CN111679296B - Single-star positioning method based on GP iterative extrapolation - Google Patents
Single-star positioning method based on GP iterative extrapolation Download PDFInfo
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Abstract
A single-star positioning method based on GP iterative 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-star positioning process by adopting the traditional Doppler positioning method. The invention optimizes the traditional GP iterative extrapolation algorithm and Doppler single-star positioning algorithm. The signal received by the satellite is reasonably extrapolated by adopting fractional Fourier transform, so that the signal is more limited in a transform domain, and a proper filter is favorably selected and a better extrapolation effect is achieved. Meanwhile, under the requirement of ensuring certain positioning precision, more Doppler information is obtained by applying signal extrapolation, the complexity problem of the positioning process is alleviated, the time of single-satellite positioning and the satellite flight time are reduced, and the satellite flight cost is reduced. The invention can be applied to single star positioning.
Description
Technical Field
The invention belongs to the field of signal processing, and particularly relates to a single-star positioning method based on GP (Gerchberg-Papoulis) iterative extrapolation.
Background
The single-star positioning has wide application in military target positioning and civil navigation, and the signal extrapolation has important significance for communication, radar, and recovery of voice and video signals. In the single-satellite positioning, if the partial signal g (t) received by the satellite (g (t) is a partial signal in the complete signal f (t) received by the satellite above and nearby the radiation source) is used for obtaining f (t) data of the whole positioning process, the time of single-satellite positioning can be effectively shortened, and the flight cost of the satellite can be reduced.
Most signals in the field of signal processing can be mapped into a domain by some transformation where the processing of the signal can detect and extract certain properties of the signal. For the extrapolation problem of signals, the Gerchberg-Papoulis extrapolation algorithm (GP algorithm) uses the perfect orthogonality of the long sphere function in sigma-band limited space, and iterates out the signals outside the known interval by repeatedly applying Fourier transform and inverse transform and cutting off and replacing. The Sanz-Huang theory is a discrete estimation theory proposed based on the GP algorithm, so that the GP algorithm can be implemented with DFT. The method for extrapolating the discrete signal iteration can obtain the extrapolating signal with decreasing error in effective iterations, is simple and convenient to calculate, and can effectively improve the extrapolated data and signal quality.
For the single star positioning problem, r.j. Webster et al propose a method for measuring the frequency of the incoming wave signal of the target radiation source using a single observation station to achieve positioning thereof. Essentially, a single observation station locates by measuring Doppler information of the relative motion of the target and the observation station in the signal frequency. Although the single-star positioning can be completed by adopting the Doppler positioning method, the Doppler positioning method has the problems of long positioning time and long satellite flight time, and the long satellite flight time brings about the problem of high flight cost.
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-star positioning process by adopting the traditional Doppler positioning method, and provides a single-star positioning method based on GP iterative 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 surface radiation source through the original observation signals and signals obtained by extrapolation, wherein 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:
step one, when a satellite flies near the upper air of a ground surface radiation source (the specific flying position is required to be determined according to the setting of an orbit and the flying condition of an actual satellite), receiving a part of signals g (t) sent by the radiation source, and taking the g (t) as observation signals received by the satellite;
step two, performing post-zero-filling processing on the observation signal g (t) received by the satellite in the step one, namely, performing post-zero-filling on the received observation signal g (t) to a required time length, wherein the length of a zero-filling section signal can be selected to be more than 1 time of the observation signal g (t), and the signal of the zero-filling section is recorded as p (t);
initializing the signal to be extrapolated as g 0 (t)，g 0 (t) is composed of an observation signal g (t) and a zero padding section signal p (t); initializing a support set to phi 0 ，Φ 0 Is a support set containing and only containing the observation signal g (t) and the zero padding section signal p (t);
presetting the iteration number as N, and for initialized signal g to be extrapolated 0 (t) performing iterative extrapolation until a preset number of iterations is met, stopping to obtain an extrapolated signal f (t);
step four, performing time-frequency analysis on the externally deduced signal f (t) to obtain a time-frequency function of the signal f (t);
sampling the time-frequency function for M times according to the time interval delta t to obtain Doppler frequency estimated values of satellite received signals at each sampling moment respectively;
and fifthly, obtaining the position and the speed of the satellite at each sampling time according to the satellite ephemeris, and solving the position of the surface radiation source by utilizing the position and the speed of the satellite at each sampling time and the Doppler frequency estimated value of the satellite received signal at each sampling time.
Further, the specific process of the third step is as follows:
step three, initializing the signal to be extrapolated to g 0 (t) initializing a support set to Φ 0 ；
Step III, the first step in step IIIInitializing the signal to be extrapolated g 0 (t) performing p-order fractional Fourier transform to obtain signal B [ g ] 0 (t)]；
Step III, the signal B [ g ] obtained in step III 0 (t)]Filtering to obtain filtered signal Bf 0 (t)]；
Step three and four, filtering the filtered signal B [ f ] obtained in step three 0 (t)]Performing-p-order fractional Fourier transform to obtain a mapping signal f in the time domain 1 (t)；
Step three, five, initializing a support set phi 0 In, initializing the signal g to be extrapolated in the third step 0 (t) for the signal f obtained in step III and IV 1 Replacing the corresponding time domain part of (t) to obtain a signal g obtained by the first iteration in the time domain 1 (t)；
The substitution method comprises the following steps:
wherein B represents the whole time domain range;
at the same time, for initializing the support set phi 0 Updating to obtain a support set phi obtained by the first iteration 1 Support set phi 1 To include and only include signal g 1 The support set of (t);
step III, the signal g obtained in the step III and the step V 1 (t) assigning an initialization signal to be extrapolated in step three, to set the support set Φ 1 Assigning values to the initialization support set to repeat the processes from the third step to the second step until the preset iteration times N are met, stopping iteration, and obtaining a signal g obtained by the Nth iteration N (t) the signal g obtained in the nth iteration N (t) as an extrapolated signal f (t).
Further, the specific process of the fifth step is as follows:
meshing within a range of positions containing the radiation source to obtain a set of meshing points Σ, (x) k ,y k ,z k )∈∑，(x k ,y k ,z k ) Coordinates of grid points in the grid point set sigma;
wherein the intermediate variable
(x j ,y j ,z j ) Is the satellite position at the j-th sampling instant, where>
Is the satellite velocity at the j-th sampling instant, c represents the speed of light, ω j Represents the Doppler frequency estimation value of satellite received signal at the j-th sampling moment, omega is the signal frequency transmitted by the radiation source, epsilon j Vector +.>
r j Is vector->
J=1, 2, …, M;
the formula (2) is rewritten as follows:
will be
Simply denoted as g j (x k ,y k ,z k ) Then
ω j ＝ωg j (x k ,y k ,z k )+ε j (4)
The formula (4) corresponding to all M sampling points is arranged into the following matrix form:
Ω＝Gω+E (5)
wherein Ω, G, and E are all intermediate variables, and Ω= [ ω ] 1 ,ω 2 ,…,ω M ] T ，E＝[ε 1 ,ε 2 ,…,ε M ] T ，G＝[g 1 (x k ,y k ,z k ),g 2 (x k ,y k ,z k ),…,g M (x k ,y k ,z k )] T The upper corner mark T represents the transposition of the matrix;
to ensure E 2 Minimum, establish the objective function J (ω):
J(ω)＝||Ω-Gω|| 2 (5)
in the formula, |E|| represents the norm of E, let
Then
Wherein the upper subscript-1 represents the inverse of the matrix,
represents ω when the objective function J (ω) is made to take a minimum value;
substituting the coordinates of each grid point in the grid point set sigma into the formula (6) respectively, and sequentially obtaining the corresponding grid points
Will minimize the value of the objective function J (omega)>
The corresponding grid point coordinates are used as position coordinates of the radiation source.
The x-axis and the y-axis refer to three coordinate axes of a geocentric geodetic coordinate system, the geodetic coordinate system is simply called a geodetic coordinate system, the geodetic is taken as an origin O, the z-axis and the geodetic axis are parallel to each other and point to a north pole, the x-axis points to an intersection point of the primary meridian and the 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 (1000 is set in the invention and can be set by oneself).
Further, in the fourth step, the time interval Δt is 1min. The experiment is set according to actual conditions.
The beneficial effects of the invention are as follows: the invention provides a single-star positioning method based on GP iterative extrapolation, which optimizes the traditional GP iterative extrapolation algorithm and Doppler single-star positioning algorithm. Because the received signal of the satellite is a non-stationary signal, the Fourier transform is only insufficient to analyze the remarkable characteristics of the signal, the most concentrated angle of the signal can be selected to analyze by using the fractional Fourier transform, and the signal received by the satellite is reasonably extrapolated by using the fractional Fourier transform, compared with the traditional Fourier transform, the signal is more limited in the transform domain, and the selection of a proper filter and the better extrapolation effect are facilitated. Meanwhile, under the requirement of ensuring certain positioning precision, more Doppler information is obtained by applying signal extrapolation, the complexity problem of the positioning process is alleviated, the time of single-satellite positioning and the satellite flight time are reduced, and the satellite flight cost is reduced.
Drawings
FIG. 1 is a single star localization flow diagram 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 illustration of the nth extrapolation process for a sinc signal;
FIG. 4 is a flow chart of iterative extrapolation of satellite observations;
taking the example of applying fourier transform extrapolation to the sinc signal, and analogizing to the extrapolation of the satellite's observed signal. In fig. 2, the finite length observation signal G (t) is zero-padded, and is mapped to the frequency domain by discrete fourier transform DFT to obtain a frequency domain discrete signal G (ω).
Then the signals in the mapping domain after transformation are cut off by a low-pass filter, and only the signals F1 (omega) in the limited bandwidth [ -sigma, sigma ] are reserved; then, performing Inverse Discrete Fourier Transform (IDFT) on the F1 (omega), and widening the signal length to obtain a signal F1 (T), namely, obtaining an external extrapolation signal outside an observation area (-T, T), wherein the implementation process is that performing IDFT on the K-point frequency domain discrete signal F1 (omega) to obtain a time domain discrete signal F1 (T).
And then the corresponding position in the extrapolated signal is replaced by the observed signal g (T), the information of the observed signal is reserved, and the extrapolated signal outside the observation area (-T, T) is obtained. As can be seen from the flow of fig. 4, the extrapolation of the received signal of the satellite is also performed according to the flows of fig. 2 and 3, where DFT and IDFT are replaced by a p-order fractional fourier transform (FRFT) and a p-order fractional fourier transform, the expression of the p-order fractional fourier transform being:
wherein:
FIG. 5 is an extrapolation effect diagram of a satellite-simulated received signal;
according to the extrapolation flow chart of the satellite receiving signals shown in fig. 4, zero padding is performed on the satellite receiving signals, a certain number of iterations are preset, each iteration is performed on the signals, the signals are subjected to p-order fractional Fourier transform (FRFT), then pass through a low-pass filter (H), the filtered signals are subjected to p-order fractional Fourier transform (FRFT), and then the part of the signals obtained by the transformation in a supporting set phi is replaced by known observation signals, and the current iteration is finished; through continuous iteration, the signal with 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 graph of time-frequency effects of the time-frequency analysis of the extrapolated reconstructed signal;
FIG. 6 shows the time-frequency function of the extrapolated reconstructed signal, sampled at time intervals to obtain the multiple due to the relative motion between the satellite and the radiation sourceThe Doppler shift omega j As one of the input values to the doppler single star positioning algorithm.
FIG. 7 is the inverse of the frequency error measured at each grid point when the first grid division is performed on the estimated position of the radiation source;
FIG. 8 is a plot of the inverse frequency error for each grid point for a second meshing of the estimated position of the radiation source;
when the satellite receives the signal sent by the radiation source, the approximate position of the radiation source can be obtained according to the direction finding information and the like, the approximate position of the radiation source is subjected to grid division, the frequency finding error is calculated, the two divisions are selected, the grid of the second division is finer than the grid of the first division, and the position with the minimum frequency finding error is the position where the signal is located in the Doppler single star positioning algorithm, so the peak position in fig. 7 and 8 is the calculated position.
FIG. 9 a) is a schematic illustration of the actual position of the radiation source;
FIG. 9 b) is a schematic representation of the resolved final position of the radiation source;
when the grid density meets a certain requirement, outputting points which respectively obtain minimum values of the modulus values of the frequency measurement errors at two sides of the satellite understar point track, wherein the minimum values are (39.9299 DEG N,116.3879 DEG E) and (33.8964 DEG N,120.6765 DEG E). The ground radiation source set in satellite simulation software and two positioning results obtained through calculation are marked on a map, and the results are shown in fig. 9 a) and 9 b), wherein Target1 (39.9289 degrees N, +116.3883 degrees E) is the actual position of the radiation source, target3 (39.9299 degrees N,116.3879 degrees E) is the positioned radiation source position, and Target2 (33.8964 degrees N,120.6765 degrees E) is a mirror image (fuzzy) point and needs to be removed.
According to the analysis, in the Doppler single-star positioning method, GP algorithm iterative extrapolation is introduced, single-star positioning time and satellite flight cost can be reduced under the condition that the requirement of certain positioning precision is met, and time saving of the whole algorithm can be determined according to extrapolated zero-filling 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=6 min, the observation position is the first half, the duration of the zero-padding part is [ T,2T ], t=6 min;
the actual position of the radiation source is (39.9289 DEG N, +116.3883 DEG E), the longitude and latitude density of the first grid division is 1 DEG, the longitude and latitude density of the second grid division is 0.0001 DEG, the frequency measurement time interval is 60 seconds, the Doppler frequency precision is 1Hz, and the satellite position precision is 10 DEG -3 m, satellite speed accuracy 10 -3 m/s。
Detailed Description
In a first embodiment, a single-star positioning method based on iterative extrapolation of GP algorithm is described with reference to fig. 2, 3 and 4, where for more visual and clear description, taking sine signal application fourier transform extrapolation as an example, and analogizing extrapolation of received signals to satellites, it includes the following steps:
step one, zero padding is carried out on an observation signal l until the required time length is reached, and an initialization extrapolation signal g and a support set phi are obtained;
and step two, performing iterative extrapolation on the g. Presetting a certain iteration number, carrying out DFT on g when each iteration is carried out, then carrying out IDFT on the filtered signal through a low-pass filter H, replacing the part of the signal obtained by inverse transformation in the supporting set phi with a known observation signal l, updating g, and ending the current iteration; by successive iterations, a full length signal f is reconstructed.
Step three, as can be seen from fig. 4, the extrapolation of the satellite received signal is also performed according to the procedure of step two, but the DFT and IDFT need to be replaced by the FRFT of the p-order and-p-order. And then carrying out time-frequency analysis on the signal f meeting the preset iteration condition, and sampling according to a certain time interval to obtain Doppler frequencies of satellite received signals at different moments.
And fourthly, obtaining the position and the speed of the satellite at each moment according to the satellite ephemeris and the frequency measurement time, and obtaining the position with the minimum frequency measurement error as the final position of the earth surface radiation source of single-star positioning according to the Doppler single-star positioning algorithm.
Satellite ephemeris information obtained by satellite simulation software is shown in table 1:
TABLE 1
The second embodiment will be described with reference to fig. 1. The single-star positioning method based on GP algorithm iterative extrapolation in the embodiment is realized by the following steps:
A1. zero padding is carried out on the observation signal l until the required time length is reached, and an initialization extrapolation signal g and a support set phi are obtained;
A2. the total iteration times are preset, the initialized extrapolation signal g passes through an extrapolation module, and a signal f with the complete length is reconstructed through continuous iteration.
For extrapolation of the observed signal, the ith implementation procedure is:
B1. initializing an extrapolation signal to obtain g i The method comprises the steps of carrying out a first treatment on the surface of the Setting iteration times to obtain a support set phi i ；
B2. At the ith iteration, for g i Performing a p-order fractional Fourier transform FRFT to obtain a fractional domain form G i ；
B3. Will G i Through a low-pass filter H d Low-pass filtering to obtain signal
B4. For a pair of
Performing-p-order fractional Fourier transform FRFT;
B5. the signal obtained by B4 is arranged in a support set phi i The part in is replaced by g i Obtain g i+1 Ending the current iteration; if the iteration termination condition is not satisfied, returning to the step B2, otherwise, outputting a reconstruction signal f=g i+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 of single-star positioning calculation in the next step;
C2. at M sampling instants determined at C1, the position (x) of the satellite at these instants is determined from the satellite ephemeris i ,y i ,z i ) Speed and velocity
Obtaining
Combining earth reference model equations
Can get +.>
C3. According to Doppler single-star positioning algorithm, the frequency measurement error epsilon is caused i The minimum position is the final position of the earth surface radiation source positioned by the single star.
The above examples of the present invention are only for describing the calculation model and calculation flow of the present invention in detail, and are not limiting of the embodiments of the present invention. Other variations and modifications of the above description will be apparent to those of ordinary skill in the art, and it is not intended to be exhaustive of all embodiments, all of which are within the scope of the invention.
Claims (3)
1. The single-star positioning method based on GP iterative extrapolation is characterized by comprising the following steps of:
step one, 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;
step two, performing post-zero padding treatment on the observation signal g (t) received by the satellites in the step one, and marking the signal of the zero padding section as p (t);
initializing the signal to be extrapolated as g 0 (t)，g 0 (t) is composed of an observation signal g (t) and a zero padding section signal p (t); initializing a support set to phi 0 ，Φ 0 Is a support set containing and only containing the observation signal g (t) and the zero padding section signal p (t);
presetting the iteration number as N, and for initialized signal g to be extrapolated 0 (t) performing iterative extrapolation until a preset number of iterations is met, stopping to obtain an extrapolated signal f (t);
the specific process of the third step is as follows:
step three, initializing the signal to be extrapolated to g 0 (t) initializing a support set to Φ 0 ；
Step three, initializing the signal g to be extrapolated in step three 0 (t) performing p-order fractional Fourier transform to obtain signal B [ g ] 0 (t)]；
Step III, the signal B [ g ] obtained in step III 0 (t)]Filtering to obtain filtered signal Bf 0 (t)]；
Step three and four, filtering the filtered signal B [ f ] obtained in step three 0 (t)]Performing-p-order fractional Fourier transform to obtain a mapping signal f in the time domain 1 (t)；
Step three, five, initializing a support set phi 0 In, initializing the signal g to be extrapolated in the third step 0 (t) for the signal f obtained in step III and IV 1 Replacing the corresponding time domain part of (t) to obtain a signal g obtained by the first iteration in the time domain 1 (t)；
The substitution method comprises the following steps:
wherein B represents the whole time domain range;
at the same time, for initializing the support set phi 0 Updating to obtain a support set phi obtained by the first iteration 1 Support set phi 1 To include and only include signal g 1 The support set of (t);
step III, the signal g obtained in the step III and the step V 1 (t) assigning an initialization signal to be extrapolated in step three, to set the support set Φ 1 Assigning values to the initialization support set to repeat the processes from the third step to the second step until the preset iteration times N are met, stopping iteration, and obtaining a signal g obtained by the Nth iteration N (t) the signal g obtained in the nth iteration N (t) as an extrapolated signal f (t);
step four, performing time-frequency analysis on the externally deduced signal f (t) to obtain a time-frequency function of the signal f (t);
sampling the time-frequency function for M times according to the time interval delta t to obtain Doppler frequency estimated values of satellite received signals at each sampling moment respectively;
step five, obtaining the position and the speed of the satellite at each sampling time 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 time and the Doppler frequency estimated value of the satellite received signal at each sampling time;
the specific process of the fifth step is as follows:
dividing grids in a position range containing a radiation source, wherein the longitude and latitude density of grid division is 0.0001 degrees, and a grid point set sigma (x) is obtained k ,y k ,z k )∈∑，(x k ,y k ,z k ) Coordinates of grid points in the grid point set sigma;
wherein the intermediate variable
(x j ,y j ,z j ) Is the satellite position at the j-th sampling instant, where>
Is the satellite velocity at the j-th sampling instant, c represents the speed of light, ω j Represents the Doppler frequency estimation value of satellite received signal at the j-th sampling moment, omega is the signal frequency transmitted by the radiation source, epsilon j Vector +.>
r j Is vector->
J=1, 2, …, M;
the formula (2) is rewritten as follows:
will be
Simply denoted as g j (x k ,y k ,z k ) Then
ω j ＝ωg j (x k ,y k ,z k )+ε j (4)
The formula (4) corresponding to all M sampling points is arranged into the following matrix form:
Ω＝Gω+E (5)
wherein Ω, G, and E are all intermediate variables, and Ω= [ ω ] 1 ,ω 2 ,…,ω M ] T ，E＝[ε 1 ,ε 2 ,…,ε M ] T ，G＝[g 1 (x k ,y k ,z k ),g 2 (x k ,y k ,z k ),…,g M (x k ,y k ,z k )] T The upper corner mark T represents the transposition of the matrix;
to ensure E 2 Minimum, establish the objective function J (ω):
J(ω)＝||Ω-Gω|| 2 (5)
in the formula, |E|| represents the norm of E, let
Then
Wherein the upper subscript-1 represents the inverse of the matrix,
represents ω when the objective function J (ω) is made to take a minimum value;
substituting the coordinates of each grid point in the grid point set sigma into the formula (6) respectively, and sequentially obtaining the corresponding grid points
Will minimize the value of the objective function J (omega)>
The corresponding grid point coordinates are used as position coordinates of the radiation source.
2. The single-star positioning method based on GP iteration extrapolation as claimed in claim 1, wherein the preset number of iterations is N and the value of N is 1000.
3. The method for positioning a single star based on GP iteration extrapolation according to claim 2, wherein in the fourth step, the time interval Δt takes a value of 1min.
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