CN104092468B - LDPC linear programming decoding method based on acceleration alternating direction multiplier method - Google Patents

Abstract

The invention discloses an LDPC linear programming decoding method based on an acceleration alternating direction multiplier method. The LDPC linear programming decoding method mainly solves the problems that an existing message class transmission algorithm has the error floor and can be easily influenced by a short link. According to the technical scheme, the LDPC linear programming decoding method comprises the steps that first, decoding parameters are initialized; second, iteration updating is carried out on an auxiliary vector, a solution vector and a Lagrangian multiplier vector in the decoding parameters in sequence; third, acceleration processing is carried out to correct the solution vector and the Lagrangian multiplier vector; fourth, according to the corrected solution vector, infinite norms of vectors corresponding to all check nodes are calculated, and the maximum infinite norm is worked out; fifth, whether the decoding process ends or not is judged according to the obtained maximum value and the number of times of iteration; sixth, the solution vector processed through the last time of iteration is output to serve as a decoded code word. The LDPC linear programming decoding method is high in convergence rate and free of the error floor, efficiency of a decoding module in a communication system can be remarkably improved, and the LDPC linear programming decoding method is applicable to the technical field of communication.

Description

Based on the LDPC code linear programming interpretation method accelerating alternating direction multiplier method

Technical field

The invention belongs to communication technical field, particularly to a kind of interpretation method to low-density checksum LDPC code, Can be used for fiber optic communication, magnetic storage, satellite digital video and audio frequency broadcast world.

Background technology

Low density parity check code LDPC has the characteristics that check matrix is sparse, therefore flexible structure, and decoding complexity is low, There is the superperformance approaching shannon limit, be widely used in modern communicationses field, such as deep space communication, fiber optic communication etc., and quilt Various Modern communication standards are adopted, and such as 802.11n, 802.16e, 10GBASE-T etc., are that field of channel coding attracts people's attention in recent years Study hotspot.

At present, widely used interpretation method is mainly message transmission class algorithm, such as belief propagation (Belief Propagation, BP) etc..Although such method has realizes simple, the more low advantage of decoding complexity, exist and be easily subject to The shortcomings of becate affects, there is error floor, is difficult to mathematical analyses.LDPC code linear gauge based on alternating direction multiplier method ADMM Draw interpretation method to pass through to introduce auxiliary variable, devise and new be applied to LDPC code linear programming problem method for solving, effectively The shortcoming overcoming above-mentioned message transmission class algorithm.And, the interpretation method based on alternating direction multiplier method ADMM can also be abundant Using the sparse characteristic of LDPC check matrix, and there is maximum likelihood authentication feature.But traditional is linear based on ADMM It is required for executing the Euclid's project taking in a large number in the planning each iteration of interpretation method, and convergence rate is slower, because And when being applied to larger LDPC, decoding efficiency is not high.

Content of the invention

It is an object of the invention to the deficiency to above-mentioned prior art, propose a kind of based on accelerating alternating direction multiplier method LDPC code linear programming interpretation method, to reduce the multiple complexity of the time in decoding iteration, improves decoding efficiency.

Realizing the object of the invention technical scheme is：On the basis of original decoding technique, by introducing accelerating module and changing Become the renewal order of augmentation Lagrangian decomposition formula, reduce decoding iteration number of times, and avoid multiple euclidean in each iteration Projection calculates, thus improving decoding speed.Its concrete steps includes as follows：

(1) decoding initialization：

1a) binary system LDPC code C for n to code length, under additive white Gaussian noise channel, message r according to receiving obtains Obtain the coefficient gamma of linear programming object function_i=log (Pr (r_i|c_i=0)/Pr (r_i|c_i=1)), i ∈ { 1,2 ..., n }, wherein, c_iRepresent the symbol sending, Pr () represents the event occurrence rate representing in bracket；

1b) according to coefficient gamma_iBy piecewise function $x_i=\left\{\begin{array}{ccc}0,& \mathrm{if}& {\mathrm{\&gamma;}}_i\&GreaterEqual;0\\ 1,& \mathrm{if}& {\mathrm{\&gamma;}}_i<0\end{array}\right.,$ Obtain initial solution x decoding；

1c) setting accelerated factor α=1, auxiliary solution vectorIterationses k=0, setting iteration maximum times N, appearance Difference ε, to each check-node j ∈ J, arranges Lagrange multiplier y_j, auxiliary Lagrange multiplier vectorAuxiliary vector z_j、 Transmission information L_j→iIt is null vector and length is d_j, wherein, J is LDPC code check-node indexed set, d_jIt is check-node j institute The number of verification variable node；

(2) Lagrange multiplier according to kth time iterationConciliate vector x^k, calculate the auxiliary vector of+1 iteration of kth z_j：

$z_j^{k+1}={\mathrm{\&Pi;}}_{p_{d_j}}(T_j{x}^k+y_j^k),j\&Element;J,$

Wherein, T_jIt is the transition matrix being generated by check-node j,It is to be d by length_jAnd all 0-1 containing even number 1 The many cell spaces of verification that vector is constituted,Represent vectorTo the many cell spaces of verificationEurope several in Obtain project, auxiliary vectorWith Lagrange multiplier vectorContained element number is d_j, and storage verification section successively The respective value of the variable node that point j is verified；

(3) according to auxiliary vectorWith Lagrange multiplier vectorCalculate the check-node j transmission of+1 iteration of kth Information to variable node i

$L_{j\&RightArrow;i}^{k+1}={\left(z_j^{k+1}\right)}_i-{\left(y_j^k\right)}_i,j\&Element;J,i\&Element;N_c\left(j\right),$

Wherein, N_cThe indexed set of j variable node that () is verified by check-node j,WithRespectively represent kth+ 1 iteration auxiliary vectorWith kth time iteration Lagrange multiplier vectorThe corresponding value of middle variable node i；

(4) update+1 iterative solution vector x of kth^k+1The value of middle all elements；

(5) to all check-node j ∈ J, according to solution vector x^k+1Update the Lagrange multiplier vector of+1 iteration of kth

(6) pass through accelerated factor α^k+1Revise solution vector x^k+1With Lagrange multiplier vector

6a) calculate the accelerated factor of+1 iteration of kth

6b) according to accelerated factor α^k+1, first calculate the auxiliary solution vector of+1 iteration of kthUpdate solution vector again

6c) to all of check-node j ∈ J, first calculate the auxiliary Lagrange multiplier vector of+1 iteration of kthUpdate the Ge Lang multiplier vector of+1 iteration of kth again

(7) judge whether iterationses k+1 reaches iteration maximum times N, if reaching, decoding terminates, kth is changed for+1 time For solution vector x^k+1Export as translating code word, otherwise execution step (8)；

(8) according to solution vector x^k+1And auxiliary variableVector is calculated to each check-node j ∈ J's Infinite NormObtain maximum therein, if this maximum be less than tolerance ε; decode and terminate, by solution to Amount x^k+1Export as a result, otherwise k goes to step (2) after increasing 1.

The present invention decodes problem using the linear programming solving LDPC code based on alternating direction multiplier method, new by design More new regulation and introduce accelerate process operation, compared with traditional decoding based on alternating direction multiplier method, both decreased consumption When Euclid's project accelerate the convergence rate of iteration again, thus significantly reducing the time used by decoding, improve The decoding efficiency of communication system or storage system.

Brief description

Fig. 1 be the present invention realize general flow chart；

Fig. 2 is the acceleration process sub-process figure revising solution vector and Lagrange multiplier in the present invention；

Fig. 3 is the decoding simulation performance figure with the present invention to Margulis LDPC code.

Specific embodiment

The realization of the present invention is to be carried out based on the linear programming Decoding model that alternating direction multiplier method solves, and specifically describes As follows：

It is the LDPC code of n to a length, code word receive information after additive white Gaussian noise channel is vectorial r= {r₁,r₂,…,r_n, calculate log-likelihood ratioI ∈ { 1,2 ..., n }, wherein, c_iRepresent and send Symbol, Pr () represent bracket in represent event occurrence rate.

By log-likelihood ratio γ_iAs objective function coefhcient, linear programming Decoding model can be described as：

$\begin{array}{cc}\min & {\mathrm{\&Sigma;}}_{i=1}^n{\mathrm{\&gamma;}}_i{x}_i\end{array}$

$\begin{array}{cc}s.t.& T_{j}x=z_j,z_j\&Element;P_{d_j},\&ForAll;\end{array}j\&Element;J,$

Wherein, vector x={ x₁,x₂,…,x_nRepresenting the solution vector decoding, J is LDPC code check-node indexed set, d_jIt is The number of the verified variable node of check-node j, z_jBe length be d_jAuxiliary vector,It is to be d by length_jAnd all containing idol The many cell spaces of verification that several 1 0-1 vector is constituted.Transition matrix T_jGenerated by check-node j, such as check matrix Jth row h_j={ 0,1,0,1,0,1,0 }, then homography is $T_j=\left(\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\end{array}\right).$

The augmented Lagrangian of applying alternating direction multiplier method solve corresponding with above-mentioned linear programming model is writeable For：

$L_{\mathrm{\&rho;}}(x,z,y)={\mathrm{\&gamma;}}^{T}x+\frac{\mathrm{\&rho;}}{2}\underset{j\&Element;J}{\mathrm{\&Sigma;}}{\left|\right|T_{j}x-z_j+y_j\left|\right|}_2^2-\frac{\mathrm{\&rho;}}{2}\underset{j\&Element;J}{\mathrm{\&Sigma;}}{\left|\right|y_j\left|\right|}_2^2,$

Wherein, ρ is penalty factor, y_jIt is d for length_jGlug bright sub- multiplier vector, symbol | | | |₂Represent 2- norm Computing.

With reference to Fig. 1, the present invention is implemented as follows：

Step 1, decoding initialization.

Decoding needs to arrange following auxiliary variable and auxiliary vector and carry out initialization operation to it：Setting accelerated factor α=1, auxiliary solution vectorIterationses k=0, setting iteration maximum times N, tolerance ε, to each check-node j ∈ J, arranges Lagrange multiplier y_j, auxiliary Lagrange multiplier vectorAuxiliary vector z_j, transmission information L_j→iBe null vector and Length is d_j, the initial value of solution vector x is by piecewise function $x_i=\left\{\begin{array}{ccc}0,& \mathrm{if}& {\mathrm{\&gamma;}}_i\&GreaterEqual;0\\ 1,& \mathrm{if}& {\mathrm{\&gamma;}}_i<0\end{array}\right.$ It is calculated, γ_iRepresent log-likelihood ratio.

Step 2, updates auxiliary vector z_jValue.

To all of check-node j ∈ J, according to the Lagrange multiplier of kth time iterationConciliate vector x^k, calculating kth+ The auxiliary vector of 1 iteration

$z_j^{k+1}={\mathrm{\&Pi;}}_{p_{d_j}}(T_j{x}^k+y_j^k),j\&Element;J,$

Wherein, T_jIt is the transition matrix being generated by check-node j, symbolRepresent a vector to the many born of the same parents of verification BodyUpper Euclid's projection, implementing of this project can refer to document " Efficient iterative LP decoding of LDPC codes with alternating direction method of multipliers”【IEEE International Symposium of Information Theory.Jul.2013】.

Step 3, the message of the variable node i that check-node j is verified to it when calculating+1 iteration of kth：

$L_{j\&RightArrow;i}^{k+1}={\left(z_j^{k+1}\right)}_i-{\left(y_j^k\right)}_i,j\&Element;J,i\&Element;N_c\left(j\right),$

Wherein, N_cThe indexed set of j variable node that () is verified by check-node j,WithRepresent auxiliary respectively VectorWith Lagrange multiplier vectorThe corresponding value of middle variable node i.

Step 4, the transmission message obtaining according to step 3With log-likelihood ratio γ_i, when calculating renewal+1 iteration of kth All variable nodesValue：

$x_i^{k+1}={\mathrm{\&Pi;}}_{\left[\mathrm{0,1}\right]}(\frac{1}{d_i}\underset{j\&Element;N_v\left(i\right)}{\mathrm{\&Sigma;}}{L}_{j\&RightArrow;i}^{k+1}-\frac{1}{\mathrm{\&rho;}}{\mathrm{\&gamma;}}_i),i\&Element;I,$

Wherein, I is LDPC code variable node indexed set, d_iRepresent the check-node number of verification variable node i, N_vI () is The check-node indexed set of all verification variable node i, symbol Π_[0,1]() represents Europe in interval [0,1] for the scalar Project is obtained, the value after the project when the value of this scalar is more than 1 is 1, the projection when the value of this scalar is less than 0 is transported in several Value after calculation is 0, and the value after value project when interval [0,1] of scalar is constant.The optimal value of penalty factor ρ can be by imitative True experiment obtains, and its value is generally the constant in interval [2,5].

Step 5, updates the value of the Lagrange multiplier vector of+1 iteration of kth：

$y_j^{k+1}=y_j^k+T_j{x}^{k+1}-z_j^{k+1},j\&Element;J,$

Wherein, vectorAnd auxiliary vectorContained element number, transition matrix T_jLine number be check-node j institute The variable node number d of verification_j.

Step 6, by accelerated factor α^k+1Revise solution vector x^k+1With Lagrange multiplier vector

With reference to Fig. 2, the realization of this step is as follows：

6a) update accelerated factor α of+1 iteration of kth^k+1For

6b) according to accelerated factor α^k+1, first calculate auxiliary solution vectorUpdate kth+1 again The solution vector of secondary iteration

6c) to all of check-node j ∈ J, first calculate the auxiliary Lagrange multiplier vector of+1 iteration of kthUpdate the Ge Lang multiplier vector of+1 iteration of kth again

Step 7, decoding terminates to judge.

7a) judge whether to reach iteration maximum times N, if reaching, decoding terminates, and exports solution vector x^k+1As transmission Information is corresponding to translate code word, otherwise, execution step (7b)；

7b) vector is calculated to each check-node j ∈ JInfinite NormAnd obtain Maximum therein is designated asThis maximum is compared with tolerance ε：IfThen decoding terminates and exports solution vector x^k+1As sending code word, otherwise, k returns step after increasing 1 Rapid 2.

The effect of the present invention can be further illustrated by following emulation：

1. simulated conditions

The modulation system of emulation is BPSK, and channel is additive white Gaussian noise awgn channel.

The code that emulation adopts is Margulis LDPC code, and it is (2640,1320) regular code, and code check isIts row is again 6, row are 3 again.

Emulation setting tolerance ε is 10^-5, penalty factor ρ be 5, maximum iteration time N is set to 60,200,600.

2. emulation content

Under Gaussian channel, respectively with the interpretation method pair of existing BP interpretation method, ALP interpretation method and the present invention The Margulis LDPC code error-correcting performance of code check is emulated, and result is as shown in figure 3, in figure gives 5 curves, wherein：

Represent under additive white Gaussian noise channel with circular curve, the interpretation method of the present invention sets greatest iteration time Number N is 60 error-correcting performance simulation curve；

Foursquare curve is carried to represent under additive white Gaussian noise channel, the interpretation method of the present invention sets greatest iteration Times N is 200 error-correcting performance simulation curve；

Curve with triangle represents under additive white Gaussian noise channel, and the interpretation method of the present invention sets greatest iteration Times N is 600 error-correcting performance simulation curve；

Pentagonal curve is carried to represent under additive white Gaussian noise channel, with the error-correcting performance of existing ALP interpretation method Simulation curve；

Carry criss-cross curve to represent under additive white Gaussian noise channel, imitated with the error-correcting performance of existing BP interpretation method True curve.

As seen from Figure 3, the present invention can significantly improve entangling of interpretation method of the present invention by increasing maximum iteration time N Wrong performance, but after increasing to 200 obtain gain very limited, that is, when maximum iteration time is for N=600 with existing The error-correcting performance of ALP interpretation method is closely.

The present invention compared with existing BP interpretation method, signal to noise ratio be less than 2.5dB when, interpretation method error-correcting performance of the present invention Poor, but the error-correcting performance slope of curve of the present invention is larger, and that is, with the increase of signal to noise ratio, error-correcting performance raising becomes apparent from.? When signal to noise ratio is more than 2.6dB, error-correcting performance is substantially better than BP interpretation method.Meanwhile, error-correcting performance be 10^-5When, BP decoding side Method occurs in that error floor phenomenon, and now error-correcting performance increases with signal to noise ratio and improves slowly, and interpretation method of the present invention This error floor phenomenon does not occur.

The foregoing is only presently preferred embodiments of the present invention, not in order to limit the present invention, all spirit in the present invention and Within principle, any modification, equivalent substitution and improvement done etc., should be included in the range of the comprising of the present invention.

Claims (2)

1. a kind of LDPC code linear programming interpretation method based on acceleration alternating direction multiplier method, comprises the steps：

(1) decoding initialization：

1a) binary system LDPC code C for n to code length, under additive white Gaussian noise channel, obtains line according to message r receiving The coefficient gamma of property object of planning function_i=log (Pr (r_i|c_i=0)/Pr (r_i|c_i=1)), i ∈ { 1,2 ..., n }, wherein, c_iTable Show the symbol of transmission, Pr () represents the event occurrence rate representing in bracket；

1b) according to coefficient gamma_iBy piecewise functionObtain initial solution x decoding；

1c) setting accelerated factor α=1, auxiliary solution vectorIterationses k=0, setting iteration maximum times N, tolerance ε, to each check-node j ∈ J, arranges Lagrange multiplier y_j, auxiliary Lagrange multiplier vectorAuxiliary vector z_j, transmission Information L_j→iIt is null vector and length is d_j, wherein, J is LDPC code check-node indexed set, d_jIt is that check-node j is verified The number of variable node；

(2) Lagrange multiplier according to kth time iterationConciliate vector x^k, calculate the auxiliary vector z of+1 iteration of kth_j：

$z_j^{k+1}={\mathrm{\&Pi;}}_{p_{d_j}}\left(T_j{x}^k+y_j^k\right),j\&Element;J,$

Wherein, T_jIt is the transition matrix being generated by check-node j,It is to be d by length_jAnd all containing even number 1 0-1 vector The many cell spaces of verification being constituted,Represent vectorTo the many cell spaces of verificationEuclid projection Computing, auxiliary vectorWith Lagrange multiplier vectorContained element number is d_j, and store check-node j institute successively The respective value of the variable node of verification；

(3) according to auxiliary vectorWith Lagrange multiplier vectorThe check-node j calculating+1 iteration of kth passes to change The information of amount node i

$L_{j\&RightArrow;i}^{k+1}={\left(z_j^{k+1}\right)}_i-{\left(y_j^k\right)}_i,j\&Element;J,i\&Element;N_c\left(j\right),$

Wherein, N_cThe indexed set of j variable node that () is verified by check-node j,WithExpression kth+1 time respectively Iteration auxiliary vectorWith kth time iteration Lagrange multiplier vectorThe corresponding value of middle variable node i；

(4) update+1 iterative solution vector x of kth^k+1The value of middle all elements；

$x_i^{k+1}={\mathrm{\&Pi;}}_{\&lsqb;0,1\&rsqb;}\&lsqb;\frac{1}{d_i}(\underset{j\&Element;N_v\left(i\right)}{\&Sigma;}{L}_{j\&RightArrow;i}^{k+1}-\frac{{\mathrm{\&gamma;}}_i}{\mathrm{\&rho;}})\&rsqb;,i\&Element;I,j\&Element;J,$

Wherein, ρ is penalty factor and meets ρ ＞ 0, I is LDPC code variable node indexed set, d_iRepresent all verification variable node i Check-node number, N_vI () is the check-node indexed set of all verification variable node i, symbol ∏_[0,1]() represents one Euclid's project in interval [0,1] for the individual scalar；

(5) to all check-node j ∈ J, according to solution vector x^k+1Update the Lagrange multiplier vector of+1 iteration of kth

$y_j^{k+1}=y_j^k+T_j{x}^{k+1}-z_j^{k+1},j\&Element;J,$

Wherein,Represent the Lagrange multiplier vector of kth time iteration, T_jIt is the transition matrix being generated by check-node j, x^k+1For The solution vector of+1 iteration of kth,Represent the auxiliary vector of+1 iteration of kth；

(6) pass through accelerated factor α^k+1Revise solution vector x^k+1With Lagrange multiplier vector

6a) calculate the accelerated factor of+1 iteration of kth

6b) according to accelerated factor α^k+1, first calculate the auxiliary solution vector of+1 iteration of kthAgain Update solution vector

6c) to all of check-node j ∈ J, first calculate the auxiliary Lagrange multiplier vector of+1 iteration of kthUpdate the Ge Lang multiplier vector of+1 iteration of kth again

(7) judge whether iterationses k+1 reaches iteration maximum times N, if reaching, decoding terminates, by+1 iterative solution of kth Vector x^k+1Export as translating code word, otherwise execution step (8)；

(8) according to solution vector x^k+1And auxiliary variableVector is calculated to each check-node j ∈ JInfinite model NumberObtain maximum therein, if this maximum is less than tolerance ε, decodes and terminate, by solution vector x^k+1Make For result output, otherwise go to step (2) after k increasing 1.

2. interpretation method according to claim 1 is it is characterised in that pass through accelerated factor α in described step (6)^k+1Revise Solution vector x^k+1With Lagrange multiplier vectorIn order to accelerate the convergence rate of decoding iteration.