CN102169173B - Method for analyzing ambiguity of inclined geo-synchronization orbit synthetic aperture radar - Google Patents

Abstract

The invention relates to a method for analyzing the ambiguity of an inclined geo-synchronization orbit synthetic aperture radar, and belongs to the technical field of synthetic aperture radars. The method comprises the following steps of: calculating a Doppler frequency of a main signal area of a beam and a Doppler frequency of an ambiguous area at first; calculating deviated angles from the center of the radar beam to the x axis and the y axis of a coordinate system of a satellite main body respectively according to the obtained results; obtaining an antenna gain of a position where an ambiguous signal is placed according to the two angles; and finally calculating a ratio of total energy of the ambiguous signal to energy of a main signal so as to obtain the ambiguity ASR. In the method, such a method for approximating relevant parameters in the prior art is avoided during calculation, so calculation can be executed under complicated conditions such as an elliptical orbit, earth rotation and the like; therefore, the method is applicable to analysis of the ambiguity of high-orbit inclined geo-synchronization orbit synthetic aperture radars.

Description

A kind of method for analyzing ambiguity of inclined geo-synchronization orbit synthetic aperture radar

Technical field

The present invention relates to a kind of method for analyzing ambiguity of inclined geo-synchronization orbit synthetic aperture radar, belong to the Synthetic Aperture Radar Technique field.

Background technology

Synthetic-aperture radar (SAR) is a kind of round-the-clock, round-the-clock high-resolution microwave remote sensing imaging radar, can be installed on the flying platforms such as aircraft, satellite, spaceship, have vital role at aspects such as environmental monitoring, oceanographic observation, resource exploration, Crop Estimation, mapping and military affairs.

In the system of satellite-borne synthetic aperture radar, fuzzy noise is the important performance indexes that system designer need to be considered, also is the key factor of constraint Beam position design.Satellite-borne synthetic aperture radar fuzzy refers to be subject to from the useful signal in the observation band " pollutions " of other echoed signal of non-artificial interference, causes the phenomenon of radar image Quality Down after the signal processing.In the echoed signal of spaceborne synthetic aperture radar (SAR) system, the useful echoed signal in being with from observation, the outer unwanted echo signal of observation band also can be received machine and receive, these echoed signals can with the useful echoed signal coherence stack of target, thereby the echoed signal to target produces interference, causes the blooming of signal.Fuzzy is the intrinsic problem of synthetic-aperture radar, combines its generation reason and mainly contains 2 points: the first, and radar is operated in the pulse system; The second, there is secondary lobe in the antenna radiation pattern.Source according to fuzzy can be divided into it two classes: range ambiguity and azimuth ambiguity.

The present research to spaceborne synthetic aperture radar (SAR) system mainly is confined in the research to Low Earth Orbit Synthetic Aperture Radar satellite (LEO SAR), because in earth observation satellite, usually require satellite constant as far as possible by the height of areal in the different time, to enable to obtain identical ripple position parameter.Be the Low Earth Orbit Synthetic Aperture Radar satellite of near-circular orbit such as the spaceborne synthetic aperture radar (SAR) system of recent emission: gondola " Mediterranean periphery observation small satellite satellite base system " (Cosmo-Skymed) is comprised of 4 satellites, and adopting highly is the sun synchronization circular orbit of 620km; " land radar " track (TerraSAR-X) of Germany is the Frozen Orbit in morning and evening of the repetition of sun synchronization, and orbit altitude is the near-circular orbit of 514.8km.

But in the parameter designing of inclined geo-synchronization orbit synthetic aperture radar (GEO SAR), because orbit altitude is large (such as semi-major axis of orbit: 42164.17km), therefore track can not be approximately near-circular orbit, the geometric relationship based on areal model can not be used, the impact of elliptical orbit and earth surface curvature must be considered; And the Low Earth Orbit synthetic-aperture radar of the relative near-circular orbit of orbital velocity of inclined geo-synchronization orbit synthetic aperture radar is less, be 2750m/s such as perigean velocity, apogean velocity is 890m/s, and earth rotation impact is serious, can not ignore distance to and the orientation between coupling.If still then can introduce larger error according to the ANALYSIS OF CALCULATING inclined geo-synchronization orbit synthetic aperture radar systematic parameter of Low Earth Orbit Synthetic Aperture Radar satellite systematic parameter.And since distance to the orientation to the increasing the weight of of coupling, the cross term blur level is worsened rapidly, reach distance to the order of magnitude of blur level.Therefore need to propose a kind of inclined geo-synchronization orbit synthetic aperture radar Fuzzy Analysis and solve these problems.

Summary of the invention

The objective of the invention is to propose a kind of method for analyzing ambiguity for inclined geo-synchronization orbit synthetic aperture radar in order to remedy the deficiencies in the prior art.

A kind of method for analyzing ambiguity of inclined geo-synchronization orbit synthetic aperture radar comprises following concrete steps:

Step 1, the elliptical orbit model of setting up the satellite transit track and satellite body coordinate system, wherein satellite body coordinate system center is centroid of satellite, and the x axle deviates from the earth's core along the direction of the radius vector of satellite, the z axle points to the angular momentum direction of track, and the y axle satisfies the right-handed coordinate system criterion.

Step 2, according to the Doppler frequency f in the main signal district of formula (1) compute beam _Dc:

$f_{\mathrm{Dc}}=-\frac{2}{\mathrm{\λ}}[-{\stackrel{\·}{R}}_s\cos {\mathrm{\θ}}_R+R_s({\mathrm{\ω}}_s-{\mathrm{\ω}}_e\cos {\mathrm{\α}}_i)\sin {\mathrm{\θ}}_R\cos {\mathrm{\θ}}_A-R_s{\mathrm{\ω}}_e\sin {\mathrm{\α}}_i\cos \mathrm{\α}\sin {\mathrm{\θ}}_R\sin {\mathrm{\θ}}_A]---\left(1\right)$

Wherein, θ _R, θ _AThe angle of radar beam misalignment x axle, y axle, as shown in Figure 2, O _aBe centroid of satellite, Q is the sensing of beam center, and λ is the wavelength of satellite radar work, ω _sBe the spin velocity of satellite platform, ω _eBe rotational-angular velocity of the earth, α _iBe the orbit inclination of satellite orbit, α is the latitude argument of expression satellite position, R _sBe the size of the position vector of satellite,

Be R _sFirst time derivative.A is semi-major axis of orbit, and e is orbital eccentricity, and ω is argument of perigee, and μ is the geophysics constant.Wherein:

$R_s=\frac{a(1-e^2)}{1+e\cos (\mathrm{\α}-\mathrm{\ω})}$

${\stackrel{\·}{R}}_s=e\sin (\mathrm{\α}-\mathrm{\ω})\sqrt{\frac{\mathrm{\μ}}{a(1-e^2)}}$

$\stackrel{\·}{\mathrm{\α}}=\sqrt{\mathrm{\μa}(1-e^2)}/R_s^2---\left(2\right)$

Step 3, determine the oblique distance R of blurred signal according to formula (3):

$R=\frac{c(\mathrm{\τ}+m/\mathrm{PRF})}{2}---\left(3\right)$

Wherein, τ is that PRF is pulse repetition rate apart from the number to the confusion region for the echo time delay of observation with the main signal zone, m.

Determine θ according to formula (4) by the cosine law again _R:

$\cos {\mathrm{\θ}}_R=\frac{R_s^2+R^2-R_e^2}{2R_{s}R}---\left(4\right)$

Wherein, R _eBe earth radius.

Step 4, determine the angle theta of radar beam misalignment y axle according to the Doppler frequency of confusion region _A

The Doppler frequency F of confusion region _DcDoppler frequency f for the main signal district _DcWith n fuzzy PRF sum.

F _Dc＝f _Dc+n·PRF (5)

Find the solution about θ _AQuadratic equation with one unknown:

Xcosθ _A+Ysinθ _A＝Z (6)

Obtain angle θ _ADoppler frequency F with the confusion region _DcBetween relation

$\cos {\mathrm{\θ}}_A=\frac{\mathrm{XZ}\±\sqrt{X^2{Z}^2-(X^2+Y^2)(Z^2-Y^2)}}{X^2+Y^2}---\left(7\right)$

Wherein, X=R _s(ω _s-ω _eCos α _i) sin θ _R, Y=-R _sω _eSin α _iCos α sin θ _RWith

Step 5, according to the off-axis angle (θ of the blurred signal that obtains in the above-mentioned steps _Rθ _A), determine the antenna gain that the blurred signal position is located:

$G({\mathrm{\&theta;}}_R,{\mathrm{\&theta;}}_A)=\sin c\left(\frac{L_a}{\mathrm{\&lambda;}}(\sin {\mathrm{\&theta;}}_R\cos {\mathrm{\&theta;}}_A-\sin {\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HSA00000385601800013.png"\; state="\{\{state\}\}">R</figure-callout>}0}\cos {\mathrm{\&theta;}}_{A0})\right)\sin c\left(\frac{L_r}{\mathrm{\&lambda;}}(\sin {\mathrm{\&theta;}}_R\sin {\mathrm{\&theta;}}_A-\sin {\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HSA00000385601800013.png"\; state="\{\{state\}\}">R</figure-callout>}0}\sin {\mathrm{\&theta;}}_{A0})\right)---\left(8\right)$

Wherein, L _aFor the orientation to antenna size, L _rFor the distance to antenna size, θ _R0The θ in main signal district _R, θ _A0The θ in main signal district _A

Calculate the energy intensity of blurred signal according to the oblique distance R of confusion region:

$S_{\mathrm{mn}}(f_{\mathrm{Dc}},{\mathrm{\&tau;}}_0)=k\frac{G(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})\mathrm{\&sigma;}(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})}{R^4(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})}---\left(9\right)$

Wherein, m, n represent respectively the orientation to distance to the numbering of fuzzy region, τ ₀Be the echo time delay of observation with the main signal zone, σ is the backscattering coefficient of fuzzy region, works as m, and during n=0, obtaining is the energy intensity of observing with main signal, and other are the energy intensity of blurred signal.K is the constant component in the radar equation, with the location independent of target in the satellite platform coordinate system.

Step 6, the gross energy that calculates blurred signal and the ratio of main signal energy obtain blur level ASR:

$\mathrm{ASR}=\frac{\underset{m\&NotEqual;0,n\&NotEqual;0}{\overset{\&infin;}{\underset{m,n=-\&infin;}{\mathrm{\&Sigma;}}}}{\&Integral;}_{-\mathrm{PBW}/2}^{\mathrm{PBW}/2}G(f_{\mathrm{Dc}}+\mathrm{nPRF},{\mathrm{\&tau;}}_0+m/\mathrm{PRF})\&CenterDot;\mathrm{\&sigma;}({\mathrm{\&tau;}}_0+m/\mathrm{PRF})/R^4({\mathrm{\&tau;}}_0+m/\mathrm{PRF})\mathrm{df}}{{\&Integral;}_{-\mathrm{PBW}/2}^{\mathrm{PBW}/2}G(f_{\mathrm{Dc}},{\mathrm{\&tau;}}_0)\&CenterDot;\mathrm{\&sigma;}\left({\mathrm{\&tau;}}_0\right)/R^4\left({\mathrm{\&tau;}}_0\right)\mathrm{df}}---\left(10\right)$

Wherein, PBW is signal bandwidth.When m ≠ 0, during n=0 above-mentioned formula obtain be the distance to blur level RASR; Work as m=0, n ≠ 0 o'clock, it is that the orientation is to blur level AASR that above-mentioned formula obtains; When m ≠ 0, n ≠ 0 o'clock, it is cross term blur level XASR that above-mentioned formula obtains; Three kinds of blur level sums can obtain total blur level ASR level.

Beneficial effect

The inventive method does not adopt in computation process in the prior art correlation parameter is done the method that approximation is processed, thereby can calculate for complex conditions such as elliptical orbit and earth rotations, be very suitable for high orbit, elliptical orbit and can not ignore the blur level analysis of the satellite-borne synthetic aperture radar plateform system of earth rotation, can obtain more fast, accurately the approximate value of fuzzy analysis, satisfy the needs of systematic parameter analysis and design.

Description of drawings

Fig. 1 is elliptical orbit model and the satellite body coordinate system schematic diagram of satellite transit track;

Fig. 2 is the antenna direction schematic diagram in the satellite body coordinate system;

Fig. 3 is for using the blur level simulation result of the inventive method.

Embodiment

Below in conjunction with drawings and Examples the inventive method is described in further detail.

A kind of Fuzzy Analysis of inclined geo-synchronization orbit synthetic aperture radar comprises following concrete steps:

Step 1, the elliptical orbit model of setting up the satellite transit track and satellite body coordinate system, wherein satellite body coordinate system center is centroid of satellite, the x axle is along the direction of the radius vector of satellite, deviate from the earth's core, the z axle points to the angular momentum direction of track, the y axle satisfies the right-handed coordinate system criterion, and the Doppler frequency at compute beam center.

The elliptical orbit model of model satellite transit track and satellite body coordinate system, as shown in Figure 1.The x axle deviates from the earth's core along the direction of the radius vector of satellite, and the z axle points to the angular momentum direction of track, and the y axle satisfies the right-handed coordinate system criterion.The vector of unit length of x-axis, y-axis and z-axis is expressed as respectively u _r, u _tAnd u _pThink that herein the earth is a spherosome.

The unit vector of the beam position direction of definition antenna is Q in the satellite body coordinate system, and the angle that note departs from x axle, y axle is designated as θ _R, θ _A, as shown in Figure 2.Then the beam position Q of this moment is

Q＝-cosθ _Ru _r+sinθ _Rcosθ _Au _t-sinθ _Rsinθ _Au _p (11)

Satellite platform in the satellite body coordinate system with respect to the relative velocity vector P on ground is

$P={\stackrel{\&CenterDot;}{R}}_s{u}_r+R_s({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_e\cos {\mathrm{\&alpha;}}_i)u_t+R_s{\mathrm{\&omega;}}_e\sin {\mathrm{\&alpha;}}_i\cos \mathrm{\&alpha;}{u}_p---\left(12\right)$

Wherein, ω _sBe the spin velocity of satellite platform, ω _eBe rotational-angular velocity of the earth, α _iThe orbit inclination of satellite orbit, α is the latitude argument of expression satellite position, R _sBe the size of the position vector of satellite,

Be R _sFirst time derivative.

Beam position Q is (θ _Rθ _A) time the Doppler frequency in main signal district

$f_{\mathrm{Dc}}=-\frac{2}{\mathrm{\&lambda;}}P\&CenterDot;Q=-\frac{2}{\mathrm{\&lambda;}}[-{\stackrel{\&CenterDot;}{R}}_s\cos {\mathrm{\&theta;}}_R+R_s({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_e\cos {\mathrm{\&alpha;}}_i)\sin {\mathrm{\&theta;}}_R\cos {\mathrm{\&theta;}}_A-R_s{\mathrm{\&omega;}}_e\sin {\mathrm{\&alpha;}}_i\cos \mathrm{\&alpha;}\sin {\mathrm{\&theta;}}_R\sin {\mathrm{\&theta;}}_A]---\left(13\right)$

Wherein, the position vector of satellite size R _sAnd first time derivative

And ω _sSpin velocity ω for satellite platform _sAll can try to achieve by the elliptical orbit model at earth inertial coordinates system.

$R_s=\frac{a(1-e^2)}{1+e\cos (\mathrm{\&alpha;}-\mathrm{\&omega;})}$

${\stackrel{\&CenterDot;}{R}}_s=e\sin (\mathrm{\&alpha;}-\mathrm{\&omega;})\sqrt{\frac{\mathrm{\&mu;}}{a(1-e^2)}}$

$\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}=\sqrt{\mathrm{\&mu;a}(1-e^2)}/R_s^2---\left(14\right)$

Wherein, a is semi-major axis of orbit, and e is orbital eccentricity, and ω is argument of perigee, and μ is the geophysics constant.

Step 2, determine that the distance of echo is to the oblique distance information of blurred signal position and echo

Line of equidistance is the concentric circles centered by substar, determined that echo time delay can determine the size of oblique distance, and the size of oblique distance is only relevant with the angle that departs from the substar line.

Can determine the size of oblique distance according to the echo time,

$R=\frac{c(\mathrm{\&tau;}+m/\mathrm{PRF})}{2}---\left(15\right)$

Wherein, τ for observation with the echo time delay in main signal zone, m be apart to the confusion region number, PRF is pulse repetition rate.

Can determine θ according to the cosine law _R:

$\cos {\mathrm{\&theta;}}_R=\frac{R_s^2+R^2-R_e^2}{2R_{s}R}---\left(16\right)$

Wherein, R _eBe earth radius.

Step 3, determine that the orientation is to the position θ of blurred signal _A

The Doppler frequency F of confusion region _DcDoppler frequency f for the main signal district _DcWith n fuzzy PRF sum:

F _Dc＝f _Dc+n·PRF (17)

Wherein, f _DcBe the Doppler frequency of observation with the main signal zone.

In step 2, determined θ according to distance to the confusion region _R, equation (13) can turn to θ _AQuadratic equation with one unknown:

Xcosθ _A+Ysinθ _A＝Z (18)

Wherein, X=R _s(ω _s-ω _eCos α _i) sin θ _R, Y=-R _sω _eSin α _iCos α sin θ _RWith

Then have

Solve an equation and to get angle θ _AAnd the relation between the Echo Doppler Frequency

$\cos {\mathrm{\&theta;}}_A=\frac{\mathrm{XZ}\&PlusMinus;\sqrt{X^2{Z}^2-(X^2+Y^2)(Z^2-Y^2)}}{X^2+Y^2}---\left(19\right)$

Step 4, determine main signal and blurred signal energy intensity

Can be obtained the off-axis angle (θ of blurred signal to step 3 by step 2 _Rθ _A), then determine the antenna gain that the blurred signal position is located, for rectangle front antenna, computing method are

$G({\mathrm{\&theta;}}_R,{\mathrm{\&theta;}}_A)=\sin c\left(\frac{L_a}{\mathrm{\&lambda;}}(\sin {\mathrm{\&theta;}}_R\cos {\mathrm{\&theta;}}_A-\sin {\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HSA00000385601800013.png"\; state="\{\{state\}\}">R</figure-callout>}0}\cos {\mathrm{\&theta;}}_{A0})\right)\sin c\left(\frac{L_r}{\mathrm{\&lambda;}}(\sin {\mathrm{\&theta;}}_R\sin {\mathrm{\&theta;}}_A-\sin {\mathrm{\&theta;}}_{\mathrm{<figure-callout\; id="0"\; label="R"\; filenames="HSA00000385601800013.png"\; state="\{\{state\}\}">R</figure-callout>}0}\sin {\mathrm{\&theta;}}_{A0})\right)---\left(20\right)$

L _aFor the orientation to antenna size, L _rFor the distance to antenna size, θ _R0The θ in main signal district _R, θ _A0The θ in main signal district _A

After obtaining the antenna gain at place, blurred signal position place, the oblique distance information of the confusion region that obtains according to step 2 etc. is calculated the energy intensity of blurred signal.

$S_{\mathrm{mn}}(f_{\mathrm{Dc}},{\mathrm{\&tau;}}_0)=k\frac{G(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})\mathrm{\&sigma;}(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})}{R^4(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})}---\left(21\right)$

Wherein, m, n represent respectively the orientation to distance to the numbering of fuzzy region, τ ₀Be the echo time delay of observation with the main signal zone, σ is the backscattering coefficient of fuzzy region, works as m, and during n=0, obtaining is the energy intensity of observing with main signal, and other are the energy intensity of blurred signal.K is the constant component in the radar equation.

Step 5, calculating blur level

Blur level is a kind of tolerance of fuzzy noise in the satellite-borne synthetic aperture radar, generally represents with the dB form.

It is defined as the gross energy of blurred signal and the ratio of main signal energy, and computing method are as follows:

$\mathrm{ASR}=\frac{\underset{m\&NotEqual;0,n\&NotEqual;0}{\overset{\&infin;}{\underset{m,n=-\&infin;}{\mathrm{\&Sigma;}}}}{\&Integral;}_{-\mathrm{PBW}/2}^{\mathrm{PBW}/2}G({\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="HSA00000385601800013.png"\; state="\{\{state\}\}">f</figure-callout>}}_0+\mathrm{nPRF},{\mathrm{\&tau;}}_0+m/\mathrm{PRF})\&CenterDot;\mathrm{\&sigma;}({\mathrm{\&tau;}}_0+m/\mathrm{PRF})/R^4({\mathrm{\&tau;}}_0+m/\mathrm{PRF})\mathrm{df}}{{\&Integral;}_{-\mathrm{PBW}/2}^{\mathrm{PBW}/2}G({\mathrm{<figure-callout\; id="0"\; label="f"\; filenames="HSA00000385601800013.png"\; state="\{\{state\}\}">f</figure-callout>}}_0,{\mathrm{\&tau;}}_0)\&CenterDot;\mathrm{\&sigma;}\left({\mathrm{\&tau;}}_0\right)/R^4\left({\mathrm{\&tau;}}_0\right)\mathrm{df}}---\left(22\right)$

When m ≠ 0, during n=0, the time above-mentioned formula obtain be the distance to blur level RASR; Work as m=0, n ≠ 0 o'clock, it is that the orientation is to blur level AASR that above-mentioned formula obtains; When m ≠ 0, n ≠ 0 o'clock, it is cross term blur level XASR that above-mentioned formula obtains; Three kinds of blur level sums can obtain total blur level level.

Satellite transit is on inclination geostationary orbit (IGSO), and the mean orbit parameter is as follows:

Semi-major axis of orbit: 42164.17km

Orbit inclination: 57 degree

Excentricity: 0.07

Antenna size: diameter 30m antenna

Frequency range: L-band

Downwards angle of visibility: 4.65 degree

Utilize a kind of inclined geo-synchronization orbit synthetic aperture radar Fuzzy Analysis of the present invention, the present embodiment has been carried out emulation, the result as shown in Figure 3.

Utilize the orbit parameter of inclined geo-synchronization orbit synthetic aperture radar, blur level simulation result in the observation band that obtains as shown in Figure 3, consider the impact of bidimensional coupling, not only obtained distance to blur level and orientation to blur level, and can also obtain the simulation result of cross term blur level.

As seen utilize the inventive method, can calculate comparatively accurately the blur level result of inclined geo-synchronization orbit synthetic aperture radar, support for the inclined geo-synchronization orbit synthetic aperture radar system provides preferably.

Above-described specific descriptions; purpose, technical scheme and beneficial effect to invention further describe; institute is understood that; the above only is specific embodiments of the invention; the protection domain that is not intended to limit the present invention; within the spirit and principles in the present invention all, any modification of making, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (1)

1. a method for analyzing ambiguity of inclined geo-synchronization orbit synthetic aperture radar is characterized in that comprising the steps:

Step 1, the elliptical orbit model of setting up the satellite transit track and satellite body coordinate system, wherein satellite body coordinate system center is centroid of satellite, and the x axle deviates from the earth's core along the direction of the radius vector of satellite, the z axle points to the angular momentum direction of track, and the y axle satisfies the right-handed coordinate system criterion;

Step 2, according to the Doppler frequency f in the main signal district of formula (1) compute beam _Dc:

$f_{\mathrm{Dc}}=-\frac{2}{\mathrm{\&lambda;}}[-{\stackrel{\&CenterDot;}{R}}_s\cos {\mathrm{\&theta;}}_R+R_s({\mathrm{\&omega;}}_s-{\mathrm{\&omega;}}_e\cos {\mathrm{\&alpha;}}_i)\sin {\mathrm{\&theta;}}_R\cos {\mathrm{\&theta;}}_A-R_s{\mathrm{\&omega;}}_e\sin {\mathrm{\&alpha;}}_i\cos \mathrm{\&alpha;}\sin {\mathrm{\&theta;}}_R\sin {\mathrm{\&theta;}}_A]---\left(1\right)$

Wherein, θ _R, θ _AThe angle of radar beam misalignment x axle, y axle, O _aBe centroid of satellite, Q is the unit vector of the beam position direction of antenna, and λ is the wavelength of satellite radar work, ω _sBe the spin velocity of satellite platform, ω _eBe rotational-angular velocity of the earth, α _iBe the orbit inclination of satellite orbit, α is the latitude argument of expression satellite position, R _sBe the size of the position vector of satellite, Be R _sFirst time derivative, a is semi-major axis of orbit, e is orbital eccentricity, ω is argument of perigee, μ is the geophysics constant, wherein:

$R_s=\frac{a(1-e^2)}{1+e\cos (\mathrm{\&alpha;}-\mathrm{\&omega;})}$

${\stackrel{\&CenterDot;}{R}}_s=e\sin (\mathrm{\&alpha;}-\mathrm{\&omega;})\sqrt{\frac{\mathrm{\&mu;}}{a(1-e^2)}}$

$\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}=\sqrt{\mathrm{\&mu;a}(1-e^2)}/R_s^2---\left(2\right)$

Step 3, determine the oblique distance R of blurred signal according to formula (3):

$R=\frac{c(\mathrm{\&tau;}+m/\mathrm{PRF})}{2}---\left(3\right)$

Wherein, τ is that PRF is pulse repetition rate apart from the number to the confusion region for the echo time delay of observation with the main signal zone, m;

Determine θ according to formula (4) by the cosine law again _R:

$\cos {\mathrm{\&theta;}}_R=\frac{R_s^2+R^2-R_e^2}{{2R}_{s}R}---\left(4\right)$

Wherein, R _eBe earth radius;

Step 4, determine the angle theta of radar beam misalignment y axle according to the Doppler frequency of confusion region _A

The Doppler frequency F of confusion region _DcDoppler frequency f for the main signal district _DcWith n fuzzy PRF sum:

F _Dc＝f _Dc+n·PRF (5)

Find the solution about θ _AQuadratic equation with one unknown:

Xcosθ _A+Ysinθ _A＝Z (6)

Obtain angle θ _ADoppler frequency F with the confusion region _DcBetween relation

$\cos {\mathrm{\&theta;}}_A=\frac{\mathrm{XZ}\&PlusMinus;\sqrt{X^2{Z}^2-(X^2+Y^2)(Z^2-Y^2)}}{X^2+Y^2}---\left(7\right)$

Wherein, X=R _s(ω _s-ω _eCos α _i) sin θ _R, Y=-R _sω _eSin α _iCos α sin θ _RWith

Step 5, according to the off-axis angle (θ of the blurred signal that obtains in the above-mentioned steps _Rθ _A), determine the antenna gain that the blurred signal position is located:

$G({\mathrm{\&theta;}}_R,{\mathrm{\&theta;}}_A)=\sin c\left(\frac{L_a}{\mathrm{\&lambda;}}(\sin {\mathrm{\&theta;}}_R\cos {\mathrm{\&theta;}}_A-\sin {\mathrm{\&theta;}}_{R0}\cos {\mathrm{\&theta;}}_{A0})\right)\sin c\left(\frac{L_r}{\mathrm{\&lambda;}}(\sin {\mathrm{\&theta;}}_R\sin {\mathrm{\&theta;}}_A-\sin {\mathrm{\&theta;}}_{R0}\sin {\mathrm{\&theta;}}_{A0})\right)---\left(8\right)$

Wherein, L _aFor the orientation to antenna size, L _rFor the distance to antenna size, θ _R0The θ in main signal district _R, θ _A0The θ in main signal district _A

Calculate the energy intensity of blurred signal according to the oblique distance R of confusion region:

$S_{\mathrm{mn}}(f_{\mathrm{Dc}},{\mathrm{\&tau;}}_0)=k\frac{G(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})\mathrm{\&sigma;}(f_{\mathrm{DC}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})}{R^4(f_{\mathrm{Dc}}+m\&CenterDot;\mathrm{PRF},{\mathrm{\&tau;}}_0+\frac{n}{\mathrm{PRF}})}---\left(9\right)$

Wherein, τ ₀Be the echo time delay of observation with the main signal zone, σ is the backscattering coefficient of fuzzy region, work as m, during n=0, obtain is that observation is with the energy intensity of main signal, other are the energy intensity of blurred signal, and k is the constant component in the radar equation, with the location independent of target in the satellite platform coordinate system;

Step 6, the gross energy that calculates blurred signal and the ratio of main signal energy obtain blur level ASR:

$\mathrm{ASR}=\frac{\underset{\underset{m\&NotEqual;0,n\&NotEqual;0}{m,n=-\&infin;}}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\&Integral;}_{-\mathrm{PBW}/2}^{\mathrm{PBW}/2}G(f_{\mathrm{Dc}}+\mathrm{nPRF},{\mathrm{\&tau;}}_0+m/\mathrm{PRF})\&CenterDot;\mathrm{\&sigma;}({\mathrm{\&tau;}}_0+m/\mathrm{PRF})/R^4({\mathrm{\&tau;}}_0+m/\mathrm{PRF})\mathrm{df}}{{\&Integral;}_{-\mathrm{PBW}/2}^{\mathrm{PBW}/2}G(f_{\mathrm{Dc}},{\mathrm{\&tau;}}_0)\&CenterDot;\mathrm{\&sigma;}\left({\mathrm{\&tau;}}_0\right)/R^4\left({\mathrm{\&tau;}}_0\right)\mathrm{df}}---\left(10\right)$

Wherein, PBW is signal bandwidth, when m ≠ 0, during n=0 above-mentioned formula obtain be the distance to blur level RASR; Work as m=0, n ≠ 0 o'clock, it is that the orientation is to blur level AASR that above-mentioned formula obtains; When m ≠ 0, n ≠ 0 o'clock, it is cross term blur level XASR that above-mentioned formula obtains; Three kinds of blur level sums can obtain total blur level ASR level.

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