CN106850142A - The polar code constructing methods of the code word Optimal Distribution encoded using Homophonic under memory channel - Google Patents

The polar code constructing methods of the code word Optimal Distribution encoded using Homophonic under memory channel Download PDF

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CN106850142A
CN106850142A CN201710043128.XA CN201710043128A CN106850142A CN 106850142 A CN106850142 A CN 106850142A CN 201710043128 A CN201710043128 A CN 201710043128A CN 106850142 A CN106850142 A CN 106850142A
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polar
homophonic
yards
sequence
bit
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刘荣科
冯宝平
王闰昕
赵岭
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Beihang University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0056—Systems characterized by the type of code used
    • H04L1/0057—Block codes
    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/13—Linear codes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0056—Systems characterized by the type of code used
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0056—Systems characterized by the type of code used
    • H04L1/0064—Concatenated codes

Abstract

The invention provides a kind of polar code constructing methods of the code word Optimal Distribution encoded using Homophonic under memory channel.In the method, input message sequence meets independent and uniform distribution, the distribution characteristics for changing the input message sequence by Homophonic codings first makes it to realize that the particular probability of channel capacity is distributed under obeying its correspondence memory channel, then the list entries for the sequence that above-mentioned obedience particular probability is distributed being encoded as polar, and then polar codings are carried out, generate coding codeword.The code word obeys the optimum probability distribution under correspondence memory channel, so as to polar coding methods can reach channel capacity under realizing memory channel.The present invention carries out precoding to realize the Optimal Distribution of polar coding output codons by Homophonic yards, maintains original polar coding structures, realizes that memory channel lower channel capacity is reachable again.

Description

Polar yards of the code word Optimal Distribution encoded using Homophonic under memory channel Building method
Technical field
The invention belongs to communication channel coding techniques field, refer specifically to be encoded using Homophonic under a kind of memory channel Realize the polar code constructing methods of code word Optimal Distribution.
Background technology
Polar Codes, i.e. polarization code are 2008 by E.A kind of new channel coding for proposing.Polarization code It is designed based on channel-polarization (Channel Polarization), is that the first can be proved by strict mathematical method Reach the constructivity encoding scheme of channel capacity.Polar yards of structural texture rule, encoding and decoding complexity is low, it is easy to logical with other Letter technology carries out co-design optimization.These advantages all cause that Polar yards has good application prospect.But at present for Polar yards of research is mostly to realize that capacity is reachable in memoryless channel, and in actual communication system, due to communication environment The limitation of complicated and channel width, during information transfer, channel occurs memory characteristic unavoidably.Under memory channel, How to construct Polar yards realizes channel capacity up to just requiring study.
It is well known in the art, under memory channel,DemonstrateThe original polarization of proposition may apply to Finite state memory channel.But, although polar yards can polarize under memory channel, original polarization is constructed for reality List entries in the application of border can not realize that channel capacity is reachable.Therefore, if wanting to reach the channel under memory channel Capacity is, it is necessary to generate code word optimal sequence so that the mutual information between list entries and output sequence is maximum.Under normal circumstances, Traditional polar yards input is being uniformly distributed for i.i.d (independent same distribution), is not optimal in memory channel.For example, conventional ISI (intersymbol interference) channel, Optimal Distribution is approximate with Markov (markov) process.
In sum, original polar yards of polarization may apply to finite state memory channel, but reach channel appearance Amount is, it is necessary to generate optimal sequence, but input typically all meets being uniformly distributed for i.i.d. in practice.Therefore, it is a kind of when input is believed It is i.i.d. when being uniformly distributed to cease, and the code construction method that codeword sequence can also meet Optimal Distribution is a kind of demand.
The content of the invention
Present invention solves the technical problem that being：When input message sequence obeys independent and uniform distribution, memory how is constructed Codeword sequence makes it be issued to Optimal Distribution in the channel under channel.In order to solve the problem, the invention provides one kind memory The polar code constructing methods of the code word Optimal Distribution encoded using Homophonic under channel.
The polar code constructing methods of the code word Optimal Distribution encoded using Homophonic under memory channel of the invention, Realized using polar yards Cascade H omophonic yards, the implementation rule of cascade structure is：By the uniform information sequence of every l bit After row are encoded through Homophonic, the indefinite bit data of each code word size is exported, by each code of Homophonic coding outputs The bit data of word inserts polar yards of list entries successively, and polar yards of each frame list entries length is N.L and N are Positive integer.
Further, polar yards of the code word Optimal Distribution for being encoded using Homophonic under memory channel of the invention Building method, is achieved by the steps of：
Step 1：A group information sequence of input is read, to each information sequence bit, its corresponding probability interval is calculated, And in interval uniform design random number；If obtaining i-th bit v in information sequenceiProbability interval be J (vi), from J (vi) Middle selection random number z；
Step 2：Calculate polar coding input sequence present frame i-th bits ui,(t)Conditional probability
WhereinRepresent ui,(t)On sequenceConditional probability, sequence, it is known that calculating u respectivelyi,(t)=0 and ui,(t)=1 probable value；T is polar coding inputs The index number of sequence present frame；
Step 3：：Determine the i-th bit u of polar coding input sequence present framesi,(t)；If present bit is to freeze position, ui,(t)According to conditional probabilityCorresponding random number is taken from shared random number series to determine；If current Position is information bit, ui,(t)According to conditional probabilityImplementation rule with described cascade structure is generated；
Step 4：2 to the step that repeats the above steps 3, until the length of present encoding list entries reaches polar yards of frame It is long, polar yards of complete list entries of one frame of construction.
Advantages of the present invention is with good effect：(1) The present invention gives the Homophonic yards of volume of polar yards of cascade Code structure, precoding is carried out by Homophonic yards, makes polar yards of coding input sequence meet specific distribution to realize The code construction process of memory channel.(2) under memory channel, the code construction method that is given of the present invention by software approach come Realize, original coding structure is not changed, be easy to practical application.
Brief description of the drawings
Fig. 1 is polar yards of coding structure schematic diagram of Homophonic yards of the invention cascade；
Fig. 2 is the schematic flow sheet of polar yards of code construction method of the invention.
Specific embodiment
Below in conjunction with drawings and Examples, the present invention is described in further detail.
The present invention proposes that a kind of utilization Homophonic codings realize the polar code constructing methods of code word Optimal Distribution. In the inventive method, input message sequence meets independent and uniform distribution, is encoded by Homophonic and changes the input information sequence The distribution characteristics of row makes it to realize that the particular probability of channel capacity is distributed under obeying its correspondence memory channel, then will be above-mentioned The list entries that the sequence of particular probability distribution is encoded as polar is obeyed, and then carries out polar codings, generate coding codeword. The optimum probability that the code word generated by the inventive method is obeyed under correspondence memory channel is distributed, so as under realizing memory channel Polar coding methods can reach channel capacity.
The present invention causes that polar coding inputs sequence meets by Homophonic codings during code construction Specific probability distribution, is distributed with obtaining the particular probability of output codons of polar codings.But Homophonic codings are one Fixed length to variable-length encoding is planted, when list entries is regular length, the length of output sequence is uncertain.Due to polar yards Input is fixed long, it is impossible to which the output for directly encoding Homophonic is used as polar yards of input.It is right to need The structure of Homophonic yards of polar yards of cascade makes improvement.The list entries length that Homophonic is encoded is represented with l, is used L ' represents the code word size exported after Homophonic codings, and the polar yards of list entries length of each frame is represented with N.By l ' The data of bit insert polar yards of list entries successively.If l ' ＜ N, next do not carry out polar codings, and are directed to The uniform information sequence of next group of l bit carries out Homophonic codings, and coding result is continued into the input as polar yards Sequence, until the output bit data of Homophonic codings fills out N number of position that polar encodes frame list entries needs It is full.If it is defeated that polar is encoded a frame by the output of output length l ' ＞ N or the Homophonic coding of Homophonic codings N number of position that entering sequence needs has if unnecessary bit after filling up, then unnecessary bit is inserted into next frame polar yards successively List entries.
The present invention utilizes the implementation method of above-mentioned cascade structure, by the uniform information sequence v of every l bit1,v2,...,vl The list entries of polar yards of generationIt is referred to as a frame per N number of code word bits, N is polar yards of coded frame Long, t is frame number, t frame list entriesFig. 1 gives the code construction process.In figure In, preceding l bit information blocks are encoded to a part for the polar yards of frame of list entries first and the second frame by Homophonic.It is similar Ground, second input block is encoded into the remainder of polar yards of frame of list entries second and a part for the 3rd frame.When After the completion of a polar yards of frame of list entries is filled up by Homophonic coding output bits, polar coding inputs will be sent to End.
Utilization Homophonic codings of the invention realize the polar code constructing methods of code word Optimal Distribution, obtain polar The list entries of coding, then carries out polar codings and obtains output codons sequence according to the list entries.Polar of the invention Code constructing method main flow is as shown in Fig. 2 illustrate each to realize step below.
Step 1：Input message sequence is read, according to the probability distribution of incoming symbol, the probability interval corresponding to it is calculated, And uniformly select random number in different intervals.
One group information sequence v of input1...vl, wherein viI-th bit value in information sequence is represented, i is information sequence The bit index variables of row, vi∈ v, v are the valued space of input information.Calculate the probability interval of information sequence bit value J (v), the uniform design random number z in interval J (v).
J (v) is calculated according to the probability distribution of bit symbol, if viIt is i-th element in v, J (vi)=[QI-1, QI), wherein,PkProbability distribution in expression v corresponding to k-th element, Q0=0.The probability of all elements in v Distribution sum is 1.
V of the present inventioni∈ v={ 0,1 }, calculate J (0) and J (1) herein.If the probability distribution of bit symbol 0 and 1 is respectively P1 And P2, P1+P2=1, then J (0)=[0, P1), J (1)=[P1,P1+P2)。
Obtain viProbability interval J (vi) after, random number z is therefrom chosen, random number z is produced with random number generator.
Homophonic encoded output sequences seq is initialized as：λ represents empty sequence, the corresponding interval tables of seq It is shown as Interval range where representing sequence seq.
Step 2：Calculate the polar yards of i-th bit u of list entries present framei,(t)Conditional probabilityIts Middle t represents the index variables of present frame, and the locative indexes of i, value is the integer of 1~N.
When information sequence is when the output codons sequence length after Homophonic codings is more than polar yards of frame length, The output codons sequence can be by as the polar yards of list entries of the different frame of coding input, so needing exist for distinguishing polar The different frame sequence of code.If the Homophonic codings of previous block of information stop at polar yards of t0FrameI-th0It is individual Bit, wherein i0Represent the index value of bit, i0Span is the arbitrary integer of [1, N], uses variable t0Represent the volume of frame Number.Represent sequence Represent the polar yards of t of list entries0The i-th of frame0Individual ratio Special position, continues to generate using the inventive method
For the polar yards of i-th bit u of the current t frames of list entriesi,(t)If,Represent ui,(t)On SequenceConditional probability, sequence, it is known that calculating u respectivelyi,(t)=0 and ui,(t)=1 Probable value, that is, calculateWithSuch as formula (1), conditional probability can be converted Solved for the situation of joint probability.
Wherein：U in presentation code list entriesi,(t)OnConditional probability, character UiWithRespectively ui,(t)WithVector representation, lowercase u represent be scalar, be instantiation.Represent ui,(t)=0 time series u1,(t),u2,(t),...,ui,(t)Joint probability,Represent ui=1 time series u1,(t), u2,(t),...,ui,(t)Joint probability.Therefore can be by calculating on ui,(t)Joint probability solve
Under memory channel, using SC trellis decoding algorithms, therefore, calculating on uI, (t)State is added during joint probability Variable s.If siRepresent channel in i-th state variable at moment, s0Represent the original state of channel, sNRepresent the termination of channel State.Channel is in the i-th moment input bit ui,(t), respective channels state is designated as si。SiIt is scalar siVector representation.
The index variables t of frame is omitted in formula (2), for exampleIt is written asRepresent the u of t frames1,u2,...,ui Sequence,Represent list entriesJoint probability,Represent channel original state on list entries's Conditional probability,Represent original state s0Prior probability.
According to the recursive structure of SC,Calculating according to following formula (3), formula (4) is updated, and total number of bits N is expressed as 2M, i.e. N=2M, M are positive integer.
Wherein,Represent XOR,Represent list entries u1,u2,...,u2i,Represent sequenceMiddle index is The item of even number,Represent sequenceMiddle index is the item of odd number.
Initial valueGive current state si-1When, output codons xiWith under One state siJoint probability, Optimal Distribution of this probability to output codons that should be under memory channel.
Step 3：Calculate the i-th bit u of polar coding input sequencesi,(t)Bit information；
If current i-th bit is to freeze position；Take out corresponding from the shared random number series of polar codings sending and receiving end Random number Rj, u is determined using formula (5)i,(t)Value.Described shared random number series is generated by random number generator, random number Span is 1 to m integer, and the length of random sequence is identical with the sequence length m for freezing collection.
The span of random number is 1 to the arbitrary integer between m, if current random number Then ui,(t)=0；OtherwiseThen ui,(t)=1.
If current i-th bit is information bit；By intervalIt is divided into 2 subintervals：
WithWherein parameter
Initial value has been given by step 1, p1、p2Interval range where representing formation sequence seq.
Selection variable a ∈ { 0,1 } so that z belongs to(seq is a) to add the new sequence that variable a is formed in seq Row,Represent sequence (seq, interval range a).Renewal sequence seq be new sequence (seq, a), now ui,(t)=a.
Then judgeOrWhether corresponding J (v is belonged toi), if belonged to, The information sequence bit v of inputiHomophonic coding stop；Otherwise, the bit in follow-up polar yards of list entries Position, continues by input information viGeneration is encoded according to Homophonic to insert.
Step 4：Judge whether i is equal to N, if so, then obtaining polar yards of current t frame list entriesCarry out polar Coding, generates coding codeword, i.e. output codonsGNIt is generator matrix.Then proceed to step 5.
Otherwise, if i is less than N, make i from increasing 1, return to step 2 is continued executing with.
Step 5：Judge whether Homophonic codings stop, if so, terminating implementation procedure.Otherwise, polar yards of continuation is defeated Enter the generation of the new frame of sequence, update frame number t from increasing 1, it is 1 to update location index i, and return to step 2 is continued executing with.
Embodiment：The block length l=2 of information sequence, the polar yards of code construction embodiment of list entries frame length N=8.
Information input block length l=2, t the group information sequence of Homophonic codings in this examplePolar yards freeze collect length | Ic|=4, information bit length | I |=4.Obtained with DSMC Freeze to collect Ic={ 1,3,5,6 } and information collection I={ 2,4,7,8 }.Under memory channel, the channel of the present embodiment has two kinds of shapes State, is designated as s={ 0,1 }, and original state is set to S0=0, the Optimal Distribution of output codons is：
(px,s′|s″(0,0|0),px,s′|s″(0,0|1),px,s′|s″(1,1|0),px,s′|s″(1,1 | 1))=(p1,p2,p3,p4)
And 0≤p1,p2,p3,p4≤ 1, p1+p2+p3+p4=1, herein, x is input information bits position, and s ' represents channel Next state, s " represents the current state of channel.Here next state s ' of setting coding codeword x and channel is equal, institute With px,s′|s″(0,1|0),px,s′|s″(0,1|1),px,s′|s″(1,0|0),px,s′|s″(1,0 | 1) these four situations do not exist, can be with It is entered as 0；Freeze the used random number series R={ a, b, c, d } when collection is chosen；Because it is 4 to freeze collection length, at random Ordered series of numbers length is also 4.
Given input message sequenceProbability distribution Output sequence U ∈ { 0,1 }, probability distribution P={ q of Homophonic codings1,q2, q1+q2=1；
First, the corresponding probability interval of information sequence bit is calculated.
J (0)=[0,1/2), J (1)=[1/2,1)；It is empty sequence,
According to v1,(t)=1, interval J (1)=[1/2,1) choose random number z, z ∈ [1/2,1).
Calculate the conditional probability of present frame the 1stWithHere is specific calculating process：
According to above-mentioned recurrence formula (3), (4) calculateWith AndSo as to calculate
According to same calculating process,Value can also determine,WithCalculating terminates.
Then, current sequence position 1 belongs to and freezes to collect, so, take first value R of random number series R1,
IfThen u1,(t)=0；Otherwise u1,(t)=1.
Whether i is judged more than 8, and now i ≠ 8, then update i=i+1=2；
For the Homophonic codings of input block without stopping, then continue to generate the next bit of present frame.
Calculate the conditional probability of present frame the 2ndWithFormula in above-mentioned steps (1) conditional probability, is turned into joint probability calculation.May then pass throughWithTo solve；
Here u1Value just be result out calculated above, then can be calculated according to identical calculating processWithAnd then obtainWithResult；
Current sequence position 2 belongs to information bit, then basisWithBy interval It is divided into 2 subintervalsHereinSelection a ∈ { 0,1 } So that z belongs toThen first symbol of output sequence is a, seq=(a, seq), and carries out assignment to current information position u2,(t)=a；Then judgeWhether belong to J (1)=[1/2,1).If belonged to, v1,(t)=1 Homophonic codings stop, and subsequently then start v2,(t)=0 Homophonic codings；If be not belonging to, follow-up information The setting of the value of position continues by v1,(t)=1 Homophonic codings are produced.
Similarly, u3—u8According to above- mentioned information collection assignment is completed with the similar calculating process for freezing collection.
I=8, according toGeneration coding codeword.

Claims (7)

1. it is a kind of under memory channel using Homophonic codings code word Optimal Distribution polar code constructing methods, it is special Levy and be, realized using polar yards Cascade H omophonic yards, the implementation rule of cascade structure is：By the uniform of every l bit After information sequence is encoded through homophonic, the indefinite bit data of each code word size is exported, homophonic is encoded and is exported The bit data of each code word insert polar yards of list entries successively, the polar yards of frame length of list entries is that N, l and N are equal It is positive integer.
2. polar code constructing methods according to claim 1, it is characterised in that polar yards of described cascade During Homophonic yards, the polar yards of length of list entries is less than in the code word size l ' of Homophonic coding outputs During degree N, then polar codings are not carried out first, continuation carries out Homophonic codings to the next group of information sequence of l bit, compiles The bit data of code output continues as polar yard of input, and the output bit data encoded until Homophonic is by polar N number of position that code list entries present frame needs is filled up.
3. polar code constructing methods according to claim 1, it is characterised in that polar yards of described cascade During Homophonic yards, the polar yards of length of list entries is more than in the code word size l ' of Homophonic coding outputs During degree N, after the bit data of Homophonic coding outputs fills up N number of position of the polar yards of present frame of list entries, will Unnecessary bit continues to fill out the polar yards of next frame of list entries.
4. polar code constructing methods according to claim 1, it is characterised in that the method realizes that step includes：
Step 1：A group information sequence of input is read, to each information sequence bit, its corresponding probability interval is calculated, and Uniform design random number in interval；If obtaining i-th bit v in information sequenceiProbability interval be J (vi), from J (vi) middle choosing Take random number z；
Step 2：Calculate the polar yards of i-th bit u of list entries present framei,(t)Conditional probability
WhereinRepresent ui,(t)On sequenceConditional probability, sequence, it is known that calculating u respectivelyi,(t)=0 and ui,(t)=1 probable value；T is polar coding inputs The index number of sequence present frame；
Step 3：Determine the i-th bit u of polar coding input sequence present framesi,(t)；If i-th bit is to freeze position, ui,(t)According to bar Part probabilityCorresponding random number is taken from shared random number series to determine；If i-th bit is information bit, ui,(t)According to conditional probabilityImplementation rule with described cascade structure is generated；
Step 4：Repeat step 2 constructs polar yards of complete list entries of a frame to step 3.
5. polar code constructing methods according to claim 4, it is characterised in that in described step 2, described condition ProbabilityJoint probability is converted into solve, it is as follows：
p U i | U 1 i - 1 ( 0 | u 1 , ( t ) i - 1 ) = p U i , U 1 i - 1 ( 0 , u 1 , ( t ) i - 1 ) p U i , U 1 i - 1 ( 0 , u 1 , ( t ) i - 1 ) + p U i , U 1 i - 1 ( 1 , u 1 , ( t ) i - 1 ) p U i | U 1 i - 1 ( 1 | u 1 , ( t ) i - 1 ) = p U i , U 1 i - 1 ( 1 , u 1 , ( t ) i - 1 ) p U i , U 1 i - 1 ( 0 , u 1 , ( t ) i - 1 ) + p U i , U 1 i - 1 ( 1 , u 1 , ( t ) i - 1 )
Wherein：Character UiWithRespectively ui,(t)WithVector representation,Represent ui,(t)=0 time series u1,(t),u2,(t),...,ui,(t)Joint probability,Represent ui,(t)=1 time series u1,(t),u2,(t),..., ui,(t)Joint probability.
6. polar code constructing methods according to claim 4, it is characterised in that in described step 3, i-th bit is to freeze During position, corresponding random number R is taken from shared random number seriesjIf,Then ui,(t)=0；IfThen ui,(t)=1；M is the sequence length for freezing collection.
7. polar code constructing methods according to claim 4, it is characterised in that in described step 3, i-th bit is information During position, by the interval range of Homophonic encoded output sequences seqIt is divided into 2 subintervals：
With
Wherein, it is empty sequence in step 1 initialization sequence seq, andp1、p2Area respectively where sequence seq Between scope upper lower limit value；Intermediate parameters
Selection variable a ∈ { 0,1 } forms new sequence, and (seq a), and causes that random number z belongs toRenewal sequence seq is New sequence (seq, a), now ui,(t)=a；
Then judge nowOrWhether corresponding J (v is belonged toi), if belonged to, Information sequence bit viHomophonic coding stop；Otherwise, continue according to viCarry out Homophonic codings and produce output ratio It is special.
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