CN101257311B - Quick interpretation method of LDPC code under multiple scale - Google Patents

Abstract

The invention relates to an interpretation method of channel error correcting LDPC code. The existing interpretation method has complicated realization under multi-ary modulation mode and poor applicability. The invention firstly modifies a prior tanner map, initializes each (n, m),and then updates the verify node and sign node, finally output the result. The invention makes use of the excellent sparsity of LDPC code, carries out the iterative decoding to the LDPC code according to specific sign node initializtion scheme, specific verify node update scheme and sign node update scheme, finally obtains the codeword sequence satisfying the check matrix and finishes the decoding. The invention can reduces the decoding complexity of The LDPC code under multi-ary modulation mode, decreases the memory space during decoding and keeps favorable code error property and iterative convergence speed.

Description

A kind of method for rapidly decoding of LDPC code under multiple scale

Technical field

The invention belongs to the communications field, relate to a kind of channel error correction LDPC (low density paritycode, low-density check) Ma interpretation method, be specifically related under the multi-system modulation system fast decoding mode, still can guarantee good under the complexity that directly adopts the multi-system decoding algorithm even than based on the better decoding performance of bit decoding reducing based on symbol.

Background technology

The LDPC sign indicating number is a kind of coded system near shannon limit, and it belongs to a kind of of linear block codes, is called loe-density parity-check code.The LDPC sign indicating number was just invented by Gallager as far back as the sixties in 20th century, and had proposed a kind of decoding algorithm of probability.Yet be subjected to the restriction of hardware condition at that time, its appearance does not cause enough concerns.By 1996, be subjected to the enlightenment of Turbo code iterative decoding, people such as Mackay have rediscovered the LDPC sign indicating number, thereby have caused the research boom of a new round.

The LDPC sign indicating number be by check matrix determine block code.The characteristics of its check matrix are exactly very sparse, and this just sparse property has been brought the performance of its excellence.From the angle of decoding, because the sparse property of check matrix makes that corresponding Tanner figure is also very simple, the decoding that realization reaches actual requirement is feasible, in addition, the minimum loop of its Tanner figure is very big or do not have loop, so it is fit to iterative decoding algorithm very much.This some all brought up the success of LDPC sign indicating number.And, nowadays reached a very high research level for the interpretation method under decoding research, especially the binary system BPSK modulation of LDPC sign indicating number.Wherein, the BP algorithm is the main standards algorithm, and then based on this basic decoding algorithm, in order to reduce the decoding complexity of LDPC sign indicating number, it is the simplification decoding algorithm of representative with Min-Sum that people such as Marc P.C.Fossorier have proposed; In order to improve the decoding algorithm performance of LDPC, J.Zhang and M.Fossorier accelerate BP convergence of algorithm speed by utilizing the more information of new node, have proposed Shuffled BP algorithm.

And in the application of reality, in order to improve the transfer rate of information, modulation can not be BPSK usually, but adopts the multi-system modulation system.Consider 2 ^qThe system modulation is to the code word c=(c of encoder output ₁, c ₂..., c _N), after modulation, obtain s=(s ₁, s ₂... s _K), K=N/q wherein.2 ^qIn the system modulation, the set of originally every q bit is the channel symbol of a modulation, is designated as: x _n={ c _Nq-q+1, c _{Nq- Q+2}..., c _Nq, n=1,2 ..., K.s _nBe symbol x _nMapping point in modulation constellation is designated as

S directly imports awgn channel into.The signal phasor that receiving terminal receives is r=(r ₁, r ₂..., r _K), r wherein _n=(I _n, Q _n).I _n, Q _nAll be stochastic variable, their variance all is σ ², average is respectively

Fig. 1 is the system model figure under the multi-system modulation, can see by this figure, dope vector obtains the code word vector after through LDPC sign indicating number coding, the LDPC code coder here is not the encoder on finite field gf (q), but the encoder on finite field gf (2).Form one group for the every q of code word vector wherein and enter modulator, and become modulation symbol.Modulation symbol is input to awgn channel, and at receiving terminal, the soft information that we directly come channel transfer is input to decoder.This decoder can be based on the decoder of bit, also can be based on the decoder of modulation symbol.

For the LDPC sign indicating number under the multi-system modulation, Niclas Wiberg he thesis for the doctorate (" Codes and Decoding on General Graphs " .Ph.D.thesis,

University, S-581 83 Sweden, Department ofElectrical Engineering, 1996) lining provided the conclusion of versatility, and it is based on the general decoding algorithm of modulation symbol.Its main step is described below:

Initialization

To each symbol node x _n, calculate its local cost function, to each (n, m) right, initialization:

$Z_{\mathrm{nm}}\left(x_n\right)={\mathrm{\γ}}_n\left(x_n\right)=\frac{1}{2{\mathrm{\π\σ}}^2}\exp [-\frac{{(I_n-I_{x_n})}^2+{(Q_n-Q_{x_n})}^2}{2{\mathrm{\σ}}^2}]$ To all x _n∈ X _n(1)

Step 1 (check-node renewal)

To each check-node m and all x _n∈ X _n

$L_{\mathrm{mn}}^{\left(I\right)}\left(x_n\right)=\underset{B\left(m\right)}{\mathrm{\Σ}}\underset{n^{\′}\∈N\left(m\right)\backslash n}{\mathrm{\Π}}{Z}_{n^{\′}m}^{(I-1)}\left(x_{n^{\′}}\right)---\left(2\right)$

Wherein B (m) expression: verification formula m satisfies; N (m) n, the set that all variable nodes of linking to each other with check-node m of expression are removed node n.

Step 2 (symbol node update)

To each symbol node n, to all m ∈ M (n) and all x _n∈ X _n, calculate:

$Z_{\mathrm{nm}}^{\left(I\right)}\left(x_n\right)={\mathrm{\γ}}_n\left(x_n\right)\underset{m^{\′}\∈M\left(n\right)\backslash m}{\mathrm{\Π}}{L}_{{\mathrm{mn}}^{\′}}^{(I-1)}\left(x_n\right)---\left(3\right)$

M (n) m, the set that all check-nodes of linking to each other with variable node n of expression are removed node m.

Calculate simultaneously:

$Z_n^{\left(I\right)}\left(x_n\right)={\mathrm{\γ}}_n\left(x_n\right)\underset{m\∈M\left(n\right)}{\mathrm{\Π}}{L}_{\mathrm{mn}}^{(I-1)}\left(x_n\right)---\left(4\right)$

Step 3 (judgement)

According to

Obtain

According to Can obtain code word

If

Decoding finishes, It promptly is the code word of terminal decision.Otherwise, get back to step 1, carry out new round iteration.If through preestablishing maximum iterations, decoding does not also finish, then the code word of terminal decision as decode results or statement decoding failure.

Why iterative decoding algorithm under the binary system can extensively be used, and mainly is because under binary system, iterative decoding algorithm can carry out normalization to verification in the algorithm or symbol node.And under the multi-system modulation, by the formula that provides above as can be seen, the pairing Tanner figure of LDPC sign indicating number can change to some extent, and the node in the decoding algorithm no longer is the bit node, and is-symbol node, just a symbol in modulation constellation.Therefore in the height modulation system, as 16QAM (16APSK), 32QAM (32APSK) or the like, each symbol node can value number increase greatly, this meaning that just makes normalization calculate disappears, the renewal of each symbol node can not be transformed on the log-domain, therefore, under the multi-system modulation system, the complexity that directly adopts this general decoding algorithm to realize is too high, and application is not strong.

Summary of the invention:

The objective of the invention is to deficiency, provide the class sign indicating number of LDPC fast decoding algorithm based on symbol under the multi-system modulation, and this kind interpretation method is applicable to the LDPC sign indicating number of the overwhelming majority at current techniques.

Concrete steps of the present invention are:

(1) at first, in order to obtain the general succinct down decoded mode that is used for the LDPC sign indicating number under the multi-system modulation of a kind of form and BPSK modulation, former Tanner figure is made following modification: corresponding to two types the node of former Tanner figure, among the Tanner figure that revises two types of nodes are arranged also: wherein check-node is exactly the check-node among the former Tanner figure, symbol node x _nWith among the former Tanner figure after modulating relevant q bit set corresponding; As symbol node x _nIn a bit when in former Tanner figure, linking to each other with check-node m, symbol node x _nLink to each other with the check-node m among the amended Tanner figure;

(2) initialization

To each symbol node x _n, calculate its local cost function, to each (n, m) right, initialization:

$Z_{\mathrm{nm}}^0\left(x_n\right)={\mathrm{\γ}}_n\left(x_n\right)=\frac{{(I_n-I_{x_n})}^2+{(Q_n-Q_{x_n})}^2}{2{\mathrm{\σ}}^2}$ To all x _n∈ X _n

(3) check-node upgrades

To each check-node m, and all n ∈ N (m), the branch situation is calculated:

When all symbol nodes that link to each other with check-node all do not contain multiple limit, being updated to of check-node:

$L_{\mathrm{mn}}^{\left(I\right)}=2{\tanh }^{-1}\left[\underset{n^{\′}\∈N\left(m\right)\backslash n}{\mathrm{\Π}}\tanh \left(\frac{{\mathrm{\β}}_{n^{\′}m}^{(I-1)}}{2}\right)\right]---\left(5\right)$

When containing multiple limit in the symbol node that links to each other with check-node, being updated to of check-node:

$L_{\mathrm{mn}}^{\left(I\right)}=2{\tanh }^{-1}\left[\underset{n^{\′}\∈N\left(m\right)\backslash n}{\mathrm{\Π}}\tanh \frac{{\mathrm{\β}}_{n^{\′}m}^{(I-1)}}{2}\right]$

$\mathrm{where}{\mathrm{\β}}_{n^{\′}m}^{(I-1)}=\left(\begin{array}{c}\min \ln Z_{n^{\′}m}^{(I-1)}\left(x_{n^{\′},i_{{\mathrm{mn}}^{\′}}=\mathrm{odd}\left(1\right)}\right)\\ -\min \ln Z_{n^{\′}m}^{(I-1)}\left(x_{n^{\′},i_{{\mathrm{mn}}^{\′}}=\mathrm{even}\left(1\right)}\right)\end{array}\right)---\left(6\right)$

Expression symbol node x _nIn have the individual bit of odd number (even number) to link to each other with same check-node.

(4) symbol node update

To each symbol node n, and all m ∈ M (n), calculate:

Each symbol node n calculates

(5) declare feel output

Judgement is the same with non-improved method.

Adopt the inventive method can access than the lower complexity of decoding algorithm under the original multi-system modulation, in realization, need memory cell still less, can also obtain simultaneously than adopting traditional better decoding performance of binary decoding mode and constringency performance based on bit, be a kind of interpretation method that has very much practical prospect therefore.

The present invention at first is based on the rapid bp algorithm that general BP algorithm is derived and proposed; then by the series of algorithms derivation method of the decoding of the binary system LDPC sign indicating number under BPSK modulation; can promote and obtain Min-Sum fast; the Shuffled-BP algorithm; their most basic decoding thought is identical with the present invention too, therefore should be equally within shielded scope.

Description of drawings

Fig. 1 is encoding and decoding and modulation pattern figure under the system modulation;

Fig. 2 is the Tanner figure of the modification under the 16APSK modulation;

Fig. 3 is the tanner figure that has a double limit;

Fig. 4 is that the complexity of universal algorithm and modified model fast algorithm compares (q=4) figure;

Fig. 5 is that the complexity of universal algorithm and modified model fast algorithm compares (d (m)=4) figure;

Fig. 6 is the error performance comparison diagram of three kinds of BP algorithms;

Fig. 7 is the performance comparison diagram of two kinds of Min-Sum algorithms;

Fig. 8 is the average iterations comparison diagram of three kinds of BP algorithms;

Fig. 9 is the average iterations comparison diagram of two kinds of Min-Sum algorithms.

Embodiment:

The BP algorithm is based on the sum-product algorithm of APP criterion, therefore at document (S.M.Aji, R.J.McEliece. " The Generalized Distributive Law " .IEEETransactions on Information Theory, Vol.46, NO.2, Mar.2000:325～343) in, it satisfies the sum-product exchange semi-ring in the broad sense distributive law (GDL).Therefore can utilize GDL that the BP algorithm under the general multi-system is simplified and obtain a kind of succinct decoded mode fast.It is investigated former universal algorithm and can find, the complexity of algorithm concentrates on the step of updating of check-node, but simultaneously, very sparse of the check matrix that an outstanding performance of LDPC sign indicating number is it, therefore, for the Tanner figure after improving, investigation contains the Special matrix of such specific character: promptly have and only have a bit to link to each other with same check-node (perhaps containing the total number of the symbol node number on multiple limit much smaller than the symbol node) in the middle of the symbol node after the modulation, had so under the situation of such condition, will be very beneficial for the simplification of formula.Because the check matrix of LDPC sign indicating number is very sparse, this requirement is satisfied easily.So, at not containing multiple limit symbol node, must adopt different more new formulas with the different situations that contain multiple limit symbol node.

The check-node that does not contain multiple limit upgrades:

With 8PSK is example, supposes that the degree of a certain check-node m in new Tanner figure is 3, and the symbol node that is attached thereto is respectively n ₁, n ₂, n ₃, three bits of each symbol node correspondence and former binary code word, but have and only have a bit node to link to each other in the middle of each symbol node with identical check-node, this hypothesis can be guaranteed under the sparse property of LDPC sign indicating number.Suppose n ₁Corresponding bit is c ₁₁, c ₁₂, c ₁₃, n ₂Corresponding bit is c ₂₁, c ₂₂, c ₂₃, n ₃Corresponding bit is c ₃₁, c ₃₂, c ₃₃, suppose that check-node m corresponding check equation is:

c ₁₁+c ₂₂+c ₃₃＝0(9)

Suppose to calculate this moment

More as can be known new-type by check-node:

At this, promptly equal:

$L_{{\mathrm{mn}}_1}^{\left(I\right)}\left(1\mathrm{xx}\right)=\underset{{\mathrm{<figure-callout\; id="22"\; label="c"\; filenames="H2008100603200E00031.png"\; state="\{\{state\}\}">c</figure-callout>}}_{22}+c_{33}=1}{\mathrm{\&Sigma;}}{Z}_{n_{2}m}^{(I-1)}\left(x1x\right)Z_{n_{3}m}^{(I-1)}\left(\mathrm{xx}0\right)+\underset{{\mathrm{<figure-callout\; id="22"\; label="c"\; filenames="H2008100603200E00031.png"\; state="\{\{state\}\}">c</figure-callout>}}_{22}+c_{33}=1}{\mathrm{\&Sigma;}}{Z}_{n_{2}m}^{(I-1)}\left(x0x\right)Z_{n_{3}m}^{(I-1)}\left(\mathrm{xx}1\right)$

$L_{{\mathrm{mn}}_1}^{\left(I\right)}\left(0\mathrm{xx}\right)=\underset{{\mathrm{<figure-callout\; id="22"\; label="c"\; filenames="H2008100603200E00031.png"\; state="\{\{state\}\}">c</figure-callout>}}_{22}+c_{33}=0}{\mathrm{\&Sigma;}}{Z}_{n_{2}m}^{(I-1)}\left(x1x\right)Z_{n_{3}m}^{(I-1)}\left(\mathrm{xx}1\right)+\underset{{\mathrm{<figure-callout\; id="22"\; label="c"\; filenames="H2008100603200E00031.png"\; state="\{\{state\}\}">c</figure-callout>}}_{22}+c_{33}=0}{\mathrm{\&Sigma;}}{Z}_{n_{2}m}^{(I-1)}\left(x0x\right)Z_{n_{3}m}^{(I-1)}\left(\mathrm{xx}0\right)---\left(11\right)$

X represents that value is 0 or 1 arbitrarily.Hence one can see that by the information Λ of check-node biography to the symbol node _Mn(x _n) in fact have only two different values, so following formula can be expressed as again

$L_{{\mathrm{mn}}_1}^{\left(I\right)}\left(x_{n_1}\right)=\underset{c_{11}+{\mathrm{<figure-callout\; id="22"\; label="c"\; filenames="H2008100603200E00031.png"\; state="\{\{state\}\}">c</figure-callout>}}_{22}+c_{33}=0}{\mathrm{\&Sigma;}}{Z}_{n_{2}m}^{(I-1)}\left(x_{n_2}\right)Z_{n_{3}m}^{(I-1)}\left(x_{n_3}\right)---\left(12\right)$

Simplify above-mentioned computing with GDL this moment, and the formula that can obtain the renewal of check-node is:

$L_{\mathrm{mn}}^{\left(I\right)}\left(x_{n,i_{\mathrm{mn}}}\right)=\underset{B\left(m\right)}{\mathrm{\&Sigma;}}\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\left[\underset{(i_{\mathrm{mn}}+\mathrm{\&Sigma;}{i}_{{\mathrm{mn}}^{\&prime;}}=0)}{\mathrm{\&Sigma;}}{Z}_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^l\right)\right]---\left(13\right)$

In the following formula

The bit that links to each other with m among the expression symbol node n is i _MnThe position, and value is the situation of l (l=0,1), as long as all bits in the check equations can satisfy check equations, then pairing symbol node value also can satisfy check equations.The thought of simplifying is exactly because a check-node links to each other bit as for a bit in the symbol node

Therefore have only two kinds of possible values, with all 2 ^qIndividual x _n, according to Value be divided into two classes: make for all that X _n, the updating value of their check-node is identical, all make

X _n, the updating value of its check-node also is identical.By top formula as can be seen, if adding up and regarding single integral body as, then equation is just simplified for binary similar form, and it is carried out the simplification of log-domain:

$L_{\mathrm{mn}}^{\left(I\right)}=1n\frac{L_{\mathrm{mn}}^{\left(I\right)}\left(x_{n,i_{\mathrm{nm}}}^1\right)}{L_{\mathrm{mn}}^{\left(I\right)}\left(x_{n,i_{\mathrm{mn}}}^0\right)}=-2\tan h^{-1}\frac{\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}[1-\frac{{\mathrm{\&Sigma;Z}}_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^1\right)}{\mathrm{\&Sigma;}{Z}_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^0\right)}]}{\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}[1+\frac{{\mathrm{\&Sigma;Z}}_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^1\right)}{\mathrm{\&Sigma;}{Z}_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^0\right)}]}---\left(14\right)$

Get symbol

Represent effective bit l value that the individual symbol node of n ' links to each other with the verification formula be 1 o'clock all possible probability and, similar, definition

Then:

$L_{\mathrm{mn}}^{\left(I\right)}=2\tan h^{-1}\frac{\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}[e^{\ln \left(\frac{{\mathrm{\&phi;}}_{n^{\&prime;}m}^{l\left(1\right)}}{{\mathrm{\&phi;}}_{n^{\&prime;}m}^{l\left(0\right)}}\right)}-1]}{\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}[e^{\ln \left(\frac{{\mathrm{\&phi;}}_{n^{\&prime;}m}^{l\left(1\right)}}{{\mathrm{\&phi;}}_{n^{\&prime;}m}^{l\left(0\right)}}\right)}+1]}---\left(15\right)$

According to Jacobi's logarithmic formula:

ln(e ^x+e ^y)＝max(x，y)+ln[1+exp(-|y-x|)]

＝max(x，y)+f _c(x，y) (16)

≈max(x，y)

Can be right

Be similar to, simultaneously if to γ _n(x _n) be initialized as:

${\mathrm{\&gamma;}}_n\left(x_n\right)=\frac{{(I_n-I_{x_n})}^2+{(Q_n-Q_{x_n})}^2}{2{\mathrm{\&sigma;}}^2}---\left(17\right)$

Have so:

$\ln \left(\frac{{\mathrm{\&phi;}}_{n^{\&prime;}m}^{l\left(1\right)}}{{\mathrm{\&phi;}}_{n^{\&prime;}m}^{l\left(0\right)}}\right)\&ap;\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^1\right)-\min {\ln Z}_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^0\right)---\left(18\right)$

Order ${\mathrm{\&beta;}}_{n^{\&prime;}m}=\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^1\right)-\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}}^0\right),$ Have:

$L_{\mathrm{mn}}^{\left(I\right)}\&ap;2{\tanh }^{-1}\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\frac{(e^{{\mathrm{\&beta;}}_{n^{\&prime;}m}}-1)}{(e^{{\mathrm{\&beta;}}_{n^{\&prime;}m}}+1)}=2\tan h^{-1}\left[\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\tanh \frac{{\mathrm{\&beta;}}_{n^{\&prime;}m}}{2}\right]---\left(19\right)$

Accordingly, the renewal of symbol node also should be in log-domain:

Each symbol node n, calculate:

The implementation that contains multiple limit:

When having in the symbol node when linking to each other with same check-node, then be called and multiple limit phenomenon occurs, for the check equations that contains multiple limit, if directly use mistake more than a bit node! Do not find Reference source.Formula will certainly cause losing and mistake of information, will carry out following processing to multiple limit so, then can obtain the general fast decoding algorithm based on symbol under the multi-system modulation.

With double limit is that example is explained, as shown in Figure 3, and the B (m among the figure then ₁) should be when satisfying

c ₁₁+c ₂₁+c ₂₂+c ₃₂＝0(22)

Work as renewal The time, c ₁₁Fix, then when formula (22) can not be satisfied, all the other several bit c overturn ₂₁, c ₂₂And c ₃₂Wherein any one, then check equations m ₁Satisfy.But the interpretation method that provides is based on the decoding of symbol, so symbol node x ₂In double limit can bring The renewal mistake.If but unified consideration c ₂₁And c ₂₂Be a new bit c ₂, make c ₂Be equal to c at=1 o'clock ₂₁c ₂₂=(01 or 10), c ₂=0 is equal to c ₂₁c ₂₂=(00 or 11), check equations m so ₁Still can be satisfied.Certainly can know by inference equally more than the situation on 2 limits.

So original algorithm will be Xiao Xu in the renewal of check-node change to be being applicable to more general situation, as long as when the symbol node on multiple limit occurring containing, adopt following update mode:

$L_{\mathrm{mn}}^{\left(i\right)}=2{\tanh }^{-1}\left[\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\tanh \frac{{\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}}{2}\right]$

$\mathrm{where}{\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}=\left(\begin{array}{c}\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}=\mathrm{odd}\left(1\right)}\right)\\ -\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}=\mathrm{even}\left(1\right)}\right)\end{array}\right)---\left(23\right)$

Herein, Expression symbol node x _nIn have the odd number bit to link to each other with same check-node,

Expression symbol node x _nIn have the even number bit to link to each other with same check-node.

Fig. 2 is the example of the amended Tanner figure under the 16APSK modulation, q=4 among this figure.x _nBe 4 bit { c among the former Tanner figure _4n-3, c _4n-2, c _4n-1, c _4nSet.

Because it is similar with the LDPC sign indicating number decoding under the binary system BPSK modulation, the BP algorithm is the basis of this class decoding algorithm, based on this basic decoding algorithm, can reduce the purpose of the decoding complexity of check-node then, obtain quick Min-Sum decoding algorithm by approximate calculation; Or,, obtain the ShuffledBP algorithm by utilizing the more information of new node in order to improve the convergence rate of LDPC decoding algorithm.

Quick Min-Sum decoding scheme:

Equally, compare the method under the BPSK modulation, the further simplification that can calculate above-mentioned quick log-BP algorithm, obtain the multi-system modulation down based on the quick Min-Sum decoding algorithm of symbol: two formulas could be simplified mistake to (19) (23) formula below utilizing! Do not find Reference source.，

To two stochastic variable U, V has:

$L(U\&CirclePlus;V)=2{\tanh }^{-1}\left(\tanh \left(\frac{L\left(U\right)}{2}\right)\tanh \left(\frac{L\left(V\right)}{2}\right)\right)---\left(24\right)$

Utilize Jacobi's logarithm to get:

$L(U\&CirclePlus;V)=\mathrm{sign}\left(L\left(U\right)\right)\mathrm{sign}\left(L\left(V\right)\right)\min \left(\right|L\left(U\right)|,|L\left(V\right)\left|\right)$

$+\log (1+e^{-|L\left(U\right)+L\left(V\right)|})-\log (1+e^{-|L\left(U\right)-L\left(V\right)|})---\left(25\right)$

$\&ap;\mathrm{sign}\left(L\left(U\right)\right)\mathrm{sign}\left(L\left(V\right)\right)\min \left(\right|L\left(U\right)|,|L\left(V\right)\left|\right)$

Being updated to of check-node then:

$L_{\mathrm{mn}}^{\left(I\right)}={\mathrm{\&beta;}}_{\mathrm{nm}}^{*(I-1)}\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\mathrm{sign}\left({\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}\right)---\left(26\right)$

Herein, when all symbol nodes that link to each other with check-node all do not contain multiple limit situation,

${\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}=\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{m{n}^{\&prime;}}}^1\right)-\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{m{n}^{\&prime;}}}^0\right),$

When containing multiple limit in the symbol node that links to each other,

${\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}=\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{m{n}^{\&prime;}}=\mathrm{odd}\left(1\right)}\right)-\min \ln Z_{n^{\&prime;}m}^{(I-1)}(x_{n^{\&prime;},i_{m{n}^{\&prime;}}=\mathrm{even}\left(1\right)},$

β _Nm ^*Meaning be ${\mathrm{\&beta;}}_{\mathrm{nm}}^{*}=\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\min }\left|{\mathrm{\&beta;}}_{n^{\&prime;}m}\right|.$

Other step of updating are constant.

Quick Shuffled-BP algorithm decoding scheme:

The information that check-node has been upgraded in the utilization of Shuffled-BP algorithm is upgraded check-node afterwards, and the convergence rate that can accelerate to decipher reduces the iterations that needs, and therefore has practical occasion equally.

The quick shuffled-BP algorithm that can be obtained under the multi-system by the derivation thinking of the shuffled-BP algorithm under the BPSK has following step:

Step 1:(initialization)

To all symbol node n,, calculate all m ∈ M (n):

$Z_{\mathrm{nm}}^{\left(0\right)}\left(x_n\right)={\mathrm{\&gamma;}}_n\left(x_n\right)=\frac{{(I_n-I_{x_n})}^2+{(Q_n-Q_{x_n})}^2}{{2\mathrm{\&sigma;}}^2}---\left(27\right)$

Step 2:

For all symbol node 1≤n≤N,

A) level is upgraded, and the m ∈ M (n) for all has,

$L_{\mathrm{mn}}^{\left(I\right)}\&ap;2\mathrm{tam}{h}^{-1}\left\{\begin{array}{c}\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n.n^{\&prime;}<n}{\mathrm{\&Pi;}}\tanh \frac{{\mathrm{\&beta;}}_{n^{\&prime;}m}^{\left(I\right)}}{2}\\ \&CenterDot;\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n.n^{\&prime;}>n}{\mathrm{\&Pi;}}\tanh \frac{{\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}}{2}\end{array}\right\}---\left(28\right)$

Wherein:

B) the vertical renewal, the m ∈ M (n) for all is updated to:

To each symbol node n, and all m ∈ M (n), calculate:

Each symbol node n, calculate:

Step 3:(judgement)

Judgement is the same with improved Log-BP method.

Complexity compares:

Compare with the decoding algorithm under original multi-system is modulated, the fast decoding algorithm after the improvement greatly reduces the complexity of decoding.Fig. 4 has provided in iteration, spend check-node for d (m) under the multi-system modulation general-purpose algorithm and the complexity under the modified model fast algorithm relatively.From figure, can find out intuitively, no matter be follow-on rapid bp algorithm or follow-on quick Min-Sum algorithm, under the certain situation of q, (equal 4 with q among the figure and obtain matched curve), along with the raising of the degree of check-node, the reduction meeting of complexity than former universal algorithm all the more obviously.Can find out intuitively then that in Fig. 5 under the certain situation of the degree of check-node (d (m)=4 herein), along with the continuous increase of q, the reduction of complexity also is significant all the more.Therefore, when under multi-system modulation the LDPC sign indicating number being deciphered, a series of fast decoding algorithms provided by the present invention are very effective.Error performance compares:

Obtain the error performance of different decoding algorithms under the awgn channel by Computer Simulation, adopt the short frame coding mode of the LDPC that provides in the DVB-S2 standard, length is 16200, and code check is 1/3, and maximum iteration time is made as 50.The modified model rapid bp algorithm that is based on symbol substitution that Fig. 6 provides, the quick shuffled-BP algorithm of modified model and based on the error performance of the binary system BP decoding algorithm of bit decoding.Can see that rapid bp algorithm can access under identical signal to noise ratio than based under the decoding algorithm of bit better decoding performance, the especially high s/n ratio, effect is more obvious; And the shuffled-BP algorithm has been accelerated the convergence rate of upgrading owing to improved the renewal of check-node, therefore can access than the better error performance of BP algorithm.

Fig. 7 provides is based on symbol and BP based on bit, and two of four kinds of algorithms of Min-Sum are compared.Can see, the same with the situation under the BPSK modulation, the Min-Sum algorithm also must need certain error performance as cost when having simplified computational complexity greatly, and be the Min-Sum algorithm equally, remain based on the decoding algorithm of symbol and be better than based on the decoding algorithm of bit.

What Fig. 8 provided is three kinds of needed average iterationses of BP algorithm.Can see that the required average iterations of modified model rapid bp algorithm lacks than the average iterations of binary radix in the BP of bit algorithm.This convergence that modified model rapid bp algorithm has been described is better.In the iterative decoding of reality, the decoding speed that improves algorithm than theory estimate will be good many.And the shuffled-BP algorithm since better utilization the characteristic of upgrading, accelerated the convergence rate of decoding, therefore the process of decoding is able to shortening again, iterations also further reduces.

Fig. 9 has then provided the average iterations of two kinds of Min-Sum algorithms, can restrain faster based on the decoding algorithm of symbol, and the iterations that needs also lacks than the radix-2 algorithm based on bit.

Can reach a conclusion by above performance simulation result and analysis of complexity, modified model fast decoding algorithm based on symbol can access than the lower complexity of decoding algorithm under the original multi-system modulation, in realization, need memory cell still less, can also obtain simultaneously than adopting traditional better decoding performance of binary decoding mode and constringency performance based on bit, be a kind of interpretation method that has very much practical prospect therefore.

Claims (1)

1. the method for rapidly decoding of a LDPC code under multiple scale is characterized in that the concrete steps of this method are:

(1) former Tanner figure is made amendment, contain two types of nodes among the amended Tanner figure: symbol node and check-node; Wherein check-node is exactly the check-node among the former Tanner figure, symbol node x _nWith among the former Tanner figure after modulating relevant q bit set corresponding; As symbol node x _nIn a bit when in former Tanner figure, linking to each other with check-node m, symbol node x _nLink to each other with the check-node m among the amended Tanner figure;

(2) to each symbol node x _n, calculate its local cost function, to each (n, m) right, initialization:

$Z_{\mathrm{nm}}^{\left(0\right)}\left(x_n\right)={\mathrm{\&gamma;}}_n\left(x_n\right)=\frac{{(I_n-I_{x_n})}^2+{(Q_n-Q_{x_n})}^2}{2{\mathrm{\&sigma;}}^2}$ To all x _n∈ X _n

(3) to each check-node m, and all n ∈ N (m):

When all symbol nodes that link to each other with check-node all do not contain multiple limit, being updated to of check-node:

$L_{\mathrm{mn}}^{\left(I\right)}=2{\tanh }^{-1}\left[\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\tanh \left(\frac{{\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}}{2}\right)\right]$

When containing multiple limit in the symbol node that links to each other with check-node, being updated to of check-node:

$L_{\mathrm{mn}}^{\left(I\right)}=2{\tanh }^{-1}\left[\underset{n^{\&prime;}\&Element;N\left(m\right)\backslash n}{\mathrm{\&Pi;}}\tanh \frac{{\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}}{2}\right]$

$\mathrm{where}{\mathrm{\&beta;}}_{n^{\&prime;}m}^{(I-1)}=\left(\begin{array}{c}\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}=\mathrm{odd}\left(1\right)}\right)\\ -\min \ln Z_{n^{\&prime;}m}^{(I-1)}\left(x_{n^{\&prime;},i_{{\mathrm{mn}}^{\&prime;}}=\mathrm{even}\left(1\right)}\right)\end{array}\right)$

Expression symbol node x _nIn have the odd number bit to link to each other with same check-node,

Expression symbol node x _nIn have the even number bit to link to each other with same check-node;

(4) to each symbol node n, and all m ∈ M (n), calculate:

Each symbol node n, calculate:

(5) judgement output.

Images