CN112752339A - Fingerprint database updating method based on large-scale MIMO single-station system - Google Patents

Fingerprint database updating method based on large-scale MIMO single-station system Download PDF

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CN112752339A
CN112752339A CN202011605236.XA CN202011605236A CN112752339A CN 112752339 A CN112752339 A CN 112752339A CN 202011605236 A CN202011605236 A CN 202011605236A CN 112752339 A CN112752339 A CN 112752339A
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CN112752339B (en
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王霄峻
孙伟光
陈晓曙
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Southeast University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W64/00—Locating users or terminals or network equipment for network management purposes, e.g. mobility management
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
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Abstract

The invention discloses a fingerprint database updating method based on a large-scale MIMO single station system, which comprises the steps of firstly designing an ADCPM fingerprint matrix for subsequent fingerprint matching; preliminarily storing the fingerprint matrix and sparsely storing the fingerprint matrix through a ternary table; proposing a probability-based fingerprint update criterion for fingerprint library updates; dividing the positioning area based on CAOA; updating a fingerprint database by using CAOA-PFUC; and finally, carrying out online stage fingerprint positioning on the finely classified fingerprint matrix by a weighted KNN method. The invention can effectively improve the positioning precision under the condition of the change of the scattering environment.

Description

Fingerprint database updating method based on large-scale MIMO single-station system
Technical Field
The invention relates to a fingerprint database updating method based on a large-scale MIMO single station system, and belongs to the technical field of signal and information processing.
Background
Due to security and business requirements, the problem of how to obtain accurate Location information of a target has attracted a lot of attention in the industry and academia, which is called Location-Based Service (LBS). The mature LBS technology is a Global Positioning System (GPS) technology originated for military applications, but the GPS is affected by environmental noise, multipath interference, non-line-of-sight (nlcs) paths, and the like, and not only has low Positioning accuracy but also has high GPS power consumption. Conventional positioning methods based on wireless networks generally assume that a wireless Signal propagates along a Line-of-Sight (LOS), and then measurements of received Signal energy (RSS), Angle of Arrival (AOA), and Time of Arrival (TOA) are co-located by a plurality of Base Stations (BSs). However, with the development of cities and the arrival of the 5G era, the daily environment of people is mostly complex, the propagation path of wireless signals is non Line-of-Sight (NLOS), which causes the positioning performance to be greatly reduced, and the cooperation of multiple BSs causes additional load and delay.
In order to overcome the problems faced by the conventional wireless positioning technology, the fingerprint positioning technology has been widely studied, and there are two types of fingerprints generally adopted, one is to use the received signal strength RSS as the fingerprint, and the other is to use the multipath characteristics between the mobile terminal and the BS as the fingerprint, such as AOA, channel impulse response CIR, channel state information CSI and power delay profile PDP. At present, a fingerprint positioning method based on a terminal mainly focuses on improvement of positioning accuracy, however, under the condition that a scattering environment changes, the positioning accuracy is reduced, and research on the problem is rare.
Disclosure of Invention
The purpose of the invention is as follows: in order to solve the problem that the positioning accuracy is reduced under the condition of scattering environment change in the existing fingerprint positioning method, the invention provides a fingerprint database updating method based on a large-scale MIMO single-station system, and the updating criterion based on probability and CAOA clustering are combined, so that more efficient and stable fingerprint positioning is realized.
The technical scheme is as follows: in order to achieve the purpose, the invention adopts the technical scheme that:
a fingerprint database updating method based on a large-scale MIMO single-station system comprises the following steps:
step1, constructing an ADCPM fingerprint matrix based on a large-scale MIMO single station system: ADCPM fingerprint matrix between kth user and base station BS
Figure BDA0002873033810000011
Wherein E indicates an expectation, which indicates a Hadamard product,
Figure BDA0002873033810000012
is a channel response matrix, Z, between the k-th user and the BSk *Represents an angular delay channel response matrix, V ∈ Nt×NtFor DFT phase shift PS-DFT matrix, U belongs to NL×NgIs a DFT unitary matrix.
The DFT phase shift PS-DFT matrix V satisfies:
Figure BDA0002873033810000021
the DFT unitary matrix U satisfies:
Figure BDA0002873033810000022
Figure BDA0002873033810000023
row α and column β elements of V,
Figure BDA0002873033810000024
phi row of U
Figure BDA0002873033810000025
Column element, NtNumber of antennas of BS, NLIs the number of symbols of OFDM, NgIs the number of cyclic prefixes.
Figure BDA0002873033810000026
For the k-th user to BS overall channel frequency response CFR matrix,
Figure BDA0002873033810000027
is the CFR matrix on the l-th subcarrier, l is 0,1L-1，NPIs the total number of paths, αp,kFor the complex channel gain, θ, of the k-th user on the p-th pathp,kFor the angle of arrival AOA of the kth user on the p-th path,
Figure BDA00028730338100000210
for propagation delay, τp,kTOA, T for the p-th pathsFor the minimum sampling interval of a massive MIMO single station system,
Figure BDA0002873033810000028
representing a real space, e (θ)p,k) Representing the array response vector, and j represents the imaginary unit.
And 2, dividing a target area to be positioned into uniform grids, taking grid vertexes in the target area to be positioned as reference points, and storing the ADCPM fingerprint matrix of each reference point and the corresponding position coordinates thereof into a database to obtain a fingerprint database F-DB, namely the ADCPM fingerprint database.
Step3, adopting a ternary table TT method to convert the fingerprint matrix F of the ith reference pointiIs compressed into
Figure BDA0002873033810000029
With combined use of FTTSubstituting F in fingerprint databaseiWherein F isTTRepresents the ADCPM fingerprint library after compression, i is 1,2, N is the number of reference points, v1,...,vMIs FiMaximum M elements of (1), row1,...,rowMIs v is1,...,vMAt FiLine coordinate of (1), col1,...,colMIs v is1,...,vMAt FiColumn coordinates of (1).
Step4, setting an updating threshold based on the probability: defining the fingerprint distance d before and after the change of the wireless environment as
Figure BDA0002873033810000031
In the formula, X and Y respectively represent two-dimensional fingerprint matrixes before and after the change of the wireless environment, XijIndicating the fingerprint before change, YijRepresenting the changed fingerprint, | · | | non-woven phosphor2Representing the matrix 2-norm, Δ d1Is a very small positive number. Converting the fingerprint distance d into corresponding probability value through a function, wherein the specific mapping function is
Figure BDA0002873033810000032
p represents the fingerprint similarity probability, lambda is a preset parameter, the reference point fingerprints before and after the wireless environment changes are given, the corresponding fingerprint similarity probability is calculated, and then whether the reference point fingerprint needs to be updated or not is determined through an updating threshold.
Step5, CAOA clustering: CAOA of ADCPM fingerprint in the angular domain
Figure BDA0002873033810000033
Considering the angular spread, the practical range is [ alpha ]m-Δα,Δm+Δα]Wherein Δ α represents a maximum angle offset value, Fm,jAnd representing the ADCPM matrix array element, calculating corresponding CAOA for each reference point, and taking the CAOA as the cluster center of the reference point.
And 6, selecting a plurality of anchor points in a sub-area corresponding to the clustering center, then judging the similarity probability of the fingerprints of the anchor points, and if the similarity probability meets the updating condition, updating the fingerprints.
And 7, estimating the position of the point to be measured by using a weighted KNN matching algorithm.
Preferably: fingerprint database in step2
Figure BDA0002873033810000034
Wherein the fingerprint database F-DB represents ADCPM fingerprint database, Loci(xi,yi) As position coordinates of the ith reference point, FiAn adpcm fingerprint matrix of the ith reference point, where i is 1, 2.
Preferably: the method for the ternary table TT in the step3 comprises the following steps: the first M elements with the maximum in the acquisition matrix are stored in a table, the ternary table is composed of 3 rows and M columns, the first row stores the first M maximum elements, and the second row and the third row respectively store row and column values of the first row of elements in the original matrix.
Preferably: the value of M is compressed by the matrix
Figure BDA0002873033810000041
And (6) determining.
Preferably: in step 7, estimating the position of the point to be measured by using a weighted KNN matching method:
Figure BDA0002873033810000042
wherein the content of the first and second substances,
Figure BDA0002873033810000043
k is the number of reference point fingerprints with the highest similarity to the fingerprints of the points to be measured, which are obtained by screening the weighted KNN matching algorithm,
Figure BDA0002873033810000044
to screen the position coordinates of the k-th reference point fingerprint,
Figure BDA0002873033810000045
is composed of
Figure BDA0002873033810000046
Corresponding weight, dkIs the Euclidean distance, delta d, between the fingerprint matrix of the k-th reference point and the real-time fingerprint matrix of the point to be measured2Is a very small positive number.
Compared with the prior art, the invention has the following beneficial effects:
for the scene of the change of the scatterer, the CAOA-PFUC algorithm can effectively judge the change condition of the real-time fingerprint, balance the consumption of the algorithm and the positioning precision and achieve a better positioning result.
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FIG. 1 is a flow chart of the present invention.
Fig. 2 is a distribution diagram of channel strength over an angular domain.
Detailed Description
The present invention is further illustrated by the following description in conjunction with the accompanying drawings and the specific embodiments, it is to be understood that these examples are given solely for the purpose of illustration and are not intended as a definition of the limits of the invention, since various equivalent modifications will occur to those skilled in the art upon reading the present invention and fall within the limits of the appended claims.
A fingerprint database updating method based on a large-scale MIMO single station system is disclosed, as shown in figure 1, signal fingerprints of each reference point are collected firstly in an off-line stage, the signal fingerprints are stored in a fingerprint database after being preprocessed, when an unknown terminal is positioned in an on-line stage, corresponding fingerprints of a terminal to be detected are collected firstly, whether a fingerprint database is updated or not is judged by using a proposed CAOA-PFUC algorithm, finally, a position estimation is carried out by using a WKNN matching algorithm, and a result is obtained and returned to the terminal with a positioning requirement. The specific operation of each step is described in detail below.
Step 1: firstly, the invention designs an ADCPM fingerprint matrix required by subsequent fingerprint matching, wherein the fingerprint matrix is a large-scale MIMO-OFDM system based on a single base station. A single base station particularly means a base station with NtThe method comprises the steps that a uniform linear array of a root antenna is arranged, a base station and a user are located on the same horizontal plane, Channel State Information (CSI) from a terminal to the base station can be obtained through uplink channel estimation, and due to the fact that a plurality of scatterers exist in the coverage range of a target area, wireless signals propagate along multipath, and therefore the CSI contains multipath information of a scattering environment.
The specific ADCPM calculation steps are as follows:
step 11: calculating the CIR of the user k on the p-th path by using the formula (1) as follows:
Figure BDA0002873033810000051
where CIR represents the channel impulse response, αp,k∈CN(0,θp,k) Is the complex channel gain, CN (0, theta), of the kth user on the p-th pathp,k) Mean 0, real and imaginary part statistically independent, and variance each thetap,kComplex Gaussian distribution of/2, θp,kE (0, π) is the corresponding AOA, AOA denotes the angle of arrival, e (θ)p,k) For the array response vector on the p-th path, dp,kFor the physical distance, λ, from the transmitting antenna to the first receiving antenna on the p-th pathcIs the carrier wavelength, j represents the imaginary unit. Assuming that the distribution of scatterers in the target area is independent and random,the above parameters are independent of each other. If the signal phase of the first antenna is taken as the reference phase, the response vector of the antenna array is:
Figure BDA0002873033810000052
wherein N istDenotes the number of antenna elements and T denotes transposition.
Step 12: after obtaining the CIR, the CIR contains the multipath characteristics of the antenna domain, and can be mapped to a corresponding angle domain through DFT, which represents discrete fourier transform. Considering that the TOA of each path is different from each other, the TOA represents the arrival time, and the CIR of the kth user can be expressed as the sum of CIRs of all paths, that is:
Figure BDA0002873033810000053
wherein the content of the first and second substances,
Figure BDA0002873033810000054
TOA for each path, v is the speed of light, NpThe total number of paths is indicated. Delta (. tau. -)p,k) Representing samples for impulse functions, ap,kRepresenting the complex channel gain of the kth user on the p-th path.
Setting the minimum sampling interval of the massive MIMO single station system as TsAnd sampling the array output, then Tc＝NLTsIs the symbol interval of OFDM, NLIndicating the number of symbols for OFDM. T isg＝NgTsFor cyclic prefix interval, assume TgIs much greater than taup,k. The bandwidth of each subcarrier is delta f-1/NcTsThe frequency of the l-th subcarrier is flL Δ f. Through OFDM modulation, a frequency selective fading channel caused by multipath propagation can be converted into a frequency flat channel, and the TOA of each path is obtained through time domain sampling.
Step 13: for the ith subcarrier, fourier transform is performed on the CIR to obtain a corresponding channel frequency response CFR, that is, the CFR is equal to the sum of the time domain CIRs of all paths with different time delays:
Figure BDA0002873033810000061
wherein the content of the first and second substances,
Figure BDA0002873033810000066
in order to be able to delay the propagation time,
Figure BDA0002873033810000067
representing the integer closest to x. The overall CFR matrix for user k to base station BS is made up of CFRs on all subcarriers, i.e.
Figure BDA0002873033810000062
CFR denotes the channel frequency response.
Step 14: after the CFR is obtained, the CFR matrix describes the characteristics of the channel between the kth user and the BS in the space-frequency domain. For a large-scale MIMO-OFDM system, the BS can obtain higher multipath resolution in an angle-time delay domain, so that the BS is converted into a matrix on the angle-time delay domain, and a response characteristic matrix is extracted to serve as a positioning fingerprint. The matrix is converted from the space-frequency domain to the angle-time delay domain by DFT transformation, and the channel response matrix is expressed as
Figure BDA0002873033810000063
Wherein V is Nt×NtFor a DFT phase shift (PS-DFT) matrix, PS-DFT represents a phase shift discrete Fourier transform, U ∈ NL×NgThe DFT unitary matrix respectively satisfies the following expressions:
Figure BDA0002873033810000064
the V matrix and the U matrix respectively convert H into HkMapping onto angular and time-delay domains, thus [ Z ]k]i,jRepresenting the ith AOA and the jth TOA on the angle-delay domain matrixA channel complex gain value.
ADCPM fingerprint matrix between the kth user and the base station BS is
Figure BDA0002873033810000065
Step 2: and sparsely storing the obtained ADCPM fingerprint matrix through a ternary table, wherein the specific ternary table storage mode is as follows:
step 21: firstly, preliminarily storing the obtained ADCPM matrix in a fingerprint database. The key task of the off-line stage is to establish a fingerprint database to prepare for fingerprint matching of the on-line stage. Therefore, a plurality of reference points are selected in a target area to be positioned, and a fingerprint matrix of each reference point is extracted.
The method comprises the following specific steps: firstly, determining a division interval s (unit m), and dividing a target area to be positioned into uniform grids by the interval s, wherein a grid vertex in the target area to be positioned serves as a reference point. Here, the smaller the spacing s, the more reference points are generated.
Calculating ADCPM fingerprint matrix F from ith reference point to BSi(i ═ 1, 2.., N). Storing the fingerprint matrix and the position coordinates of each reference point into a database to obtain a final fingerprint database F-DB:
Figure BDA0002873033810000071
wherein, Loci(xi,yi) Is the position coordinate of the ith reference point, and N is the number of the reference points.
Step 22: and sparsely storing the fingerprint matrix through a ternary table. Because of the sparsity of the fingerprint matrix, a matrix compression algorithm may be employed before the fingerprint is stored in the database. Fingerprint preprocessing is carried out by adopting a Ternary Table (TT) algorithm. In order to effectively compress the fingerprint matrix, in the process of creating the ternary table, the ADCPM matrix is sorted, and the first M elements with the largest size in the acquisition matrix are stored in the table. The ternary table consists of 3 rows and M columns, the first row stores the first M largest elements, the second row and the third rowThe row stores the row and column values of the first row element in the original matrix, wherein the value of M is compressed by the matrix
Figure BDA0002873033810000072
And (6) determining. The compressed fingerprint is represented in the form of a ternary table:
Figure BDA0002873033810000073
wherein v is1,...,vMIs FiMaximum M elements of (1), row1,...,rowMIs v is1,...,vMAt FiLine coordinate of (1), col1,...,colMIs v is1,...,vMAt FiColumn coordinates of (1).
Step 23: in constructing fingerprint database, use FTTInstead of the original matrix, it is stored in a database together with the position coordinates and other information. If the ternary table is to be restored to the matrix form, the element values are filled into the positions corresponding to the fingerprint matrix according to the coordinates of the second row and the third row, and other positions are quickly filled by using a zero padding method.
Step 3: setting an updating threshold based on probability, and specifically implementing the following steps:
step 31: defining fingerprint distances d corresponding to the wireless environment before and after change, wherein the fingerprint distances are expressed as follows:
Figure BDA0002873033810000081
wherein X and Y represent two-dimensional fingerprint matrix before and after wireless environment change, respectivelyijIndicating the fingerprint before change, YijRepresenting the changed fingerprint, | · | | non-woven phosphor2Representing the matrix 2-norm, Δ d1Is to prevent a very small positive number with a denominator of 0, typically Δ d1Take 1 e-5.
Step 32: converting the fingerprint distance d into corresponding probability value through a function, wherein the specific mapping function is
Figure BDA0002873033810000082
Step 33: giving reference point fingerprints before and after the wireless environment changes, and calculating corresponding fingerprint similarity probability so as to determine whether the reference point fingerprints need to be updated or not through a threshold value in the following process.
Step 4: CAOA clustering: the scattering environment of each path from the user terminal to the BS is mainly determined by scatterers near the user terminal, and most of the energy of the signal after propagating through the wireless channel is distributed on only a few multipath components, as shown in fig. 2. The CAOA of ADCPM fingerprint in the angular domain is:
Figure BDA0002873033810000083
considering the angular spread, the practical range is [ alpha ]m-Δα,αm+Δα]Wherein F ism,jAnd expressing the ADCPM matrix array element, wherein delta alpha expresses the maximum angle offset value, and for each reference point, calculating corresponding CAOA, wherein the CAOA expresses the central arrival angle and is used as the cluster center of the reference point.
Step 5: and selecting a plurality of anchor points in a sub-area corresponding to the clustering center, then judging the similarity probability of the fingerprints of the anchor points, and if the similarity probability meets the updating condition, updating the fingerprints.
Step 6: : and carrying out fingerprint positioning on the finely classified fingerprint matrixes at the online stage by a weighted KNN method. The weighted KNN algorithm considers the distance between the terminal to be measured and the reference point, the higher the similarity of the point to be measured and the fingerprint, the closer the physical distance should be theoretically, the greater the contribution to the position estimation result, the higher weight should be given, and the corresponding formula is:
Figure BDA0002873033810000084
wherein
Figure BDA0002873033810000085
Wherein d iskADCPM fingerprint matrix of the screened kth reference point and the fingerprint matrix to be testedEuclidean distance between the real-time adpcm fingerprint matrices of points. Δ d2Is a very small positive number, Δ d, in order to prevent the denominator from being 02Generally 1 e-5. w is akIs a weighting factor for each location.
Compared with the traditional fingerprint positioning method, the method can effectively improve the positioning accuracy under the condition of the change of the scattering environment.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (5)

1. A fingerprint database updating method based on a massive MIMO single station system is characterized by comprising the following steps:
step1, constructing an ADCPM fingerprint matrix based on a large-scale MIMO single station system: ADCPM fingerprint matrix between kth user and base station BS
Figure FDA0002873033800000011
Wherein E indicates an expectation, which indicates a Hadamard product,
Figure FDA0002873033800000012
is a channel response matrix, Z, between the k-th user and the BSk *Represents an angular delay channel response matrix, V ∈ Nt×NtFor DFT phase shift PS-DFT matrix, U belongs to NL×NgIs DFT unitary matrix;
the DFT phase shift PS-DFT matrix V satisfies:
Figure FDA0002873033800000013
the DFT unitary matrix U satisfies:
Figure FDA0002873033800000014
Figure FDA0002873033800000015
row α and column β elements of V,
Figure FDA0002873033800000016
phi row of U
Figure FDA0002873033800000017
Column element, NtNumber of antennas of BS, NLIs the number of symbols of OFDM, NgIs the number of cyclic prefixes;
Figure FDA0002873033800000018
for the k-th user to BS overall channel frequency response CFR matrix,
Figure FDA0002873033800000019
is the CFR matrix on the l-th subcarrier, l is 0,1L-1，NPIs the total number of paths, αp,kFor the complex channel gain, θ, of the k-th user on the p-th pathp,kFor the angle of arrival AOA of the kth user on the p-th path,
Figure FDA00028730338000000110
for propagation delay, τp,kTOA, T for the p-th pathsFor the minimum sampling interval of a massive MIMO single station system,
Figure FDA00028730338000000111
representing a real space, e (θ)p,k) Representing the array response vector, j representing the imaginary unit;
step2, dividing a target area to be positioned into uniform grids, taking grid vertexes in the target area to be positioned as reference points, and storing an ADCPM fingerprint matrix of each reference point and corresponding position coordinates thereof into a database to obtain a fingerprint database F-DB, namely the ADCPM fingerprint database;
step3, adopting a ternary table TT method to convert the fingerprint matrix F of the ith reference pointiIs compressed into
Figure FDA0002873033800000021
With combined use of FTTSubstituting F in fingerprint databaseiWherein F isTTRepresents the ADCPM fingerprint library after compression, i is 1,2, N is the number of reference points, v1,...,vMIs FiMaximum M elements of (1), row1,...,rowMIs v is1,...,vMAt FiLine coordinate of (1), col1,...,colMIs v is1,...,vMAt FiColumn coordinates of (1);
step4, setting an updating threshold based on the probability: defining the fingerprint distance d before and after the change of the wireless environment as
Figure FDA0002873033800000022
In the formula, X and Y respectively represent two-dimensional fingerprint matrixes before and after the change of the wireless environment, XijIndicating the fingerprint before change, YijRepresenting the changed fingerprint, | · | | non-woven phosphor2Representing the matrix 2-norm, Δ d1Is a very small positive number; converting the fingerprint distance d into corresponding probability value through a function, wherein the specific mapping function is
Figure FDA0002873033800000023
p represents the fingerprint similarity probability, lambda is a preset parameter, the reference point fingerprints before and after the change of the wireless environment are given, the corresponding fingerprint similarity probability is calculated, and then whether the reference point fingerprints need to be updated or not is determined through an updating threshold;
step5, CAOA clustering: CAOA of ADCPM fingerprint in the angular domain
Figure FDA0002873033800000024
Considering the angular spread, the practical range is [ alpha ]m-Δα,αm+Δα]Wherein, Δ αIndicates the maximum angular offset value, Fm,jRepresenting ADCPM matrix array elements, calculating corresponding CAOA for each reference point, and taking the CAOA as the cluster center of the reference point;
step6, selecting a plurality of anchor points in a sub-area corresponding to the clustering center, then judging the similarity probability of the fingerprints of the anchor points, and if the similarity probability meets the updating condition, updating the fingerprints;
and 7, estimating the position of the point to be measured by using a weighted KNN matching algorithm.
2. The massive MIMO single station system-based fingerprint database updating method according to claim 1, wherein: fingerprint database in step2
Figure FDA0002873033800000031
Wherein the fingerprint database F-DB represents ADCPM fingerprint database, Loci(xi,yi) As position coordinates of the ith reference point, FiAn adpcm fingerprint matrix of the ith reference point, where i is 1, 2.
3. The massive MIMO single station system based fingerprint database updating method as claimed in claim 2, wherein: the method for the ternary table TT in the step3 comprises the following steps: the first M elements with the maximum in the acquisition matrix are stored in a table, the ternary table is composed of 3 rows and M columns, the first row stores the first M maximum elements, and the second row and the third row respectively store row and column values of the first row of elements in the original matrix.
4. The massive MIMO single station system-based fingerprint database updating method as claimed in claim 3, wherein: the value of M is compressed by the matrix
Figure FDA0002873033800000032
And (6) determining.
5. The massive MIMO single station system-based fingerprint database updating method as claimed in claim 4, wherein: using weighted KN in step 7The N matching method estimates the positions of the points to be measured as follows:
Figure FDA0002873033800000033
wherein the content of the first and second substances,
Figure FDA0002873033800000034
k is the number of reference point fingerprints with the highest similarity to the fingerprints of the points to be measured, which are obtained by screening the weighted KNN matching algorithm,
Figure FDA0002873033800000035
to screen the position coordinates of the k-th reference point fingerprint,
Figure FDA0002873033800000036
is composed of
Figure FDA0002873033800000037
Corresponding weight, dkIs the Euclidean distance, delta d, between the fingerprint matrix of the k-th reference point and the real-time fingerprint matrix of the point to be measured2Is a very small positive number.
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