CN110809247A - OFDM frequency domain error estimation and positioning precision evaluation method for indoor Wi-Fi positioning - Google Patents

OFDM frequency domain error estimation and positioning precision evaluation method for indoor Wi-Fi positioning Download PDF

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CN110809247A
CN110809247A CN201911097878.0A CN201911097878A CN110809247A CN 110809247 A CN110809247 A CN 110809247A CN 201911097878 A CN201911097878 A CN 201911097878A CN 110809247 A CN110809247 A CN 110809247A
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cfo
matrix
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CN110809247B (en
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周牧
张振亚
何维
谢良波
聂伟
田增山
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Chongqing University of Post and Telecommunications
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W4/00—Services specially adapted for wireless communication networks; Facilities therefor
    • H04W4/30—Services specially adapted for particular environments, situations or purposes
    • H04W4/33—Services specially adapted for particular environments, situations or purposes for indoor environments, e.g. buildings
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L27/00—Modulated-carrier systems
    • H04L27/26—Systems using multi-frequency codes
    • H04L27/2601—Multicarrier modulation systems
    • H04L27/2647—Arrangements specific to the receiver only
    • H04L27/2655—Synchronisation arrangements
    • H04L27/2668—Details of algorithms
    • H04L27/2669—Details of algorithms characterised by the domain of operation
    • H04L27/2672—Frequency domain
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
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Abstract

The invention discloses an OFDM frequency domain error estimation and positioning precision evaluation method for indoor Wi-Fi positioning. The method firstly proposes the expression of antenna receiving signals with Frequency domain errors caused by Carrier Frequency Offset (CFO), selects angle analysis from the Frequency domain, processing the unknown parameter vector to obtain the parameter to be estimated, calculating the FIM of the parameter to be estimated, further deducing an estimation error closed expression of the CFO, establishing a relation between a signal amplitude value and a time delay and a target coordinate to be estimated, deducing a closed expression of an indoor Wi-Fi positioning error bound based on signal State Information (CSI) under the CFO, and finally, the influence of different factors on the indoor CSI-based Wi-Fi positioning accuracy under CFO is analyzed, in addition, the method also avoids the problem that the Probability Density Function (PDF) cannot be obtained when the Cramer-Rao Lower Bound (CRLB) is solved in the time domain. The method can provide reference basis when designing the positioning system, and evaluate the positioning performance of the system, so as to optimize the positioning system and improve the positioning precision.

Description

OFDM frequency domain error estimation and positioning precision evaluation method for indoor Wi-Fi positioning
Technical Field
The invention belongs to an indoor positioning technology, and particularly relates to an OFDM frequency domain error estimation method for indoor Wi-Fi positioning and a positioning precision evaluation method thereof.
Background
With the rapid development of electronic technology and communication industry, people's life style is moving towards intellectualization and convenience, and more aware of the importance of Location information, Location-based Service (LBS) technology is on the move. Currently, the Global Positioning System (GPS) and the cellular base station Positioning System belong to two most mature outdoor Positioning systems, which can provide accurate position information for outdoor users. In contrast, the complexity of the indoor environment, as well as the effects of personnel walking and obstructions on signal propagation, can result in indoor users not being able to stably receive signals from satellites and cellular base stations. Therefore, a series of studies on indoor positioning are carried out by many scholars, and various indoor positioning systems, such as an indoor bluetooth, Infrared (IR), ZigBee, Ultra Wide Band (UWB), Wi-Fi positioning system, etc., are proposed according to different Signal characteristics, wherein an indoor Wi-Fi positioning system based on Received Signal Strength (RSS) is gradually becoming the mainstream of the indoor positioning system due to the characteristics of Wide Signal distribution and simple deployment.
Compared with the indoor Wi-Fi positioning method based on RSS, the Channel State Information (CSI) includes finer granularity and more diversified physical layer information in the signal transmission process, so the indoor Wi-Fi positioning method based on CSI generally has higher positioning accuracy and the positioning result is more stable. When positioning is performed using CSI, a transmitting end transmits data in parallel on a plurality of Orthogonal subcarriers by using an Orthogonal Frequency Division Multiplexing (OFDM) technique and demodulates the data at a receiving end, but orthogonality of carriers at the receiving end is difficult to be guaranteed due to a Frequency domain error (CFO) caused by Carrier Frequency Offset (CFO), and thus performance of an indoor Wi-Fi positioning method based on CSI is degraded due to an influence of Inter-Carrier interference (ICI). The influence of frequency domain errors on the positioning performance needs to be analyzed, and the accuracy and the robustness of the positioning system are improved by compensating the frequency domain errors.
In order to solve the problems, the method provides an OFDM frequency domain error estimation and positioning accuracy evaluation method for indoor Wi-Fi positioning, and provides a signal transmission model under the condition that signal propagation delay, path loss, multipath effect and frequency domain errors caused by CFO are considered, then a CFO estimation error Bound, namely a Cramer-Rao Lower Bound (CRLB), is deduced from the angle of a frequency domain, and a closed expression of the indoor Wi-Fi positioning error Bound based on CSI under CFO is deduced, and finally, the influence of different factors on the indoor Wi-Fi positioning accuracy based on CSI is analyzed and used as a quantitative reference index for analyzing and designing an indoor Wi-Fi positioning system based on CSI signals.
Disclosure of Invention
The invention aims to provide an OFDM frequency domain error estimation method for indoor Wi-Fi positioning and a positioning precision evaluation method thereof. According to the method, an estimation error closed expression of the CFO is deduced from the angle of a frequency domain, the estimation of a frequency domain error (namely the CFO) is facilitated, meanwhile, a closed expression of a CSI positioning error boundary under the CFO is deduced, the influence of different factors on the positioning accuracy of the indoor Wi-Fi based on the CSI under the CFO is analyzed, and the problem that a Probability Density Function (PDF) cannot be obtained when the CRLB is solved in a time domain is also avoided. The method can provide reference basis when designing the positioning system, and evaluate the positioning performance of the system, so as to optimize the positioning system and improve the positioning precision.
The invention relates to an OFDM frequency domain error estimation and positioning accuracy evaluation method for indoor Wi-Fi positioning, which comprises the following steps:
step one, assuming that a received signal waveform of an antenna in indoor Wi-Fi positioning based on CSI can be represented as:
Figure BDA0002268905420000021
wherein PN represents the total number of paths through which the signal propagates, s (t) represents the transmit waveform, a(p)Representing the amplitude, tau, of the signal of the p-th path(p)Represents the signal time delay of the p-th path, and z (t) represents noise.
Step two, the CSI mainly includes signal amplitude and phase information of the received signal converted by an Analog-to-digital converter (ADC), and a signal waveform of the received signal converted by the ADC is represented as:
Figure BDA0002268905420000031
where n represents the number of samples, L represents the total number of samples, and T represents the sampling period.
Step three, interference exists on CSI positioning by multipath signals in the positioning process, in order to achieve a better positioning effect, only direct paths are considered, and the direct signal waveform received by the mth antenna is represented as:
rm(nT)＝ams(nT-τm)+z(nT),m＝1,…,N， (3)
where N represents the total number of antennas.
Step four, analyzing the CFO estimation error from the angle of the frequency domain, wherein the step four comprises the following steps:
step four (one), according to fig. 1, the CFO can be regarded as a fixed frequency offset Δ f of the signal in the frequency domain, so that the waveform of the signal affected by the CFO received by the mth antenna can be represented as:
gm(nT)＝ams(nT-τm)ej2πkε/LT+z(nT),n＝1,…,L， (4)
wherein, amRepresenting the amplitude, τ, of the signal arriving at the m-th antennamRepresenting the time delay of the signal arriving at the mth antenna,
Figure BDA0002268905420000032
denotes normalized CFO, fcRepresenting the carrier frequency.
Step four (two), performing L-point Fourier transform on the signal expression under the CFO in the step four (one) to obtain:
Figure BDA0002268905420000033
wherein η (k) denotes a noise power spectrum subject to Gaussian distribution, Gm(k) Dependent on the variables ε, amAnd τmObtaining the unknown parameter vector theta ═ epsilon, am,τm]。
Step four (three), all G is represented by vector Xm(k)：
X＝[Gm(0),…,Gm(L-1)]T， (6)
Step four (IV), the expectation of X is expressed as:
Figure BDA0002268905420000034
step five, calculating a Fisher information matrix for theta according to a formula (7), and specifically comprising the following steps:
step five (one), respectively calculating:
Figure RE-GDA0002299868020000041
Figure RE-GDA0002299868020000042
Figure RE-GDA0002299868020000043
step five (two), solving the jth element of the ith row of the Fisher information matrix of theta by using a formula (8),
Figure BDA0002268905420000044
wherein, thetai，θjRespectively representing the relevant parameters epsilon and a in the unknown parameter vector thetam，τmOne of them. The unknown parameter vector θ includes three related parameters, so a fisher information matrix of 3 × 3 is obtained:
Figure BDA0002268905420000045
wherein the content of the first and second substances,
Figure BDA0002268905420000046
sixthly, estimating the error bound of the epsilon by using the Matrix dimensionality reduction property (representing a high-dimensional Matrix by a low-dimensional Matrix and keeping the key Information quantity unchanged) of Equivalent FIM (Equivalent Fisher Information Matrix, EFIM), and specifically comprising the following steps of:
step six (one), partitioning the FIM of NxN:
Figure BDA0002268905420000052
wherein A ∈ Rn×n，B∈Rn×(N-n)，C∈R(N-n)×(N-n)。
Step six (two), the EFIM about θ can be obtained from equation (8):
Figure BDA0002268905420000053
wherein, depending on the nature of the EFIM,
Figure BDA0002268905420000054
and step six (three), estimating the error bound of epsilon, wherein A is expressed as 1X 1 matrix I1×1To matrix IθPartitioning to obtain:
Figure BDA0002268905420000055
Figure BDA0002268905420000056
Figure BDA0002268905420000057
step six (four), calculating an inverse matrix of the EFIM to obtain an estimation error bound of the CFO:
Figure BDA0002268905420000058
wherein the content of the first and second substances,
Figure BDA0002268905420000061
Figure BDA0002268905420000062
step seven, coordinate information about the target to be measured needs to be introduced in order to obtain a positioning estimation error bound under the CFO, and the method specifically comprises the following steps:
step seven (one), order
Figure BDA0002268905420000063
Wherein (x, y) represents the true position of the target, (x)m,ym) Indicating the position of the mth antenna,the distance between the target to be measured and the mth antenna is shown, and c represents the speed of light.
Step seven (two), according to the attenuation model of the amplitude
Figure BDA0002268905420000065
In free spaceFor example, the amplitude of the mth antenna is simplified to be
Figure BDA0002268905420000066
Wherein, a0Representing the signal reference amplitude at 1m from the antenna.
Step seven (three), and combining tau in step seven (one) and step seven (two)mAnd amSubstituting equation (5) gives:
Figure BDA0002268905420000067
from this, it can be seen that the original unknown parameter vector θ ═ epsilon, a for the target and mth antenna positionm,τm]Becomes a new unknown parameter vector theta ═ epsilon, x, y]。
Step seven (four), all G is represented by vector Xm(k)：
X＝[G1(0),…,G1(L-1),…,Gm(0),…,Gm(L-1)]T， (17)
Step seven (five), the expectation of X is expressed as:
Figure BDA0002268905420000068
step eight, calculating a Fisher information matrix for theta' by a formula (18), and specifically comprising the following steps:
step eight (one), calculating the partial derivative of each element in mu and respectively calculating
Figure BDA0002268905420000071
Figure BDA0002268905420000072
Figure BDA0002268905420000073
Figure BDA0002268905420000074
Wherein m is 1, …, N, k is 0, …, L-1,
Figure BDA0002268905420000075
Figure BDA0002268905420000076
and step eight (two), solving the jth element of the ith row of the Fisher information matrix of theta' by using a formula (8), wherein thetai，θjRespectively representing one of the relevant parameters x, y, epsilon in the parameter theta' to be estimated. The parameter θ' to be estimated includes three related parameters, so a fisher information matrix of 3 × 3 is obtained:
Figure BDA0002268905420000077
wherein the content of the first and second substances,
Figure BDA0002268905420000078
Figure BDA0002268905420000081
step nine, in order to solve the positioning estimation error bound under the CFO, estimating the positioning error bound by using the Matrix dimension reduction property (a high-dimensional Matrix is represented by a low-dimensional Matrix, and the amount of key Information is unchanged) of an equivalent FIM (equivalent fisher Information Matrix, EFIM), specifically including the following steps:
step nine (one), order
Figure BDA0002268905420000083
Can be combined withθ′The method is simplified as follows:
Figure BDA0002268905420000084
step nine (two), utilizing step six (one) and step six (two), and converting 3X 3Iθ′Partitioning, estimating x, y, A can be expressed as 2 x 2 order matrix, and for matrix IθPartitioning to obtain:
Figure BDA0002268905420000085
Figure BDA0002268905420000086
step nine (three), calculating inverse matrix of EFIM to obtain positioning estimation error bound under CFO
Figure BDA0002268905420000091
Figure BDA0002268905420000092
Wherein the content of the first and second substances,
Figure BDA0002268905420000093
Figure BDA0002268905420000095
step ten, defining OFDM frequency domain error estimation for indoor Wi-Fi positioning and an evaluation criterion of positioning accuracy of the OFDM frequency domain error estimation.
Advantageous effects
The invention provides an antenna received signal waveform expression under a frequency domain error caused by CFO, selects an angle analysis from a frequency domain to determine an unknown parameter vector, processes the unknown parameter vector by a method of calculating CRLB (cross correlation block) through the frequency domain to obtain a parameter to be estimated, and further obtains an estimation error of the CFO and an indoor Wi-Fi positioning estimation error bound based on CSI under the CFO by calculating FIM of the parameter to be estimated. The invention provides a CFO estimation error and a closed expression of the positioning estimation error of the indoor Wi-Fi positioning based on the CSI under a frequency domain error caused by CFO, and analyzes the influence of different factors on the CFO estimation error and the indoor Wi-Fi positioning accuracy based on the CSI. The method can provide reference basis when designing the positioning system, compensate frequency domain errors in the system, and evaluate the positioning performance of the system so as to optimize the positioning system and improve the positioning precision.
Drawings
FIGS. 1a and 1b are schematic diagrams of CFO effects;
FIG. 2 is a CFO estimation error bound, wherein the influence of L and ρ on the CFO estimation error bound is changed respectively;
FIG. 3 is a positioning estimation error bound under CFO, wherein the influence of L and ρ on the CFO positioning estimation error bound is changed respectively;
FIG. 4 is a CFO positioning estimation error bound, the effect of changing N on the CFO positioning estimation error bound;
FIG. 5 evaluation criteria of CFO estimation error and positioning accuracy under CFO;
detailed description of the preferred embodiments
The technical scheme of the invention is further described in detail by combining the attached drawings:
as shown in fig. 1 to 4, an OFDM frequency domain error estimation method for indoor Wi-Fi positioning and a positioning accuracy evaluation method thereof specifically include the following steps:
step one, assuming that a received signal waveform of an antenna in indoor Wi-Fi positioning based on CSI can be represented as:
Figure BDA0002268905420000101
wherein PN represents the total number of paths through which the signal propagates, s (t) represents the transmit waveform, a(p)Representing the amplitude, tau, of the signal of the p-th path(p)Represents the signal time delay of the p-th path, and z (t) represents noise.
Step two, the CSI mainly includes signal amplitude and phase information of the received signal converted by an Analog-to-digital converter (ADC), and a signal waveform of the received signal converted by the ADC is represented as:
where n represents the number of samples, L represents the total number of samples, and T represents the sampling period.
Step three, interference exists on CSI positioning by multipath signals in the positioning process, in order to achieve a better positioning effect, only direct paths are considered, and the direct signal waveform received by the mth antenna is represented as:
rm(nT)＝ams(nT-τm)+z(nT),m＝1,…,N， (3)
where N represents the total number of antennas.
Step four, analyzing CFO estimation errors from the angle of a frequency domain, wherein the step four comprises the following steps:
step four (one), according to fig. 1, the CFO can be regarded as a fixed frequency offset Δ f of the signal in the frequency domain, so that the waveform of the signal affected by the CFO received by the mth antenna can be represented as:
gm(nT)＝ams(nT-τm)ej2πkε/LT+z(nT),n＝1,…,L， (4)
wherein, amRepresenting the amplitude, τ, of the signal arriving at the m-th antennamRepresenting the time delay of the signal arriving at the mth antenna,
Figure BDA0002268905420000111
denotes normalized CFO, fcRepresenting carrier frequency
Step four (two), performing L-point Fourier transform on the signal expression under the CFO in the step four (one) to obtain:
Figure BDA0002268905420000112
wherein η (k) denotes a noise power spectrum subject to Gaussian distribution, Gm(k) Dependent on the variables ε, amAnd τmObtaining the unknown parameter vector theta ═ epsilon, am,τm]。
Step four (three), all G is represented by vector Xm(k)：
X＝[Gm(0),…,Gm(L-1)]T， (6)
Step four (IV), the expectation of X is expressed as:
step five, calculating a Fisher information matrix for theta according to a formula (7), and specifically comprising the following steps:
step five (one), respectively calculating:
Figure RE-GDA0002299868020000114
Figure RE-GDA0002299868020000121
step five (two), solving the jth element of the ith row of the Fisher information matrix of theta by using a formula (8),
Figure BDA0002268905420000123
wherein, thetai，θjRespectively representing relevant parameters epsilon and a in the parameter theta to be estimatedm，τmOne of them. The parameter θ to be estimated includes three related parameters, so a fisher information matrix of 3 × 3 is obtained:
Figure BDA0002268905420000124
wherein the content of the first and second substances,
Figure BDA0002268905420000125
sixthly, estimating the error bound of the epsilon by using the Matrix dimensionality reduction property (representing a high-dimensional Matrix by a low-dimensional Matrix and keeping the key Information quantity unchanged) of Equivalent FIM (Equivalent Fisher Information Matrix, EFIM), and specifically comprising the following steps of:
step six (one), partitioning the FIM of NxN:
Figure BDA0002268905420000131
wherein A ∈ Rn×n，B∈Rn×(N-n)，C∈R(N-n)×(N-n)。
Step six (two), the EFIM about θ can be obtained from equation (8):
Figure BDA0002268905420000132
wherein, depending on the nature of the EFIM,
Figure BDA0002268905420000133
and step six (three), estimating the error bound of epsilon, wherein A is expressed as a matrix I of 1 multiplied by 11×1To matrix IθPartitioning to obtain:
Figure BDA0002268905420000134
Figure BDA0002268905420000135
Figure BDA0002268905420000136
step six (four), calculating an inverse matrix of the EFIM to obtain an estimation error bound of the CFO:
Figure BDA0002268905420000137
wherein the content of the first and second substances,
Figure BDA0002268905420000138
Figure BDA0002268905420000139
step seven, coordinate information about the target to be measured needs to be introduced in order to obtain a positioning estimation error bound under the CFO, and the method specifically comprises the following steps:
step seven (one), orderWherein (x, y) represents the true position of the target, (x)m,ym) Indicating the position of the mth antenna,
Figure BDA0002268905420000141
the distance between the target to be measured and the mth antenna is shown, and c represents the speed of light.
Step seven (two), according to the attenuation model of the amplitude
Figure BDA0002268905420000142
Taking a signal propagation model in free space as an example, the amplitude of the m-th antenna is simplified into
Figure BDA0002268905420000143
Wherein, a0Representing the signal reference amplitude at 1m from the antenna.
Step seven (three), and combining tau in step seven (one) and step seven (two)mAnd amSubstituting equation (5) gives:
Figure BDA0002268905420000144
from this, it can be seen that the original unknown parameter vector θ ═ epsilon, a for the target and mth antenna positionm,τm]Becomes a new unknown parameter vector theta ═ epsilon, x, y]。
Step seven (four), all G is represented by vector Xm(k)：
X＝[G1(0),…,G1(L-1),…,Gm(0),…,Gm(L-1)]T， (17)
Step seven (five), the expectation of X is expressed as:
Figure BDA0002268905420000145
step eight, calculating a Fisher information matrix for theta' by a formula (18), and specifically comprising the following steps:
step eight (one), calculating the partial derivative of each element in mu and respectively calculating
Figure BDA0002268905420000146
Figure BDA0002268905420000147
Figure BDA0002268905420000152
Wherein m is 1, …, N, k is 0, …, L-1,
Figure BDA0002268905420000153
Figure BDA0002268905420000154
step eight (two), solving the jth element of the ith row of the Fisher information matrix of theta' by using a formula (10), wherein thetai，θjRespectively representing one of the relevant parameters x, y, epsilon in the parameter theta' to be estimated. The unknown parameter vector θ' includes three related parameters, so a fisher information matrix of 3 × 3 is obtained:
Figure BDA0002268905420000155
wherein the content of the first and second substances,
Figure BDA0002268905420000156
step nine, in order to solve the positioning estimation error bound under the CFO, estimating the positioning error bound by using the Matrix dimension reduction property (a high-dimensional Matrix is represented by a low-dimensional Matrix, and the amount of key Information is unchanged) of an equivalent FIM (equivalent fisher Information Matrix, EFIM), specifically including the following steps:
step nine (one), order
Figure BDA0002268905420000161
Figure BDA0002268905420000162
Can be combined withθ′The method is simplified as follows:
Figure BDA0002268905420000163
step nine (two), utilizing step six (one) and step six (two), and converting 3X 3Iθ′For blocking, since we need to estimate x and y, A can be expressed as a matrix of 2 × 2 order, for matrix IθPartitioning to obtain:
Figure BDA0002268905420000164
Figure BDA0002268905420000165
Figure BDA0002268905420000166
step nine (three), calculating inverse matrix of EFIM to obtain positioning estimation error bound under CFO
Figure BDA0002268905420000167
Figure BDA0002268905420000168
Wherein the content of the first and second substances,
Figure BDA0002268905420000169
Figure BDA0002268905420000171
Figure BDA0002268905420000172
tenth, estimating OFDM frequency domain errors for indoor Wi-Fi positioning and an evaluation criterion of positioning accuracy thereof, wherein the ten frequency domain error estimation and the evaluation criterion of the positioning accuracy thereof are as follows:
1. with the increase of L, the CFO estimation error is reduced, and the positioning precision under CFO is improved.
2. As ρ increases, the CFO estimation error increases, and the positioning accuracy under CFO decreases.
3. As N increases, the positioning accuracy under CFO increases.

Claims (3)

1. An OFDM frequency domain error estimation and positioning accuracy evaluation method for indoor Wi-Fi positioning is characterized by comprising the following steps:
step four, analyzing the CFO estimation error from the angle of the frequency domain according to the direct signal waveform expression received by the mth antenna in the step three, and specifically comprising the following steps:
step four (one), according to fig. 1, the CFO can be regarded as a fixed frequency offset Δ f of the signal in the frequency domain, so that the waveform of the signal affected by the CFO received by the mth antenna can be represented as:
gm(nT)＝ams(nT-τm)ej2πkε/LT+z(nT),n＝1,…,L， (4)
wherein, amRepresenting the amplitude, τ, of the signal arriving at the m-th antennamRepresenting the time delay of the signal arriving at the mth antenna,denotes normalized CFO, fcRepresenting the carrier frequency.
Step four (two), performing L-point Fourier transform on the signal expression under the CFO in the step four (one) to obtain:
Figure RE-FDA0002299868010000012
wherein η (k) denotes a noise power spectrum subject to Gaussian distribution, Gm(k) Dependent on the variables ε, amAnd τmThe unknown parameter vector theta ═ epsilon, a can be obtainedm,τm]。
Step four (three), all G is represented by vector Xm(k)：
X＝[Gm(0),…,Gm(L-1)]T， (6)
Step four (IV), the expectation of X is expressed as:
Figure RE-FDA0002299868010000013
step five, calculating a Fisher information matrix for theta by a formula (7), and specifically comprising the following steps:
step five (one), respectively calculating:
Figure RE-FDA0002299868010000014
Figure RE-FDA0002299868010000021
Figure RE-FDA0002299868010000022
step five (two), solving the jth element of the ith row of the Fisher information matrix of theta by using a formula (8),
wherein, thetai，θjRespectively representing the relevant parameters epsilon and a in the unknown parameter vector thetam，τmOne of them. The unknown parameter vector θ includes three related parameters, so a fisher information matrix of 3 × 3 is obtained:
Figure RE-FDA0002299868010000024
wherein the content of the first and second substances,
Figure RE-FDA0002299868010000025
and sixthly, estimating the error bound of the epsilon by using the Matrix dimensionality reduction property (a high-dimensional Matrix is represented by a low-dimensional Matrix and the key Information quantity is unchanged) of an Equivalent FIM (Equivalent Fisher Information Matrix, EFIM).
Step seven, in order to obtain a positioning estimation error bound under the CFO, coordinate information about the target to be measured needs to be introduced, and the method specifically comprises the following steps:
step seven (one), order
Figure RE-FDA0002299868010000031
Wherein (x, y) represents the true position of the target, (x)m,ym) Indicating the position of the mth antenna,
Figure RE-FDA0002299868010000032
the distance between the target to be measured and the mth antenna is shown, and c represents the speed of light.
Step seven (two), according to the attenuation model of the amplitude
Figure RE-FDA0002299868010000033
Taking a signal propagation model in free space as an example, the amplitude of the m-th antenna is simplified into
Figure RE-FDA0002299868010000034
Wherein, a0Representing the signal reference amplitude at 1m from the antenna.
Step seven (three), and combining tau in step seven (one) and step seven (two)mAnd amSubstituting equation (5) gives:
Figure RE-FDA0002299868010000035
from this, it can be seen that the original unknown parameter vector θ ═ epsilon, a for the target and mth antenna positionm,τm]Becomes a new unknown parameter vector theta ═ epsilon, x, y]。
Step seven (four), all G is represented by vector Xm(k)：
X＝[G1(0),…,G1(L-1),…,Gm(0),…,Gm(L-1)]T， (17)
Step seven (five), the expectation of X is expressed as:
Figure RE-FDA0002299868010000036
step eight, calculating a Fisher information matrix for theta' by a formula (18), and specifically comprising the following steps:
step eight (one), calculating the partial derivative of each element in mu and respectively calculating
Figure RE-FDA0002299868010000041
Figure RE-FDA0002299868010000042
Wherein m is 1, …, N, k is 0, …, L-1,
Figure RE-FDA0002299868010000044
step eight (two), solving the jth element of the ith row of the Fisher information matrix of theta' by using a formula (10), wherein thetai，θjRespectively representing one of the relevant parameters x, y, epsilon in the parameter theta' to be estimated. The unknown parameter vector θ' includes three related parameters, so a fisher information matrix of 3 × 3 is obtained:
Figure RE-FDA0002299868010000045
wherein the content of the first and second substances,
Figure RE-FDA0002299868010000046
step nine, in order to solve the positioning estimation error bound under the CFO, the positioning error bound is estimated by using the Matrix dimensionality reduction property (a high-dimensional Matrix is represented by a low-dimensional Matrix, and the key information amount is unchanged) of an Equivalent FIM (Equivalent FisherInformation Matrix, EFIM).
Step ten, defining the frequency domain error estimation in the indoor Wi-Fi positioning based on OFDM and the evaluation criterion of the positioning estimation error.
2. The method of claim 1, wherein the sixth step comprises the following steps:
step six (one), partitioning the FIM of NxN:
wherein A ∈ Rn×n，B∈Rn×(N-n)，C∈R(N-n)×(N-n)。
Step six (two), the EFIM about θ can be obtained from equation (8):
Figure FDA0002268905410000052
wherein, depending on the nature of the EFIM,
Figure FDA0002268905410000053
and step six (three), estimating the error bound of epsilon, wherein A is expressed as a matrix I of 1 multiplied by 11×1To matrix IθPartitioning to obtain:
Figure FDA0002268905410000054
Figure FDA0002268905410000055
Figure FDA0002268905410000056
step six (four), calculating an inverse matrix of the EFIM to obtain an estimation error bound of the CFO:
Figure FDA0002268905410000057
wherein the content of the first and second substances,
Figure FDA0002268905410000058
Figure FDA0002268905410000059
3. the method of claim 1, wherein the ninth step comprises the following steps:
step nine (one), order
Figure FDA0002268905410000061
Figure FDA0002268905410000062
Can be combined withθ′The method is simplified as follows:
Figure FDA0002268905410000063
step nine (two), utilizing step six (one) and step six (two), and converting 3X 3Iθ′Partitioning is performed, and for estimation of x, y, A may be expressed as a 2 × 2 matrix, for matrix IθPartitioning to obtain:
Figure FDA0002268905410000065
Figure FDA0002268905410000066
step nine (three), calculating inverse matrix of EFIM to obtain positioning estimation error bound under CFO
Figure FDA0002268905410000067
Wherein the content of the first and second substances,
Figure FDA0002268905410000069
Figure FDA0002268905410000071
Figure FDA0002268905410000072
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