CN105071900B - A kind of accurate method and system for solving multiple grid successive minima amount problem - Google Patents
A kind of accurate method and system for solving multiple grid successive minima amount problem Download PDFInfo
- Publication number
- CN105071900B CN105071900B CN201510439710.9A CN201510439710A CN105071900B CN 105071900 B CN105071900 B CN 105071900B CN 201510439710 A CN201510439710 A CN 201510439710A CN 105071900 B CN105071900 B CN 105071900B
- Authority
- CN
- China
- Prior art keywords
- matrix
- vector
- sgn
- idx
- indicate
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Classifications
-
- 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
- H04L1/0045—Arrangements at the receiver end
- H04L1/0047—Decoding adapted to other signal detection operation
-
- 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
- H04L1/0045—Arrangements at the receiver end
- H04L1/0054—Maximum-likelihood or sequential decoding, e.g. Viterbi, Fano, ZJ algorithms
Landscapes
- Engineering & Computer Science (AREA)
- Computer Networks & Wireless Communication (AREA)
- Signal Processing (AREA)
- Artificial Intelligence (AREA)
- Complex Calculations (AREA)
- Mobile Radio Communication Systems (AREA)
Abstract
The present invention provides a kind of accurate method and system for solving multiple grid successive minima amount problem.The beneficial effects of the present invention are: the present invention accurately solves the method for multiple grid successive minima amount problem when being applied to when compeling in whole linear receiver of mimo wireless communication system, this method can find optimal coefficient matrix, so that it is guaranteed that best receptivity can be obtained by compeling whole linear receiver.Compared with existing method for precisely solving, present method avoids exhaustions, therefore can greatly reduce computation complexity and the requirement to storage.Furthermore, when method of the invention is applied to compel whole linear receiver, the baseband system of plural number will not needed to be transformed into equivalent real number system, the number for the improvement sphere decoder that method of the invention uses needed for enabling to reduces half, it to further improve computational efficiency, therefore is a kind of more practicable algorithm.
Description
Technical field
The present invention relates to fields of communication technology, more particularly to a kind of accurate method for solving multiple grid successive minima amount problem
And system.
Background technique
Lattice (lattice) theory is the study on classics field in geometric number theory.In recent years, case theory is in multiple-input and multiple-output
It is widely used in (Multiple-Input Multiple-Output, MIMO) wireless communication system, such as uses ball
(Sphere Decoding, SD) algorithm is decoded to realize that maximum likelihood receives, with lattice reduction (Lattice basis
Reduction, LR) algorithm improves linear receiver and successive interference cancellation (Successive Interference
Cancellation, SIC) receiver receptivity, etc..It is directed to mimo system recently, has scholar to propose a kind of novel
Compel whole (Integer-Forcing, IF) linear receiver, and demonstrate this receiver and can obtain than existing others
The all good receptivity of linear receiver, even SIC receiver.In the implementation process for compeling whole linear receiver, root is needed
Coefficient matrix is selected according to current channel conditions and system mode, and existing research is it has been proved that when to maximize system
When reachable transmission rate is target, optimal coefficient matrix is needed by solving the problems, such as most short Independent Vector group (shortest
Independent vectors problem, SIVP) or successive minima amount problem (successive minima problem,
SMP it) obtains.
Traditionally for lattice research be all carry out in real number field, but with case theory in a wireless communication system must
More and more applications are arrived, case theory is gradually extended to complex field, because existing for the related algorithm of multiple grid construction
There can be higher computational efficiency in.
The relevant background knowledge of one, case theory:
1.1 real number lattice:
One m ties up real number fieldOn lattice (referred to as real number lattice) be one groupUpper linear independent base vector
{g1,...,gmWhole integer coefficient linear combination set, be denoted as:
We are matrixIt is called the base or a generator matrix of this lattice.'s
Any one lattice vector can enough linear equations uniquely indicate: v=Gu, whereinIt is v
Coefficient vector, subscript ()TIndicate transposition.If the QR for obtaining G is decomposed: G=QR, wherein Q is an orthogonal matrix, and R is
The upper triangular matrix that one diagonal element is positive, we sayWithIt is of equal value, because after the former may be considered
Person is obtained by rotation in space.
Successive minima amount (successive minima): latticeA successive minima amount λ of kth (1≤k≤m)kDefinition
For using origin as the centre of sphere, the minimum of the lattice vector comprising k Line independent closes the radius of ball, it may be assumed that
WhereinRepresent beOn using origin as the centre of sphere, using r as the ball that closes of radius, span () represent
It is the linear space opened by the vector for including in bracket.
Successive minima amount problem (SMP): a given m ties up latticeSMP requires to find the vector of one group of Line independent
{v1,v2,...,vmMake | | vk| |=λk, 1≤k≤m.||vk| | expression is vkLength.
Most short Independent Vector group problem (SIVP): a given m ties up latticeSIVP requires to find one group of Line independent
Vector { v1,v2,...,vmMake | | vk||≤λm, 1≤k≤m.
For any one latticeThe accurate solution of both of these problems all certainly exists.And from the definition of the two
It can be seen that the accurate solution of SMP is centainly also the accurate solution of SIVP.
1.2 multiple grids:
M ties up complex fieldOn lattice (referred to as multiple grid) definition it is consistent in form with the definition of real number lattice, area
It is not only that the base vector { g of multiple grid1,...,gmBeUpper linear independent one group of vector, and the coefficient of lattice vector
It is Gaussian integer, i.e., in formula (1) Likewise, such as
The QR that fruit obtains generator matrix G is decomposed: G=QR, wherein Q is a unitary matrice, R be a diagonal element be positive real number, other
Element is the upper triangular matrix of any plural number,WithIt is also of equal value.The successive minima amount of multiple grid, SMP and
The definition of SIVP can directly extend from the definition of real number lattice to be obtained, and difference is only that will beRather thanOn account for,
Such as " Line independent " refers toUpper Line independent, " dimension " refer toA dimension.For arbitrary plural number
The accurate solution of lattice, its SMP and SIVP also certainly exists, and the accurate solution of SMP is centainly also the accurate solution of SIVP.
Two, compel whole linear receiver:
2.1 System describes:
Consideration one is equipped with ntRoot transmitting antenna, nrThe baseband model of the mimo system of root receiving antenna, and assume that it is passed through
Quasi-static, non-frequency-selective fading channel is gone through, we can use a complex matrixIndicate channel, and
This mimo system is regarded as a complex linear system.In transmitting terminal, all transmitting antennas all use a same trellis code
(lattice code) codebook Code word be all real number latticeIn vector.L (1≤l≤
nt) at root transmitting antenna, the encoder of trellis code is two independent information vectorsWithIt is mapped to one respectivelyIn
Two code wordsWithThey collectively form complex base band signalAnd made by n times channel
With being launched.Furthermore by pairScaling so that each transmitting signal meet power limit E | | xl||2}
=nP.Assuming that channel remains unchanged in the emission process of single code word, then using by n times channel, receiving end is received
Complex base band signalIt can indicate are as follows:
YC=HCXC+ZC (3)
WhereinIt is transmitting signal matrix,It is additive white Gaussian noise
Matrix, it is assumed that HCAnd ZCEach element be mean value is 0, variance is 1 Cyclic Symmetry multiple Gauss variable.
2.2 compel whole linear reception for equivalent real system:
Since the research of traditionally plaid matching is carried out in real number field, it is generally used in existing literature by formula
(3) method for being converted into equivalent real number system carries out compeling whole linear reception in real number field.Enable Nt=2 × nt, Nr=2 × nr, formula
(3) n shown inr×ntComplex linear system can be equivalent to a Nr×NtSolid linear system:
Simply it is expressed as
Y=HX+Z (5)
The transmitting signal matrix of this equivalent real systemChannel matrixReceive signal square
Battle arrayObviously compared with plural system (3), the dimension of these matrixes all increases one times.
Compeling whole linear receiver need to be according to current one reversible INTEGER MATRICES of system state selectionWith
One mapping matrixSince the row vector of X is all real number latticeLattice vector, by the definition of lattice it is found that AX
Row vector centainly stillLattice vector.The target for compeling whole linear receiver is exactly the concurrently output B from mapping matrixIFY
Every a line in restore AX row given by lattice vector.And since A is reversible, as long as AX can correctly restore, it will be able to
Therefrom obtain original transmitted signal X.So that optimal coefficient matrix A and the optimal mapping square of mimo system acquisition maximum transmission rate
Battle array BIFIt needs to obtain by following step:
It decomposes to obtain by choleskyWhereinIt is under one
Triangle battle array；
LTIt regards a generator matrix as, finds latticeThe accurate solution of upper SIVP or SMP, with this accurate solution
(NtThe lattice vector of a Line independent) integer quotient constitute optimal coefficient matrix A；
Select optimal mapping matrix:
Therefore, in order to enable optimal receptivity can be obtained by compeling whole linear reception, it would be desirable to accurately solve real number
LatticeSIVP or SMP.
2.3 compel whole linear reception for plural system:
If we design directly in complex field compels whole linear receiver, need to just be selected according to current system mode
Select a reversible Gauss integral matrixWith a plural mapping matrixA at this timeCXC's
The real and imaginary parts of row vector are allVector.Compel whole linear receiver concurrently from BIF,CYCEvery a line in restore ACXC
Row given by vector, then therefrom restore original transmitted signal XC.Mimo system is made to obtain maximum transmission rate at this time
Optimal coefficient matrix ACWith optimal mapping matrix BIF,CIt needs to obtain by following step:
It decomposes to obtain by choleskyIt is wherein one
Inferior triangular flap, ()HWhat is indicated is conjugate transposition；
?It regards generator matrix as, finds multiple gridSIVP or SMP accurate solution, it is accurate with this
Solve (ntThe lattice vector of a Line independent) Gaussian integer coefficient constitute optimal coefficient matrix AC；
Select optimal mapping matrix:
If it is possible to accurately solve multiple gridSMP, also allow for compeling whole linear reception and obtain most
Good receptivity.Target of the invention is exactly to construct the algorithm that can accurately solve multiple grid SMP.
SMP is the basic problem of case theory, and the method for current solution SMP can be divided into two classes, and one kind is that approximation is asked
The method of solution, another kind of is the method accurately solved.
Approximate solution method: the approximate solution method of SMP is primarily referred to as lattice reduction algorithm.For given latticeThe target of lattice reduction is to find more shorter than the base vector length in G, relative orthogonal from each other a base.Cause
It is centainly independent from each other for base vector, therefore the obtained reduction base of lattice reduction algorithm can be used as the approximate solution of SMP.
Lattice reduction algorithm the most famous includes Minkowski reduction algorithm, Hermite-Korkine-Zolotareff (HKZ) rule
About algorithm, Lenstra-Lenstra-Lov á sz (LLL) reduction algorithm etc..Since the research of traditionally plaid matching is all in real number field
Upper progress, therefore most of lattice reduction algorithm is proposed both for real number lattice, be can only operate in real number field, is used for
When compeling whole linear receiver, then the baseband system plural number is needed to be converted into equivalent real number system.As lattice reduction exists
More and more applications are obtained in mimo system, researcher gradually starts directly to construct lattice reduction calculation for multiple grid
Method.For example, LLL reduction algorithm has had been extended to multiple grid, and it is named as Complex-LLL (CLLL) reduction algorithm；And
And it has been proved that CLLL can reduce the calculation amount of nearly half than LLL specification.With lattice reduction algorithm come approximate solution SMP
The advantages of be that computation complexity is relatively low；The disadvantage is that being only able to find approximate solution, and when the dimensional comparison of lattice is high or uses
Lattice reduction comparison between the standards it is loose when (as used LLL/CLLL specification), their obtained approximate solutions are probably and accurately
The gap of solution is larger.Therefore, when they are for it is not ideal enough to will lead to receptivity when compeling whole linear receiver.
Method for precisely solving: current existing method for precisely solving all uses the method for exhaustion.Specifically, for one
A m ties up latticeIt is understood that its m-th of successive minima amount λmIt is certain to be not more than longest base vector in generator matrix G
Length is denoted as r.The way of exhaustion is handleWhole lattice vectors of the middle length no more than r are all found, then take certain row
Sequence compares method and therefrom finds out that one group of length is most short and vector of Line independent.The advantages of this method is can centainly to find
The accurate solution of SMP, but defect is also clearly: the complexity of this method is very high, while the memory space needed is also very
Greatly.Therefore for actual system, the method for this exhaustion is unpractiaca.
Summary of the invention
In order to solve the problems in the existing technology, the present invention provides a kind of accurate solution multiple grid successive minima amounts
The method of problem.
The present invention provides a kind of accurate methods for solving multiple grid successive minima amount problem, include the following steps:
Step 1: to given generator matrix GCCLLL specification is carried out, the new base that specification is obtained directly is assigned to GC, rule
The unimodular matrix about obtained is assigned to T；
Step 2: constructionIsomorphism real number lattice generator matrixAnd to G
QR decomposition is carried out, G=QR is obtained；
Step 3: 2 ..., m successively carries out operations described below for k=1:
(1) with R and [uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)] it is input, with | | gC,idx(k)| | it is initial search radius,
It is found with subalgorithm GSVPC and meets rank ([uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)U])=2 × (k-1)+1 conditions it is most short
The coefficient vector of lattice vector, and this coefficient vector is returned to uR,2×(k-1)+1；
(2) u is utilizedR,2×(k-1)+1Directly constructAnd uC,k←
uR,2×(k-1)+1(1:m)+i·uR,2×(k-1)+1(m+1:2m)；
Step 4: returning to U ← T [uC,1uC,2…uC,m]。
As a further improvement of the present invention, the subalgorithm GSVPC includes the following steps:
Step 1: enabling u ← 02m×1、o←12m×1、W0←||gC,idx(k)||2、uR,2×(k-1)+1←eidx(k)；Enable k ← 1, uk←uk
+ok, ok←-ok-sgn*(ok)；
Step 2: calculatingIf Wnew< W0, execute step 3；It is no to then follow the steps 5；Step 3: if k
≠ 1, then enable k ← k-1, Wk←Wnew, ok←sgn*(ck-uk)；If k
=1, execute step 4；
Step 4: calculating K ← rank ([uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)U]), if K=2 × (k-1)+1, enables
uR,2×(k-1)+1←u、W0←Wnew, then enable k ← k+1, uk←uk+ok, ok←-ok-sgn*(ok), return step 2；If K ≠ 2 ×
(k-1)+1, then directly enable uk←uk+ok, ok←-ok-sgn*(ok) and return step 2；
Step 5: if k=2m, terminator simultaneously exports uR,2×(k-1)+1；If k ≠ 2m, k ← k+1, u are enabledk←uk+ok, ok
←-ok-sgn*(ok) and return step 2.
As a further improvement of the present invention, the method for accurately solving multiple grid successive minima amount problem be applied to compel it is whole
In linear receiver.
The present invention also provides a kind of accurate systems for solving multiple grid successive minima amount problem, comprising:
Preprocessing module: for given generator matrix GCCLLL specification is carried out, the new base that specification is obtained directly is assigned
To GC, the unimodular matrix that specification is obtained is assigned to T；
Generation module: for constructingIsomorphism real number lattice generator matrixAnd QR decomposition is carried out to G, obtain G=QR；
Processing module: for successively carrying out operations described below for k=1,2 ..., m:
(1) with R and [uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)] it is input, with | | gC,idx(k)| | it is initial search radius,
It is found with subalgorithm GSVPC and meets rank ([uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)U])=2 × (k-1)+1 conditions it is most short
The coefficient vector of lattice vector, and this coefficient vector is returned to uR,2×(k-1)+1；
(2) u is utilizedR,2×(k-1)+1Directly constructAnd uC,k←
uR,2×(k-1)+1(1:m)+i·uR,2×(k-1)+1(m+1:2m)；
Return module: for returning to U ← T [uC,1uC,2…uC,m]。
As a further improvement of the present invention, the subalgorithm GSVPC includes the following steps:
Step 1: enabling u ← 02m×1、o←12m×1、W0←||gC,idx(k)||2、uR,2×(k-1)+1←eidx(k)；Enable k ← 1, uk←uk
+ok, ok←-ok-sgn*(ok)；
Step 2: calculatingIf Wnew< W0, execute step 3；It is no to then follow the steps 5；Step 3: if k
≠ 1, then enable k ← k-1, Wk←Wnew, ok←sgn*(ck-uk)；If k
=1, execute step 4；
Step 4: calculating K ← rank ([uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)U]), if K=2 × (k-1)+1, enables
uR,2×(k-1)+1←u、W0←Wnew, then enable k ← k+1, uk←uk+ok, ok←-ok-sgn*(ok), return step 2；If K ≠ 2 ×
(k-1)+1, then directly enable uk←uk+ok, ok←-ok-sgn*(ok) and return step 2；
Step 5: if k=2m, terminator simultaneously exports uR,2×(k-1)+1；If k ≠ 2m, k ← k+1, u are enabledk←uk+ok, ok
←-ok-sgn*(ok) and return step 2.
The present invention also provides one kind to compel whole linear receiver, compels to run the accurate solution in whole linear receiver at this
The system of multiple grid successive minima amount problem.
The beneficial effects of the present invention are: the method that the present invention accurately solves multiple grid successive minima amount problem is urgent when being applied to
When in whole linear receiver, this method can find optimal coefficient matrix, so that it is guaranteed that compeling whole linear receiver can obtain
Obtain best receptivity.Compared with existing method for precisely solving, present method avoids exhaustions, therefore can be very big
Ground reduces computation complexity and the requirement to storage.In addition, when method of the invention is applied to compel whole linear receiver, it will not
It needs the baseband system plural number to be transformed into equivalent real number system, expands as original two so as to avoid the dimension of system
Times, therefore compared with the method for the accurate solution successive minima amount problem for using same thought but being made for real number lattice, this hair
The number for the improvement sphere decoder that bright method uses needed for enabling to reduces half, to further improve calculating effect
Rate, therefore be a kind of more practicable algorithm.
Detailed description of the invention
Fig. 1 is the flow diagram of subalgorithm GSVPC.
Specific embodiment
The invention discloses a kind of accurate methods for solving multiple grid successive minima amount problem.
The Integral Thought of the algorithm for the accurate solution multiple grid SMP that the one, present invention is constructed
It is contemplated that a generator matrix isMultiple grid, it is assumed that its m
Successive minima amount is { λ1,λ2,...,λm, and assume { vC,1,vC,2,...,vC,mBe exactly one groupUpper Line independent, length
Degree is respectively equal to the lattice vector of this m successive minima amount.From the definition of SMP it is found that { vC,1,vC,2,...,vC,mThis can be passed through
The process of sample is found:
It finds firstOne most short non-vanishing vector, be denoted as vC,1；
Then k=2,3 ..., m are found according to thisIn one with { vC,1,vC,2,...,vC,k-1?Upper linear independent most short amount, is denoted as vC,k。
In case theory, claim multiple gridIt is with generator matrix
Real number latticeIsomorphism.This is because each lattice vector is provided by a unique linear equation, and just
As shown in formula (3) and formula (4), any one complex linear equation can be with a real number linear equation equivalent.From gradually
The definition of minimum is it is found that real number latticeShare 2m successive minima amount.ConsiderA vector vC=GCuC,
WhereinObviously existUpper coefficient vector is iuCLattice vector and vCIt is linear not independent.But if enabling
So with uR,1And uR,2For two lattice vector vs of coefficient vectorR,1=GuR,1And vR,2=GuR,2?On be then
Line independent, and | | vR,1| |=| | vC| |=| | ivC| |=| | vR,2||.Therefore, we it follows that2m successive minima amount be centainly equal to { λ1,λ1,λ2,λ2,...,λm,λm, and if we { vC,1,
vC,2,...,vC,k-1M Gaussian integer coefficient vector all in accordance with uCAnd uR,1、uR,2Relationship construct 2m integer quotient to
Amount, then they are correspondingLattice vector just giveOne of SMP accurate solution.In other words, only
Want us that can findAs soon as SMP accurate solution in m vector in odd positions, it is straight above-mentioned relation to be passed through
It connects to obtain the vector on neighbouring even-numbered position, can also obtainSMP accurate solution.
Therefore, we can find multiple grid by following stepsSMP accurate solution:
It is obtained according to formula (8)Isomorphism real number lattice generator matrix G；
It findsOne most short non-vanishing vector, be denoted as vR,1=GuR,1, and according to the relationship of formula (9) and formula (10)
Directly obtain vR,2=GuR,2；
Successively k=2,3 ..., m are foundIn one with { vR,1,vR,2,...,vR,2×(k-2)+1,
vR,2×(k-1)?Upper linear independent most short amount, is denoted as vR,2×(k-1)+1=GuR,2×(k-1)+1, and according to formula (9) formula
(10) relationship directly obtains vR,2k；
Finally m vector { v in odd positionsR,1,vR,3,...,vR,2×(m-1)+1It is converted into corresponding multiple grid
Vector { vC,1,vC,2,...,vC,k-1}。
Find vR,1The problem of be actually solve real number latticeMost short Vector Problem (shortest vector
Problem, SVP), it is known that SVP can effectively be solved with the sphere decoder based on Schnorr-Euchner (SE) enumeration strategy
Certainly.And kth (k=2,3 ..., m) step thereafter can be regarded asGeneral most short Vector Problem
(generalized shortest vector problem, GSVP), that is, the SVP with specified conditions limitation.It is specific next
It says, kth step finds vR,2×(k-1)+1The problem of namely to findIn one meet condition
rank([uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)U])=2 × (k-1)+1 (11)
Most short lattice vector v=Gu, rank () indicates to ask the operation of rank of matrix in bracket in formula, and this problem can be with
It is solved by improving sphere decoder.
Two, solve the improvement sphere decoder of kth step GSVP:
Sphere decoder is introduced first solves SVP (that is, finding uR,1) process.Due toWithIt is equivalent
(It is that the upper triangular matrix that QR is decomposed is carried out to G), sphere decoder is solvingSVP when will
Equally considerTo the particularity using R in structure.If we can determineMost short non-vanishing vector
Length be centainly no more than r1, then we ask for the vector v of searchingR,1One is positioned at using origin as the centre of sphere, with r1For radius
In spheric region.
For lattice vector v=Ru, its length square is equal toWherein vkIt is kth (1≤k of v
≤ 2m) a element；Utilize the upper triangle characteristic of R, vkIt can be expressed as vk=rk,k(uk-ck), it is therein
Completely by uk+1,...,u2mDetermine (c2m=0), ri,jWhat is indicated is the element in R on the i-th row, jth column.Cause
We can construct one 2m layers of search tree, (2m layers) of top corresponding u for this2m, (the 1st layer) of the bottom corresponding u1, appoint
The node of one layer of meaning all meets restrictive condition:
In formulaIt is k layers of pervious accumulation weight.That is, if
uk+1,...,u2mIt has been determined that, then in order to meet length limitation, ukIt is only possible in (ck-γk,ck+γk) value in this section.
Sphere decoder from top to down searches these coefficients on this search tree by the way of depth-first
Rope.And in kth, (k=2m, 2m-1 ..., 1) layer, SE enumeration strategy defines will be according to away from ckSequence from the near to the remote, also
It is to make partial weightIncremental sequence scans for the integral point in above-mentioned section, to guarantee to find first
Lattice vector length it is shorter；Concrete implementation method is in the amount of bias o by a variationkIt is (specific to carry out command deployment
Method will provide in algorithm description).When the search arrival bottom, also means that and have found length less than current search radius
A lattice vector coefficient vector u, as long as it is stored as u by sphere decoder then u is not null vectorR,1One time
Vector is selected, and search radius is reduced into the length of its correspondence lattice vector.If again can not find in current region of search
As soon as shorter non-vanishing vector, then the candidate vector being eventually found is returned as uR,1。
Therefore the solution of kth step GSVP can be realized by carrying out simple improvement to the above process: whenever search
When finding coefficient u of the length less than the lattice vector of current search radius, sphere decoder calculating matrix [u is all enabledR,1uR,2…
uR,2×(k-2)+1uR,2×(k-1)U] order, only when order is equal to 2 × (k-1)+1, u is just stored as u by algorithmR,2×(k-1)+1One
Candidate vector, and search radius be reduced into it correspond to lattice vector length.And if order is not equal to 2 × (k-1)+1, just
Give up this u, and is continued searching in current region of search.If again can not find in current search region one it is shorter
And can satisfy order be 2 × (k-1)+1 condition lattice vector when, just the candidate vector being eventually found is returned to
uR,2×(k-1)+1.It is seen that the sphere decoder for solving SVP can regard a kind of special circumstances of this innovatory algorithm as, because
This uR,1It can also be found with this innovatory algorithm.
Finally, it should be noted that how to be the suitable initial search radius of improvement sphere decoder selection of each step.According to
According to formula (8) we it is seen that the length of 2m base vector in G is actually equal to GCIn m base vector length two
Secondary repetition, therefore GCThe length (according to sequence from small to large) of m base vector can serve as λ1,λ2,...,λmIt is upper
Limit, thus also it is then used as the initial search radius of this m times improvement sphere decoder.In addition, before solving SMP we
It can be found by the CLLL algorithm of low complex degreeThe shorter reduction base of a length, to reduce as best one can
Improve the search range of sphere decoder.
Three, are summarized:
The method for the accurate solution multiple grid successive minima amount problem that the present invention constructs is named as SMPC method and (calculated by we
Method SMPC).Multiple grid is tieed up for mWhen solving SMP, algorithm SMPC is needed using the m improvement based on SE enumeration strategy
Sphere decoder (is named as subalgorithm GSVPC).What algorithm SMPC was finally returned that is a coefficient matrixMake
Obtain GCThe column vector of U is exactlyAs soon as SMP accurate solution, their length correspond toM successive minima
Measure { λ1,λ2,...,λm}.We summarize algorithm SMPC and subalgorithm GSVPC according to this below.
In the description of algorithm SMPC, we indicate G with idx (k)CThe short base vector of middle kth index (1≤k≤m,
1≤idx(k)≤m)、||gC,idx(k)| | indicate the length of this base vector,It indicatesA standard base of the i-th dx (k)；For vector u, what u (p:q) was indicated is that its p-th of element is constituted to q-th of element
Vector.
The present invention accurately solves the method (algorithm SMPC) of multiple grid successive minima amount problem, includes the following steps:
The first step (pretreatment): to given generator matrix GCCLLL specification is carried out, the new base that specification is obtained directly is assigned
To GC, the unimodular matrix that specification is obtained is assigned to T.
Step 2: being constructed according to formula (8)Isomorphism real number lattice generator matrix G, and to G carry out QR decomposition,
Obtain G=QR.
Step 3: 2 ..., m successively carries out operations described below for k=1:
(1) with R and [uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)] it is input, with | | gC,idx(k)| | it is initial search radius,
The coefficient vector of the most short lattice vector of the condition of satisfaction (11) is found with algorithm GSVPC, and this coefficient vector is returned to
uR,2×(k-1)+1；
(2) relationship according to formula (9) formula (10) directly constructs uR,2kAnd uC,k, i.e.,
uC,k←uR,2×(k-1)+1(1:m)+i·uR,2×(k-1)+1(m+1:2m) (15)
Step 4: returning to U ← T [uC,1uC,2…uC,m]。
It is in initial search radius | | gC,idx(k)| | spheric region in subalgorithm GSVPC to uR,2×(k-1)+1It scans for
Process be summarized as follows, flow chart is presented in Fig. 1.In the de-scription, 02m×1Indicate be a length be 2m full 0 to
Amount, 12m×1What is indicated is complete 1 vector that a length is 2m；Define bias vectorDefined function
Subalgorithm GSVPC specifically comprises the following steps:
The first step (initialization): u ← 0 is enabled2m×1、o←12m×1、W0←||gC,idx(k)||2、uR,2×(k-1)+1←eidx(k)；Enable k
← 1, uk←uk+ok, ok←-ok-sgn*(ok)。
Step 2: calculatingIf Wnew< W0, execute third step；Otherwise the 5th step is executed.
Step 3: enabling k ← k-1, W if k ≠ 1k←Wnew,
ok←sgn*(ck-uk)；If k=1, the 4th step is executed.
Step 4: calculating K ← rank ([uR,1uR,2…uR,2×(k-2)+1uR,2×(k-1)u]).If K=2 × (k-1)+1, enables
uR,2×(k-1)+1←u、W0←Wnew, then enable k ← k+1, uk←uk+ok, ok←-ok-sgn*(ok), return to second step.If K ≠ 2 ×
(k-1)+1, then directly enable uk←uk+ok, ok←-ok-sgn*(ok) and return to second step.
Step 5: terminator simultaneously exports u if k=2mR,2×(k-1)+1；If k ≠ 2m, k ← k+1, u are enabledk←uk+ok,
ok←-ok-sgn*(ok) and return to second step.
The present invention constructs a kind of accurate method (algorithm SMPC) for solving multiple grid successive minima amount problem, of the invention
Method can be used for finding the optimal coefficient matrix for compeling whole linear receiver, so that it is guaranteed that compeling whole linear receiver can obtain
Best receptivity.Compared with existing method for precisely solving, method (algorithm SMPC) of the invention avoids exhaustion, therefore
Computation complexity and the requirement to storage can be greatly reduced.In addition, when method (algorithm SMPC) of the invention is applied to compel
When whole linear receiver, the baseband system of plural number will not needed to be transformed into equivalent real number system, so as to avoid system
Dimension expands as original twice.If complex baseband system is transformed into equivalent real number system, reuse for real number lattice
The algorithm of the accurate solution successive minima amount problem made finds optimal coefficient matrix, then will need in total using 2ntIt is secondary to change
Goal decoding algorithm (ntIt is transmitting antenna number), and when use method (algorithm SMPC) of the invention, it only needs using n in totaltIt is secondary
Improve sphere decoder.Therefore, method (algorithm SMPC) of the invention can further increase computational efficiency, be a kind of more real
The feasible algorithm in border.
The above content is a further detailed description of the present invention in conjunction with specific preferred embodiments, and it cannot be said that
Specific implementation of the invention is only limited to these instructions.For those of ordinary skill in the art to which the present invention belongs, exist
Under the premise of not departing from present inventive concept, a number of simple deductions or replacements can also be made, all shall be regarded as belonging to of the invention
Protection scope.
Claims (2)
1. a kind of accurate method for solving multiple grid successive minima amount problem, which comprises the steps of:
Step 1: to given generator matrix GCCLLL specification is carried out, the new base that specification is obtained directly is assigned to GC, specification is obtained
To a unimodular matrix be assigned to T；
Step 2: constructionIsomorphism real number lattice generator matrixAnd G is carried out
QR is decomposed, and obtains G=QR,It indicates by complex matrix GCThe matrix that the real part of corresponding position element is constituted,It indicates
By complex matrix GCThe matrix that the imaginary part of corresponding position element is constituted；
Step 3: 2 ..., m successively carries out operations described below, and m indicates generator matrix G for k=1CDimension, that is, GC's
Column vector number:
(1) with R and [uR,1,uR,2,…,uR,2×(k-2)+1,uR,2×(k-1)] it is input, with | | gC,idx(k)| | it is initial search radius,
It is found with subalgorithm GSVPC and meets rank ([uR,1,uR,2,…,uR,2×(k-2)+1,uR,2×(k-1), u])=2 × (k-1)+1 conditions
The coefficient vector of most short lattice vector, and this coefficient vector is returned to uR,2×(k-1)+1, idx (k) expression GCThe short basal orientation of middle kth
The index of amount, 1≤k≤m, 1≤idx (k)≤m, gC,idx(k)Indicate GCThe short base vector of middle kth, | | gC,idx(k)| | indicate its length
Degree；
(2) u is utilizedR,2×(k-1)+1Directly constructAnd uC,k←uR,2×(k-1)+1(1:
m)+i·uR,2×(k-1)+1(m+1:2m), whereinIndicate imaginary unit；
Step 4: returning to U ← T [uC,1,uC,2,…,uC,m]；
What the accurate method for solving multiple grid successive minima amount problem was applied to mimo wireless communication system compels whole linear reception
When in machine, by channel matrix HC, transmission power P and number of transmission antennas ntObtain generator matrix GCProcess are as follows: pass through
Cholesky decomposes to obtainWhereinIt is an inferior triangular flap, ()H
Indicate conjugate transposition, it willAs generator matrix GC；Find out multiple gridNamelySuccessive minima amount problem
Accurate solution after, obtain Optimal matrix acquire ACAnd BIF, CProcess are as follows: by the Gaussian integer coefficient matrix U in above-mentioned 4th step
As optimal coefficient matrix AC, and optimal mapping matrix is calculated
The subalgorithm GSVPC includes the following steps:
Step 1: enabling u ← 02m×1、o←12m×1、W0←||gC,idx(k)||2、uR,2×(k-1)+1←eidx(k)；Enable k ← 1, uk←uk+ok,
ok←-ok-sgn*(ok)；It indicatesA standard base of the i-th dx (k)；ukIndicate candidate
K-th of element of coefficient vector u, 1≤k≤2m, okIt indicates for controlling ukThe amount of bias of one variation of search, according to ok←-
ok-sgn*(ok) be calculated, wherein function sgn*(ok) is defined as:
Step 2: calculatingIf Wnew< W0, execute step 3；It is no to then follow the steps 5；Indicate the pervious accumulation weight of kth layer；Indicate the partial weight of kth layer, wherein vkIt is
K-th of element of lattice vector v=Ru, 1≤k≤2m, according to formula vk=rk,k(uk-ck) calculate, wherein
Step 3: if k ≠ 1, enabling k ← k-1, Wk←Wnew, ok←
sgn*(ck-uk)；If k=1, step 4 is executed；rk,lWhat is indicated is to be located at row k, the element on l column in upper triangular matrix R, 1
≤k≤l≤2m；
Step 4: calculating K ← rank ([uR,1,uR,2,…,uR,2×(k-2)+1,uR,2×(k-1), u]), if K=2 × (k-1)+1, enables
uR,2×(k-1)+1←u、W0←Wnew, then enable k ← k+1, uk←uk+ok, ok←-ok-sgn*(ok), return step 2；If K ≠ 2 ×
(k-1)+1, then directly enable uk←uk+ok, ok←-ok-sgn*(ok) and return step 2；
Step 5: if k=2m, terminator simultaneously exports uR,2×(k-1)+1；If k ≠ 2m, k ← k+1, u are enabledk←uk+ok, ok←-
ok-sgn*(ok) and return step 2.
2. a kind of accurate system for solving multiple grid successive minima amount problem characterized by comprising
Preprocessing module: for given generator matrix GCCLLL specification is carried out, the new base that specification is obtained directly is assigned to GC,
The unimodular matrix that specification is obtained is assigned to T；
Generation module: for constructingIsomorphism real number lattice generator matrixAnd
QR decomposition is carried out to G, obtains G=QR,It indicates by complex matrix GCThe matrix that the real part of corresponding position element is constituted,It indicates by complex matrix GCThe matrix that the imaginary part of corresponding position element is constituted；
Processing module: for successively carrying out operations described below for k=1,2 ..., m, m indicates generator matrix GCDimension, also
It is GCColumn vector number:
(1) with R and [uR,1,uR,2,…,uR,2×(k-2)+1,uR,2×(k-1)] it is input, with | | gC,idx(k)| | it is initial search radius,
It is found with subalgorithm GSVPC and meets rank ([uR,1,uR,2,…,uR,2×(k-2)+1,uR,2×(k-1), u])=2 × (k-1)+1 conditions
The coefficient vector of most short lattice vector, and this coefficient vector is returned to uR,2×(k-1)+1, idx (k) expression GCThe short basal orientation of middle kth
The index of amount, 1≤k≤m, 1≤idx (k)≤m, gC,idx(k)Indicate GCThe short base vector of middle kth, | | gC,idx(k)| | indicate its length
Degree；
(2) u is utilizedR,2×(k-1)+1Directly constructAnd uC,k←uR,2×(k-1)+1(1:
m)+i·uR,2×(k-1)+1(m+1:2m), whereinIndicate imaginary unit；Return module: for returning to U ← T [uC,1,
uC,2,…,uC,m]；
The accurate system for solving multiple grid successive minima amount problem is run in compeling whole linear receiver, by channel matrix HC, transmitting
Power P and number of transmission antennas ntObtain generator matrix GCProcess are as follows: decompose to obtain by CholeskyWhereinIt is an inferior triangular flap, ()HIndicate conjugate transposition, it willAs generator matrix GC；Find out multiple gridNamelySuccessive minima amount problem accurate solution after, obtain
Optimal matrix is taken to acquire ACAnd BIF,CProcess are as follows: the Gaussian integer coefficient matrix U for obtaining return module is as optimal coefficient
Matrix AC, and optimal mapping matrix is calculatedThe subalgorithm
GSVPC includes the following steps:
Step 1: enabling u ← 02m×1、o←12m×1、W0←||gC,idx(k)||2、uR,2×(k-1)+1←eidx(k)；Enable k ← 1, uk←uk+ok,
ok←-ok-sgn*(ok)；It indicatesA standard base of the i-th dx (k)；ukIndicate candidate
K-th of element of coefficient vector u, 1≤k≤2m, okIt indicates for controlling ukThe amount of bias of one variation of search, according to ok←-
ok-sgn*(ok) be calculated, wherein function sgn*(ok) is defined as:
Step 2: calculatingIf Wnew< W0, execute step 3；It is no to then follow the steps 5；Indicate the pervious accumulation weight of kth layer；Indicate the partial weight of kth layer, wherein vkIt is
K-th of element of lattice vector v=Ru, 1≤k≤2m, according to formula vk=rk,k(uk-ck) calculate, wherein
Step 3: if k ≠ 1, enabling k ← k-1, Wk←Wnew, ok←
sgn*(ck-uk)；If k=1, step 4 is executed；rk,lWhat is indicated is to be located at row k, the element on l column in upper triangular matrix R, 1
≤k≤l≤2m；
Step 4: calculating K ← rank ([uR,1,uR,2,…,uR,2×(k-2)+1,uR,2×(k-1), u]), if K=2 × (k-1)+1, enables
uR,2×(k-1)+1←u、W0←Wnew, then enable k ← k+1, uk←uk+ok, ok←-ok-sgn*(ok), return step 2；If K ≠ 2 ×
(k-1)+1, then directly enable uk←uk+ok, ok←-ok-sgn*(ok) and return step 2；
Step 5: if k=2m, terminator simultaneously exports uR,2×(k-1)+1；If k ≠ 2m, k ← k+1, u are enabledk←uk+ok, ok←-
ok-sgn*(ok) and return step 2.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201510439710.9A CN105071900B (en) | 2015-07-24 | 2015-07-24 | A kind of accurate method and system for solving multiple grid successive minima amount problem |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201510439710.9A CN105071900B (en) | 2015-07-24 | 2015-07-24 | A kind of accurate method and system for solving multiple grid successive minima amount problem |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN105071900A CN105071900A (en) | 2015-11-18 |
| CN105071900B true CN105071900B (en) | 2019-08-16 |
Family
ID=54501196
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201510439710.9A Active CN105071900B (en) | 2015-07-24 | 2015-07-24 | A kind of accurate method and system for solving multiple grid successive minima amount problem |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN105071900B (en) |
Families Citing this family (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107147606B (en) * | 2017-05-05 | 2020-06-09 | 哈尔滨工业大学 | Lattice reduction assisted linear detection method in generalized spatial modulation |
| CN107835064B (en) * | 2017-10-19 | 2020-09-29 | 北京科技大学 | Physical layer network coding transmission method based on 1-bit feedback and lattice division |
| CN108900448B (en) * | 2018-06-27 | 2022-08-23 | 重庆邮电大学 | Low-complexity packet decoding method based on MIMO system |
Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101917368A (en) * | 2010-07-30 | 2010-12-15 | 北京邮电大学 | Lattice reduction-based MIMO detection soft output method |
| WO2011092697A1 (en) * | 2010-01-28 | 2011-08-04 | Ramot At Tel-Aviv University Ltd. | Transmission system with isi channel and method of operating thereof |
| CN102882659A (en) * | 2012-10-11 | 2013-01-16 | 河海大学 | Data transmission method of self-adaption multiple input multiple output (MIMO) transmission system |
-
2015
- 2015-07-24 CN CN201510439710.9A patent/CN105071900B/en active Active
Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2011092697A1 (en) * | 2010-01-28 | 2011-08-04 | Ramot At Tel-Aviv University Ltd. | Transmission system with isi channel and method of operating thereof |
| CN101917368A (en) * | 2010-07-30 | 2010-12-15 | 北京邮电大学 | Lattice reduction-based MIMO detection soft output method |
| CN102882659A (en) * | 2012-10-11 | 2013-01-16 | 河海大学 | Data transmission method of self-adaption multiple input multiple output (MIMO) transmission system |
Non-Patent Citations (2)
| Title |
|---|
| A fast lattice reduction algorithm for SIC-based MIMO detection;Zhiyong Chen et al;《Communications(ICC),2015 IEEE International Conference on》;20150608;全文 |
| Integer-Forcing MIMO Linear Receivers Based on Lattice Reduction;amin.sakzad et al;《IEEE Transactions on Wireless Communications》;20130912;全文 |
Also Published As
| Publication number | Publication date |
|---|---|
| CN105071900A (en) | 2015-11-18 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| RU2475982C1 (en) | Method and system of preliminary coding and method to build code book of preliminary coding | |
| CN103941220B (en) | The outer target Wave arrival direction estimating method of a kind of grid based on sparse reconstruct | |
| CN105071900B (en) | A kind of accurate method and system for solving multiple grid successive minima amount problem | |
| CN107634787A (en) | A kind of method of extensive MIMO millimeter wave channel estimations | |
| CN109613473A (en) | The relatively prime linear array angle estimating method of expansion based on sparsity | |
| CN103888217A (en) | Sphere decoding detection method and device | |
| CN108650056A (en) | A kind of mixed iteration detection method in extensive mimo system uplink | |
| CN107566305A (en) | A kind of millimeter-wave systems channel estimation methods of low complex degree | |
| CN106789770A (en) | The nonopiate multiple-user access methods of extensive MIMO based on compressed sensing | |
| CN110174658A (en) | Wave arrival direction estimating method based on one dimensionality reduction model and matrix completion of order | |
| CN105634568A (en) | LLR calculation method based on large-scale MIMO system signal detection | |
| CN106452533B (en) | A kind of receiving end antenna selecting method based on maximum capacity | |
| CN113872655B (en) | Multicast beam forming rapid calculation method | |
| CN105187102A (en) | Low-complexity zero-forcing pre-coding method based on over relaxation iteration | |
| CN105978835A (en) | Low-complexity distributive interference aligning method | |
| CN111988069B (en) | Large-scale MIMO generalized eigenvector structure precoding solving method and device | |
| CN105721028A (en) | Downlink beam forming method and device of VLC (Visible Light Communication) MU-MISO (Multiple Users-Multiple Input-Single Output) system and VLC (Visible Light Communication) MU-MISO (Multiple Users-Multiple Input-Single Output) system | |
| CN110635833B (en) | Power distribution method and device based on deep learning | |
| CN105119692B (en) | A kind of accurate method and system for solving real number lattice successive minima amount problem | |
| CN110213186A (en) | The angle of arrival and signal-noise ratio estimation method of ESPRIT based on mixed structure | |
| CN106941393B (en) | LDPC interpretation method and system based on SOR | |
| CN105376185A (en) | Constant modulus blind equalization processing method based on optimization of DNA shuffled frog leaping algorithm in communication system | |
| CN115802466A (en) | Combined power distribution and phase shift design method based on distributed RIS (RIS) assisted multi-user system | |
| CN107342798B (en) | Method and device for determining codebook | |
| CN103794219B (en) | A kind of Codebook of Vector Quantization based on the division of M code word generates method |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| C06 | Publication | ||
| PB01 | Publication | ||
| C10 | Entry into substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |