CN101030789A - Method for constructing orthogonal code - Google Patents
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- CN101030789A CN101030789A CNA2007100905085A CN200710090508A CN101030789A CN 101030789 A CN101030789 A CN 101030789A CN A2007100905085 A CNA2007100905085 A CN A2007100905085A CN 200710090508 A CN200710090508 A CN 200710090508A CN 101030789 A CN101030789 A CN 101030789A
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Abstract
The invention is used in any CDMA and spread spectrum based radio communication system. It comprises two steps: in the first, looking for or constructing a code-group S having n different letters and being in n code length; then, looking for or constructing n sub-codes whose quadrature are accordance with specific requirement each other; replacing n letters in the code-group S with n sub-codes to get the desired binary or non-binary quadrature code.
Description
Technical field:
The present invention designs a kind of orthogonal code construction method, can be used for the multiple field of information technology, especially for any radio digital communication system that contains code division multiple access (CDMA) and spread spectrum.
Background technology:
Orthogonal code is an important theme of wireless communication field.Along with going deep into of social informatization degree, people are also more and more higher to the requirement of wireless communication technology.CDMA technology is the effective means that solves contradiction between limited radio spectrum resources and the magnanimity message capacity demand.In any cdma system, each user has own distinctive spectrum-spreading address code, for mutual identification.For reducing mutual interference, should keep good orthogonality between each spectrum-spreading address code as far as possible.There are several technology can generate quadrature or quasiorthogonal code at present, as m sequence, Gold sequence and Walsh sign indicating number.These technology are used widely in modern digital communication systems.Certainly, these sign indicating numbers are also imperfect in some aspects, as quantity, system, distribution, the correlation of sign indicating number, or the like.Generally, the m sequence is under the situation that code length is fixed, and the sign indicating number that acquisition has good orthogonality in a large number is difficult.The Gold sequence derives from the m sequence, has good relatively orthogonality.These two kinds of sign indicating numbers are often used as the scrambler in the cdma system because autocorrelation is outstanding.The Walsh sign indicating number is completely orthogonal sign indicating number, is often used as the address code in the cdma system.Should be noted that all these three kinds of sign indicating numbers all are towards binary, and all present certain specific distribution.If we have higher or requirement more flexibly at the aspects such as orthogonality, quantity, system or distribution of sign indicating number, to realize the more powerful communication system of performance, we are difficult to utilize above-mentioned technology to generate the coding that satisfies the demands.Seeking well behaved orthogonal code is a major issue of digital communication and other areas of information technology.
Summary of the invention:
The present invention has provided a kind of orthogonal code coding method, utilizes this coding method can construct the sign indicating number more outstanding than Walsh code performance.Certain this coding method is not limited in and makes up the binary orthogonal that is similar to the Walsh sign indicating number.
The construction method of this orthogonal code is such.At first, consider n different literals { C
1, C
2, C
3..., C
nThe n bit code that constitutes, each literal just occurs once like this, and obviously they can constitute n (n-1) (n-2) ... 21 is n! Individual different sign indicating number.Then, seek or construct such code character S, the code distance of wherein any two codings of requirement is minimum to be d, promptly for whole n position, has d locational literal difference at least, has n-d locational literal identical at most.When n is bigger, n! Be a very googol, qualified code character is impossible simply search for by computer programming.The sufficient code character S of structure quantity may need to use the relevant mathematical theorem that the inventor finds.Suppose to have in the S code character m qualified sign indicating number.The upper limit M of m draws easily.For code distance d, M=n (n-1) ... (n-d).
At last, on this S code character basis, construct binary orthogonal code like this: at first, the length of employing Walsh sign indicating number or individual quadrature each other of other any way generation n or transothogonal is consistent to be the subcode { s of L
1, s
2..., s
n, use s then
iReplace the C in the S code character of front
iLiteral promptly gets orthogonal code sets O finally.The code length of code character O expands to nL, and quantity still is m.
Obviously, if d approaches n, the orthogonality between any two group codings is rather desirable, because macroscopic view, whole n positions between any two codings of code character S have only on seldom n-d the position literal identical, and the individual locational literal of other d is all inequality.Microcosmic, the subcode s on these different d macro positions
iOrthogonal or transothogonal.If the different part of literal is considerably beyond the identical part of literal, perhaps s
iTransothogonal very strong, total effect all will be to obtain coding with good orthogonality.
The two-stage construction method of above-mentioned binary orthogonal obviously can be used to make up the orthogonal code of non-binary or other distribution pattern, as long as with subcode s
iThe distribution that is built into needed nonbinary form and guarantees the subcode summation meets the requirements and gets final product.
Embodiment:
Typically, orthogonal code is used for the strong noise communication technology such as cdma system, and digital copyright protection technology such as digital fingerprint etc.The key index of weighing the orthogonal code performance comprises recall rate and false drop rate.Know that easily these two indexs have correlation---higher recall rate must cause higher false drop rate.Under the concurrent condition of equal extent noise circumstance and equivalent amount orthogonal code, when false drop rate is identical, have the orthogonal code of higher recall rate, its performance is even more ideal.Perhaps when recall rate is identical, have the orthogonal code of low false drop rate, its performance is even more ideal.
A concrete orthogonal code structure example is as follows.Make text type n=4{C
1, C
2, C
3, C
4, the long L=3 of subcode.According to preceding method, when d=n-1, can obtain code length is that 43=12, quantity are the orthogonal code of M=43=12.Claim that this group orthogonal code is ST2 sign indicating number (name is from accurate 2 heavy transitive group Sharp 2-fold Transitive Group).When d=n-2, can obtain that code length is similarly 43=12 but quantity increases to the orthogonal code of M=432=24.Claim that this group quasiorthogonal code is ST3 sign indicating number (name is from accurate 3 heavy transitive group Sharp 3-fold Transitive Group).Obviously the orthogonality of ST3 sign indicating number is weaker than the ST2 sign indicating number.In this example, select the ST2 sign indicating number.Mathematical theorem according to the inventor finds for the ST2 sign indicating number, only need utilize 2 generator＜g
1, g
2, g
1=(C
1) (C
2C
3C
4), g
2=(C
1C
2) (C
3C
4), carry out in-place computation, just can generate whole 12 orthogonal codes of ST2 sign indicating number:
C
1C
2C
3C
4，
C
1C
3C
4C
2，
C
1C
4C
2C
3，
C
2C
1C
4C
3，
C
2C
3C
1C
4，
C
2C
4C
3C
1，
C
3C
1C
2C
4，
C
3C
2C
4C
1，
C
3C
4C
1C
2，
C
4C
1C
3C
2，
C
4C
2C
1C
3，
C
4C
3C
2C
1
Can see between any two codings has only a literal identical at most.
In second step, count n=4 and subcode length L=3 according to literal, structure subcode s
iRequirement for subcode is: need the orthogonality that keeps as well as possible each other, and the weight sum of subcode satisfies particular requirement, for example average in the case be 0 requirement (for average be that 0 Walsh sign indicating number compares).Consider that code length is 4, quantity is 4 Walsh sign indicating number, each yard remove the first element just in time can satisfy above about the requirement of subcode:
s
1：?1 1 1
s
2：-1 1 -1
s
3：?1 -1 -1
s
4：-1 -1 1
With concrete subcode s
iReplace C
i, it is as follows finally to encode:
1 1 1 -1 1 -1 1 -1 -1- 1 -1 1
1 1 1 1 -1 -1 -1 -1 1- 1 1 -1
1 1 1 -1 -1 1 -1 1 -1 1 -1 -1
-1 1 -1 1 1 1 -1 -1 1 1 -1 -1
-1 1 -1 -1 -1 1 1 -1 -1 1 1 1
-1 1 -1 1 -1 -1 1 1 1 -1 -1 1
1 -1 -1 1 1 1 -1 1 -1 -1 -1 1
1 -1 -1 -1 1 -1 -1 -1 1 1 1 1
1 -1 -1 -1 -1 1 1 1 1 -1 1 -1
-1 -1 1 1 1 1 1 -1 -1 -1 1 -1
-1 -1 1 1 -1 -1 -1 1 -1 1 1 1
-1 -1 1 -1 1 -1 1 1 1 1 -1 -1
In order to compare, selecting same code length is 12 Walsh sign indicating number.The Walsh sign indicating number is derived from 12 rank Hadamard matrixes:
1 1 1 1 1 1 1 1 1 1 1 1
1 -1 1 -1 1 1 1 -1 -1 -1 1 -1
1 -1 -1 1 -1 1 1 1 -1 -1 -1 1
1 1 -1 -1 1 -1 1 1 1 -1 -1 -1
1 -1 1 -1 -1 1 -1 1 1 1 -1 -1
1 -1 -1 1 -1 -1 1 -1 1 1 1 -1
1 -1 -1 -1 1 -1 -1 1 -1 1 1 1
1 1 -1 -1 -1 1 -1 -1 1 -1 1 1
1 1 1 -1 -1 -1 1 -1 -1 1 -1 1
1 1 1 1 -1 -1 -1 1 -1 -1 1 -1
1 -1 1 1 1 -1 -1 -1 1 -1 -1 1
1 1 -1 1 1 1 -1 -1 -1 1 -1 -1
In order to keep the consistency of weight, remove first row of top Walsh code table, it is 12 Walsh sign indicating number sign indicating number in contrast that 11 code lengths are arranged like this.Have identical code length and weight with the ST2 sign indicating number that construct the front.
Under identical concurrent quantity and Noise Criterion, the false detection rate and the recall rate of test ST2 sign indicating number and Walsh sign indicating number.Typically, the concurrent quantity that can make orthogonal code is 5, and Noise Criterion＜=8 obtain following result:
| False drop rate | ST2 sign indicating number recall rate | Walsh sign indicating number recall rate | False drop rate | ST2 sign indicating number recall rate | Walsh sign indicating number recall rate |
| … | … | … | 0.11 | 0.363 | 0.344 |
| 0.001 | 0.016 | 0.014 | 0.12 | 0.384 | 0.363 |
| 0.002 | 0.026 | 0.023 | 0.13 | 0.402 | 0.381 |
| 0.003 | 0.034 | 0.030 | 0.14 | 0.420 | 0.399 |
| 0.004 | 0.042 | 0.037 | 0.15 | 0.437 | 0.415 |
| 0.005 | 0.049 | 0.043 | 0.16 | 0.453 | 0.432 |
| 0.006 | 0.055 | 0.049 | 0.17 | 0.469 | 0.447 |
| 0.007 | 0.062 | 0.055 | 0.18 | 0.484 | 0.463 |
| 0.008 | 0.068 | 0.061 | 0.19 | 0.499 | 0.478 |
| 0.009 | 0.073 | 0.065 | 0.20 | 0.514 | 0.493 |
| 0.01 | 0.078 | 0.071 | 0.21 | 0.528 | 0.507 |
| 0.02 | 0.124 | 0.114 | 0.22 | 0.542 | 0.521 |
| 0.03 | 0.161 | 0.149 | 0.23 | 0.555 | 0.534 |
| 0.04 | 0.194 | 0.180 | 0.24 | 0.568 | 0.547 |
| 0.05 | 0.225 | 0.209 | 0.25 | 0.580 | 0.559 |
| 0.06 | 0.251 | 0.235 | 0.26 | 0.592 | 0.572 |
| 0.07 | 0.277 | 0.260 | 0.27 | 0.604 | 0.583 |
| 0.08 | 0.300 | 0.283 | 0.28 | 0.616 | 0.595 |
| 0.09 | 0.322 | 0.304 | 0.29 | 0.628 | 0.606 |
| 0.10 | 0.344 | 0.324 | 0.30 | 0.639 | 0.617 |
Can see that the performance of ST2 sign indicating number is better than the Walsh sign indicating number under identical code length and weight.This is not astonishing, because the ST2 sign indicating number of being constructed is each other except completely orthogonal part, other parts can reach transothogonal, and the Walsh sign indicating number of identical code length and weight only can be accomplished complete quadrature.And the quantity of ST2 sign indicating number is also Duoed 1 than the Walsh sign indicating number.
Above-mentioned construction method is typical.For the orthogonal code of other code length, also make up in a manner described, its performance is better than the Walsh sign indicating number of equal code length.Can predict the structure of the subcode of orthogonal, accurate quadrature or transothogonal and to choose be the main aspect that influences this encode final form and performance.The orthogonal coding of being constructed is a high flexible, needs not to be binary system.In addition, the several generators of this employing generate the make of encoding by in-place computation, are easy to physically realize by the mode of high-velocity electrons circuit.
The building method of code character S:
In order to obtain orthogonal code as much as possible, need construct code character S dexterously.The structure of code character S relates to a new Combinational Mathematics problem: for the code length that is made of n different literals is the coding of n, if require the minimum d of being of the code distance of any two codings (promptly for whole n position, have d locational literal difference at least, have n-d the locational literal identical at most), can be there how many codings to satisfy this requirement at most? how to construct these sign indicating numbers?
The upper bound M of group/cording quantity is easy to draw.For the code distance minimum is d, M=n (n-1) ... (n-d).Is the problems referred to above can change into: at specific code length that n, text type are that n, code distance minimum are that group/cording quantity can reach upper bound M under the condition of d? and how to construct? at this this problem is called weak upper bound encoded question.Strong upper bound encoded question is: as M=n (n-1) ... what can the maximum quantity of coding reach when quantity (n-d) can not reach? and how to construct?
Utilize the knowledge of permutation group and accurate multiple transitive group aspect, the inventor has solved weak upper bound encoded question, thereby makes orthogonal code construction method proposed by the invention that a solid foundation arranged.
At first, with C
1C
2C
nAll code words of form (altogether n! Individual) set regard as act on Ω 1,2 ..., the n unit symmetric group on the n}.Element among the Ω is called a little.The codeword set that satisfies code distance d is certain subclass of n unit symmetric group.Especially, permutation group G is the subgroup of n unit symmetric group.We represent in-place computation with *.
In order to understand the settling mode of upper bound encoded question, we need the notion of track, number of times and transitive group in the permutation group.For the permutation group G that acts on the Ω, an equivalence class constitutes a track.The number of the point of actual change is called the number of times of this group in the permutation group.If G has only a track on Ω, promptly Ω itself claims that then G is the transitive group on the Ω.Generally, transitive group means that any point on the Ω all has an opportunity to be replaced as any another one point under the effect of in permutation group certain element g, perhaps is replaced as point own.
Notion to above-mentioned transitive group is expanded, and can obtain the notion of multiple transitive group.If any two k order subset (i of unit to Ω
1..., i
k) and (j
1..., j
k), exist g ∈ G that these two subclass are replaced mutually, claim that then G is the heavy transitive group of k.
Lemma: establishing G is heavy transitive group, then n (n-1) of k on Ω ... (n-k+1) divide exactly the rank of G.The teaching materials of the visible any finite group/finite permutation group of the proof of this lemma, for example " the finite group guiding " of Xu Ming sunlight, Science Press, 2001.
Can obtain the notion of the heavy transitive group of accurate k by the heavy transitive group of k: if the rank of the heavy transitive group G of k are n (n-1) just ... (n-k+1), claim that then G is the heavy transitive group of accurate k.
(this is that the inventor proposes) theorem: accurately the heavy transitive group of k is that our needed group/cording quantity reaches the set that code distance between the upper bound and any two codings is at least the coding of d=n-k+1 just.
Proof:
At first, accurately to retransmit and pass order of a group be that number of elements is n (n-1) to k ... (n-k+1), code distance is that the quantity of the last boundary coding of d=n-k+1 is M=n (n-1) ... (n-d)=n (n-1) ... (n-k+1), their quantity just equates.
We use reduction to absurdity now, suppose that the heavy transitive group of this accurate k is not the code set of our needed the sort of character, and promptly we can find at least two group codings, have more than n-d=k-1 locational literal identical between them.Might as well suppose to have between this two group coding k locational literal identical.
Selected this k position: p
1, p
2..., p
k(they are n the k in the position)
And corresponding character: a
1, a
2..., a
k(we have two groups of such word sequences at least according to hypothesis)
According to the definition of the heavy transitive group of k, last any two the order subset (i of Ω
1..., i
k) and (j
1..., j
k) change under the effect of certain displacement of this transitive group of all having an opportunity.Specifically, p
1Can become n kind literal (comprising itself), then p
2Can become n-1 kind literal ..., p
kBecome and to become n-k+1 kind literal.Notice that it is n (n-1) just that order of a group is passed in accurate k re-transmission ... (n-k+1), this means (p
1, p
2..., p
k) become certain concrete (x
1, x
2..., x
k) can only once therefore, two groups of identical word sequence (a be arranged
1, a
2..., a
k) be impossible, otherwise (the p that will account for
1, p
2..., p
k) become the chance of other certain word sequence.Card is finished ■
Accurately the code set of heavy transitive group of k and the high diversity factor that we look for has confidential relation, the true not so difficult understanding that this can prove, but in the past as yet the someone notice.The weak upper bound encoded question that proposes like this, above just fundamentally is resolved.Construct the heavy transitive group of accurate k and can obtain our needed code character S.
From application point of view, it should be noted that such mathematics fact: infinite a plurality of accurate 2 heavy and 3 heavy transitive groups are arranged, and (their number of times is respectively p
mAnd p
m+ 1, p is a prime number), but accurate 4 heavy and 5 heavy transitive groups only respectively have one (also having only one or two not even accurately), and their rank are respectively M
11=111098 and M
12=12111098 (because n too little, orthogonality is obviously undesirable) then have been proved to be more than or equal to 6 heavy transitive groups and have not existed.
Generally speaking, accurate 2 group/cording quantities heavy or that 3 heavy transitive groups are provided are more sufficient.At this moment the group/cording quantity maximum can reach M=n (n-1) (n-2), and when n was bigger, the numerical value of M can satisfy most of actual demands.
Claims (3)
1, a kind of orthogonal code construction method is characterized in that the branch two-stage makes up.First order step is as follows: at first, consider n different literals { C
1, C
2, C
3..., C
nThe n bit code that constitutes, each literal just occurs once, and obviously they can constitute n (n-1) (n-2) ... 21 is n! Individual different sign indicating number.Then, seek or construct such code character S, the code distance of wherein any two codings of requirement is minimum to be d, promptly for whole n position, has d locational literal difference at least, has n-d locational literal identical at most.
2, second level step, on the described code character S of claim 1 basis, the final binary orthogonal of structure like this: at first, adopting Walsh sign indicating number, m sequence or other any way to generate n, quadrature, transothogonal or quasi-orthogonal length are consistent each other is the subcode { s of L
1, s
2..., s
n, require the weight sum of subcode to satisfy final certain weights requirement of encoding.Use s then
iReplace the C in the S code character of front
iLiteral promptly gets orthogonal code sets O finally.The code length of code character O expands to nL, and quantity is identical with code character S.
3, second level step, on the described code character S of claim 1 basis, construct final nonbinary orthogonal code like this: at first, adopting any way to generate n, quadrature, transothogonal or quasi-orthogonal length are consistent each other is the subcode { s of L
1, s
2..., s
n, require the distribution sum of subcode to satisfy final specific distribution requirement of encoding.Use s then
iReplace the C in the S code character of front
iLiteral promptly gets orthogonal code sets O finally.The code length of code character O expands to nL, and quantity is identical with code character S.
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|---|---|---|---|
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|---|---|---|---|
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105281861A (en) * | 2014-06-09 | 2016-01-27 | 航天恒星科技有限公司 | Multisystem modulation code block and modulation code generation method |
| CN111614373A (en) * | 2020-05-20 | 2020-09-01 | 北京升哲科技有限公司 | Spread spectrum signal transmission method, spread spectrum signal reception method, spread spectrum signal transmission device, spread spectrum signal reception device, and spread spectrum signal reception medium |
| CN111931891A (en) * | 2020-10-13 | 2020-11-13 | 北京博大格林高科技有限公司 | Method for constructing anti-counterfeiting graphic code by using novel orthogonal code, anti-counterfeiting graphic code and generation device |
-
2007
- 2007-04-11 CN CNA2007100905085A patent/CN101030789A/en active Pending
Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105281861A (en) * | 2014-06-09 | 2016-01-27 | 航天恒星科技有限公司 | Multisystem modulation code block and modulation code generation method |
| CN111614373A (en) * | 2020-05-20 | 2020-09-01 | 北京升哲科技有限公司 | Spread spectrum signal transmission method, spread spectrum signal reception method, spread spectrum signal transmission device, spread spectrum signal reception device, and spread spectrum signal reception medium |
| CN111614373B (en) * | 2020-05-20 | 2021-08-10 | 北京升哲科技有限公司 | Spread spectrum signal transmission method, spread spectrum signal reception method, spread spectrum signal transmission device, spread spectrum signal reception device, and spread spectrum signal reception medium |
| CN111931891A (en) * | 2020-10-13 | 2020-11-13 | 北京博大格林高科技有限公司 | Method for constructing anti-counterfeiting graphic code by using novel orthogonal code, anti-counterfeiting graphic code and generation device |
| CN111931891B (en) * | 2020-10-13 | 2021-08-24 | 北京博大格林高科技有限公司 | Method for constructing anti-counterfeiting graphic code by using novel orthogonal code, anti-counterfeiting graphic code and generation device |
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