CN112835116A - Method and system for judging function of geosynchronous orbit spin stabilization space target - Google Patents

Method and system for judging function of geosynchronous orbit spin stabilization space target Download PDF

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CN112835116A
CN112835116A CN202011562435.7A CN202011562435A CN112835116A CN 112835116 A CN112835116 A CN 112835116A CN 202011562435 A CN202011562435 A CN 202011562435A CN 112835116 A CN112835116 A CN 112835116A
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王阳
杜小平
李智
徐灿
周海俊
马志昊
宋一铄
段永胜
邢维艳
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Peoples Liberation Army Strategic Support Force Aerospace Engineering University
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Abstract

The invention discloses a method and a system for judging the function of a geosynchronous orbit spin stabilized space target, wherein the method comprises the following steps: the method comprises the following steps: judging whether the ground luminosity curve of the target has a sine structure or not, if not, initially inverting the rotating speed of the target by using a phase dispersion minimization method, and then entering the third step; if the structure has a sine structure, entering the step two; step two: judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, and if the ground luminosity curve of the target is continuous and the sampling rate is constant, primarily inverting the target rotating speed by using discrete Fourier transform; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis; step three: checking a preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target; step four: and judging the function of the target according to the actual rotating speed of the target in the step three. The invention effectively identifies the GEO target.

Description

Method and system for judging function of geosynchronous orbit spin stabilization space target
Technical Field
The invention belongs to the technical field of space target detection and identification, and particularly relates to a function judgment method and system for a geosynchronous orbit spin-stabilized space target.
Background
The ground-based optical observation system is affected by the atmosphere, the distance and the resolution, so that high-resolution imaging of high-Orbit space targets such as Geostationary Earth Orbit (GEO) targets is difficult to perform, and only a few light spots with pixel brightness changes can be presented. At present, a ground-based optical observation system can only give position, speed and brightness information of a target, and an effective means for carrying out feature recognition on a GEO target is not available. Therefore, although the ground-based telescope acquires massive photometric data, a large amount of time-sensitive data is not effectively utilized, and the data is not converted into effective information.
Disclosure of Invention
The technical problem solved by the invention is as follows: the method and the system for judging the function of the geosynchronous orbit spin stabilization space target are provided, and the GEO target is effectively subjected to characteristic identification.
The purpose of the invention is realized by the following technical scheme: a method for judging the function of a target in a geosynchronous orbit spin stabilization space, which comprises the following steps: the method comprises the following steps: judging whether the ground luminosity curve of the target has a sine structure or not, if not, initially inverting the rotating speed of the target by using a phase dispersion minimization method, and then entering the third step; if the structure has a sine structure, entering the step two; step two: judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, if so, primarily inverting the target rotating speed by using discrete Fourier transform, and then entering the third step; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis, and then entering the third step; step three: checking a preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target; step four: and judging the function of the target according to the actual rotating speed of the target in the step three.
In the fourth step, the target with the actual rotating speed of 6 rpm is taken as the early warning satellite, the target with the actual rotating speed of 50 rpm is taken as the communication satellite, the target with the actual rotating speed of 100 rpm is taken as the meteorological satellite, and the other rotating speeds are taken as the space debris.
In the method for determining a function of a geosynchronous orbit spin-stabilized space object, in the second step, the discrete fourier transform includes:
preset x (n) is a finite sequence, i.e.:
Figure BDA0002860795120000021
the discrete fourier transform of x (n) is:
Figure BDA0002860795120000022
wherein ,
Figure BDA0002860795120000023
n is the number of transform points, N is the length of discrete Fourier transform interval, X (k) is the discrete Fourier transform of finite length sequence x (N), k represents the kth point of Fourier transform, WNIs an intermediate variable.
In the method for judging the function of the target in the geosynchronous orbit spin stabilization space, in the second step, the least square spectrum analysis is realized by solving a Lomb-Scargle periodogram, and the method comprises the following steps:
presetting the observation time number M and the time tiCorresponding observed value is hiAnd if so, the mean value and variance of the observed value are as follows:
Figure BDA0002860795120000024
Figure BDA0002860795120000025
the time delay τ is defined as:
Figure BDA0002860795120000026
the Lomb-Scargle periodogram is:
Figure BDA0002860795120000031
wherein ,
Figure BDA0002860795120000032
is the mean value of the observed values, i is the observed time number, σ is the variance of the observed values, τ is the time delay, PNAnd (omega) is a Lomb-Scargle periodogram.
A function determination system for a target in a geosynchronous orbital spin stabilized space, comprising: the first module is used for judging whether the ground luminosity curve of the target has a sine structure or not, and if not, preliminarily inverting the rotating speed of the target by using a phase dispersion minimization method; the second module is used for judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, and if the ground luminosity curve of the target is continuous and the sampling rate is constant, the rotating speed of the target is preliminarily inverted by using discrete Fourier transform; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis; the third module is used for verifying the preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target; and the fourth module is used for judging the function of the target according to the actual rotating speed of the target in the third module.
In the function determination system for the target in the geosynchronous orbit spinning stable space, the target with the actual rotating speed of 6 rpm is the early warning satellite, the target with the actual rotating speed of 50 rpm is the communication satellite, the target with the actual rotating speed of 100 rpm is the meteorological satellite, and other rotating speeds are space fragments.
In the system for determining a function of a geosynchronous orbit spin-stabilized space object, the discrete fourier transform includes:
preset x (n) is a finite sequence, i.e.:
Figure BDA0002860795120000033
the discrete fourier transform of x (n) is:
Figure BDA0002860795120000034
wherein ,
Figure BDA0002860795120000035
n is the number of transform points, N is the length of discrete Fourier transform interval, X (k) is the discrete Fourier transform of finite length sequence x (N), k represents the kth point of Fourier transform, WNIs an intermediate variable.
In the system for determining a function of a target in a geosynchronous orbit spin stabilization space, the least square spectrum analysis is implemented by solving a Lomb-Scargle periodogram, and the method comprises the following steps:
presetting the observation time number M and the time tiCorresponding observed value is hiAnd if so, the mean value and variance of the observed value are as follows:
Figure BDA0002860795120000041
Figure BDA0002860795120000042
the time delay τ is defined as:
Figure BDA0002860795120000043
the Lomb-Scargle periodogram is:
Figure BDA0002860795120000044
wherein ,
Figure BDA0002860795120000045
is the mean value of the observed values, i is the observed time number, σ is the variance of the observed values, τ is the time delay, PNAnd (omega) is a Lomb-Scargle periodogram.
Compared with the prior art, the invention has the following beneficial effects:
(1) the invention is beneficial to improving the application benefit of the ground luminosity data. Currently, although a ground-based telescope acquires a large amount of photometric data, a large amount of time-sensitive data is not effectively utilized, and the data is not converted into effective information. The purpose of photometric observation is to identify the target, the invention can break the gap between the characteristics of the spin-stabilized GEO target and photometric data, and convert the photometric data into the knowledge "gold block".
(2) The invention is helpful for improving the observation capability of the foundation optical system. Due to the influence of distance and atmosphere, the difficulty of identifying a target in a high-orbit space by a ground-based optical observation system is higher, the traditional detection means and method cannot meet the requirement more and more, and equipment transformation or development is a process with a longer period.
(3) The method can provide new clues for space application such as space target threat assessment and the like. The target function is an important clue for target threat assessment, the method can determine the function of the GEO spin stabilization target based on the ground luminosity data, and has important significance for solving the problem of target threat assessment in space situation perception.
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Various other advantages and benefits will become apparent to those of ordinary skill in the art upon reading the following detailed description of the preferred embodiments. The drawings are only for purposes of illustrating the preferred embodiments and are not to be construed as limiting the invention. Also, like reference numerals are used to refer to like parts throughout the drawings. In the drawings:
FIG. 1 is a flow chart of a method for determining a function of a geosynchronous orbit spin stabilized space object according to an embodiment of the present invention;
FIG. 2 is a schematic illustration of a ground based photometric curve for a spin stabilized GEO target provided by an embodiment of the present invention;
FIG. 3 is a schematic diagram of the result of a discrete Fourier transform of a ground based photometric curve provided by an embodiment of the present invention;
FIG. 4 is a graphical illustration of the phase folding results of the photometric curve at a 4.17s folding period as provided by an embodiment of the present invention;
FIG. 5 is a graphical illustration of the phase folding results of the photometric curve at an 8.34s folding period as provided by an embodiment of the present invention.
Detailed Description
Exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be embodied in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art. It should be noted that the embodiments and features of the embodiments may be combined with each other without conflict. The present invention will be described in detail below with reference to the embodiments with reference to the attached drawings.
The photometric characteristics of a spatial object generally refer to the observed luminance of the object as a function of time, usually constant with a photometric curve. The luminosity curve is a time-varying curve of the brightness of the target observed by an observer, and the brightness of the target is generally expressed by a sight star or the like. Since the target apparent star, etc., is a function of the features of the target size, orientation, and surface properties, these features can be inverted by photometric curves.
GEO targets typically exhibit some regularity in their photometric curve in the time domain due to geostationary. The GEO target mainly has two stabilization modes of spin stabilization and three-axis stabilization, wherein the luminosity curve of the spin-stabilized target has certain periodicity, and the period is not changed basically along with time. The period of the spin-stabilized target photometric curve is independent of the shape of the target itself, and is only dependent on the actual rotation period of the target, which is typically equal to the target photometric curve period or 2 times the photometric curve period.
Fig. 1 is a flowchart of a method for determining a function of a geosynchronous orbit spin-stabilized space object according to an embodiment of the present invention. As shown in fig. 1, the method comprises the steps of:
the method comprises the following steps: judging whether the ground luminosity curve of the target has a sine structure or not, if not, initially inverting the rotating speed of the target by using a phase dispersion minimization method, and then entering the third step; if the structure has a sine structure, entering the step two;
step two: judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, if so, primarily inverting the target rotating speed by using discrete Fourier transform, and then entering the third step; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis, and then entering the third step;
step three: checking a preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target;
step four: and judging the function of the target according to the actual rotating speed of the target in the step three.
In the fourth step, the target with the actual rotating speed of 6 revolutions per minute is the early warning satellite, the target with the actual rotating speed of 50 revolutions per minute is the communication satellite, the target with the actual rotating speed of 100 revolutions per minute is the meteorological satellite, and other rotating speeds are the space debris.
Preliminary inversion of spin-stabilized GEO target rotational speed. For a photometric curve with continuous sampling and constant sampling rate, carrying out inversion by using discrete Fourier transform; for a ray profile with a non-constant sampling rate or with gaps, performing inversion using least squares spectral analysis; for photometric curves with non-sinusoidal structures, which fourier transform and least squares spectral analysis do not handle well, the inversion is performed using a phase dispersion minimization technique.
And (5) verifying the rotation speed of the spin-stabilized GEO target. The actual spin period of the object is determined using phase folding. Unlike the principle of discrete fourier transform, least squares spectral analysis, the frequency components calculated by minimizing the phase dispersion may not be the frequency components with a large proportion in the photometric curve. In addition, even if the maximum frequency component in the frequency spectrum is determined, the component is not necessarily equal to the actual rotational frequency of the target. In order to determine the actual rotation frequency of the target, it is ensured that the frequency components obtained by discrete fourier transform, least squares spectral analysis and phase dispersion minimization correspond to the photometric curve characteristics, and the photometric curve period is verified by a phase folding method.
And judging the function of the spin-stabilized GEO target. On the basis of the reversal of the target rotation speed, the function of the GEO target is further determined based on the correlation between the target spin cycle and the target function.
The invention relates to a specific implementation method of technical links such as discrete Fourier transform, least square spectrum analysis, phase dispersion minimization, phase folding, GEO target spin cycle and function correlation analysis and the like, which comprises the following steps:
1) discrete Fourier transform
For a photometric curve that is sampled continuously and at a constant sampling rate, discrete fourier transforms can be used to convert the discretely sampled time domain photometric curve into power spectra of the various frequency components of the curve, thereby identifying photometric curve periods. Let x (n) be a finite sequence, i.e.:
Figure BDA0002860795120000071
then the discrete Fourier transform of x (n)
Figure BDA0002860795120000072
wherein ,
Figure BDA0002860795120000081
2) least squares spectral analysis
In the case of a light curve having a non-constant sampling rate or gap, a least squares spectral analysis may be used to generate a power spectrum of the photometric curve. The method is similar to Fourier analysis, is a method for estimating a frequency spectrum based on least square fitting of a sine curve and a data sample, and can identify the hidden periodic signal frequency in non-equidistant sampling data. Least square spectrum analysis can be realized by solving a Lomb-Scargle periodogram, and assuming that the number of observation moments is M and the moment t is tiCorresponding observed value is hiAnd if so, the mean value and variance of the observed value are as follows:
Figure BDA0002860795120000082
Figure BDA0002860795120000083
the time delay τ is defined as:
Figure BDA0002860795120000084
the Lomb-Scargle periodogram is:
Figure BDA0002860795120000085
3) minimization of phase dispersion
For photometric curves with non-sinusoidal structures where fourier transform and least squares spectra do not work well, a phase dispersion minimization technique can be used to determine the curve period. The analysis code for minimizing phase dispersion is available at http:// www.stellingwerf.com/rfs-bin/index. cgigition ═ PageView & id ═ 29 website. The method searches for all possible cycles in the curve. During the retrieval, the entire photometric curve is divided into several parts according to the test period, and these parts are layered on top of each other. Thereafter, the stacked data is further subdivided into a series of data boxes. The variance of all data in these bins is calculated and compared to the total variance of the data set to produce a value between 0 and 1. Where 0 represents the best match and 1 represents the least match.
4) Phase folding
The frequency component calculated by the phase dispersion minimization may not be the frequency component having a large proportion in the photometric curve. In addition, even if the maximum frequency component in the frequency spectrum is determined, the component is not necessarily equal to the actual rotational frequency of the target. In order to determine the actual rotation frequency of the target and ensure that the frequency components obtained by discrete Fourier transform, least squares spectral analysis and phase dispersion minimization correspond to the photometric curve characteristics, the invention checks the photometric curve period by phase folding by taking the idea of phase dispersion minimization as a reference.
5) GEO target spin period and functional correlation analysis
And analyzing based on open source databases such as a UCS satellite database, an STK software satellite database and the like, wherein 554 GEO targets still working in orbit are obtained in total by 4 months and 1 day 2020. Among them, there were 14 spin-stabilized targets, as shown in table 1. The rotation speed of the GEO spinning stabilization target and the functions of the GEO spinning stabilization target have strong relevance, the rotation speed of the early warning satellite is 6 revolutions per minute, the rotation speed of the communication satellite is 50 revolutions per minute, and the rotation speed of the meteorological satellite is 100 revolutions per minute.
The embodiment of the invention provides an example for judging the GEO spin stabilization target function based on the ground-based photometric data. In this embodiment, a ground luminosity curve of a spin-stabilized GEO target shown in fig. 2 is taken as an example, and a determination process and a result of a target function are described according to the method provided by the present patent, where the method specifically includes the following steps:
step one, preliminarily determining the rotating speed of the spin stabilization GEO target. Firstly, a proper target rotating speed inversion method is selected according to the characteristics of the luminosity curve, and whether the sampling of the luminosity curve is continuous or not and whether the sampling rate is constant or not influence the selection of the inversion method. For the photometric curve shown in fig. 1, the sampling is continuous and the sampling rate is constant, so the target rotation speed inversion is performed using the discrete fourier transform, and the result is shown in fig. 3. It can be seen that the peak is greatest at a frequency of 0.24Hz, and that there is also a peak at 0.12Hz, with the two frequencies being exactly twice the relationship. Since the actual rotation period of the target is equal to the target photometric curve period or 2 times the photometric curve period, it can be inferred that the actual rotation period of the target at this time is 4.17 seconds or 8.34 seconds and the rotation speed is 14.4 rpm or 7.2 rpm.
TABLE 1 Global on-orbit operational GEO spin stabilization target information
Figure BDA0002860795120000101
And step two, checking the rotating speed of the spin stabilization GEO target. After the first step, the target rotation speed may be determined to be 14.4 rpm or 7.2 rpm, and assuming that the target rotation frequency is 0.24Hz and the corresponding period is 4.17s, the luminance curve is phase-folded and normalized based on the period, and the folding result of the luminance curve in one period is shown in fig. 4. It can be seen that there are two peak amplitudes of the folded data, which are about 0.4 and 0.9, respectively, indicating that 4.17s is half of the target actual period. Assuming that the rotation period of the object is 8.34s, the luminance profile is phase-folded based on this period, and the result is shown in fig. 5. As can be seen from the figure, the folding data of each part are basically overlapped in one period, so that 8.34s is closer to the rotation period of the target, and therefore the actual rotation speed of the target should be 7.2 rpm.
And step three, judging the function of the spin stabilization GEO target. Through the second step, the rotating speed of the target is 7.2 revolutions per minute, and the target is known not to be a meteorological satellite, a communication satellite or an early warning satellite based on the relevance between the GEO target spinning period and the target function shown in the table 1, so that the target can be judged to be a space debris.
The embodiment also provides a function determination system for a geosynchronous orbit spin stabilized space target, which comprises: the first module is used for judging whether the ground luminosity curve of the target has a sine structure or not, and if not, preliminarily inverting the rotating speed of the target by using a phase dispersion minimization method; the second module is used for judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, and if the ground luminosity curve of the target is continuous and the sampling rate is constant, the rotating speed of the target is preliminarily inverted by using discrete Fourier transform; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis; the third module is used for verifying the preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target; and the fourth module is used for judging the function of the target according to the actual rotating speed of the target in the third module.
The invention is beneficial to improving the application benefit of the ground luminosity data. Currently, although a ground-based telescope acquires a large amount of photometric data, a large amount of time-sensitive data is not effectively utilized, and the data is not converted into effective information. The purpose of photometric observation is to identify the target, the invention can break the gap between the characteristics of the spin-stabilized GEO target and photometric data, and convert the photometric data into the knowledge "gold block".
The invention is helpful for improving the observation capability of the foundation optical system. Due to the influence of distance and atmosphere, the difficulty of identifying a target in a high-orbit space by a ground-based optical observation system is higher, the traditional detection means and method cannot meet the requirement more and more, and equipment transformation or development is a process with a longer period.
The method can provide new clues for space application such as space target threat assessment and the like. The target function is an important clue for target threat assessment, the method can determine the function of the GEO spin stabilization target based on the ground luminosity data, and has important significance for solving the problem of target threat assessment in space situation perception.
Although the present invention has been described with reference to the preferred embodiments, it is not intended to limit the present invention, and those skilled in the art can make variations and modifications of the present invention without departing from the spirit and scope of the present invention by using the methods and technical contents disclosed above.

Claims (8)

1. A method for judging the function of a target in a geosynchronous orbit spin stabilization space, which is characterized by comprising the following steps:
the method comprises the following steps: judging whether the ground luminosity curve of the target has a sine structure or not, if not, initially inverting the rotating speed of the target by using a phase dispersion minimization method, and then entering the third step; if the structure has a sine structure, entering the step two;
step two: judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, if so, primarily inverting the target rotating speed by using discrete Fourier transform, and then entering the third step; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis, and then entering the third step;
step three: checking a preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target;
step four: and judging the function of the target according to the actual rotating speed of the target in the step three.
2. The method of determining a function of a geosynchronous orbit spin-stabilized space object according to claim 1, wherein: in the fourth step, the target with the actual rotating speed of 6 revolutions per minute is the early warning satellite, the target with the actual rotating speed of 50 revolutions per minute is the communication satellite, the target with the actual rotating speed of 100 revolutions per minute is the meteorological satellite, and other rotating speeds are the space debris.
3. The method of determining a function of a geosynchronous orbit spin-stabilized space object according to claim 1, wherein: in step two, the discrete fourier transform comprises:
preset x (n) is a finite sequence, i.e.:
Figure FDA0002860795110000011
the discrete fourier transform of x (n) is:
Figure FDA0002860795110000012
wherein ,
Figure FDA0002860795110000013
n is the number of transform points, N is the length of discrete Fourier transform interval, X (k) is the discrete Fourier transform of finite length sequence x (N), k represents the kth point of Fourier transform, WNIs an intermediate variable.
4. The method of determining a function of a geosynchronous orbit spin-stabilized space object according to claim 1, wherein: in step two, the least square spectrum analysis is implemented by solving a Lomb-Scargle periodogram, which comprises the following steps:
presetting the observation time number M and the time tiCorresponding observed value is hiAnd if so, the mean value and variance of the observed value are as follows:
Figure FDA0002860795110000021
Figure FDA0002860795110000022
the time delay τ is defined as:
Figure FDA0002860795110000023
the Lomb-Scargle periodogram is:
Figure FDA0002860795110000024
wherein ,
Figure FDA0002860795110000025
is the mean value of the observed values, i is the observed time number, σ is the variance of the observed values, τ is the time delay, PNAnd (omega) is a Lomb-Scargle periodogram.
5. A function decision system for a target in a spin stabilization space of a geosynchronous orbit, comprising:
the first module is used for judging whether the ground luminosity curve of the target has a sine structure or not, and if not, preliminarily inverting the rotating speed of the target by using a phase dispersion minimization method;
the second module is used for judging whether the ground luminosity curve of the target is continuous or the sampling rate is constant, and if the ground luminosity curve of the target is continuous and the sampling rate is constant, the rotating speed of the target is preliminarily inverted by using discrete Fourier transform; if the ground luminosity curve of the target is discontinuous or the sampling rate is not constant, primarily inverting the target rotating speed by using least square spectrum analysis;
the third module is used for verifying the preliminary inversion result of the target rotating speed by using a phase folding method to obtain the actual rotating speed of the target;
and the fourth module is used for judging the function of the target according to the actual rotating speed of the target in the third module.
6. The system for determining a function of a geosynchronous orbit spin-stabilized space object according to claim 5, wherein: the target with the actual rotating speed of 6 revolutions per minute is an early warning satellite, the target with the actual rotating speed of 50 revolutions per minute is a communication satellite, the target with the actual rotating speed of 100 revolutions per minute is a meteorological satellite, and other rotating speeds are space debris.
7. The system for determining a function of a geosynchronous orbit spin-stabilized space object according to claim 5, wherein: the discrete fourier transform includes:
preset x (n) is a finite sequence, i.e.:
Figure FDA0002860795110000031
the discrete fourier transform of x (n) is:
Figure FDA0002860795110000032
wherein ,
Figure FDA0002860795110000033
n is the number of transform points, N is the length of discrete Fourier transform interval, X (k) is the discrete Fourier transform of finite length sequence x (N), k represents the kth point of Fourier transform, WNIs an intermediate variable.
8. The system for determining a function of a geosynchronous orbit spin-stabilized space object according to claim 5, wherein: the least square spectrum analysis is realized by solving a Lomb-Scargle periodogram, which comprises the following steps:
presetting the observation time number M and the time tiCorresponding observed value is hiThen the mean sum of the observed valuesThe observed variance is:
Figure FDA0002860795110000034
Figure FDA0002860795110000041
the time delay τ is defined as:
Figure FDA0002860795110000044
the Lomb-Scargle periodogram is:
Figure FDA0002860795110000042
wherein ,
Figure FDA0002860795110000043
is the mean value of the observed values, i is the observed time number, σ is the variance of the observed values, τ is the time delay, PNAnd (omega) is a Lomb-Scargle periodogram.
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