CN111679296A - Single-satellite positioning method based on GP iteration extrapolation - Google Patents

Single-satellite positioning method based on GP iteration extrapolation Download PDF

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CN111679296A
CN111679296A CN202010470043.1A CN202010470043A CN111679296A CN 111679296 A CN111679296 A CN 111679296A CN 202010470043 A CN202010470043 A CN 202010470043A CN 111679296 A CN111679296 A CN 111679296A
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CN111679296B (en
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房宵杰
王迦祺
沙学军
梅林�
李卓明
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Harbin Institute of Technology
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Abstract

A single-satellite positioning method based on GP iteration extrapolation belongs to the field of signal processing. The invention solves the problems of long positioning time, long satellite flight time and high flight cost in the single-satellite positioning process by adopting the traditional Doppler positioning method. The method optimizes the traditional GP iteration extrapolation algorithm and the Doppler single-star positioning algorithm. The signals received by the satellite are reasonably extrapolated by adopting fractional Fourier transform, so that the signals are more band-limited in a transform domain, and a proper filter is selected and a better extrapolation effect is achieved. Meanwhile, under the requirement of ensuring certain positioning accuracy, more Doppler information is obtained by applying signal extrapolation, the complexity problem of the positioning process is relieved, the single-satellite positioning time and the satellite flight time are reduced, and the satellite flight cost is reduced. The invention can be applied to single-satellite positioning.

Description

Single-satellite positioning method based on GP iteration extrapolation
Technical Field
The invention belongs to the field of signal processing, and particularly relates to a single-satellite positioning method based on GP (Gerchberg-Papoulis) iterative extrapolation.
Background
Single-satellite positioning has wide application in military target positioning and civil navigation, and signal extrapolation has important significance for communication, radar, and recovery of voice and video signals. In single-satellite positioning, if partial signals g (t) (g (t) received by a satellite are partial signals in complete signals f (t)) received by the satellite near a radiation source, f (t) data of the whole positioning process is obtained, the time of single-satellite positioning can be effectively shortened, and the flight cost of the satellite is reduced.
Most signals in the signal processing domain can be mapped into a domain by some transform, and processing the signal in the transformed domain can detect and extract some properties of the signal. For the extrapolation problem of signals, the Gerchberg-Papoulis extrapolation algorithm (GP algorithm for short) utilizes the complete orthogonality of a long sphere function in a sigma-band limited space, and iterates out signals outside a known interval by repeatedly applying Fourier transform and inverse transform, and truncating and replacing. The Sanz-Huang theory is based on a discrete estimation theory proposed by the GP algorithm, so that the GP algorithm can be implemented with DFT. The method for iterative extrapolation of discrete signals can obtain extrapolated signals with decreasing errors in effective iterations, is simple and convenient to calculate, and can effectively improve extrapolated data and signal quality.
For the single satellite positioning problem, R.J.Webster et al propose a method for measuring the frequency of an incoming wave signal of a target radiation source by using a single observation station to realize positioning. The single observation station is used for positioning by measuring Doppler information of relative movement of a target and the observation station in signal frequency. Although single-satellite positioning can be completed by adopting the doppler positioning method, the doppler positioning method has the problems of long required positioning time and long satellite flight time, and the problem of high flight cost caused by the long satellite flight time.
Disclosure of Invention
The invention aims to solve the problems of long positioning time, long satellite flight time and high flight cost in the single-satellite positioning process by adopting the traditional Doppler positioning method, and provides a single-satellite positioning method based on GP iteration extrapolation.
The invention reasonably extrapolates part of observation signals received by a satellite receiver based on the idea of GP extrapolation algorithm, and positions a ground surface radiation source through the original observation signals and the signals obtained by extrapolation, the flow of the signal extrapolation positioning algorithm is shown in figure 1, and the technical scheme adopted for solving the technical problems is as follows:
the method comprises the following steps that firstly, when a satellite flies near a ground surface radiation source (the specific flying position needs to be determined according to the setting of an orbit and the flying condition of an actual satellite), a part of signals g (t) sent by the radiation source are received, and g (t) is used as observation signals received by the satellite;
secondly, carrying out post zero filling processing on the observation signals g (t) received by the satellite in the first step, namely, carrying out post zero filling on the received observation signals g (t) for a required time length, wherein the length of a signal in a zero filling section can be selected to be more than 1 time of the observation signals g (t), and recording the signal in the zero filling section as p (t);
step three, initializing the signal to be extrapolated to be g0(t)，g0(t) is composed of observation signal g (t) and zero-padding segment signal p (t); initializing the support set to Φ0，Φ0Is a support set containing and only containing observation signal g (t) and zero-padding segment signal p (t);
presetting iteration number as N, and for initialized signal g to be extrapolated0(t) carrying out iterative extrapolation until a preset iteration number is met, and stopping extrapolation to obtain an extrapolated signal f (t);
fourthly, performing time-frequency analysis on the externally pushed signal f (t) to obtain a time-frequency function of the signal f (t);
sampling the Doppler frequency of the time-frequency function for M times according to the time interval delta t to respectively obtain Doppler frequency estimated values of satellite receiving signals at each sampling moment;
and fifthly, obtaining the position and the speed of the satellite at each sampling moment according to the satellite ephemeris, and solving the position of the earth surface radiation source by utilizing the position and the speed of the satellite at each sampling moment and the Doppler frequency estimation value of the satellite receiving signal at each sampling moment.
Further, the specific process of the third step is as follows:
step three, initializing the signal to be extrapolated to be g0(t) initializing the support set to Φ0；
Step three, initializing the signal g to be extrapolated in the step three0(t) performing a fractional Fourier transform of order p, the signal obtained after the fractional Fourier transform of order p being B g0(t)]；
Step three, the signal B [ g ] obtained in the step three0(t)]Filtering to obtain filtered signal B [ f0(t)]；
Step three and four, the filtered signal B [ f ] obtained in the step three is subjected to filtering0(t)]Carrying out-p fractional Fourier transform to obtain a mapping signal f in the time domain1(t)；
Step three and five, initializing a support set phi0In step three, a signal g to be extrapolated is initialized0(t) for the signal f obtained in step three or four1(t) replacing the corresponding time domain part to obtain a signal g obtained by the first iteration in the time domain1(t)；
The alternative method comprises the following steps:
Figure BDA0002514005490000038
wherein B represents the entire time domain;
at the same time, to the initialized support set phi0Updating to obtain a support set phi obtained by the first iteration1Support set phi1Is composed of only signal g1(t) the supporting set;
step three six, the signal g obtained in the step three five is used1(t) assigning to the initialized signal to be extrapolated in step three-one, and assigning the support set phi1Assigning values to the initialized support set to repeat the process from the third step to the third step until the preset stack is metStopping iteration when the generation times are N, and obtaining a signal g obtained by the Nth iterationN(t) comparing the signal g obtained in the Nth iterationN(t) as the extrapolated signal f (t).
Further, the specific process of the step five is as follows:
the grid is divided over a range of positions including the radiation source to obtain a set of grid points ∑, (x)k,yk,zk)∈∑，(xk,yk,zk) Grid point coordinates within the set of grid points ∑;
Figure BDA0002514005490000031
wherein the intermediate variable
Figure BDA0002514005490000032
(xj,yj,zj) Is the position of the satellite at the jth sample time,
Figure BDA0002514005490000033
is the velocity of the satellite at the jth sampling instant, c represents the speed of light, ωjAn estimate of the doppler frequency representing the signal received by the satellite at the jth sampling instant, ω being the frequency of the signal emitted by the radiation source,jfor frequency error, vector, of jth sampling instant
Figure BDA0002514005490000034
rjIs a vector
Figure BDA0002514005490000035
Figure BDA0002514005490000035
Figure BDA0002514005490000035
1,2, …, M;
rewrite equation (2) to the following form:
Figure BDA0002514005490000036
will be provided with
Figure BDA0002514005490000037
Abbreviated as gj(xk,yk,zk) Then, then
ωj＝ωgj(xk,yk,zk)+j(4)
And (3) arranging the formula (4) corresponding to all M sampling points into a matrix form as follows:
Ω＝Gω+E (5)
wherein Ω, G and E are intermediate variables, and Ω ═ ω1,ω2,…,ωM]T，E＝[1,2,…,M]T，G＝[g1(xk,yk,zk),g2(xk,yk,zk),…,gM(xk,yk,zk)]TThe superscript T represents the transpose of the matrix;
to ensure the E luminance2At a minimum, an objective function J (ω) is established:
J(ω)＝||Ω-Gω||2(5)
wherein | | | E | | | represents the norm of E, such that
Figure BDA0002514005490000041
Then
Figure BDA0002514005490000042
In the formula, the upper corner mark-1 represents the inverse of the matrix,
Figure BDA0002514005490000043
represents ω when the objective function J (ω) takes a minimum value;
the coordinates of each grid point in the grid point set ∑ are respectively substituted into the formula (6), and the coordinates corresponding to each grid point are sequentially obtained
Figure BDA0002514005490000044
Will minimize the value of the objective function J (ω)
Figure BDA0002514005490000045
Corresponding toThe grid point coordinates serve as position coordinates of the radiation source.
The x axis, the y axis and the z axis refer to three coordinate axes of a geocentric coordinate system, the geocentric coordinate system is simply called as a geocentric coordinate system, the geocentric coordinate system is a coordinate system with an origin O, the z axis and the geocentric axis are parallel to point to a north pole, the x axis points to an intersection point of an initial meridian and an equator, the y axis is perpendicular to an xOz plane, and the x axis, the y axis and the z axis form a right-hand coordinate system.
Further, the preset iteration number is N, and the value of N is 1000 times (the preset value of the present invention is 1000, which can be set by itself).
Furthermore, in the fourth step, the time interval Δ t is 1 min. The experiment is set according to actual conditions.
The invention has the beneficial effects that: the invention provides a single-satellite positioning method based on GP iterative extrapolation, which optimizes the traditional GP iterative extrapolation algorithm and the Doppler single-satellite positioning algorithm. Because the received signal of the satellite is a non-stationary signal, the significant characteristics of the non-stationary signal cannot be analyzed only by Fourier transform, the angle with the most concentrated signal can be selected for analysis by applying fractional Fourier transform, and the signal received by the satellite is reasonably extrapolated by adopting fractional Fourier transform, so that the signal is more band-limited in a transform domain compared with the traditional Fourier transform, and a proper filter is selected favorably and a better extrapolation effect is achieved. Meanwhile, under the requirement of ensuring certain positioning accuracy, more Doppler information is obtained by applying signal extrapolation, the complexity problem of the positioning process is relieved, the single-satellite positioning time and the satellite flight time are reduced, and the satellite flight cost is reduced.
Drawings
FIG. 1 is a flow chart of single-star positioning based on iterative extrapolation of the GP algorithm;
FIG. 2 is a schematic diagram of a first extrapolation process for a sinc signal;
FIG. 3 is a schematic diagram of an nth extrapolation process for a sinc signal;
FIG. 4 is a flow chart of iterative extrapolation of satellite observation signals;
taking the example of applying fourier transform extrapolation to a sinc signal, and analogy is to extrapolation of the observed signal to a satellite. In fig. 2, zero padding is performed on the finite length observation signal G (t), and the signal is mapped into the frequency domain through discrete fourier transform DFT, so as to obtain a frequency domain discrete signal G (ω).
Then, the signal in the transformed mapping domain is cut off by a low-pass filter, and only the signal F1 (omega) in the limited bandwidth [ -sigma, sigma ] is reserved; and then performing Inverse Discrete Fourier Transform (IDFT) on the F1 (omega), widening the signal length, obtaining a signal F1(T), namely, an extrapolated signal appears outside an observation region (-T, T), and performing IDFT on the K-point frequency domain discrete signal F1 (omega) to obtain a time domain discrete signal F1 (T).
And replacing the corresponding position in the extrapolated signal with the observed signal g (T), and keeping the information of the observed signal while obtaining the extrapolated signal outside the observed region (-T, T). As can be seen from the flow of fig. 4, the extrapolation of the received signal of the satellite is also according to the flows of fig. 2 and 3, and it is necessary to replace DFT and IDFT by fractional fourier transform of order p (FRFT) and fractional fourier transform of order-p, where the fractional fourier transform of order p is expressed as:
Figure BDA0002514005490000051
wherein:
Figure BDA0002514005490000052
FIG. 5 is a graph of the extrapolation effect of a simulated satellite received signal;
according to the extrapolation flow chart of the satellite receiving signals shown in fig. 4, firstly, zero padding is carried out on the satellite receiving signals, a certain iteration number is preset, in each iteration, p-order fractional fourier transform (FRFT) is firstly carried out on the signals, then, a low-pass filter (H) is carried out on the signals, then, p-order fractional Fourier transform (FRFT) is carried out on the filtered signals, then, the part of the signals obtained by transformation in a support set phi is replaced by known observation signals, and the current iteration is finished; through continuous iteration, a signal with a complete length is reconstructed. In fig. 5, the first half is the known observed signal and the second half is the extrapolated signal.
FIG. 6 is a time-frequency effect graph after time-frequency analysis of an extrapolated reconstructed signal;
FIG. 6 shows the time-frequency function of the extrapolated and reconstructed signal, and the Doppler shift ω generated by the relative motion between the satellite and the radiation source can be obtained by setting a certain time interval for samplingjAs one of the input values to the doppler single star positioning algorithm.
FIG. 7 is the reciprocal of the frequency error at each grid point when the grid is first divided for the estimated position of the radiation source;
FIG. 8 is the reciprocal of the frequency error at each grid point when the grid is divided for the second time for the estimated position of the radiation source;
when the satellite receives a signal from the radiation source, the approximate position of the radiation source can be obtained according to direction finding information and the like, the approximate position of the radiation source is subjected to grid division, and the frequency measurement error is calculated, wherein twice division is selected, the grid of the second time is finer than that of the first time, and the position with the minimum frequency measurement error is the positioned position in the doppler single satellite positioning algorithm, so the peak position in fig. 7 and 8 is the calculated position.
Fig. 9a) is a schematic view of the actual position of the radiation source;
FIG. 9b) is a schematic view of the calculated final position of the radiation source;
when the grid density meets a certain requirement, outputting points which enable the modulus value of the frequency measurement error to obtain the minimum value on two sides of the satellite subsatellite point track respectively, wherein the points are (39.9299 degrees N, 116.3879 degrees E) and (33.8964 degrees N, 120.6765 degrees E). The ground radiation source set in the satellite simulation software and two positioning results obtained through calculation are marked on a map, and the results are shown in fig. 9a) and fig. 9b), Target1(39.9289 ° N, +116.3883 ° E) is the actual position of the radiation source, Target3(39.9299 ° N, 116.3879 ° E) is the position of the positioned radiation source, and Target2(33.8964 ° N, 120.6765 ° E) is a mirror image (fuzzy) point and needs to be removed.
Through the analysis, the GP algorithm iterative extrapolation is introduced into the Doppler single-satellite positioning method, so that the single-satellite positioning time and the flight cost of the satellite can be reduced under the condition of meeting the requirement of certain positioning precision, and the time saved by the whole algorithm can be determined according to the extrapolated zero padding length and time; and the complexity of the positioning process can be reduced.
The parameters of the simulation process are set as follows:
the duration of the original signal g (T) is [0, T ], T ═ 6min, the observed position is the first half, the duration of the zero padding portion is [ T,2T ], T ═ 6 min;
the actual position of the radiation source is (39.9289 degrees N, +116.3883 degrees E), the longitude and latitude density of the first grid division is 1 degree, the longitude and latitude density of the second grid division is 0.0001 degree, the frequency measurement time interval is 60 seconds, the Doppler frequency precision is 1Hz, and the satellite position precision is 10 degrees-3m, satellite velocity accuracy of 10-3m/s。
Detailed Description
First embodiment, a single satellite positioning method based on GP algorithm iterative extrapolation is described with reference to fig. 2, fig. 3 and fig. 4, where for the sake of more intuitive and clear description, a sinc signal is extrapolated by fourier transform, and the method is analogous to extrapolation of a received signal to a satellite, and includes the following steps:
step one, carrying out zero filling on an observation signal l to a required time length to obtain an initialized extrapolation signal g and a support set phi;
and step two, performing iterative extrapolation on g. Presetting certain iteration times, performing DFT on g during each iteration, then performing low-pass filter H, then performing IDFT on the filtered signal, replacing the part of the signal obtained by inverse transformation in the support set phi with a known observation signal l, updating g, and finishing the current iteration; through continuous iteration, a signal f with a complete length is reconstructed.
Step three, as can be seen from fig. 4, the extrapolation of the received signal of the satellite is also according to the procedure of step two, but the DFT and IDFT need to be replaced by FRFT of order p and order-p. And then, performing time-frequency analysis on the signal f meeting the preset iteration condition, and sampling according to a certain time interval to obtain the Doppler frequency of the satellite receiving signal at different moments.
And step four, obtaining the position and the speed of each moment of the satellite according to the satellite ephemeris and the frequency measurement time, and enabling the position with the minimum frequency measurement error to be the final position of the earth surface radiation source of the single-satellite positioning according to the Doppler single-satellite positioning algorithm.
The satellite ephemeris information obtained by the satellite simulation software is shown in table 1:
TABLE 1
Figure BDA0002514005490000071
Second embodiment this embodiment will be described with reference to fig. 1. In the embodiment, a single-satellite positioning method based on GP algorithm iterative extrapolation is implemented by the following steps:
A1. zero filling is carried out on the observation signal l until the required time length is reached, and an initialized extrapolation signal g and a support set phi are obtained;
A2. presetting the total iteration times, and reconstructing a signal f with the complete length by continuously iterating the initialized extrapolated signal g through an extrapolation module.
For the extrapolation of the observed signal, the ith implementation is:
B1. initializing the extrapolated signal to obtain gi(ii) a Setting iteration times to obtain a support set phii；
B2. At the i-th iteration, for giPerforming fractional Fourier transform (FRFT) of order p to obtain fractional domain form Gi；
B3. G is to beiThrough a low-pass filter HdObtaining the signal by low-pass filtering
Figure BDA0002514005490000081
B4. To pair
Figure BDA0002514005490000082
Performing-p fractional Fourier transform (FRFT);
B5. the signals obtained by B4 are arranged in a support set phiiInner part is replaced by giTo obtain gi+1The current iteration is finished; if the iteration end condition is not met, returning to the step B2, otherwise, outputting the repeatForm signal f ═ gi+1。
C1. And carrying out time-frequency analysis on the output reconstruction signal f. Setting a time interval, and sampling the Doppler frequency for M times to be used as one of input values for the next step of single-satellite positioning calculation;
C2. at the M sampling times determined by C1, the positions (x) of the satellites at these times are determined from the satellite ephemerisi,yi,zi) And speed
Figure BDA0002514005490000083
To obtain
Figure BDA0002514005490000084
Equation of reference model combined with earth surface
Figure BDA0002514005490000085
Can obtain
Figure BDA0002514005490000086
C3. According to the Doppler single star positioning algorithm, the frequency measurement error is causediThe minimum position is the final position of the earth surface radiation source positioned by the single star.
Figure BDA0002514005490000087
The above-described calculation examples of the present invention are merely to explain the calculation model and the calculation flow of the present invention in detail, and are not intended to limit the embodiments of the present invention. It will be apparent to those skilled in the art that other variations and modifications of the present invention can be made based on the above description, and it is not intended to be exhaustive or to limit the invention to the precise form disclosed, and all such modifications and variations are possible and contemplated as falling within the scope of the invention.

Claims (5)

1. A single-satellite positioning method based on GP iterative extrapolation is characterized by comprising the following steps:
firstly, when a satellite flies above a ground surface radiation source, receiving a signal g (t) sent by the radiation source, and taking the g (t) as an observation signal received by the satellite;
secondly, carrying out post zero filling processing on observation signals g (t) received by the satellite in the first step, and recording signals of a zero filling section as p (t);
step three, initializing the signal to be extrapolated to be g0(t)，g0(t) is composed of observation signal g (t) and zero-padding segment signal p (t); initializing the support set to Φ0，Φ0Is a support set containing and only containing observation signal g (t) and zero-padding segment signal p (t);
presetting iteration number as N, and for initialized signal g to be extrapolated0(t) carrying out iterative extrapolation until a preset iteration number is met, and stopping extrapolation to obtain an extrapolated signal f (t);
fourthly, performing time-frequency analysis on the externally pushed signal f (t) to obtain a time-frequency function of the signal f (t);
sampling the Doppler frequency of the time-frequency function for M times according to the time interval delta t to respectively obtain Doppler frequency estimated values of satellite receiving signals at each sampling moment;
and fifthly, obtaining the position and the speed of the satellite at each sampling moment according to the satellite ephemeris, and solving the position of the earth surface radiation source by utilizing the position and the speed of the satellite at each sampling moment and the Doppler frequency estimation value of the satellite receiving signal at each sampling moment.
2. The GP iterative extrapolation-based single satellite positioning method according to claim 1, wherein the specific process of the third step is as follows:
step three, initializing the signal to be extrapolated to be g0(t) initializing the support set to Φ0；
Step three, initializing the signal g to be extrapolated in the step three0(t) performing a fractional Fourier transform of order p, the signal obtained after the fractional Fourier transform of order p being B g0(t)]；
Step three, the signal B [ g ] obtained in the step three0(t)]Filtering to obtain filtered signal B [ f0(t)]；
Step three and four, the filtered signal B [ f ] obtained in the step three is subjected to filtering0(t)]Carrying out-p fractional Fourier transform to obtain a mapping signal f in the time domain1(t)；
Step three and five, initializing a support set phi0In step three, a signal g to be extrapolated is initialized0(t) for the signal f obtained in step three or four1(t) replacing the corresponding time domain part to obtain a signal g obtained by the first iteration in the time domain1(t)；
The alternative method comprises the following steps:
Figure FDA0002514005480000021
wherein B represents the entire time domain;
at the same time, to the initialized support set phi0Updating to obtain a support set phi obtained by the first iteration1Support set phi1Is composed of only signal g1(t) the supporting set;
step three six, the signal g obtained in the step three five is used1(t) assigning to the initialized signal to be extrapolated in step three-one, and assigning the support set phi1Assigning values to the initialization support set to repeat the processes from the third step to the third step until the preset iteration number N is met, and stopping iteration to obtain a signal g obtained by the Nth iterationN(t) comparing the signal g obtained in the Nth iterationN(t) as the extrapolated signal f (t).
3. The GP iterative extrapolation-based single satellite positioning method according to claim 2, wherein the concrete process of the step five is as follows:
the grid is divided over a range of positions including the radiation source to obtain a set of grid points ∑, (x)k,yk,zk)∈∑，(xk,yk,zk) Is a netGrid point coordinates within grid point set ∑;
Figure FDA0002514005480000022
wherein the intermediate variable
Figure FDA0002514005480000023
(xj,yj,zj) Is the position of the satellite at the jth sample time,
Figure FDA0002514005480000024
is the velocity of the satellite at the jth sampling instant, c represents the speed of light, ωjAn estimate of the doppler frequency representing the signal received by the satellite at the jth sampling instant, ω being the frequency of the signal emitted by the radiation source,jfor frequency error, vector, of jth sampling instant
Figure FDA0002514005480000025
rjIs a vector
Figure FDA0002514005480000026
1,2, …, M;
rewrite equation (2) to the following form:
Figure FDA0002514005480000027
will be provided with
Figure FDA0002514005480000028
Abbreviated as gj(xk,yk,zk) Then, then
ωj＝ωgj(xk,yk,zk)+j(4)
And (3) arranging the formula (4) corresponding to all M sampling points into a matrix form as follows:
Ω＝Gω+E (5)
wherein Ω, G and E are intermediate variables, and Ω ═ ω1,ω2,…,ωM]T，E＝[1,2,…,M]T，G＝[g1(xk,yk,zk),g2(xk,yk,zk),…,gM(xk,yk,zk)]TThe superscript T represents the transpose of the matrix;
to ensure the E luminance2At a minimum, an objective function J (ω) is established:
J(ω)＝||Ω-Gω||2(5)
wherein | | | E | | | represents the norm of E, such that
Figure FDA0002514005480000031
Then
Figure FDA0002514005480000032
In the formula, the upper corner mark-1 represents the inverse of the matrix,
Figure FDA0002514005480000033
represents ω when the objective function J (ω) takes a minimum value;
the coordinates of each grid point in the grid point set ∑ are respectively substituted into the formula (6), and the coordinates corresponding to each grid point are sequentially obtained
Figure FDA0002514005480000034
Will minimize the value of the objective function J (ω)
Figure FDA0002514005480000035
The corresponding grid point coordinates are used as the position coordinates of the radiation source.
4. The single-satellite positioning method based on GP iterative extrapolation according to claim 3, wherein the preset number of iterations is N, and the value of N is 1000.
5. The single-satellite positioning method based on GP iterative extrapolation according to claim 4, wherein in step four, the time interval Δ t is 1 min.
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