CN103051587A - Method for constructing high-dimensional constellation maps on basis of grids and by applying optimal planning - Google Patents

Method for constructing high-dimensional constellation maps on basis of grids and by applying optimal planning Download PDF

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CN103051587A
CN103051587A CN2013100173256A CN201310017325A CN103051587A CN 103051587 A CN103051587 A CN 103051587A CN 2013100173256 A CN2013100173256 A CN 2013100173256A CN 201310017325 A CN201310017325 A CN 201310017325A CN 103051587 A CN103051587 A CN 103051587A
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planisphere
constellation
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吴昊
张建秋
宋汉斌
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Fudan University
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    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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Abstract

The invention belongs to the technical field of communication, in particular to a method for constructing high-dimensional constellation maps on the basis of grids and by applying optimal planning. The high-dimensional constellation maps are designed by maximizing the constellation figure of merit. The constellation figure of merit can be resolved into the product of the coding merit of a grid and the forming merit of a constellation boundary, so the design process of the high-dimensional constellation maps constructs the process of maximizing the coding merit and the forming merit into two optimization problems and obtains a needed constellation map by solving the optimization problems with the minimum Euclidean distance and symmetry of the constellation map as constraint conditions for the two optimization problems. The method can be used as a universal method for designing constellation maps with any dimensionalities, and moreover, the design process is simple. In addition, compared with conventional high-dimensional constellation maps, the high-dimensional constellation maps provided by the invention can save more power and have a lower symbol error rate.

Description

One class is based on lattice and use optimum programming to construct the method for higher-dimension planisphere
Technical field
The invention belongs to communication technical field, be specifically related to a class based on lattice and use optimum programming to construct the method for higher-dimension planisphere.
Background technology
Nowadays, electromagnetic amplitude and phase place are modulated and are widely used for transmission of information.And in order further to improve the performance of wireless communication system, increasing research [1]-[6] are placed on attentiveness and utilize the polarization of ele parameter to come on the benefit that carry information brings.The people such as Andrews point out that in document [1] polarization of electromagnetic wave can bring the extra degree of freedom in wireless communication system, thereby improve power system capacity.The polarization diversity of antenna has been discussed in document [2]-[4], and this class technology is to utilize a kind of approach of polarization of ele information.The people such as Nehorai have proposed electromagnetic amplitude, phase place and its two polarization parameters first in document [5] can modulate transmission of information simultaneously.Document [6] has proposed a kind of novel three-dimensional adjustment method of in wireless communication system electromagnetic wave amplitude, auxiliary polarizing angle and polarization phases angle being modulated based on the mathematic(al) representation of external electromagnetic field.To sum up, the design of higher-dimension planisphere has just had practical significance.
It is that amplitude and the phase place of modulated electromagnetic wave come transmission information that people know quadrature amplitude modulation (quadrature amplitude modulation, QAM).Among the qam constellation figure, each constellation point formation has repeatedly grid shape, and this grid shape is very common in the planisphere design.Lattice can be used for defining the above-mentioned grid-like point that is as a kind of mathematic(al) structure.Therefore, we can design planisphere on the basis of case theory.Usually, construct the method for planisphere based on case theory and at first can choose a kind of lattice, and then choose one and comprise and specify number of constellation points purpose border.Elaborate in document [7]-[9] about lattice with based on the character of the planisphere of lattice.Document [7] has proposed this concept of gain index (the constellation figure of merit, CFM) of planisphere, and it has reflected the efficient based on the planisphere of lattice.When the constellation dimension of plot is not too large (three-dimensional or four-dimensional), the planisphere with larger gain index has very low error sign ratio.The people such as Proakis have proved that in document [9] gain index of planisphere can resolve into coding gain and the planisphere border shaping gain of lattice.Therefore, maximize respectively coding gain and shaping gain and can produce the planisphere with maximum CFM value.
Summary of the invention
The object of the invention is to propose a class based on lattice and use optimum programming to construct the method for higher-dimension planisphere, such planisphere of constructing has the characteristics such as power is saved, error sign ratio is low.
The method of the structure higher-dimension planisphere that the present invention proposes, coding gain by maximizing respectively lattice and the shaping gain on planisphere border, so that the gain index of higher-dimension planisphere reaches maximum, thereby make the higher-dimension planisphere that finally constructs have the advantage that power is saved, error sign ratio is low.
The structure of this class higher-dimension planisphere proposed by the invention mainly is divided into two parts: the one, nChoosing of dimension lattice, another is nChoosing of dimension border.
One nLattice in the dimension space are R n In a discrete point set, and this point set has the group structure [9] under the vector addition.Usually, one nThe lattice of dimension ΛCan by nThe base vector of individual linear independence
Figure 759213DEST_PATH_IMAGE001
Represent, so any point in the lattice x Can be expressed as:
， (1)
Wherein
Figure 518407DEST_PATH_IMAGE003
Another kind of definition nWei Ge ΛMethod, be by one n* nThe generator matrix of dimension G , G In each behavior base vector
Figure 241513DEST_PATH_IMAGE001
After above-mentioned definition has been arranged, lattice ΛIn arbitrary lattice point x Can be expressed as
， (2)
Wherein
Figure 940664DEST_PATH_IMAGE005
One nThe dimension integer vectors.
The planisphere dimension of supposing required design is n, the first step of structure higher-dimension planisphere is selected one and is had maximum coding gain nWei Ge.Any lattice ΛCoding gain γ c( Λ) can be explained by following formula [9]:
， (3)
Wherein, d Min( Λ) the expression lattice ΛMinimum Eustachian distance [9]; V( Λ) be lattice ΛBasic volume [9]; G Be lattice ΛGenerator matrix.
Be without loss of generality, if the minimum Eustachian distance of fixed grating d MinBe 1, then we just need the coding gain of maximization lattice to obtain fine and close lattice, and further remove to construct the planisphere of required higher-dimension power saving.Because the density degree of lattice can be reflected by the determinant of its generator matrix [9], therefore, optimization problem just can be described with following formula:
Figure 875439DEST_PATH_IMAGE007
。(4)
By finding the solution this optimization problem (4), just can obtain nThe closeest lattice in the dimension space, the below can list and work as Spatial Dimension n=4,5,6 o'clock corresponding generator matrixes of the closeest lattice.
Figure 88858DEST_PATH_IMAGE008
(5)
nA zone in the dimension space RShaping gain can be explained by following formula [9]:
， (6)
Wherein V( R) be the zone RVolume (or area); Every two-dimentional average power E Avg/ 2 D Represent one nThe average energy of dimension planisphere quantizes the value behind the two-dimensional space, and it can be calculated by following formula:
Figure 39814DEST_PATH_IMAGE010
， (7)
Wherein MFor nThe number of constellation point in the dimension planisphere; x m It is an expression constellation point coordinate nDimensional vector.The shaping gain that has proved the ball-type border in the document [9] is at all nMaximum in the dimension border.Therefore, we should choose nHypersphere goes to choose constellation point from a certain given lattice as the border in the dimension space.
Suppose what we will design nConstellation point number in the dimension planisphere is MFor to choose the border fixing approaching as far as possible from giving nThe dimension hypersphere MIndividual lattice point just must be arranged the lattice point in the lattice by its mould is long first from small to large.Simultaneously, since in any lattice the mould length of lattice point all be a non-negative real discrete series (
Figure 776826DEST_PATH_IMAGE011
And ), then just can define n( r i ) be that mould length is in the lattice r i The lattice point number, and the definition n* n( r i ) the dimension matrix P ( r i ) be mould r i The point set matrix, it is respectively classified mould length as and is r i The coordinate of lattice point.After above-mentioned definition has been arranged, choose fixing from giving with hypersphere MThe process of individual constellation point can be described by following optimization problem:
， (8)
Wherein
Figure 801042DEST_PATH_IMAGE014
S That dimension is Two-dimentional selection matrix;
Figure 516374DEST_PATH_IMAGE016
With
Figure 117119DEST_PATH_IMAGE017
Represent matrix respectively S Each row and the summation of each row, col and row be the sequence number of corresponding columns and rows respectively.First constraint of above-mentioned optimization problem has guaranteed to obtain MThe point planisphere is about origin symmetry, and second and the 3rd constraint guaranteed and only had MIndividual lattice point is selected, and constellation point set matrix P ( r K ) in each lattice point at the most selected once.
Obtain in solving-optimizing problem (4) nBehind the closeest lattice, find the solution again above-mentioned integer programming problem (8) in the dimension space, just can obtain higher-dimension planisphere proposed by the invention.
Embodiment
The performance of the higher-dimension planisphere that this part will propose the present invention is assessed.
In simulation process, generation length is LThe random mark sequence be used as transmitted signal.Each symbol is taken from a given higher-dimension planisphere, and requires the minimum Eustachian distance of all planispheres d MinAll equal 1.Receiving terminal can be subject to power N 0The impact of Gauss's additive white noise, and receiving terminal carries out symbol detection with maximum likelihood detector.
We are the four-dimensional planisphere of three classes relatively, and they are respectively:
The four-dimensional planisphere of the first kind: taking from four-dimensional integer lattice and border is the three-dimensional planisphere of hypercube.
The four-dimensional planisphere of Equations of The Second Kind: the four-dimensional comprehensive cross planisphere (4-D generalized cross constellations) that document [7] proposes.
The four-dimensional planisphere of the 3rd class: four-dimensional planisphere proposed by the invention.
Table I has been listed the properties of the four-dimensional planisphere of this three class.Wherein, the severe of planisphere refers in the planisphere and is minimum Eustachian distance with arbitrary given constellation point distance d MinThe number of point; Peak-to-average power ratio refers to the ratio of peak power with the average power of planisphere.Fig. 1 provided the four-dimensional planisphere of this three class the constellation point number be 512 o'clock error sign ratio relatively.Wherein E Avg Average power for planisphere; P e The error rate for planisphere.
Can find out by above-mentioned simulation result, because this class higher-dimension planisphere proposed by the invention has larger gain index, so the planisphere of mentioning in their existing documents has less average power and lower error sign ratio.
The fundamental characteristics of different four-dimensional planispheres
Figure 846041DEST_PATH_IMAGE018
Table I
List of references
[1] M. R. Andrews, P. P. Mita and R. Carvalho, "Tripling the capacity of wireless communications using electromagnetic polarization," Nature 409, pp. 316-318, Jan. 2001.
[2] R. Compton Jr., "The tripole antenna: An adaptive array with full polarization flexibility," IEEE Trans. Antennas Propag., vol.29, no.6, pp. 944- 952, Nov. 1981.
[3] A. M. D. Turkmani, A. A. Arowojolu, P. A. Jefford and C. J. Kellett, "An experimental evaluation of the performance of two-branch space and polarization diversity schemes at 1800 MHz," IEEE Trans. Veh. Technol., vol.44, no.2, pp. 318-326, May 1995.
[4] B. Lindmark and M. Nilsson, "On the available diversity gain from different dual-polarized antennas," IEEE J. Sel. Areas Commun., vol.19, no.2, pp. 287-294, Feb. 2001.
[5] A. Nehorai and E. Paldi, "Vector-sensor array processing for electromagnetic source localization," IEEE Trans. Signal Process., vol.42, no.2, pp. 376-398, Feb. 1994.
[6] Song Hanbin. three-dimensional modulator-demodulator doctorate paper. Shanghai: electronic engineering of Fudan University, 2012.
[7] G. D. Forney, Jr and L. -F. Wei, "Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations," IEEE J. Sel. Areas Commun., vol.7, no.6, pp. 877-892, Aug. 1989.
[8] G. D. Forney, Jr and L. -F. Wei, "Multidimensional constellations. II. Voronoi constellations," IEEE J. Sel. Areas Commun., vol.7, no.6, pp. 941-958, Aug. 1989.
J. G. Proakis and M. Salehi, "Optimum Receivers for AWGN Channels," in Digital Communications, 5th ed., New York: McGraw Hill International Editions, 2008, pp. 160-289.。

Claims (1)

1. a class is characterized in that concrete steps are based on lattice and use optimum programming to construct the method for higher-dimension planisphere:
(1) establishes one nLattice in the dimension space are R n In a discrete point set, and this point set has the group structure under the vector addition, one nThe lattice of dimension ΛBy nThe base vector of individual linear independence Represent any point in the lattice x Can be expressed as:
Figure 407051DEST_PATH_IMAGE002
， (1)
Wherein
Figure 630222DEST_PATH_IMAGE003
Another kind of definition nWei Ge ΛMethod, be by one n* nThe generator matrix of dimension G , G In each behavior base vector
Figure 163972DEST_PATH_IMAGE004
, lattice ΛIn arbitrary lattice point x Can be expressed as:
Figure 310526DEST_PATH_IMAGE005
， (2)
Wherein
Figure 5949DEST_PATH_IMAGE006
One nThe dimension integer vectors;
Any lattice ΛCoding gain γ c( Λ) explained by following formula:
Figure 247575DEST_PATH_IMAGE007
， (3)
Wherein, d Min( Λ) the expression lattice ΛMinimum Eustachian distance [9]; V( Λ) be lattice ΛBasic volume [9]; G Be lattice ΛGenerator matrix;
The minimum Eustachian distance of fixed grating d MinBe 1, so by finding the solution following optimization problem:
Figure 850595DEST_PATH_IMAGE008
(4)
Obtain nThe closeest lattice in the dimension space;
(2) establish nA zone in the dimension space RShaping gain explained by following formula:
Figure 494065DEST_PATH_IMAGE009
， (5)
Wherein V( R) be the zone RVolume or area; Every two-dimentional average power E Avg/ 2 D Represent one nThe value of average power behind the two-dimensional space of dimension planisphere; Choose nHypersphere goes to choose constellation point from a certain given lattice as the border in the dimension space;
If design nConstellation point number in the dimension planisphere is MFor to choose the border fixing approaching as far as possible from giving nThe dimension hypersphere MIndividual lattice point is arranged the lattice point in the lattice by its mould is long first from small to large; Because the mould length of lattice point all is a non-negative real discrete series in any lattice:
Figure 829232DEST_PATH_IMAGE010
And , definition n( r i ) be that mould length is in the lattice r i The lattice point number, and the definition n* n( r i ) the dimension matrix P ( r i ) be mould r i The point set matrix, it is respectively classified mould length as and is r i The coordinate of lattice point; So choose fixing from giving with hypersphere MThe process of individual constellation point can be described by following optimization problem:
Figure 105809DEST_PATH_IMAGE012
(6)，
Wherein
Figure 869366DEST_PATH_IMAGE013
S That dimension is
Figure 641013DEST_PATH_IMAGE014
Two-dimentional selection matrix;
Figure 591651DEST_PATH_IMAGE016
With Represent matrix respectively S Each row and the summation of each row, col and row be the sequence number of corresponding columns and rows respectively;
Obtain in solving-optimizing problem (4) nBehind the closeest lattice, find the solution again above-mentioned integer programming problem (6) in the dimension space, just obtain required higher-dimension planisphere.
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CN104320372A (en) * 2014-10-28 2015-01-28 广东顺德中山大学卡内基梅隆大学国际联合研究院 Method for establishing communication system three-dimensional planisphere
CN104410596A (en) * 2014-12-19 2015-03-11 中国地质大学（武汉） Enhancement-mode six-dimensional 32PSK modulation and demodulation methods
CN106130949A (en) * 2016-08-12 2016-11-16 上海交通大学 Higher-dimension constellation generates method and system
CN108052560A (en) * 2017-12-04 2018-05-18 四川理工学院 A kind of data analysis processing method of data analysis processing method and employment trend data based on colleges and universities' data
CN109075869A (en) * 2016-09-30 2018-12-21 华为技术有限公司 A kind of data modulation method and encoder

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CN101582754A (en) * 2008-05-05 2009-11-18 财团法人工业技术研究院 Method for transmitting data to reduce average power ratio and device
CN102822816A (en) * 2010-03-30 2012-12-12 国际商业机器公司 An efficient homomorphic encryption scheme for bilinear forms

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US5706312A (en) * 1994-10-11 1998-01-06 Lucent Technologies Inc. Trellis coded modulation employing lower dimensionality convolutional encoder
CN1604516A (en) * 2004-11-11 2005-04-06 上海交通大学 Allowable code word lattice point search based vertical layered space-time code spherical decoding method
CN101582754A (en) * 2008-05-05 2009-11-18 财团法人工业技术研究院 Method for transmitting data to reduce average power ratio and device
CN102822816A (en) * 2010-03-30 2012-12-12 国际商业机器公司 An efficient homomorphic encryption scheme for bilinear forms

Cited By (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN104320372A (en) * 2014-10-28 2015-01-28 广东顺德中山大学卡内基梅隆大学国际联合研究院 Method for establishing communication system three-dimensional planisphere
CN104320372B (en) * 2014-10-28 2017-10-10 广东顺德中山大学卡内基梅隆大学国际联合研究院 A kind of method for constructing communication system three-dimensional constellation map
CN104410596A (en) * 2014-12-19 2015-03-11 中国地质大学（武汉） Enhancement-mode six-dimensional 32PSK modulation and demodulation methods
CN104410596B (en) * 2014-12-19 2017-06-09 中国地质大学（武汉） A kind of modulation and demodulation method of enhanced sextuple 32PSK
CN106130949A (en) * 2016-08-12 2016-11-16 上海交通大学 Higher-dimension constellation generates method and system
CN106130949B (en) * 2016-08-12 2019-05-10 上海交通大学 Higher-dimension constellation generation method and system
CN109075869A (en) * 2016-09-30 2018-12-21 华为技术有限公司 A kind of data modulation method and encoder
CN109075869B (en) * 2016-09-30 2020-03-20 华为技术有限公司 Data modulation method and encoder
CN108052560A (en) * 2017-12-04 2018-05-18 四川理工学院 A kind of data analysis processing method of data analysis processing method and employment trend data based on colleges and universities' data

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