS decoding method according to the present invention. As shown in FIG.
Input the reliability matrix and the target value of the multiplicity calculation, represent the target value by a monotone function of one independent variable, and use the characteristic that the target value is a monotone function of the independent variable to Calculating an independent variable value corresponding to a target value by performing a fast iteration on the target value, and calculating a multiplicity matrix based on the reliability matrix and the calculated independent variable value;
Calculating a coefficient of a bivariate polynomial based on the calculated redundancy matrix to obtain a bivariate polynomial, and decomposing the obtained bivariate polynomial to obtain a candidate codeword for list decoding; Including.

here,
When the target value is the number of interpolation points for calculating the degree of redundancy, the fast iteration specifically includes:

  First, for the independent variable, an upper limit value that is larger than the independent variable value corresponding to the target value and a lower limit value that is smaller than the independent variable value corresponding to the target value are taken, and then, between the upper limit value and the lower limit value. Calculate the objective function value of one point, determine whether the objective function value of the point is smaller than the number of interpolation points, and if it is smaller, update the lower limit value with the independent variable value of the point, and if not smaller, The upper limit value is updated with the independent variable value of the point, and then it is determined whether the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If so, return to the calculation of the objective function value for the point between the upper and lower limits and then repeat.

  If the target value is a cost value, the fast iteration specifically includes:

  First, for the independent variable, an upper limit value that is larger than the independent variable value corresponding to the target value and a lower limit value that is smaller than the independent variable value corresponding to the target value are taken, and then, between the upper limit value and the lower limit value. The objective function value of one point is calculated, and it is determined whether the objective function value of the point is smaller than the cost value. If it is smaller, the lower limit value is updated with the independent variable value of the point. Update the upper limit value with the independent variable value of, and then determine whether the difference between the upper limit value and the lower limit value is less than a preset threshold value, if small, terminate the fast iteration, if not, Return to the calculation of the objective function value for the point between the upper and lower limits and then repeat.

  Next, performance examples of the present invention will be described.

  In order to verify the effectiveness of the multiplicity calculation according to the present method, the simulation is performed with respect to the calculation speed of the algorithm on the Pentium (registered trademark) 4 Windows (registered trademark) XP platform by programming in C ++. Good. In order to eliminate the correlation between the execution result of the algorithm and the device, the ratio of the required execution time between the method submitted to Koetter and the present invention will be considered.

In order to accurately calculate the execution time, the test target algorithm may be called a plurality of times to obtain the average time. In the simulation, the setting of the number of calls is based on the principle that the obtained execution time guarantees a confidence level of 99%. By selecting two RS codes, one is defined in the finite field GF (256) (255, 144, 112), and the other is defined in the finite field GF (64) (63, 32). , 32). A simulation is performed using the two codes, and the execution speed is observed. A simulation is performed using an Additive White Gaussian Noise (AWGN) channel, and the result when the signal-to-noise ratio (Eb / N0) is 10 dB is observed. For each received symbol, closest to the received point

  Put the corresponding probabilities of the parameter points into the confidence matrix.

  Since the first and second embodiments correspond to different target values of the same algorithm, only the execution times of the improved second embodiment and the second case of the prior art are compared.

For list decoding, the number of codewords is

If not exceeding, according to the definition of Koetter etc.

Is the information length and the target value

May be defined as follows. Parameters

And set. Some parameter pairs

  Select to perform simulation. The simulation results are as shown in Table 1. The method of the present invention has the same inputs and outputs as the conventional algorithm such as Koeter.

Table 1 shows the results of the execution times of some RS codes, and the column S-R represents the time ratio of the redundancy calculation between the algorithm such as Koeter and the present invention.

  As can be seen from Table 1, the execution speed of the method of the present invention is much higher than the conventional algorithm. The present invention guarantees that the algebraic soft-decision decoding performance of the RS code does not change and reduces the implementation complexity.

 As described above, the target value of the redundancy calculation step of the algebraic soft decision decoding of the RS code can be represented by a monotonically increasing function or a monotonically decreasing function of an independent variable.

 The present invention introduces an increasing function characteristic into the algebraic soft decision decoding redundancy calculation of the RS code, and adopts the bisection method or the golden division method to find a suitable initial value, thereby making the redundancy matrix faster. Can be calculated. Since the increase function characteristic is introduced, the required result can be obtained with only a small number of iteration steps by using the dichotomy method or the golden section method with a fast convergence speed. This avoids the disadvantage that the amount of calculation by the sequential search of the method such as Koetter is avoided, and the complexity of the redundancy calculation is extremely reduced.

  The above are only preferred embodiments of the present invention and do not limit the protection scope of the present invention. Various modifications, equivalent switching, improvements and the like made within the spirit and principle of the present invention should all be included in the protection scope of the present invention.

It is a flowchart which shows the example of the calculation method of the duplication degree in RS decoding of this invention. It is a flowchart of the calculation method of the duplication degree in RS decoding by 1st Example of this invention. It is a flowchart of the calculation method of the duplication degree in RS decoding by 2nd Example of this invention. It is a figure which shows the example of selection of the upper limit and lower limit in 1st Example of this invention. It is a figure which shows the example of selection of the upper limit and lower limit in 2nd Example of this invention. It is a block diagram which shows the example of the calculation apparatus of the duplication degree in RS decoding by this invention. It is a block diagram which shows the example of RS decoder by this invention. 5 is a flowchart illustrating an example of an RS decoding method according to the present invention.

Explanation of symbols

  DESCRIPTION OF SYMBOLS 600 ... Multiplicity calculation apparatus, 601 ... Input module, 602 ... Target value display module, 603 ... Multiplicity matrix calculation module, 700 ... Decoder of RS code, 701 ... Multiplicity calculation apparatus, 702 ... Bivariate polynomial interpolation Device, 703... Two-variable polynomial decomposition device

Claims (15)

A method for calculating the degree of duplication in RS decoding,
Inputting a confidence matrix and a target value for calculating the multiplicity, and expressing the target value by a monotonically increasing function or a monotonically decreasing function of one independent variable;
Using the characteristic that the target value is a monotonically increasing function or a monotonically decreasing function of the independent variable, the independent variable value corresponding to the target value is calculated by performing fast iteration on the independent variable, and the reliability matrix and Calculating a multiplicity matrix based on the calculated independent variable values;
The calculation method of the duplication degree in RS decoding characterized by including these. Performing fast iteration on the independent variable is
Fast iteration over the independent variables with a bisection algorithm or golden section algorithm,
The method of calculating the degree of duplication in RS decoding according to claim 1, wherein: The target value is the number of interpolation points of the multiplicity calculation, and the fast iteration is
Step A1 that takes an upper limit value that is larger than the independent variable value corresponding to the target value and a lower limit value that is smaller than the independent variable value corresponding to the target value for the independent variable;
Calculates one objective function value between the upper limit value and the lower limit value, and determines whether the objective function value at that point is smaller than the number of interpolation points. If the value is updated and not smaller, step B1 for updating the upper limit value with the independent variable value of the point;
It is determined whether or not the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the fast iteration is terminated. If not, step C1 returns to step B1.
The method of calculating the degree of duplication in RS decoding according to claim 1, wherein: The lower limit is

And the upper limit is

And

Here, N is the code length of the RS code, s is the number of interpolation points,

Is the modulation order,
The method for calculating the degree of duplication in RS decoding according to claim 3. The target value is a cost value, and the fast iteration is
Step A2 for taking an upper limit value larger than the independent variable value corresponding to the target value and a lower limit value smaller than the independent variable value corresponding to the target value for the independent variable;
The objective function value at one point between the upper limit value and the lower limit value is calculated, and it is determined whether the objective function value at the point is smaller than the cost value. Updating, if not small, step B2 for updating the upper limit value with the independent variable value of the point;
It is determined whether the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the fast iteration is terminated. If not smaller, step C2 returns to step B2.
The method of calculating the degree of duplication in RS decoding according to claim 1, wherein: The lower limit is

And the upper limit is

And

Here, N is the code length of the RS code, C is the cost value,

Is the modulation order,

Is the element of the i-th row and j-th column of the reliability matrix,
The method for calculating the degree of redundancy in RS decoding according to claim 5.  The method of calculating the degree of duplication in RS decoding according to any one of claims 1 to 6, wherein the reliability matrix is a reliability complete matrix or a reliability sparse matrix. An apparatus for calculating the degree of redundancy in RS decoding,
An input module for receiving a reliability matrix and a target value for calculating the degree of overlap;
A target value display module for expressing the target value by a monotone function of one independent variable;
Using the characteristic that the target value is a monotonic function of an independent variable, the independent variable value corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the independent variable value calculated with the reliability matrix A redundancy matrix calculation module for calculating a redundancy matrix based on
An apparatus for calculating the degree of duplication in RS decoding, comprising:   9. The RS decoding according to claim 8, wherein the target value display module represents the target value as a monotonically increasing function of one independent variable, or represents the target value as a monotone decreasing function of one independent variable. A device for calculating the degree of duplication in computerization.  The apparatus according to claim 8, wherein the input module receives a reliability complete matrix or a reliability sparse matrix.   The RS decoding according to any one of claims 8 to 10, wherein the multiplicity matrix calculation module performs high-speed iteration on the independent variable by a dichotomy algorithm or a golden section algorithm. A device for calculating the degree of duplication in computerization. An RS code decoder comprising:
Calculate the multiplicity matrix based on the input reliability matrix and the target value of the multiplicity calculation, where the target value is represented by a monotone function of one independent variable, and the target value is a monotone function of the independent variable Using the characteristics, the independent variable corresponding to the target value is calculated by performing high-speed iteration on the independent variable, and the multiplicity matrix is calculated based on the calculated independent variable value and the reliability matrix. A multiplicity calculator,
A bivariate polynomial interpolator for obtaining a bivariate polynomial by calculating a coefficient of the bivariate polynomial based on the calculated multiplicity matrix;
A bivariate polynomial decomposition apparatus that decomposes the obtained bivariate polynomial to obtain a candidate codeword for list decoding;
An RS code decoder comprising: An RS code decoding method comprising:
Input the reliability matrix and the target value of the multiplicity calculation, represent the target value by a monotone function of one independent variable, and use the characteristic that the target value is a monotone function of the independent variable to Calculating an independent variable value corresponding to a target value by performing high-speed iteration on, and calculating a multiplicity matrix based on the reliability matrix and the calculated independent variable value;
Calculating a coefficient of a bivariate polynomial based on the calculated redundancy matrix to obtain a bivariate polynomial, decomposing the obtained bivariate polynomial to obtain a list decoding candidate codeword;
An RS code decoding method characterized by comprising: The target value is the number of interpolation points of the multiplicity calculation, and the fast iteration is
Step A3 for taking an upper limit value larger than the independent variable value corresponding to the target value and a lower limit value smaller than the independent variable value corresponding to the target value for the independent variable;
Calculates one objective function value between the upper limit value and the lower limit value, and determines whether the objective function value at that point is smaller than the number of interpolation points. If the value is updated and not smaller, step B3 for updating the upper limit value with the independent variable value of the point;
It is determined whether the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the fast iteration is terminated. If not, step C3 returns to step B3.
The RS code decoding method according to claim 13, comprising: The target value is a cost value, and the fast iteration is
Step A4 for taking an upper limit value larger than the independent variable value corresponding to the target value and a lower limit value smaller than the independent variable value corresponding to the target value for the independent variable;
The objective function value at one point between the upper limit value and the lower limit value is calculated, and it is determined whether the objective function value at the point is smaller than the cost value. Updating, if not smaller, step B4 for updating the upper limit value with the independent variable value of the point;
It is determined whether the difference between the upper limit value and the lower limit value is smaller than a preset threshold value. If the difference is smaller, the fast iteration is terminated. If not, step C4 returns to step B4.
The RS code decoding method according to claim 13, comprising:

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