CN1697330A - Method of using algorithm of adding edge one by one for encoding and decoding low-density parity check code in multiple system of carry digit - Google Patents

Abstract

The method includes encoding part and corresponding decoding part. PEG design method is adopted to complete design for encoding codes of multiple scale system low-density parity check (LDPC). Using equivalent transformation and combining it with Fourier transformation realizes decoding codes of LDPC in multiple scale system. The invention improves performance of error rate of LDPC code in multiple scale system evidently, and reduces complexity of decoding codes. Moreover, loss of performance is in acceptable range. In condition of transmitting middle - short frames, the invention is superior to performance of traditional random structured LDPC code.

Description

Add the multi-system encoding and decoding low-density parity check code method of limit algorithm one by one

Technical field

The invention belongs to digital communication technology field, be specifically related to a kind of based on multi-system low-density checksum (LDPC) code coding/decoding method that adds limit (PEG) algorithm one by one.

Background technology

Shannon had proposed famous channel coding theorem first in his laying a foundation property paper " mathematical principle of communication " in 1948: be the communication channel of C to any channel capacity, if the desired transmission rate R of communication system is less than C, then there is a coding method, fully big and when adopting maximum-likelihood decoding as code length n, it is arbitrarily small that the error rate of system can reach.His theorem has three primary conditions:

1) adopts random coded;

2) code length n → ∞, promptly code length is tending towards unlimited;

3) best maximum likelihood (ML) method is adopted in decoding.

In the research and development process of chnnel coding, be main developing direction with latter two condition basically.Owing to be difficult to realize the randomness coding method in practice, and when code length is very big, adopt the maximum-likelihood decoding algorithm to be difficult to realize.And 1993 by Berrou C, Glavieux A, Thitimajshima propose new channel coding schemes Turbo code, obtain almost decoding performance near the Shannon theoretical limit thereby then used the stochastic behaviour of coding and iterative decoding well.About the research of Turbo code coding and iterative decoding becomes the hot subject that channel coding theorem is studied soon, and in 3GPP, be applied.

Iterative decoding algorithm has better performance again when having reduced complexity, its principle is exactly to carry out hard decision by the iteration of external information after satisfying condition.

The error correcting code theoretical developments is to today, particularly after the nineties, become the main research object of industry based on the efficient error correcting code of random coded and iterative decoding, new error correcting system based on graph theory is flourish, and Here it is with tanner figure is the method for visualizing of the base growth code Design of getting up: factor graph and the iterative decoding algorithm and the sum-product algorithm that transmit based on information on the limit among the figure.

The LDPC sign indicating number is to propose (R.G.Gallager. low density parity check code [J] .IRE Trans.Info.Theory1962IT-8 (1): 1-28) by Gallager in 1962.Do not cause at that time that people noted too much, the nineties in 20th century, the someone finds that the LDPC sign indicating number adopts the sum-product algorithm iterative decoding can reach the performance of nearly shannon limit when long frame.So the LDPC sign indicating number becomes new research focus rapidly.

The LDPC sign indicating number is based on the parity check code of sparse matrix, and the design principle of matrix H is the relation of adjusting between code word bits and the check bit, thereby reaches the optimized purpose of information flow in the decode procedure.Therefore, the design of matrix H is most important.Usually H adopts random configuration and eliminates the mode that length is 4 ring, and X.Y.Hu etc. have proposed a kind of sub-optimal method---PEG (Progressive Edge--Growth) method (X.-Y.Hu that can construct long as far as possible ring, E.Eleftheriou, and D.-M.Arnold; " the Taylor diagram that the limit increases one by one " IEEE Proc.Globecom ' 2001; San Antorio, TX, Nov.2001.).Its simulation result shows: based on good than based on random fashion of the LDPC sign indicating number of PEG algorithm, especially in short-and-medium frame.But prior art only discloses the binary system LDPC sign indicating number based on the PEG algorithm.

Above-mentioned binary system LDPC sign indicating number can also generally change into finite field gf (q) (q=2 ^p, p ∈ Z ⁺) on the m-ary LDPC sign indicating number.Its bilateral figure and binary system LDPC sign indicating number similar, different is the variable point has q may value, the constrained of checkpoint is also more complicated in addition.David J.C.Mackay discovers, the performance outline of some m-ary LDPC sign indicating number is better than binary system LDPC sign indicating number, code check (the M.C.Davey that is improved simultaneously, low density parity check code [J] .IEEE Commun.Lett. on the D.J.C.Mackay.Q unit territory, 1998,2 (6): 165-167.).

The m-ary LDPC sign indicating number can adopt the decoding of BP algorithm, but complexity is very high like this, Richardson and Urbanke suggestion adopt the decoding of FT (Fourier transform) algorithm to reduce decoding complexity (T.Richardson and R.Urbanke. adopts the capacity .Submitted to IEEE Transactions onInformation Theory of the low density parity check code of reliability pass-algorithm decoding, 1998.).For example, the Fourier transform of GF (2) superior function f is: F ⁰=f ⁰+ f ¹, F ¹=f ⁰-f ¹.GF (2 ^p) on Fourier transform can regard a succession of 2 system conversion as, each has p dimension.Therefore, the Fourier transform on the GF (4) is:

F ⁰＝[f ⁰+f ¹]+[f ²+f ³]

F ¹＝[f ⁰-f ¹]+[f ²-f ³]

F ²＝[f ⁰+f ¹]-[f ²+f ³]

F ³＝[f ⁰-f ¹]-[f ²-f ³]

The form of inverse fourier transform is identical with Fourier transform, just will remove last 2 more ^p

From the situation of above-mentioned prior art, very poor by the H matrix stability of random fashion structure for the m-ary LDPC sign indicating number, cause corresponding code performance to alternate betwwen good and bad thus, so all adopt expectation to measure its performance usually.And adopting the complexity of BP algorithm decoding too high, this has limited its application in practical communication system.We wish to adopt more stable H matrix, thereby obtain the LDPC sign indicating number of stable performance, adopting the decoding algorithm of suboptimum simultaneously is that cost reduces decoding complexity with certain performance loss, makes encoding and decoding algorithm guarantee that being easy to hardware under certain performance advantage prerequisite simultaneously realizes.

Summary of the invention

The present invention is directed to the prior art above shortcomings, purpose is to provide a kind of m-ary LDPC code coding/decoding method based on the PEG algorithm, employing PEG design is finished the code Design to the m-ary LDPC sign indicating number, adopt equivalent transformation, and combine with Fourier transform and to realize the decoding of m-ary LDPC sign indicating number, significantly improve the bit error rate performance of m-ary LDPC sign indicating number, reduce the complexity of decoding, and performance loss can accepted in the scope, under the short-and-medium frame transmission conditions of use in digital communication system, significantly be superior to the performance of the m-ary LDPC sign indicating number of conventional random configuration.

Technical scheme of the present invention is as follows:

One, coded portion:

Know all that as the researcher in this field H matrix stability of random configuration is very poor in the LDPC sign indicating number, so can not guarantee that the H performance that constructs is all fine at every turn.Obtained stable in the m-ary LDPC sign indicating number code Design and the Tanner figure long length of minimum ring so we introduce the PEG algorithm.The PEG algorithm is a kind of structure algorithm of the Tanner figure of long ring as far as possible.The mode that it adopts the limit to add one by one, application drawing is discussed knowledge, the limit is joined among the Tanner figure one by one go.A given symbol node s _j, suppose that its degree is ds _j, adding ds so _jShould follow during the bar limit: new limit of every interpolation should make among the current Tanner figure by node s _jThe long maximized principle of minimum ring.

Coding step is as follows:

1, utilize PEG to construct binary H earlier.

Suppose the H matrix that will construct m * n dimension, Tanner figure so correspondingly has m variable nodes and n parity check nodes.We are the Tanner seal that (V, E), V is fixed point set and V=V _c∪ V _s, V _c={ c ₀, c ₁..., c _M-1Be the set of checkpoint, V _s={ s ₀, s ₁..., S _N-1It is the set of variable point.E is limit collection and E=V _c* V _s, the current h that only works as _Ij≠ 0 o'clock, limit (c _i, s _j) ∈ E, h _IjIt is the capable j column element of i of H.Variable point s _jDegree be designated as ds _j, the set that all limits associated therewith constitute is designated as Es _j, E=Es is then arranged ₀∪ ES ₁∪ ... ∪ Es _N-1With s _jRelated k bar limit is designated as E _Sj ^k, 0≤k≤ds _j-1.As shown in Figure 1, we are s _jThe checkpoint set that can arrive in the l layer on its expansion tree is designated as N _Sj ^l, supplementary set correspondingly is designated as

Then have ${\stackrel{\‾}{N}}_{s_j}^l=V_c\backslash N_{s_j}^l.$

The PEG algorithm is as follows:

for?j＝0：n-1

for?k＝0：ds _j-1

if?k＝0

Limit (c _i, s _j) compose and give E _Sj ⁰, c _iBe current Tanner figure Es ₀∪ Es ₁∪ ... ∪ Es _J-1The parity check nodes of moderate minimum.

else

In current Tanner figure from variable nodes s _jExpand a l layer tree up to satisfying But

Perhaps N _Sj ^lGesture no longer continue to increase but still less than M.We are limit (c then _i, s _j) compose and give E _Sj ^k, c _iBe from In a minimum checkpoint of select degree.

end

end

When two circulations were finished, Tanner figure had just built up, and also is that corresponding H has constructed.

It should be noted that we are selecting c _iThe time, be chosen in that parity check nodes of current figure moderate minimum, be that the degree of parity check nodes distributes even as far as possible among the Tanner figure that generates thus in order to allow.If the minimum nodes of a plurality of degree are arranged, this schemes often to occur when initial at structure Tanner.To this, two kinds of processing modes are arranged: 1) select one of them randomly, 2) according to these nodes at c ₀, c ₁..., c _M-1In order, select that node of subscript minimum all the time.What here, we adopted is first kind of mode.

2, then the nonzero element among the H is used randomly 1,2 ..., the element substitution among the q-1} has obtained stable and more excellent H.

for?i＝0：m-1

for?j＝0：n-1

if(h _ij≠0)

h _ij＝rand_int(q-1)+1

end

end

Wherein, function rand_int (q) produces an integer between 0～q-1 at random.

3, decomposition can obtain generator matrix G according to LU, and information bit S and generator matrix G multiply each other and can obtain code word U then.That is U=S  G.Multiplication is followed the multiplication rule on the GF (q).

Two, decoding part:

The employing said method has obtained the m-ary LDPC sign indicating number based on the PEG algorithm, and we wish that decoding algorithm is accurate and simple simultaneously.

Suppose that C is a code word allowable, c _jBe j component of code word.We go up at GF (2) earlier and analyze $\underset{j}{\mathrm{\Σ}}{h}_{\mathrm{ij}}{c}_j=0$ (h _Ij≠ 0, h _IjBe the capable j column element of i among the H).Because h _Ij=1, so have $\underset{j}{\mathrm{\Σ}}{c}_j=0$ (j is the nonzero element subscript of every row among the H).And on GF (q), because h _Ij∈ 1,2 ..., so q-1} is can not be with h _IjSave.If we make c ' _j=h _Ijc _j(multiplication is followed the multiplication rule on the GF (q)) then has $\underset{j}{\mathrm{\Σ}}{c^{\′}}_j=0$ (j is the nonzero element subscript of every row among the H).So just nonzero element among the H has been transferred in the code word the contribution of each verification, the equivalent H of this moment is exactly a binary H.Like this, we just can decipher according to the mode of similar binary decoding, and Equivalent Thought is come therefrom.

The BP algorithm is one for binary system LDPC sign indicating number well to be selected, but for the m-ary LDPC sign indicating number, its algorithm complex is too high (with q ²Be directly proportional).We propose a kind of equivalent transformation according to above-mentioned Equivalent Thought, and it combines with the FT conversion and can realize decoding.Order

Be the code word vector that sends, decoding is exactly to seek to satisfy $H\hat{x}=\hat{0}$ Be zero vector.During concrete realization of decoding, equivalence is conciliate equivalence and is all carried out at probabilistic information, and transformation for mula is:

$q_{\mathrm{mn}}^{\′a}=\mathrm{ET}\left[q_{\mathrm{mn}}^a\right]=q_{\mathrm{mn}}^{a\÷h_{\mathrm{mn}}}$ (equivalent transformation)

$r_{\mathrm{mn}}^a=\mathrm{IET}\left[r_{\mathrm{mn}}^{\′a}\right]=r_{\mathrm{mn}}^{\′(a\⊗h_{\mathrm{mn}})}$ (contrary equivalent transformation)

Wherein, " ÷ " is the inverse operation (introducing for convenience) of "  ", a=0, and 1,2 ..., q-1.q _Mn ^a(q _Mn ^{' a}) expression ${\hat{x}}_n=a$ Probability, r _Mn ^a(r _Mn ^{' a}) expression ought ${\hat{x}}_n=a$ The time m the probability that verification is satisfied,

For

N component, h _MnThe element of the capable n row of m among the expression H.ET represents equivalent transformation, and IET represents contrary equivalent transformation.

Concrete decoding algorithm performing step is as follows:

1, initialization

According to channel model q _Mn ^aBe initialized as f _n ^a, promptly work as ${\hat{x}}_n=a$ The time likelihood value.Make N (m) :={ n:h _Mn≠ 0} represents to participate in the set of the variable point of m verification, M (n) :={ m:h _Mn≠ 0} represents the set of n the check that variable point participated in.

2, carry out ET (equivalent transformation) and FT (Fourier transform)

$q_{\mathrm{mn}}^{\′a}=q_{\mathrm{mn}}^{a\÷h_{\mathrm{mn}}}$

$Q_{\mathrm{mn}}^{\′a}=\mathrm{FT}[q_{\mathrm{mn}}^{\′0},q_{\mathrm{mn}}^{\′1},...,q_{\mathrm{mn}}^{\′q-1}]$

3, (contrary equivalent transformation upgrades r to carry out IFT (inverse fourier transform) and IFT _Mn ^a)

$r_{\mathrm{mn}}^{\′a}=\mathrm{IFT}[\left(\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′0}\right),...,\left(\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′q-1}\right)]$

$r_{\mathrm{mn}}^a=r_{\mathrm{mn}}^{\′(a\⊗h_{\mathrm{mn}})}$

4., upgrade q _Mn ^a

$q_{\mathrm{mn}}^a={\mathrm{\α}}_{\mathrm{mn}}{f}_n^a\underset{j\∈M\left(n\right)\backslash m}{\mathrm{\Π}}{r}_{\mathrm{jn}}^a$

α wherein _MnBe normalization factor, satisfy ${\mathrm{\Σ}}_{a=0}^{q-1}{q}_{\mathrm{mn}}^a=1$

5., calculate q _n ^a

$q_n^a={\mathrm{\α}}_n{f}_n^a\underset{j\∈M\left(n\right)}{\mathrm{\Π}}{r}_{\mathrm{jn}}^a$

α wherein _nBe normalization factor, satisfy ${\mathrm{\Σ}}_{a=0}^{q-1}{q}_n^a=1.$

6, decoding judgement:

${\hat{x}}_n=\underset{a}{\arg \max }{f}_n^a\underset{j\∈M\left(n\right)}{\mathrm{\Π}}{r}_{\mathrm{jn}}^a$

If $H\hat{x}=\hat{0}$ Perhaps iterations has reached default maximum, and then algorithm stops, and continues iteration otherwise got back to for the 2nd step.

Above-mentioned steps has clearly illustrated the decode procedure of m-ary LDPC sign indicating number.

We have built emulation platform, referring to Fig. 2 and Fig. 3, compared on the GF (4) respectively based on the performance of the sign indicating number of PEG algorithm and random configuration mode, and all based on the PEG algorithm, during identical frame length, the performance of 4 system LDPC sign indicating numbers and 2 system LDPC sign indicating numbers.All code checks all are 1/2.By Fig. 2. as seen: under awgn channel, better than the LDPC code performance of random fashion coding based on the 4 system LDPC sign indicating numbers of PEG.As seen from Figure 3, equally based on the PEG algorithm, 4 system LDPC sign indicating numbers are better than 2 system LDPC code performance outlines.

Though we have only provided the simulation result (as Fig. 2 and Fig. 3) on the GF (4), decoding algorithm is at GF (2 ^p) on be general, other m-ary LDPC sign indicating numbers also are suitable for.

Invention advantage or good effect:

Confirm by a large amount of computer simulation experiments and theory analysis, utilize the PEG algorithm can significantly improve the bit error rate performance of m-ary LDPC sign indicating number, and the decoded mode that adopts equivalent transformation to combine with Fourier transform can reduce decoding complexity greatly, and performance loss can accepted in the scope.It has remarkable improvement than the performance of the LDPC sign indicating number of the random configuration of routine, and this advantage is very obvious under short-and-medium frame transmission situation.This shows and adopts such sign indicating number type to have important practical significance in the digital communication system of reality, in following digital communication system, under the short-and-medium frame transmission conditions, is a better choice based on the m-ary LDPC sign indicating number of PEG algorithm.

Description of drawings

Fig. 1 .s _jExpansion tree go up the parity check nodes that can arrive in the l layer and gather N _Sj ^l

Fig. 2.: based on the ber curve of the quaternary LDPC sign indicating number of PEG algorithm and random configuration mode, frame length is respectively 1000,4000,20000bits.

Fig. 3: based on the ber curve of binary system LDPC sign indicating number under the identical frame length of PEG algorithm and quaternary LDPC sign indicating number, frame length is for being 1000,4000bits.

Embodiment

Below we are example with a concrete coding and decode procedure, further describe the realization of the inventive method.

We are example with (3,6) on the GF (4) sign indicating number.For the ease of understanding, can adopt the H of random fashion earlier, understand decoding and how to realize.Then the H of random fashion is changed into the H that is constructed by the PEG mode, other places are just the same.

Suppose that code word size is N, then H is the matrix of one M * N, M=N/2.h _IjBe the capable j column element of i among the H.Function rand_int (n) produces 0---random integers between the n-1.Generator matrix is designated as G, and code word is designated as

Information bit is designated as computing such as the table 1 on the  .GF (4), and division table wherein is according to the multiplication table structure, and division is the inverse operation of multiplication.

0 1 2 3

0 0 1 2 3

1 1 0 3 2

2 2 3 0 1

3 3 2 1 0

Addition table on table 1.1 GF (4)

0 1 2 3

0 0 0 0 0

1 0 1 2 3

2 0 2 3 1

3 0 3 1 2

Multiplication table on table 1.2 GF (4)

÷ 0 1 2 3

0 × × × ×

1 0 1 2 3

2 0 3 1 2

3 0 2 3 1

Division table on table 1.3 GF (4), row is a dividend, row be place's number we produce binary H earlier, then at random usefulness of ' 1 ' among the H 1,2, an element substitution among the 3}

for?i＝0：M-1

for?j＝0：N-1

if(h _ij＝1)

h _ij＝rand_int(3)+1

end

end

We utilize LU to decompose and find G then, utilize formula then

$\hat{u}=\hat{s}\⊗G$

Obtain code word.

We adopt binary channel to transmit code word.With the code word on the GF (4)

Be mapped as the code word on the GF (2) Send into awgn channel after the BPSK modulation, the noise variance of channel is σ ²

Be channel output, y[i] be its i component.P0[2*N), p1[2*N] store code word respectively

In each component be 0 or be 1 posterior probability: pn0[N], pn1[N], pn2[N], pn3[N] store code word respectively

In each component be 0,1,2,3 posterior probability.That is:

for?i＝0：2*N-1

p1[i]＝1/(1+exp(-2*y[i]/σ ²))

p0[i]＝1-p1[i]

end

With binary code word

The posterior probability inverse mapping become quaternary code word Posterior probability.

for?i＝0：N-1

pn0[i]＝p0[2*i]*p0[2*i+1]

pn1[i]＝p0[2*i]*p1[2*i+1]

pn2[i]＝p1[2*i]*p0[2*i+1]

pn3[i]＝p1[2*i]*p1[2*i+1]

end

Below just can decipher with the mode that equivalent transformation combines with Fourier transform.

^*Initialization

for?m＝0：M-1

for?n＝N(m)

$q_{\mathrm{mn}}^0=\mathrm{pn}0\left[n\right]$

$q_{\mathrm{mn}}^1=\mathrm{pn}1\left[n\right]$

$q_{\mathrm{mn}}^2=\mathrm{pn}2\left[n\right]$

$q_{\mathrm{mn}}^3=\mathrm{pn}3\left[n\right]$

end

end

^*The beginning iterative decoding, iter_num is the maximum iteration time of presetting.

for?c＝1：iter_num

for?m＝0：M-1

/ * * * * * * * calculates Q _Mn ^{' a}* * * * * * */

for?n＝N(m)

/ * * * * * * * * equivalent transformation * * * * * * * */

$q_{\mathrm{mn}}^{\′0}=q_{\mathrm{mn}}^{0\÷h_{\mathrm{mn}}}$

$q_{\mathrm{mn}}^{\′1}=q_{\mathrm{mn}}^{1\÷h_{\mathrm{mn}}}$

$q_{\mathrm{mn}}^{\′2}=q_{\mathrm{mn}}^{2\÷h_{\mathrm{mn}}}$

$q_{\mathrm{mn}}^{\′3}=q_{\mathrm{mn}}^{3\÷h_{\mathrm{mn}}}$

/ * * * * * * * * * Fourier transform * * * * * * * * * */

$Q_{\mathrm{mn}}^{\′0}=[q_{\mathrm{mn}}^{\′0}+q_{\mathrm{mn}}^{\′1}]+[q_{\mathrm{mn}}^{\′2}+q_{\mathrm{mn}}^{\′3}]$

$Q_{\mathrm{mn}}^{\′1}=[q_{\mathrm{mn}}^{\′0}-q_{\mathrm{mn}}^{\′1}]+[q_{\mathrm{mn}}^{\′2}-{q\′}_{\mathrm{mn}}^3]$

$Q_{\mathrm{mn}}^{\′2}=[q_{\mathrm{mn}}^{\′0}+q_{\mathrm{mn}}^{\′1}]-[q_{\mathrm{mn}}^{\′2}+q_{\mathrm{mn}}^{\′3}]$

$Q_{\mathrm{mn}}^{\′3}=[q_{\mathrm{mn}}^{\′0}-q_{\mathrm{mn}}^{\′1}]-\[q_{\mathrm{mn}}^{\′2}-q_{\mathrm{mn}}^{\′3}\]$

end

/ * * * * * * * * * calculates r _Mn ^a* * * * * * * */

for?n＝N(m)

/ * * * * * * * * * * * * * inverse Fourier transform * * * * * * * * */

$r_{\mathrm{mn}}^{\′0}=\left(\right[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′0}+\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′1}]+[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′2}+\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′3}\left]\right)/4$

$r_{\mathrm{mn}}^{\′1}=\left(\right[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′0}-\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′1}]+[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′2}-\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′3}\left]\right)/4$

$r_{\mathrm{mn}}^{\′2}=\left(\right[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′0}+\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′1}]-[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′2}+\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′3}\left]\right)/4$

$r_{\mathrm{mn}}^{\′3}=\left(\right[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′0}-\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′1}]-[\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′2}-\underset{j\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Q}_{\mathrm{mj}}^{\′3}\left]\right)/4$

The contrary equivalent transformation * * * * * * * * * * of/* * * * * * * */

$r_{\mathrm{mn}}^0=r_{\mathrm{mn}}^{\′(0\⊗h_{\mathrm{mn}})}$

$r_{\mathrm{mn}}^1=r_{\mathrm{mn}}^{\′(1\⊗h_{\mathrm{mn}})}$

$r_{\mathrm{mn}}^2=r_{\mathrm{mn}}^{\′(2\⊗h_{\mathrm{mn}})}$

$r_{\mathrm{mn}}^3=r_{\mathrm{mn}}^{\′(3\⊗h_{\mathrm{mn}})}$

end

end

/ * * * * * * * upgrades q _Mn ^a, the judgement of do decoding simultaneously * * * * * * */

for?n＝0：N-1

for?m＝M(n)

$q_{\mathrm{mn}}^0={\mathrm{\&alpha;}}_{\mathrm{mn}}\mathrm{pn}0\left[n\right]\underset{j\&Element;M\left(n\right)\backslash m}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="0"\; label="jn"\; filenames="A20051005710500161.png"\; state="\{\{state\}\}">jn</figure-callout>}}^0$

$q_{\mathrm{mn}}^1={\mathrm{\&alpha;}}_{\mathrm{mn}}\mathrm{pn}1\left[n\right]\underset{j\&Element;M\left(n\right)\backslash m}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="1"\; label="jn"\; filenames="A20051005710500161.png,A20051005710500171.png"\; state="\{\{state\}\}">jn</figure-callout>}}^1$

$q_{\mathrm{mn}}^2={\mathrm{\&alpha;}}_{\mathrm{mn}}\mathrm{pn}2\left[n\right]\underset{j\&Element;M\left(n\right)\backslash m}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="2"\; label="jn"\; filenames="A20051005710500171.png"\; state="\{\{state\}\}">jn</figure-callout>}}^2$

$q_{\mathrm{mn}}^3={\mathrm{\&alpha;}}_{\mathrm{mn}}\mathrm{pn}3\left[n\right]\underset{j\&Element;M\left(n\right)\backslash m}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="3"\; label="jn"\; filenames="A20051005710500171.png"\; state="\{\{state\}\}">jn</figure-callout>}}^3$

/ * * * * * is α wherein _MnSatisfy ${\mathrm{\&Sigma;}}_{a=0}^3{q}_{\mathrm{mn}}^a=1$ ********/

/ * * * * * * * * * * calculates q _n ^a* * * * * * * * */

${\mathrm{<figure-callout\; id="0"\; label="q\; n"\; filenames="A20051005710500161.png"\; state="\{\{state\}\}">q</figure-callout>}}_n^{}={\mathrm{\&alpha;}}_n\mathrm{pn}0\left[n\right]\underset{j\&Element;M\left(n\right)}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="0"\; label="jn"\; filenames="A20051005710500161.png"\; state="\{\{state\}\}">jn</figure-callout>}}^0$

${\mathrm{<figure-callout\; id="1"\; label="q\; n"\; filenames="A20051005710500161.png,A20051005710500171.png"\; state="\{\{state\}\}">q</figure-callout>}}_n^{}={\mathrm{\&alpha;}}_n\mathrm{pn}1\left[n\right]\underset{j\&Element;M\left(n\right)}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="1"\; label="jn"\; filenames="A20051005710500161.png,A20051005710500171.png"\; state="\{\{state\}\}">jn</figure-callout>}}^1$

${\mathrm{<figure-callout\; id="2"\; label="q\; n"\; filenames="A20051005710500171.png"\; state="\{\{state\}\}">q</figure-callout>}}_n^{}={\mathrm{\&alpha;}}_n\mathrm{pn}2\left[n\right]\underset{j\&Element;M\left(n\right)}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="2"\; label="jn"\; filenames="A20051005710500171.png"\; state="\{\{state\}\}">jn</figure-callout>}}^2$

${\mathrm{<figure-callout\; id="3"\; label="q\; n"\; filenames="A20051005710500171.png"\; state="\{\{state\}\}">q</figure-callout>}}_n^{}={\mathrm{\&alpha;}}_n\mathrm{pn}3\left[n\right]\underset{j\&Element;M\left(n\right)}{\mathrm{\&Pi;}}{r}_{\mathrm{<figure-callout\; id="3"\; label="jn"\; filenames="A20051005710500171.png"\; state="\{\{state\}\}">jn</figure-callout>}}^3$

/ * * * * * * is α wherein _nSatisfy ${\mathrm{\&Sigma;}}_{a=0}^3{q}_n^a=1$ *******/

/ * * * * * * decoding judgement * * * * * * */

${\hat{x}}_n=\underset{a}{\arg \max }{q}_n^a$

end

end

If/* * * * * * * then jumps out circulation successfully decoded, otherwise continue iteration up to circulation end * * * * * * */

$\mathrm{if}(H\hat{x}=\hat{0})$

break；

end

Provided the coding and decoding false code of the LDPC sign indicating number on the GF (4) above.As long as certain basis is arranged, will be easy to correct GF (4) and other GF (2 of simulating according to false code for binary system LDPC sign indicating number ^m) the emulation of LDPC sign indicating number on awgn channel on the territory.

Claims (2)

1, adds the multi-system encoding and decoding low-density parity check code method of limit algorithm one by one, comprise coded portion and corresponding decoding part, method is to adopt that to add the limit one by one be that the PEG algorithm carries out the coding that the multi-system low-density checksum is the LDPC sign indicating number, obtain stable and the long long Taner figure of minimum ring, adopt equivalent transformation simultaneously and combine and carry out the decoding of m-ary LDPC sign indicating number with Fourier transform.

2, the m-ary LDPC code coding/decoding method of PEG algorithm according to claim 1 is characterized in that:

Coding step is as follows:

(1) utilize PEG to construct binary matrix H earlier:

(2) then the nonzero element among the H is used randomly 1,2 ..., the element substitution among the q-1} has obtained stable and more excellent H;

(3) decompose according to LU and obtain generator matrix G, suppose that information bit is designated as S, code word is designated as U, then has U=S  G multiplication to follow multiplication rule on the GF (q);

The decoding step is as follows:

(1) initialization

According to channel model q _Mn ^αBe initialized as f _n ^α, that is, and the likelihood value when =α;

Make N (m) :={ n: h _Mn≠ 0}, expression participates in the set of the variable point of m verification, M (n) :={ m: h _Mn≠ 0},

Represent the set of n the check that variable point participated in;

(2) carrying out ET is that equivalent transformation and FT are Fourier transform

$q_{\mathrm{mn}}^{{}^{\&prime;}\mathrm{\&alpha;}}=q_{\mathrm{mn}}^{{\mathrm{\&alpha;}\&divide;h}_{\mathrm{mn}}}$

$Q_{\mathrm{mn}}^{{}^{\&prime;}\mathrm{\&alpha;}}=\mathrm{FT}[q_{\mathrm{mn}}^{{}^{\&prime;}0},q_{\mathrm{mn}}^{{}^{\&prime;}1},...,q_{\mathrm{mn}}^{{}^{\&prime;}q-1}]$

(3) carrying out IFT is the promptly contrary equivalent transformation of inverse fourier transform and IET, to calculate r _Mn ^α

$r_{\mathrm{mn}}^{{}^{\&prime;}\mathrm{\&alpha;}}=\mathrm{IFT}\left[\right(\underset{j\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}{Q}_{\mathrm{mj}}^{{}^{\&prime;}0}),...,\left(\underset{j\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}{Q}_{\mathrm{mj}}^{{}^{\&prime;}q-1}\right)]$

$r_{\mathrm{mn}}^{\mathrm{\&alpha;}}=r_{\mathrm{mn}}^{{}^{\&prime;}(\mathrm{\&alpha;}\&CircleTimes;h_{\mathrm{mn}})}$

(4) upgrade q _Mn ^α

$q_{\mathrm{mn}}^{\mathrm{\&alpha;}}={\mathrm{\&alpha;}}_{\mathrm{mn}}{f}_n^{\mathrm{\&alpha;}}\underset{j\&Element;M\left(n\right)\backslash m}{\mathrm{\&Pi;}}{r}_{\mathrm{jn}}^{\mathrm{\&alpha;}}$

α wherein _MnSatisfy ${\mathrm{\&Sigma;}}_{\mathrm{\&alpha;}=0}^{q-1}{q}_{\mathrm{mn}}^{\mathrm{\&alpha;}}=1$

(5) calculate q _n ^α

$q_n^{\mathrm{\&alpha;}}={\mathrm{\&alpha;}}_n{f}_n^{\mathrm{\&alpha;}}\underset{j\&Element;M\left(n\right)}{\mathrm{\&Pi;}}{r}_{\mathrm{jn}}^{\mathrm{\&alpha;}}$

α wherein _nSatisfy ${\mathrm{\&Sigma;}}_{\mathrm{\&alpha;}=0}^{q-1}{q}_n^{\mathrm{\&alpha;}}=1;$

(6) decoding judgement:

${\hat{x}}_n=\underset{\mathrm{\&alpha;}}{\arg \max }{q}_n^{\mathrm{\&alpha;}}$

If $H\hat{x}=\hat{0}$ Perhaps iterations has reached default maximum, and then algorithm stops, and continues iteration otherwise got back to for (2) step.

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