Method, system and apparatus for computing a lower bound on time of arrival positioning error

Abstract

The application discloses a kind of based on time of arrival positioning error lower bound calculation method, system and device, the method includes: S1, establishing positioning system model;S2, design changes prior distribution encoder;S3, based on positioning system model and encoder calculation time positioning error lower bound.The application can realize conversion ranging error, based on the ranging error of conversion calculates positioning error lower bound, improves the precision of positioning.

Description

Technical Field

[0001] This invention relates to the field of positioning error lower bound calculation, and in particular to a method, system and apparatus for calculating positioning error lower bound based on time of arrival. Background Technology

[0002] With the development of advanced wireless communication technologies, 6G cellular networks are expected to support a wider range of services, such as integrated communication and sensing services. Commercial cellular networks can be used as radar for ubiquitous positioning and ancillary user services. In such applications, 6G base stations can serve as ubiquitous anchor points (positioning reference points) as they are expected to be densely deployed. Furthermore, the millimeter-wave carriers used in 6G will accommodate more bandwidth for high-precision sensing. Against this backdrop, the development of new technologies to further improve positioning efficiency is significant and represents a currently popular research area.

[0003] Typically, location services can be provided using methods such as Time of Arrival (TOA) and Received Signal Strength (RSS). RSS offers excellent positioning accuracy in non-line-of-sight (NLS) scenarios; however, it is highly sensitive to the distance between the anchor point and the user. Therefore, it may not be suitable for long-distance positioning. If the frequency bandwidth allocated to the positioning signal is sufficiently wide, the TOA method generally achieves higher accuracy gains than other methods; currently, widely used broadband or ultra-wideband positioning employs the TOA method. However, utilizing signals from non-line-of-sight paths within the TOA has always been considered a challenging task in positioning problems.

[0004] In non-line-of-sight (TOA) scenarios, TOA positioning leads to ranging errors, which are random numbers. Without prior information about these errors, a lower bound for the error based on maximum likelihood (ML) estimation has been derived and used to measure the accuracy of TOA positioning. Furthermore, a lower bound for the error based on maximum a posteriori (MAP) estimator has also been proposed. However, current research on the lower bound of MAP-based TOA positioning errors is limited. The mathematical expressions derived in current studies only apply when the ranging error follows a minority distribution, such as a Gaussian distribution. Existing methods also assume that the ranging follows a Gaussian distribution; however, the lower bound obtained under this assumption is not optimal. Summary of the Invention

[0005] The purpose of this invention is to provide a method, system, and apparatus for calculating the lower bound of positioning error based on time of arrival, aiming to solve the problem of calculating the lower bound of positioning error based on time of arrival.

[0006] This invention provides a method for calculating the lower bound of positioning error based on time of arrival, comprising:

[0007] S1. Establish a positioning system model;

[0008] S2. Design an encoder that alters the prior distribution;

[0009] S3. Calculate the lower bound of the time positioning error based on the positioning system model and encoder.

[0010] The present invention also provides a calculation system based on the lower bound of the time-of-arrival positioning error, comprising:

[0011] Establish a module for building the positioning system model;

[0012] Transformation module: Used to design encoders that alter prior distributions;

[0013] Calculation module: Used to calculate the lower bound of time positioning error based on the positioning system model and encoder.

[0014] This invention also provides a computing device based on the lower bound of the time-of-arrival positioning error, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the computer program, when executed by the processor, implements the steps of the above method.

[0015] This invention also provides a computer-readable storage medium storing an information transmission implementation program, which, when executed by a processor, implements the steps of the above-described method.

[0016] By employing the embodiments of the present invention, the present invention can realize the transformation of ranging error, calculate the lower bound of positioning error based on the transformed ranging error, and improve the positioning accuracy.

[0017] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, and in order to make the above and other objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention are described below. Attached Figure Description

[0018] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0019] Figure 1 This is a flowchart of the calculation method for the lower bound of positioning error based on arrival time according to an embodiment of the present invention;

[0020] Figure 2This is a schematic diagram of a calculation system based on the lower bound of the positioning error based on the time of arrival, according to an embodiment of the present invention.

[0021] Figure 3 This is a schematic diagram of a calculation device based on the lower bound of the positioning error based on the time of arrival, according to an embodiment of the present invention. Detailed Implementation

[0022] The technical solution of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] Method Implementation Examples

[0024] According to embodiments of the present invention, a method for calculating the lower bound of positioning error based on time of arrival is provided. Figure 1 This is a flowchart of the calculation method for the lower bound of positioning error based on arrival time according to an embodiment of the present invention, as shown below. Figure 1 As shown, it specifically includes:

[0025] S1. Establish a positioning system model;

[0026] S2. Design an encoder that alters the prior distribution;

[0027] S3. Calculate the lower bound of time positioning error based on the positioning system model and encoder.

[0028] S1 specifically includes: establishing 2N anchor points to provide positioning services; the first N anchor points provide non-line-of-sight signals, and the last N anchor points provide line-of-sight signals; the location of the ground user is p. u =[xy] T The position of the i-th ground anchor point is p. i =[x i y i ] T , 1≤i≤2N, and the time delay for receiving the signal from the anchor point at the user is defined as follows:

[0029]

[0030] Among them, l i It is the ranging error, when 1≤i≤N. i >0, when N < i ≤ 2N i =0; c = 3 * 10^8 m / s is the speed of light, and the baseband of the received signal is represented as:

[0031] r i (t)=A i s(t-τi)+ni , 1≤i≤2N (2)

[0032] Where A i s(t) is the signal amplitude, s(t) is the known signal waveform, and n is the signal amplitude. i It is additive white Gaussian noise with a power spectral density of N0 / 2 during the transmission process of the signal from the i-th anchor point.

[0033] S2 specifically includes: based on the positioning system model, the parameter vector to be estimated is:

[0034] θ=[xy l1 l2 … l N ] T (3)

[0035] The received signals from different anchor points are independent of each other, and the likelihood function of the maximum likelihood estimation is expressed as:

[0036]

[0037] The joint prior distribution of non-line-of-sight ranging errors is:

[0038] Φ(l)=φ(l1)φ(l2)…φ(l N ) (5)

[0040] The joint probability density function is then:

[0041] f(r, θ) ∝ m(r|θ)·Φ(l) (6)

[0043] Based on the maximum a posteriori estimation and the definition of the Bayesian information matrix, the equivalent calculation method for the Bayesian information matrix is ​​as follows:

[0044]

[0045] Among them, J D It is the Fisher information matrix, used to calculate the covariance matrix of the maximum likelihood estimator, J. Φ J′ is the second moment of the first derivative of the logarithm of the prior distribution. Φ The first moments of the second derivative of the logarithm of the prior distribution are:

[0046]

[0047]

[0048]

[0049] The lower bound of the mean squared error (MSE) for user location estimation is obtained by calculating the inverse of the Bayesian information matrix, i.e.,

[0050] MSE≥(B θ -1 ) 11 +(B θ -1 ) 22 (11) Formula (7) is applicable when the ranging error follows any form of prior distribution and has generality. If the ranging error is Gaussian distribution, the encoder is used to convert it to exponential distribution or Erlang distribution.

[0051] The mathematical representation of the encoder is as follows:

[0052]

[0053] When the ranging error is converted to an exponential distribution,

[0054]

[0055] When the ranging error is converted to an Erlang distribution, it becomes:

[0056]

[0057] Where Φ⁻¹(x) represents the inverse function of Φ(x).

[0058] The Bayesian information matrices corresponding to the exponential and Erlang distributions of the ranging error are as follows:

[0059] (1) Exponential distribution

[0060] li (1≤i≤N) follow an independent exponential distribution with probability density function:

[0061] φ(l i )=λ i exp(-λ i l i ), 1≤i≤N (12)

[0062] Where λ is the rate parameter of the exponential distribution, substituting equation (12) into equation (5) yields the joint prior distribution of the ranging error. Based on equation (7), substituting the joint prior distribution into equations (9) and (10) yields the Bayesian information matrix corresponding to the ranging error following an exponential distribution. That is:

[0063] B θ exp =J D +3J Φ exp (13)

[0064] in,

[0065]

[0066] Erlang distribution:

[0067] l i (1≤i≤N) follow an independent Erlang distribution with the probability density function:

[0068]

[0069] Where ε and k are the rate parameter and order of the Erlang distribution, respectively. Substituting equation (15) into equation (5) yields the joint prior distribution of the ranging error. Based on equation (7), substituting the joint prior distribution into equations (9) and (10) yields the Bayesian information matrix corresponding to the ranging error following the Erlang distribution. That is,

[0070] B θ erlg =J D +3J Φ erlg +2J Φ ' erlg (16)

[0071] in,

[0072]

[0073]

[0074]

[0075]

[0076] Γ(x) represents the gamma function with respect to x. Equations (13) and (16) can be used to calculate the Bayesian information matrix when the ranging error follows an exponential distribution and an El-Rang distribution, respectively, thus obtaining the lower bound of the MSE for user location estimation. We design a sampling distribution to sample the ranging error, which can achieve the transformation of the sample distribution. We call this design an encoder. The principle of this encoder is that when the ranging error follows a Gaussian distribution, sampling can transform the prior distribution from a Gaussian distribution to an exponential or El-Rang distribution. However, it should be noted that when setting up this encoder, the variance of the exponential or El-Rang distribution should not be set to infinitely small in order to obtain higher accuracy. Although the samples after the encoder will contain more information about the estimator, when finally obtaining the estimated value of the ranging error, the samples after the encoder need to be restored to the original distribution. Too small a variance will lead to too low a resolution in the restoration, thus reducing the positioning accuracy. Therefore, the actual application scenario should be considered when selecting the variance. Let Φgss, Φ, Φexp, and Φerlg represent the Gaussian probability distribution function, target probability distribution function, exponential probability distribution function, and Erlang probability distribution function related to the ranging error, respectively. The encoder is defined as g, and its mathematical representation is as follows:

[0077]

[0078] Taking the exponential distribution as an example,

[0079]

[0080] For the Erlang distribution, it is:

[0081]

[0082] Where Φ⁻¹(x) represents the inverse function of Φ(x).

[0083] The beneficial effects of this invention are as follows:

[0084] An equivalent calculation formula for the Bayesian Information Matrix (BIM) based on the Maximum A posteriori estimator (MAP) is proposed. This calculation method is applicable when the ranging error follows an arbitrary form of prior distribution.

[0085] Based on the above calculation method, we assume that the ranging error follows an exponential distribution and an Erlang distribution respectively, and calculate the corresponding BIM to obtain the lower bound of the mean square error (MSE) for TOA positioning. We then compare the performance with the existing method, which follows a Gaussian distribution, and numerical and simulation verification show that the proposed exponential distribution and Erlang distribution outperform the existing method.

[0086] An encoder is proposed to transform the prior distribution of ranging error. If the ranging error follows a Gaussian distribution, the encoder transforms the Gaussian distribution into an exponential or Erlang distribution, thereby improving the positioning accuracy of TOA in non-line-of-sight scenarios.

[0087] System Implementation Examples

[0088] According to embodiments of the present invention, a calculation system based on the lower bound of the positioning error based on the time of arrival is provided. Figure 3 This is a schematic diagram of a calculation system based on the lower bound of the positioning error based on the time of arrival, according to an embodiment of the present invention. Figure 3 As shown, it specifically includes:

[0089] A calculation system based on the lower bound of the time-of-arrival positioning error includes:

[0090] Establish a module for building the positioning system model;

[0091] Transformation module: Used to design encoders that change prior distributions;

[0092] Calculation module: Used to calculate the lower bound of time positioning error based on the positioning system model and encoder.

[0093] The module is specifically used to: establish 2N anchor points to provide positioning services. The first N anchor points provide non-line-of-sight signals, and the last N anchor points provide line-of-sight signals. The location of the ground user is p. u =[xy] T The position of the i-th ground anchor point is p. i =[x i y i ] T , 1≤i≤2N, and the time delay for receiving the signal from the anchor point at the user is defined as follows:

[0094]

[0095] Among them, l i It is the ranging error, when 1≤i≤N. i >0, when N < i ≤ 2N i =0; c = 3 * 10^8 m / s is the speed of light, and the baseband of the received signal is represented as:

[0096] r i (t)=A i s(t-τi)+n i , 1≤i≤2N (2)

[0097] Where A i s(t) is the signal amplitude, s(t) is the known signal waveform, and n is the signal amplitude. iIt is additive white Gaussian noise with a power spectral density of N0 / 2 during the transmission process of the signal from the i-th anchor point.

[0098] The calculation module is specifically used for: based on the positioning system model, the parameter vector to be estimated is:

[0099] θ=[xy l1 l2 … l N ] T (3)

[0101] The received signals from different anchor points are independent of each other, and the ML likelihood function is expressed as:

[0102]

[0103] The joint prior distribution of non-line-of-sight ranging errors is:

[0104] Φ(l)=φ(l1)φ(l2)…φ(l N ) (5)

[0106] The joint probability density function is then:

[0107] f(r,θ)∝m(r|θ)·Φ(l) (6)

[0108] Based on the maximum a posteriori estimation and the definition of the Bayesian information matrix, the equivalent calculation method for the Bayesian information matrix is ​​as follows:

[0109]

[0110] Among them, J D It is the Fisher information matrix, used to calculate the covariance matrix of the maximum likelihood estimator, J. Φ J′ is the second moment of the first derivative of the logarithm of the prior distribution. Φ The first moments of the second derivative of the logarithm of the prior distribution are:

[0111]

[0112]

[0113]

[0114] The lower bound of the mean squared error (MSE) for user location estimation is obtained by calculating the inverse of the Bayesian information matrix, i.e.,

[0115] MSE≥(B θ -1 ) 11 +(B θ -1 )22 (11) Formula (7) is applicable when the ranging error follows any form of prior distribution and has generality. If the ranging error is Gaussian distribution, the encoder is used to convert it to exponential distribution or Erlang distribution.

[0116] The mathematical representation of the encoder is as follows:

[0117]

[0118] When the ranging error is converted to an exponential distribution,

[0119]

[0120] When the ranging error is converted to an Erlang distribution, it becomes:

[0121]

[0122] Where Φ -1 (x) represents the inverse function of Φ(x).

[0123] The embodiments of the present invention are system embodiments corresponding to the above method embodiments. The specific operation of each module can be understood by referring to the description of the method embodiments, and will not be repeated here.

[0124] Device Example 1

[0125] This invention provides a calculation device based on the lower bound of the positioning error according to the time of arrival, such as... Figure 3 As shown, it includes: a memory 30, a processor 32, and a computer program stored on the memory 30 and executable on the processor 32. When the computer program is executed by the processor, it implements the steps in the above method embodiments.

[0126] Device Example 2

[0127] This invention provides a computer-readable storage medium storing an information transmission implementation program, which, when executed by a processor 32, implements the steps described in the above method embodiments.

[0128] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions to the technical solutions of the embodiments of the present invention do not cause the essence of the corresponding technical solutions to deviate from the scope of the present solution.

Claims

1. A method for calculating the lower bound of positioning error based on time of arrival, characterized in that, include: S1. Establish a positioning system model; S2. Design an encoder that alters the prior distribution; S3. Calculate the lower bound of time positioning error based on the positioning system model and encoder; S1 specifically includes: establishing 2N anchor points to provide positioning services, with the first N anchor points providing non-line-of-sight signals and the last N anchor points providing line-of-sight signals, and the location of the ground user being p. u =[xy] T The position of the i-th ground anchor point is p. i =[x i y i ] T For any 1 ≤ i ≤ 2N, the time delay for receiving the signal from the anchor point at the user is defined as follows: （1） where, is the ranging error, when 1 ≤ i ≤ N > 0, when N < i ≤ 2N = 0; c = 3*10^8 m / s is the speed of light, and the baseband representation of the received signal is: (2) Where A i s(t) is the signal amplitude, s(t) is the known signal waveform, and n is the signal amplitude. i It is additive white Gaussian noise with a power spectral density of N0 / 2 during the transmission process of the signal from the i-th anchor point; S2 specifically includes: based on the positioning system model, the parameter vector to be estimated is: (3) The received signals from different anchor points are independent of each other, and the likelihood function of the maximum likelihood estimation is expressed as: (4) The joint prior distribution of non-line-of-sight ranging errors is: (5) The joint probability density function is then: (6) Based on the maximum a posteriori estimation and the definition of the Bayesian information matrix, the equivalent calculation method for the Bayesian information matrix is ​​as follows: (7) Among them, J D It is the Fisher information matrix, used to calculate the covariance matrix of the maximum likelihood estimator. It is the second moment of the first derivative of the logarithm of the prior distribution. The first moments of the second derivative of the logarithm of the prior distribution are: (8) (9) (10) The lower bound of the mean squared error (MSE) for user location estimation is obtained by calculating the inverse of the Bayesian information matrix, i.e., (11), Formula (7) is applicable when the ranging error follows any form of prior distribution and has generality. If the ranging error is Gaussian distribution, the encoder is used to convert it to exponential distribution or Erlang distribution. The encoder is mathematically represented as follows: (21) When the ranging error is converted to an exponential distribution, (22) When the ranging error is converted to an Erlang distribution, it becomes: (23) Where Φ -1 (x) represents the inverse function of Φ(x).

2. A calculation system based on the lower bound of the time-of-arrival positioning error, characterized in that, include: Establish a module for building the positioning system model; Transformation module: Used to design encoders that change prior distributions; Calculation module: used to calculate the lower bound of time positioning error based on the positioning system model and encoder; The establishment module is specifically used to: establish 2N anchor points to provide positioning services, with the first N anchor points providing non-line-of-sight signals and the last N anchor points providing line-of-sight signals, and the location of the ground user being p. u =[xy] T The position of the i-th ground anchor point is p. i =[x i y i ] T For any 1 ≤ i ≤ 2N, the time delay for receiving the signal from the anchor point at the user is defined as follows: （1） Among them, is the ranging error, when 1 ≤ i ≤ N > 0, when N < i ≤ 2N = 0; c = 3 * 10^8 m / s is the speed of light, and the baseband representation of the received signal is: (2) Where A i s(t) is the signal amplitude, s(t) is the known signal waveform, and n is the signal amplitude. i It is additive white Gaussian noise with a power spectral density of N0 / 2 during the transmission process of the signal from the i-th anchor point; The calculation module is specifically used for: based on the positioning system model, the parameter vector to be estimated is: (3) The received signals from different anchor points are independent of each other, and the ML likelihood function is expressed as: (4) The joint prior distribution of non-line-of-sight ranging errors is: (5) The joint probability density function is then: (6) Based on the maximum a posteriori estimation and the definition of the Bayesian information matrix, the equivalent calculation method for the Bayesian information matrix is ​​as follows: (7) Among them, J D It is the Fisher information matrix, used to calculate the covariance matrix of the maximum likelihood estimator. It is the second moment of the first derivative of the logarithm of the prior distribution. The first moments of the second derivative of the logarithm of the prior distribution are: (8) (9) (10) The lower bound of the mean squared error (MSE) for user location estimation is obtained by calculating the inverse of the Bayesian information matrix, i.e., (11), Formula (7) is applicable when the ranging error follows any form of prior distribution and has generality. If the ranging error is Gaussian distribution, the encoder is used to convert it to exponential distribution or Erlang distribution. The encoder is mathematically represented as follows: (21) When the ranging error is converted to an exponential distribution, (22) When the ranging error is converted to an Erlang distribution, it becomes: (23) Where Φ -1 (x) represents the inverse function of Φ(x).

3. A computing device based on the lower bound of the positioning error according to the time of arrival, characterized in that, include: The memory, the processor, and the computer program stored in the memory and executable on the processor, wherein the computer program, when executed by the processor, implements the steps of the calculation method based on the lower bound of the positioning error based on the time of arrival as described in claim 1.

4. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores an implementation program for information transmission, which, when executed by a processor, implements the steps of the calculation method for the lower bound of the positioning error based on the time of arrival as described in claim 1.

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