CN102811065A - Mini-sum decoding correcting method based on linear minimum mean error estimation - Google Patents

Mini-sum decoding correcting method based on linear minimum mean error estimation Download PDF

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CN102811065A
CN102811065A CN2012102820792A CN201210282079A CN102811065A CN 102811065 A CN102811065 A CN 102811065A CN 2012102820792 A CN2012102820792 A CN 2012102820792A CN 201210282079 A CN201210282079 A CN 201210282079A CN 102811065 A CN102811065 A CN 102811065A
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苏凯雄
吴子静
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Fuzhou University
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Abstract

The invention relates to a mini-sum decoding correcting method based on linear minimum mean error estimation. The method comprises the steps of establishing a model on a check message amplitude by a linear minimum mean error estimation method, and accelerating the determination of an estimation parameter by a golden section search algorithm to enable the estimation value approach to the check message amplitude in an error back propagation (BP) method; and revising the estimation parameter by taking the influence of the iterative times on the estimation parameter into consideration. According to the mini-sum decoding correcting method, a fixed estimation parameter is applied to different signal to noise ratios, so as to ensure the decoding performance and to reduce the expense of the hardware; the low density parity check (LDPC) code is decoded after the estimation parameter is obtained. The method not only ensures an excellent decoding performance but guarantees a rapid calculation of the estimation parameter; and is low in decoding complexity and simple in implementation of the hardware.

Description

Modified minimum and decoding method based on linear minimum mean square error estimation
Technical Field
The invention relates to the technical field of LDPC coding, in particular to a correction minimization and decoding method based on linear minimum mean square error estimation, which is used for standards such as ground digital multimedia television broadcasting DTMB, second-generation satellite digital video broadcasting DVB-S2, IEEE802.11n, IEEE802.16e, CCSDS and the like.
Background
The general soft-decision decoding method of the LDPC codes is established on the basis of a Belief Propagation (BP) algorithm, and improves the degree of confidence by transmitting iteration of an external message between a variable node and a check node, thereby achieving the purpose of decoding. However, the operation of checking the node message processing by the BP algorithm is too complex, and the hardware implementation cost is large. The min-sum algorithm is a simplification of the BP algorithm, and in check node message processing, the function operation in the BP algorithm is replaced by the minimum value, so that the operation amount is greatly reduced, but the decoding performance of the algorithm has about 0.5 to 1dB loss. In order to improve the decoding performance of the minimum sum algorithm without increasing the operation amount, at present, there are two kinds of correction algorithms of the minimum sum, namely a Normalized BP-Based algorithm and an Offset BP-Based algorithm.
The correction method of the Normalized BP-Based algorithm comprises the following steps:
Figure 2012102820792100002DEST_PATH_IMAGE002
，
Figure 2012102820792100002DEST_PATH_IMAGE004
(11)
the correction method of the Offset BP-Based algorithm comprises the following steps:
， (12)
in formulae (11) and (12), L1For checking the amplitude, L, of messages in the BP algorithm2To minimize the sum of the amplitudes of the check messages in the algorithm,is an estimate of the amplitude of the check message.
Both of the above correction algorithms improve decoding performance by introducing a correction factor (a or b). However, the obtained correction factor value has a certain difference from the actually required value, so the mean square error of the check message amplitude estimation value is larger, and the decoding performance has a certain difference from the BP algorithm.
In practical engineering application, a Monte Carlo method is also adopted to determine the correction factor of the correction factor, so that a more accurate estimation can be obtained, but a large amount of calculation simulation experiments are needed, so that the calculation amount is greatly increased.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a correction minimum and decoding method based on linear minimum mean square error estimation.
In order to achieve the purpose, the technical scheme of the invention is as follows:
definition ofC i Representation and variable nodeiA set of connected check nodes that are,R j representing and checking nodesjA collection of connected variable nodes that are connected,C i \jmeans for removingjExtrinsic and variable nodesiA set of connected check nodes that are,R j \imeans for removingiThe set of variable nodes that are externally connected to the check nodes,L(r ji ) Representing check nodesjTo variable nodesiThe external information of (a) is received,L(q ij ) Representing variable nodesiTo check nodesjThe external information of (a) is received,crepresents a codeword;L 1the check message amplitude in the BP algorithm is given by the following value:
(1)
wherein (A), (B), (C), (D), (C), (l-1) represents the formulal-1) iterations;
wherein；L 2 The check message amplitude in the min-sum algorithm is given by the following values:
(2)
the method comprises the following steps:
step 1, establishing an estimation model of the check message amplitude: based on a linear minimum mean square error estimation model and in combination with the principle that the sign of the corrected check message is unchanged, an estimation model shown as the formula (3) is established for the check message amplitude:
Figure 2012102820792100002DEST_PATH_IMAGE018
(3)
whereina、bThe estimated parameters to be calculated;
step 2, calculating the mean square error function
Figure 2012102820792100002DEST_PATH_IMAGE020
Minimum estimated parametera、b：
Step 2.1, assume boundary
Figure 2012102820792100002DEST_PATH_IMAGE022
Is a constant
Figure 2012102820792100002DEST_PATH_IMAGE024
When is coming into contact with
Figure 2012102820792100002DEST_PATH_IMAGE026
Then, the available estimation parameters are:
Figure 2012102820792100002DEST_PATH_IMAGE028
(4)
wherein cov (L 1,L 2) To representL 1AndL 2the covariance of (a) of (b),D(L 2) To representL 2The variance of (a) is determined,E(L 1)、E(L 2) Respectively representL 1、L 2The mean square error of (d);
step 2.2, comparison of the calculated
Figure 593044DEST_PATH_IMAGE022
And
Figure 510185DEST_PATH_IMAGE024
whether or not they are equal, if
Figure 2012102820792100002DEST_PATH_IMAGE030
Obtained bya、bIf not, discarding; if it is
Figure 2012102820792100002DEST_PATH_IMAGE032
Then obtaina、bThe correct value of (d);
step 2.3, adopting a golden section search algorithm, iterating according to the steps 2.1 and 2.2, and quickly determining the boundarykThen, the estimated parameters are obtained according to the formula (4)a、b；
And 3, correcting the estimation parameters according to the iteration times:
carrying out weighted average processing on the estimation parameters obtained from the previous n iterations according to the formula (5) to obtain the corrected estimation parametersa、bAnd using fixed estimation parameters for each subsequent iteration:
Figure 2012102820792100002DEST_PATH_IMAGE034
(5)
wherein,a i is shown asiEstimated parameters obtained by sub-iterationa，λ i To representa i The weighted average coefficient of (a) is,b i is shown asiEstimated parameters obtained by sub-iterationb，μ i To representb i The weighted average coefficient of (2);
step 4, adopting the same fixed estimation parameter for different signal-to-noise ratios;
and 5, after obtaining the required estimation parameters, decoding the LDPC code according to the following steps:
step 5.1, calculating the channel to transmit to the variable nodeiInitial probability likelihood ratio messageL(P i )Then all variable nodes are calculatediTo check nodesjBelong toC i Initial message of (2):
Figure 2012102820792100002DEST_PATH_IMAGE036
(6)
step 5.2, carrying out iterative processing according to the following steps (1), (2) and (3):
(1) check node message processing
Calculating all check nodesjDirection variable nodeiBelong toR j \iThe message of (2):
Figure 2012102820792100002DEST_PATH_IMAGE038
(7)
(2) variable node message processing
Computing all variable nodesiForward check nodejBelong toC i \jThe message of (2):
Figure 2012102820792100002DEST_PATH_IMAGE040
(8)
(3) decoding decision
For all variable nodesiCalculating a hard decision message:
(9)
the codeword is then:
(10)
step 5.3, performing iterative calculation according to the step 5.2 until a stop condition is metOr the iteration times reach the maximum iteration times, the iteration calculation is finished, otherwise, the iteration is continued; wherein,
Figure DEST_PATH_IMAGE048
a check matrix representing the LDPC code,
Figure DEST_PATH_IMAGE050
the resulting code words are represented as being decoded,represents a transpose of the matrix if
Figure 978338DEST_PATH_IMAGE046
The solved codeword is correct.
The method has the advantages that the estimation parameters are obtained by utilizing a linear minimum mean square error estimation method, the check message amplitude in the minimum sum algorithm approaches the check message amplitude in the BP algorithm, the influence of iteration times is considered, and the obtained estimation parameters are further corrected, so that the decoding performance of the method is better than that of the minimum sum algorithm and the traditional correction algorithm thereof. Meanwhile, the estimation parameters in the invention have definite calculation expressions under the condition of boundary determination, a large amount of simulation calculation is not needed, and the golden section search algorithm further accelerates the determination of the boundary value, so that the operation complexity is far lower than that of the BP algorithm, and the hardware implementation is simpler. And because the average iteration number of the invention is reduced, the total calculation amount of the decoding is lower than that of the traditional decoding method, and the decoding time delay is lower. Because fixed estimation parameters are adopted for different signal-to-noise ratios, the hardware overhead is further reduced.
Drawings
FIG. 1 is a work flow diagram of the method of the present invention.
Fig. 2 is a diagram of a simulated communication system according to an embodiment of the present invention.
FIG. 3 shows k anda graph of the relationship (c).
FIG. 4 is a graph of parameters a and b versus iteration number in an embodiment of the present invention.
Fig. 5 is a bit error rate graph of various decoding algorithms in an embodiment of the invention.
FIG. 6 is a graph of estimated parameters a and b versus SNR in an embodiment of the present invention.
Detailed Description
For convenience of description, reference will be made to symbols relating to the method of the present invention. The LDPC code can be represented by two methods, a check matrix H and a Tanner graph, which are in one-to-one correspondence. The columns in the check matrix correspond to variable nodes in the graph, usingiRepresents; and the rows in the check matrix correspond to check nodes in the graph, usingjAnd (4) showing. When in check matrixjGo to the firstiColumn element 1, the second in Tanner graphiIndividual variable node and the firstjThere is a connected edge between check nodes.
C i Representation and variable nodeiA set of connected check nodes that are,R j representing and checking nodesjA collection of connected variable nodes that are connected,C i \jmeans for removingjExtrinsic and variable nodesiA set of connected check nodes that are,R j \imeans for removingiThe set of variable nodes that are externally connected to the check nodes,L(r ji ) Representing check nodesjTo variable nodesiThe external information of (a) is received,L(q ij ) Representing variable nodesiTo check nodesjThe external information of (a) is received,crepresents a codeword;L 1the check message amplitude in the BP algorithm is given by the following value:
Figure 435864DEST_PATH_IMAGE012
(1)
wherein (A), (B), (C), (D), (C), (l-1) represents the formulal-1) iterations;
wherein
Figure 250236DEST_PATH_IMAGE014
；L 2 The check message amplitude in the min-sum algorithm is given by the following values:
Figure 885748DEST_PATH_IMAGE016
(2)
the invention relates to a minimum correction and decoding method based on linear minimum mean square error estimation, which is shown in figure 1 and comprises the following steps:
step 1, establishing an estimation model of the check message amplitude: based on a linear minimum mean square error estimation model and in combination with the principle that the sign of the corrected check message is unchanged, an estimation model shown as the formula (3) is established for the check message amplitude:
Figure 152781DEST_PATH_IMAGE018
(3)
whereina、bThe estimated parameters to be calculated;
step 2, calculating the estimation parameter which minimizes the mean square error functiona、b：
Step 2.1, assume boundary
Figure 351681DEST_PATH_IMAGE022
Is a constant
Figure 82877DEST_PATH_IMAGE024
When is coming into contact withThen, the available estimation parameters are:
Figure 830570DEST_PATH_IMAGE028
(4)
wherein cov (L 1,L 2) To representL 1AndL 2the covariance of (a) of (b),D(L 2) To representL 2The variance of (a) is determined,E(L 1)、E(L 2) Respectively representL 1、L 2The mean square error of (d);
step 2.2, comparing whether the calculated sums are equal or not, and if so, obtaininga、bIf not, discarding; if so, obtaina、bThe correct value of (d);
step 2.3, adopting a golden section search algorithm, iterating according to the steps 2.1 and 2.2, and quickly determining the boundarykThen, the estimated parameters are obtained according to the formula (4)a、b；
And step 3: and correcting the estimation parameters according to the iteration times:
because the decoding algorithm of the LDPC needs to be subjected to multiple iterative computations, the correct code word can be finally solved, and the estimation parameters required in the iterative computation process are different. According to the relation between the iteration times and the estimation parameters, the estimation parameters obtained from the previous n iterations are subjected to weighted average processing according to the formula (5), namely, the estimation parameters are corrected according to the formula (5) and are used in each iteration process in the future, so that the hardware is simple to realize, and the decoding performance is ensured. Since the messages output by the check nodes during the first iterations have a large impact on the overall decoding performance, the first iterations are weighted more heavily, and only the first 3 to 4 iterations are generally considered.
Figure 567582DEST_PATH_IMAGE034
(5)
Wherein,a i is shown asiEstimated parameters obtained by sub-iterationa，λ i To representa i The weighted average coefficient of (a) is,b i is shown asiEstimated parameters obtained by sub-iterationb，μ i To representb i The weighted average coefficient of (2);
step 4, considering that the uncertainty of the signal-to-noise ratio estimation causes the reduction of decoding performance and the hardware overhead, the same fixed estimation parameter is adopted for different signal-to-noise ratios;
and 5, after obtaining the required estimation parameters, decoding the LDPC code according to the following steps:
step 5.1, calculating the channel to transmit to the variable nodeiInitial probability likelihood ratio messageL(P i )Then all variable nodes are calculatediTo check nodesjBelong toC i Initial message of (2):
Figure 904016DEST_PATH_IMAGE036
(6)
step 5.2, carrying out iterative processing according to the following steps (1), (2) and (3):
(1) check node message processing
Calculating all check nodesjDirection variable nodeiBelong toR j \iThe message of (2):
Figure 333861DEST_PATH_IMAGE038
(7)
wherein,lis shown aslPerforming secondary iteration; (l-1) represents the formulal-1) iterations;
(2) variable node message processing
Computing all variable nodesiForward check nodejBelong toC i \jThe message of (2):
(8)
(3) decoding decision
For all variable nodesiCalculating a hard decision message:
Figure 850610DEST_PATH_IMAGE042
(9)
the codeword is then:
Figure 556398DEST_PATH_IMAGE044
(10)
step 5.3, performing iterative calculation according to the step 5.2 until a stop condition is met
Figure 891564DEST_PATH_IMAGE046
Or the iteration times reach the maximum iteration times, the iteration calculation is finished, otherwise, the iteration is continued; wherein,
Figure 620486DEST_PATH_IMAGE048
a check matrix representing the LDPC code,
Figure 699300DEST_PATH_IMAGE050
the resulting code words are represented as being decoded,
Figure 10327DEST_PATH_IMAGE052
represents a transpose of the matrix ifThe solved codeword is correct.
By the method, decoding of the LDPC code can be realized. It can be seen from the above steps that the estimation parameters of the present invention have definite calculation expressions under the condition of boundary determination, a large amount of simulation calculation is not needed, and the golden section search algorithm further accelerates the determination of the boundary values, so that the estimation parameters are obtained quickly and simply. The decoding performance and algorithm complexity analysis generated by the present invention will be given in conjunction with the following example.
The invention is further described with reference to the following figures and specific embodiments.
In the example, the LDPC (7493,3048) code under the GB20600 standard of digital television terrestrial broadcasting in China is adopted, and the simulation result is completed under an MATLAB simulation platform. The simulation parameters are as follows: the LDPC code rate is 0.4, the Gaussian white noise channel is used for BPSK modulation, the frame number is 200 frames, and the maximum iteration number of decoding is 20. The simulated communication system architecture is shown in figure 2.
1. And establishing an estimation model of the amplitude of the check message, and obtaining the value of the estimation parameter.
In thatSNR=2.2dBWhen the temperature of the water is higher than the set temperature,kand
Figure DEST_PATH_IMAGE056
the relationship of (c) is shown in fig. 3. As can be seen from the figure, there is one and only onekCan satisfy
Figure 732612DEST_PATH_IMAGE032
The conditions of (1). Boundary ofkThe determination of (2) can be quickly obtained by adopting a golden section searching algorithm. Suppose thatkFrom 0 to 0.5, takekHas an accuracy of 0.0001, and only 18 searches are needed to obtain the boundarykHas a value of 0.3318. According to the formula (4), aa=0.66497，b=0.22064, at this time
Figure 615118DEST_PATH_IMAGE032
And the condition is met, namely, the value of the unique estimation parameter is obtained.
2. And correcting the estimation parameters according to the iteration times.
The relationship between the number of iterations and the estimated parameters for different signal-to-noise ratios is given in fig. 4. According to equation (5), the estimated parameters are modified toa=0.72282，b=0.20652, and is used in each iteration later.
3. The same fixed estimation parameter is used for different signal-to-noise ratios.
In this example, the corrected estimation parameter when SNR =2.2dB is taken, so that it can be ensured that the decoding performance is not affected. I.e. the parameters are all taken for different SNR pointsa=0.72282，b=0.20652。
4. Performing LDPC decoding.
Wherein, the check node message processing is as shown in formula (11):
Figure DEST_PATH_IMAGE058
(11)
through the above steps, decoding is completed. The decoding performance of the various decoding algorithms is shown in fig. 5. In the figure, LMMSE-MinSum-1, LMMSE-MinSum-2 and LMMSE-MinSum-3 are error rate curves obtained by the method provided by the invention, and the difference is that the value of an estimation parameter is different. The estimated parameters of LMMSE-MinSum-1 are the estimated parameters obtained with the first iteration,a、bthe values of (c) correspond to those of fig. 6. The estimated parameters of LMMSE-MinSum-2 are the estimated parameters corrected by equation (5). LMMSE-MinSum-3 uses the same fixed estimation parameter for different SNRs,a=0.72282，b=0.20652。
as can be seen from FIG. 5, the decoding performance of the logarithm domain BP algorithm is best, and the bit error rate is lower than 10 already at 2.2dB-7(ii) a The decoding performance of the min-sum algorithm is obviously reduced compared with that of a logarithmic domain BP algorithm, and the error rate is lower than 10 at 3.2dB-7The difference is 1 dB; under the condition of the same error rate, the decoding performance of the traditional two modified minimum sum algorithms is improved compared with the minimum sum algorithm, and the Offset BP-Based algorithm and the Normalized BP-Based algorithm respectively reach 10 dB at 2.5dB and 2.7dB-7The error rate of (1) is 0.3dB and 0.5dB difference respectively compared with a logarithm domain BP algorithm; the LMMSE correction minimum sum algorithm provided by the invention can reach 10 error code rate at 2.3dB when the estimated parameters are not corrected-7(ii) a After the estimated parameters are corrected, the decoding performance is further improved, and the error code rate can reach 10 at 2.2dB-7The bit error rate is slightly larger than that of the logarithm domain BP algorithm when the same as that of the logarithm domain BP algorithm but the signal-to-noise ratio is less than 2.1 dB; when the same fixed estimation parameter is adopted for different SNRs, the decoding performance is hardly influenced. Therefore, the invention improves the decoding performance and reduces the hardware overhead.
From this example, it can be concluded that the decoding performance of the LDPC decoding method provided by the present invention is not only greatly superior to that of the min-sum algorithm, but also superior to that of the conventional correction algorithm, and has a steeper error rate curve than that of the BP algorithm. Meanwhile, the method has the advantages of fast calculation of the estimated parameters, low complexity of the decoding method, simple hardware implementation and the like.
The present invention has been described in detail with reference to the specific embodiments, but these are not to be construed as limiting the invention. The scope of the present invention should include those alternatives or modifications as would be apparent to those skilled in the art.

Claims (1)

1. A modified minimum sum decoding method based on linear minimum mean square error estimation is characterized in that:
definition ofC i Representation and variable nodeiA set of connected check nodes that are,R j representing and checking nodesjA collection of connected variable nodes that are connected,C i \jmeans for removingjExtrinsic and variable nodesiA set of connected check nodes that are,R j \imeans for removingiThe set of variable nodes that are externally connected to the check nodes,L(r ji ) Representing check nodesjTo variable nodesiThe external information of (a) is received,L(q ij ) Representing variable nodesiTo check nodesjThe external information of (a) is received,crepresents a codeword;L 1the check message amplitude in the BP algorithm is given by the following value:
Figure 2012102820792100001DEST_PATH_IMAGE002
(1)
wherein (A), (B), (C), (D), (C), (l-1) represents the formulal-1) iterations;
wherein；L 2 The check message amplitude in the min-sum algorithm is given by the following values:
Figure 2012102820792100001DEST_PATH_IMAGE006
(2)
the method comprises the following steps:
step 1, establishing an estimation model of the check message amplitude: based on a linear minimum mean square error estimation model and in combination with the principle that the sign of the corrected check message is unchanged, an estimation model shown as the formula (3) is established for the check message amplitude:
Figure 2012102820792100001DEST_PATH_IMAGE008
(3)
whereina、bThe estimated parameters to be calculated;
step 2, calculating the mean square error function
Figure 2012102820792100001DEST_PATH_IMAGE010
Minimum estimated parametera、b：
Step 2.1, assume boundary
Figure 2012102820792100001DEST_PATH_IMAGE012
Is a constant
Figure 2012102820792100001DEST_PATH_IMAGE014
When is coming into contact withThen, the available estimation parameters are:
(4)
wherein cov (L 1,L 2) To representL 1AndL 2the covariance of (a) of (b),D(L 2) To representL 2The variance of (a) is determined,E(L 1)、E(L 2) Respectively representL 1、L 2The mean square error of (d);
step 2.2, comparison of the calculated
Figure 429699DEST_PATH_IMAGE012
Andwhether or not they are equal, if
Figure 2012102820792100001DEST_PATH_IMAGE020
Obtained bya、bIf not, discarding; if it isThen obtaina、bThe correct value of (d);
step 2.3, adopting golden section search algorithm, and iterating according to the steps 2.1 and 2.2Fast determination of boundarieskThen, the estimated parameters are obtained according to the formula (4)a、b；
And 3, correcting the estimation parameters according to the iteration times:
carrying out weighted average processing on the estimation parameters obtained from the previous n iterations according to the formula (5) to obtain the corrected estimation parametersa、bAnd using fixed estimation parameters for each subsequent iteration:
Figure 2012102820792100001DEST_PATH_IMAGE024
(5)
wherein,a i is shown asiEstimated parameters obtained by sub-iterationa，λ i To representa i The weighted average coefficient of (a) is,b i is shown asiEstimated parameters obtained by sub-iterationb，μ i To representb i The weighted average coefficient of (2);
step 4, adopting the same fixed estimation parameter for different signal-to-noise ratios;
and 5, after obtaining the required estimation parameters, decoding the LDPC code according to the following steps:
step 5.1, calculating the channel to transmit to the variable nodeiInitial probability likelihood ratio messageL(P i )Then all variable nodes are calculatediTo check nodesjBelong toC i Initial message of (2):
Figure 2012102820792100001DEST_PATH_IMAGE026
(6)
step 5.2, carrying out iterative processing according to the following steps (1), (2) and (3):
(1) check node message processing
Calculating all check nodesjDirection variable nodeiBelong toR j \iThe message of (2):
(7)
(2) variable node message processing
Computing all variable nodesiForward check nodejBelong toC i \jThe message of (2):
Figure 2012102820792100001DEST_PATH_IMAGE030
(8)
(3) decoding decision
For all variable nodesiCalculating a hard decision message:
Figure 2012102820792100001DEST_PATH_IMAGE032
(9)
the codeword is then:
Figure 2012102820792100001DEST_PATH_IMAGE034
(10)
step 5.3, performing iterative calculation according to the step 5.2 until a stop condition is met
Figure 2012102820792100001DEST_PATH_IMAGE036
Or the iteration times reach the maximum iteration times, the iteration calculation is finished, otherwise, the iteration is continued; wherein,
Figure 2012102820792100001DEST_PATH_IMAGE038
a check matrix representing the LDPC code,
Figure 2012102820792100001DEST_PATH_IMAGE040
the resulting code words are represented as being decoded,
Figure DEST_PATH_IMAGE042
represents a transpose of the matrix ifThe solved codeword is correct.
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