CN102331577A - Improved NCS (Nonlinear Chirp Scaling) imaging algorithm suitable for geosynchronous orbit (GEO) SAR (Synthetic Aperture Radar) - Google Patents

Abstract

The invention relates to an improved NCS (Nonlinear Chirp Scaling) imaging algorithm suitable for a geosynchronous orbit (GEO) SAR (Synthetic Aperture Radar), belonging to the technical field of SAR imaging. The NCS imaging algorithm is improved by establishing a curved locus signal model for replacing an equivalent linear model in an original NCS imaging algorithm and evaluating a two-dimensional resolving spectrum expression suitable for the NCS imaging algorithm on the basis of the established curved locus signal model. Compared with the prior art, the improved NCS imaging algorithm has the advantages that: a novel curved locus model suitable for the GEO SAR is obtained with a high-order Taylor expansion method, and the defects of large error of a near equivalent linear model, completely unavailable application of a far equivalent linear model and the like of the GEO SAR can be overcome by adopting the locus model; and meanwhile, a two-dimensional resolving spectrum suitable for the NCS algorithm is obtained on the basis of the equivalent linear model, and various compensation functions of the NCS algorithm can be resolved by using the spectrum, so that the requirement on large-scene imaging of the GEO SAR is met.

Description

A kind of improvement NCS imaging algorithm that is applicable to geostationary orbit SAR

Technical field

The present invention relates to a kind of improvement NCS imaging algorithm, particularly a kind of be applicable to geostationary orbit (GEO) SAR improvement NCS imaging algorithm, belong to the synthetic aperture radar (SAR) technical field of imaging.

Background technology

Present SAR satellite is low orbit satellite, and orbit altitude is no more than 1, and 000km to being generally 3 to 5 days the coverage cycle of particular locality, also needs at least 1 day time when carrying out orbit maneuver; Therefore, low, solution of emergent event long problem retardation time of low rail SAR life period resolution.A kind of effective ways of head it off are geostationary orbit synthetic-aperture radar (GEO SAR) satellites, and this is to operate in 36, the SAR satellite on the 000km height geostationary orbit; This geostationary orbit is not a geostationary orbit, and it has certain angle of inclination, and its sub-satellite track is ' 8 ' font, can obtain the relative motion with terrain object thus, realizes the two-dimensional SAR imaging.

Present SAR imaging algorithm all is based on the foundation of low rail (LEO) situation; Defective is: LEO SAR is general, and the synthetic aperture time ratio is shorter; The track of satellite flight can be similar to equivalent straight line model; But the synthetic aperture time generally reaches up to a hundred seconds in GEO SAR, thereby the equivalent straight line model that low rail SAR imaging algorithm is relied on was because of losing efficacy the aperture time of GEO SAR overlength.The NCS algorithm is a kind of outstanding large scene imaging algorithm, but it is based on equivalent straight line model, therefore is difficult to directly apply to GEO SAR.To above situation, we have proposed the requirement that a kind of improved NCS imaging algorithm goes to realize the imaging of GEO SAR large scene.

Summary of the invention

The objective of the invention is to have proposed a kind of improvement NCS imaging algorithm that is applicable to geostationary orbit SAR in order to realize the imaging of GEO SAR large scene.

The present invention realizes through following technical scheme.

A kind of improvement NCS imaging algorithm that is applicable to geostationary orbit SAR of the present invention; Its improvements are two parts: the one, and set up the serpentine track signal model and replace the equivalent straight line model in the former NCS imaging algorithm; The 2nd, and on the basis of setting up the serpentine track signal model, obtain the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm, the detailed process of two parts is respectively:

1) traditional NCS algorithm is based on equivalent straight line model; But equivalent straight line model error ratio occurs than problems such as big even inefficacies in GEO SAR; Therefore need set up a kind of serpentine track signal model and go the true oblique distance between approximate satellite and the target historical, the process of setting up the serpentine track signal model is:

Definition satellite and the target in each pulse repetition time (PRT) coordinates were

and

satellite and target slant range between the real history is expressed as

$R_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{gn}}\left|\right|---\left(1\right)$

Formula (1) is carried out obtaining the serpentine track model after the Taylor expansion

R _n＝R+k ₁·t _a+k ₂·t _a ²+k ₃·t _a ³+k ₄·t _a ⁴+… (2)

T wherein _aFor the orientation to the time, R, k ₁, k ₂, k ₃And k ₄Be R _nThe Taylor expansion coefficient on 0 to 4 rank, k wherein ₁, k ₂, k ₃And k ₄The formula of embodying is respectively:

$k_1=k_{10}+{\stackrel{.}{k}}_1\·(R-R_0)---\left(3\right)$

$k_2=k_{20}+{\stackrel{.}{k}}_2\·(R-R_0)---\left(4\right)$

$k_3=k_{30}+{\stackrel{.}{k}}_3\·(R-R_0)---\left(5\right)$

$k_4=k_{40}+{\stackrel{.}{k}}_4\·(R-R_0)---\left(6\right)$

In formula (3)～(6), k ₁₀～k ₄₀, The formula that embodies be respectively:

$k_{10}={\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T/\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|---\left(7\right)$

$k_{20}=\frac{{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}---\left(8\right)$

$k_{30}=\frac{{\stackrel{\→}{b}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·{\stackrel{\→}{a}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T}{6\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}+\frac{{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^3}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^5}$ (9)

$-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}$

$k_{40}=\frac{{\stackrel{\→}{d}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·{\stackrel{\→}{b}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T}{24\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}+\frac{{\left|\right|{\stackrel{\→}{a}}_{s0}\left|\right|}^2}{8\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{k_2^2+2\·k_1\·k_3}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}---\left(10\right)$

${\stackrel{.}{k}}_1=\frac{v_{s0x}}{r_{s0x}}-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{{R_0}^2}---\left(11\right)$

${\stackrel{\·}{k}}_2=\frac{a_{s0x}\·{R_0}^2-r_{s0x}\·({\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2)}{2\·{R_0}^2\·r_{s0x}}-\frac{v_{s0x}\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{{R_0}^2\·r_{s0x}}+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^4}---\left(12\right)$

${\stackrel{\·}{k}}_3=\frac{b_{s0x}\·{R_0}^2-r_{s0x}\·[{\stackrel{\→}{b}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·{\stackrel{\→}{a}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T]}{6\·{R_0}^2\·r_{s0x}}+$

$\frac{3\·v_{s0x}\·{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^2-5\·r_{s0x}\·{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^3}{2\·{R_0}^6\·r_{s0x}}-\frac{v_{s0x}\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+a_{s0x}\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^2\·r_{s0x}}---\left(13\right)$

$+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^4}-\frac{v_{s0x}\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{R_0}^2\·r_{s0x}}+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{R_0}^4}$

${\stackrel{.}{k}}_4=\frac{d_{s0x}\·{R_0}^2-r_{s0x}\·[{\stackrel{\→}{d}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+4\·{\stackrel{\→}{b}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T]}{24\·{R_0}^2\·r_{s0x}}-\frac{{\left|\right|{\stackrel{\→}{a}}_{s0}\left|\right|}^2}{8\·{R_0}^2}$ (14)

$+\frac{k_{20}\·{\stackrel{\·}{k}}_2+k_{30}\·{\stackrel{\·}{k}}_1+k_{10}\·{\stackrel{\·}{k}}_3}{r_{s0x}}+\frac{k_{20}^2+2\·k_{10}\·k_{30}}{2\·{R_0}^2}$

In formula (7)～formula (14),

With Represent satellite at aperture center position vector constantly,

With Represent respectively satellite aperture center constantly velocity, acceleration, acceleration vector and add acceleration vector, R ₀Expression satellite and reference point target are at aperture center distance constantly, r _S0x, a _S0x, b _S0x, v _S0xAnd d _S0xBe respectively

With

Distance under the scene coordinate system is to component;

2) based on serpentine track, the process of obtaining the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm is:

The processing of NCS algorithm begins from two-dimensional frequency, and the two-dimensional analysis spectrum expression formula of therefore trying to achieve under the serpentine track is particularly important; The serpentine track that is proposed is the high-order Taylor expansion, so after utilizing the progression inversion principle to try to achieve site in the phasing, the spectrum expression formula that obtains does

$S(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{2\·k_1}{c}\·\left(f_r{+f}_c\right)]\·\exp (-j\·\mathrm{\π}\·\frac{f_r^2}{K_r})$

$\exp \{j\·2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{2\·(f_r+f_c)}{c}\·R\\ +\frac{1}{4\·k_2}\·\left(\frac{c}{2\·(f_r+f_c)}\right)\·{(f_a+\frac{2\·k_1}{c}\·(f_r+f_c))}^2\\ +\frac{k_3}{8\·{k_2}^3}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^2\·{(f_a+\frac{2\·k_1}{c}\·(f_r+f_c))}^3\\ +\frac{9\·k_3^2-4\·k_2\·k_4}{64\·k_2^5}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^3\·{(f_a+\frac{2\·k_1}{c}\·(f_r+f_c))}^4\end{array}\right]\}---\left(15\right)$

Wherein, f _rAnd f _aBe respectively the distance to the orientation to frequency, u _r() and u _a() be respectively the distance to the orientation to envelope, k _rFor the distance to the frequency modulation rate, c is the light velocity, f _cBe the radar carrier frequency;

Formula (15) can not directly be used in the NCS algorithm, needs further to derive, and draws

$\frac{1}{f_r+f_c}=\frac{1}{f_c}[1-\frac{f_r}{f_c}+{\left(\frac{f_r}{f_c}\right)}^2-{\left(\frac{f_r}{f_c}\right)}^3+\·\·\·]$

${\left(\frac{1}{f_r+f_c}\right)}^2=\frac{1}{f_c^2}[1-\frac{2\·f_r}{f_c}+3\·{\left(\frac{f_r}{f_c}\right)}^2-4\·{\left(\frac{f_r}{f_c}\right)}^3+\·\·\·]---\left(16\right)$

${\left(\frac{1}{f_r+f_c}\right)}^3=\frac{1}{f_c^3}[1-\frac{3\·f_r}{f_c}+6\·{\left(\frac{f_r}{f_c}\right)}^2-10\·{\left(\frac{f_r}{f_c}\right)}^3+\·\·\·]$

Utilize formula (16),, draw GEO SAR 2-d spectrum and be through after deriving:

$S(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{{2k}_1}{c}(f_r+f_c)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$ (17)

$\·\exp [-j\·2\·\mathrm{\π}\·b(f_a,f_r)]\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_a\right)}\·f_r]\·\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{K_s(f_a,R)}]\·\exp [j\·{\mathrm{\φ}}_3(f_a,R)\·f_r^3]$

Formula (17) is further described as follows:

2.1 the ф on formula (17) equal sign right side _Az(f _a, R) be the orientation to modulating function, the formula of embodying does

${\mathrm{\φ}}_{\mathrm{az}}(f_a,R)=[\frac{k_1}{2\·k_2}+\frac{3\·k_1^2\·k_3}{8\·k_2^3}+\frac{k_1^3\·(9\·k_3^2-4\·k_2\·k_4)}{16\·k_2^5}]\·f_a+$

$[\frac{\mathrm{\λ}}{8\·k_2}+\frac{3\·\mathrm{\λ}\·k_1\·k_3}{16\·k_2^3}+\frac{3\·\mathrm{\λ}\·k_1^2\·(9\·k_3^2-4\·k_2\·k_4)}{64\·k_2^5}]\·f_a^2+---\left(18\right)$

$[\frac{{\mathrm{\λ}}^2\·k_3}{32\·k_2^3}+\frac{\mathrm{\λ}\·k_1\·(9\·k_3^2-4\·k_2\·k_4)}{64\·k_2^5}]\·f_a^3+\frac{{\mathrm{\λ}}^3\·(9\·k_3^2-4\·k_2\·k_4)}{512\·k_2^5}\·f_a^4$

Because ф _Az(f _a, R) only relevant to frequency and target location with the orientation, with distance to frequency-independent, so can compensate apart from the Doppler territory;

2.2 the ф on formula (17) equal sign right side _RP(R) be excess phase after the accurate 2-d spectrum Taylor expansion, expression formula does

${\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)=\frac{k_1^2}{2\·\mathrm{\λ}\·k_2}+\frac{k_1^3\·k_3}{4\·\mathrm{\λ}\·k_2^3}+\frac{k_1^4\·(9\·k_3^2-4\·k_2\·k_4)}{32\·\mathrm{\λ}\·k_2^5}-\frac{2\·R}{\mathrm{\λ}}---\left(19\right)$

This and orientation are to frequency with apart to frequency-independent, and be relevant to the position with the distance of target, can compensate apart from the Doppler territory;

2.3 the b (f on formula (17) equal sign right side _a, f _r) the RP place migration phase place that obtains when launching for 2-d spectrum, its expression formula does

$b(f_a,f_r)=-[\frac{k_{10}^2}{2\·k_{20}\·c}+\frac{k_{10}^3\·k_{30}}{4\·k_{20}^3\·c}+\frac{k_{10}\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·c\·k_{20}^5}]\·f_r+$

$[\frac{\mathrm{\λ}}{8\·k_{20}\·f_c}+\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{16\·k_{20}^3\·f_c}+\frac{3\·\mathrm{\λ}{k}_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{64\·f_c\·k_{20}^5}]\·f_a^2\·f_r+$ (20)

$[\frac{{\mathrm{\λ}}^2\·k_{30}}{16\·k_{20}^3\·f_c}+\frac{{\mathrm{\λ}}^2\·k_{10}\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·f_c\·k_{20}^5}]\·f_a^3\·f_r+$

$\frac{3\·{\mathrm{\λ}}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{512\·f_c\·k_{20}^5}{f}_a^4\·f_r+(\frac{2\·R_0}{c}-{\stackrel{\·}{B}}_1\·R_0)\·f_r$

Wherein

${\stackrel{\·}{B}}_1=-\frac{2\·k_{10}\·k_{20}\·{\stackrel{.}{k}}_1-k_{10}^2\·{\stackrel{.}{k}}_2}{2\·c\·k_{20}^2}-\frac{(3\·k_{10}^2\·k_{30}\·{\stackrel{\·}{k}}_1+k_{10}^3\·{\stackrel{\·}{k}}_3)\·k_{20}-3\·k_{10}^3\·k_{30}\·{\stackrel{\·}{k}}_2}{4\·c\·k_{20}^4}-\frac{A_3\·{\stackrel{.}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3}{32\·c}+$

$\{-\frac{\mathrm{\λ}\·{\stackrel{.}{k}}_2}{8\·f_c\·k_{20}^2}+\frac{3\·\mathrm{\λ}\·[(k_{10}\·{\stackrel{\·}{k}}_3+k_{30}\·{\stackrel{.}{k}}_1)\·k_{20}-3\·k_{10}\·k_{30}\·{\stackrel{\·}{k}}_2]}{16\·f_c\·k_{20}^4}+\frac{3\·\mathrm{\λ}\·(2\·k_{10}\·A_3\·{\stackrel{\·}{k}}_1-k_{10}^2\·{\stackrel{\·}{A}}_3)}{64\·f_c}\}\·f_a^2\left(21\right)$

$+[\frac{{\mathrm{\λ}}^2\·(k_{20}\·{\stackrel{\·}{k}}_3-3\·k_{30}\·{\stackrel{\·}{k}}_2)}{16\·f_c\·k_{20}^4}+\frac{{\mathrm{\λ}}^2\·(A_3\·{\stackrel{\·}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3)}{32\·f_c}]\·f_a^3+\frac{3\·{\mathrm{\λ}}^2\·{\stackrel{.}{A}}_3}{512\·f_c}\·f_a^4+\frac{2}{c}$

$A_3=\frac{9\·k_{30}^2-4\·k_{20}\·k_{40}}{k_{20}^5}---\left(22\right)$

${\stackrel{.}{A}}_3=\frac{[18\·k_{30}\·{\stackrel{.}{k}}_3-4\·(k_{20}\·{\stackrel{\·}{k}}_4+k_{40}\·{\stackrel{\·}{k}}_2)]k_{20}-5\·(9\·k_{30}^2-4\·k_{20}\·k_{40})\·{\stackrel{.}{k}}_2}{k_{20}^6}---\left(23\right)$

This phase place is the part of RP place migration phase place, in traditional CS based on equivalent straight line model, NCS algorithm, does not have this, and this is distinctive in based on the CS algorithm of serpentine track, NCS algorithm, must need compensation; And because this does not have space-variant property, therefore can compensate at 2-d spectrum;

2.4 the 4th exponential term on formula (17) equal sign right side is

In,

Be range migration, M (f _a) be the migration factor and

$M\left(f_a\right)=\frac{1}{{\stackrel{.}{B}}_1\·c}---\left(24\right)$

Wherein the expression formula of is shown in (21);

Can find that this is inconsistent to the position migration for different distances, mainly be to multiply by this space-variant property of Chirp signal adjustment apart from the Doppler territory, in the NCS algorithm, needing this space-variant property of adjustment equally in the CS algorithm;

2.5 the 5th exponential term on formula (17) equal sign right side is

In

For distance to modulation item, k wherein _s(f _a, R) be new distance to frequency modulation factor, and

$\frac{1}{K_s(f_a,R)}=-[\frac{\mathrm{\λ}}{4\·k_2\·f_c^2}+\frac{3\·\mathrm{\λ}\·k_1\·k_3}{8\·k_2^3\·f_c^2}+\frac{3\·\mathrm{\λ}\·k_1^2\·(9\·k_3^2-4\·k_2\·k_4)}{32\·k_2^5\·f_c^2}]\·f_a^2-$ (25)

$[\frac{3\·{\mathrm{\λ}}^2\·k_3}{16\·k_2^3\·f_c^2}+\frac{3\·{\mathrm{\λ}}^2\·k_1\·(9\·k_3^2-4\·k_2\·k_4)}{32\·k_2^5\·f_c^2}]\·f_a^3-\frac{3\·{\mathrm{\λ}}^3\·(9\·k_3^2-4\·k_2\·k_4)}{128\·k_2^5\·f_c^2}\·f_a^4+\frac{1}{K_r}$

k _s(f _a, R) have space-variant property, in the CS algorithm, do not consider its space-variant property, it is the part of space-variant property modulation in the NCS algorithm, but it is difficult to directly use, and need be similar to for this reason:

K _s(f _a，R)＝k _s(f _a，R ₀)+Δk _s(f _a)·[τ(f _a，R)-τ(f _a，R ₀)] (26)

Wherein

$\frac{1}{K_s(f_a,R_0)}=-[\frac{\mathrm{\λ}}{4\·k_{20}\·f_c^2}+\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{8\·k_{20}^3\·f_c^2}+\frac{3\·\mathrm{\λ}\·k_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·k_{20}^5\·f_c^2}]\·f_a^2-$ (27)

$[\frac{3\·{\mathrm{\λ}}^2\·k_{30}}{16\·k_{20}^3\·f_c^2}+\frac{3\·{\mathrm{\λ}}^2\·k_{10}\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·k_{20}^5\·f_c^2}]\·f_a^3-\frac{3\·{\mathrm{\λ}}^3\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{128\·k_{20}^5\·f_c^2}\·f_a^4+\frac{1}{K_r}$

${\mathrm{\Δk}}_s\left(f_a\right)=\frac{K_s^2(f_a,R_0)\·c\·M\left(f_a\right)}{2}.$

$\left\{\begin{array}{c}-\frac{\mathrm{\λ}\·{\stackrel{\·}{k}}_2\·f_a^2}{4\·f_c^2\·k_{20}^2}+\frac{3\·\mathrm{\λ}\·[(k_{10}\·{\stackrel{\·}{k}}_3+k_{30}\·{\stackrel{\·}{k}}_1)\·k_{20}-3\·k_{10}\·k_{30}\·{\stackrel{\·}{k}}_2]}{8\·f_c^2\·k_{20}^4}\·f_a^2+\\ \frac{3\·\mathrm{\λ}\·(2\·k_{10}\·A_3\·{\stackrel{.}{k}}_1-k_{10}^2\·{\stackrel{\·}{A}}_3)}{32\·f_c^2}\·f_a^2+\frac{3\·{\mathrm{\λ}}^2\·(k_{20}\·{\stackrel{\·}{k}}_3-3\·k_{30}\·{\stackrel{\·}{k}}_2)}{16\·f_c^2\·k_{20}^4}\·f_a^3\\ +\frac{3\·{\mathrm{\λ}}^2\·(A_3\·{\stackrel{\·}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3)}{32\·f_c^2}\·f_a^3+\frac{3\·{\mathrm{\λ}}^3\·{\stackrel{\·}{A}}_3}{128\·f_c^2}{f}_a^4\end{array}\right\}---\left(28\right)$

$\mathrm{\τ}(f_a,R)=\frac{2\·R}{c\·M\left(f_a\right)}---\left(29\right)$

$\mathrm{\τ}(f_a,R_0)=\frac{2\·R_0}{c\·M\left(f_a\right)}---\left(30\right)$

According to formula (26)～(30), obtain the operations factor Y that in the NCS algorithm, uses _m(f _a), q ₂And q ₃, their expression formula is respectively (31)～(33), wherein Y _m(f _a) will be used on the other hand adjust because residual three phase errors that the adjustment of follow-up frequency modulation rate space-variant shape is introduced except that the influence of three phase places on the one hand; q ₂And q ₃Be mainly used in the space-variant property of adjustment range migration and the space-variant property of frequency modulation rate.

$Y_m\left(f_a\right)=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-0.5)}{K_s^3(f_a,R_0)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-1)}---\left(31\right)$

q ₂＝K _s(f _a，R ₀)·(M(f _ref)/M(f _a)-1) (32)

$q_3=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-1)}{2}---\left(33\right)$

2.6

in the 6th exponential term

on formula (17) equal sign right side obtains in the 2-d spectrum decoupling zero time; Relevant with distance to the cube of frequency, and

${\mathrm{\φ}}_3(f_a,R)=$

$2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{\mathrm{\λ}\·f_a^2}{8\·k_2\·f_c^3}-\frac{3\·\mathrm{\λ}\·k_1\·k_3}{16\·k_2^3\·f_c^3}\·f_a^2-\frac{3\·\mathrm{\λ}\·k_1^2\·(9\·k_3^2-4\·k_2\·k_4)}{64\·k_2^5\·f_c^3}\·f_a^2\\ -\frac{{\mathrm{\λ}}^2\·k_3\·f_a^3}{8\·k_2^3\·f_c^3}-\frac{{\mathrm{\λ}}^2\·k_1\·(9\·k_3^2-4\·k_2\·k_4)}{16\·k_2^5\·f_c^3}\·f_a^3-\frac{5\·{\mathrm{\λ}}^3\·(9\·k_3^2-4\·k_2\·k_4)}{256\·k_2^5\·f_c^3}\·f_a^4\end{array}\right]---\left(34\right)$

ф ₃(f _a, R) have space-variant property, but it with the distance to variation can ignore, generally use ф ₃(f _a, R ₀) replacement ф ₃(f _a, R), also i.e. k in (34) ₁～k ₄Use R ₀The k as a result at place ₁₀～k ₄₀, therefore, ф ₃(f _a, R ₀) expression formula do

${\mathrm{\φ}}_3(f_a,R)={\mathrm{\φ}}_3(f_a,R_0)=$

$2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{\mathrm{\λ}\·f_a^2}{8\·k_{20}\·f_c^3}-\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{16\·k_{20}^3\·f_c^3}\·f_a^2-\frac{3\·\mathrm{\λ}\·k_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{64\·k_{20}^5\·f_c^3}\·f_a^2\\ -\frac{{\mathrm{\λ}}^2\·k_{30}\·f_a^3}{8\·k_{20}^3\·f_c^3}-\frac{{\mathrm{\λ}}^2\·k_1\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{16\·k_{20}^5\·f_c^3}\·f_a^3-\frac{5\·{\mathrm{\λ}}^3\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{256\·k_{20}^5\·f_c^3}\·f_a^4\end{array}\right]---\left(35\right)$

Beneficial effect

The present invention compares with respect to prior art; It is advantageous that: the method through the high-order Taylor expansion has obtained a kind of new serpentine track model that is applicable to GEO SAR; It is bigger that this locus model can solve GEO SAR perigee equivalence straight line model error ratio, and apogee equivalence straight line model such as can not use fully at shortcoming; Simultaneously based on equivalent straight line model, obtained two frequency spectrums that a parsing is applicable to the NCS algorithm, utilize this frequency spectrum, each penalty function of NCS algorithm can be tried to achieve, and has realized the requirement of GEO SAR large scene imaging.

Description of drawings

Fig. 1 is the NCS algorithmic technique scheme implementation process flow diagram of the present invention after improving;

Fig. 2 is point target (30km ,-30km) the simulation results figure in the embodiment of the invention;

Fig. 3 is point target (30km ,-30km) the simulation results figure in the embodiment of the invention;

Fig. 4 is point target (0km, simulation results figure 0km) in the embodiment of the invention;

Fig. 5 is point target (30km, simulation results figure 30km) in the embodiment of the invention;

Fig. 6 is point target (30km, simulation results figure 30km) in the embodiment of the invention;

Embodiment

Elaborate below in conjunction with the embodiment of accompanying drawing to the inventive method.

Embodiment

Radar with certain speed flight, is launched the chirp signal earthward on geostationary orbit, and accepts the echo from ground.Echo to obtaining carries out imaging processing, proposes a kind of improved NCS algorithm of GEO SAR that is applicable to, its concrete steps are as shown in Figure 1, comprising:

1) target echo that receives of radar

Radar emission one carrier frequency is f _cLinear FM signal.After the echo process demodulation that receives, can obtain

$s(t_r,t_a)=u_r(t_r-\frac{2\·R_n}{c})\·u_a\left(t_a\right)\·\exp [j\·\mathrm{\π}\·K_r\·{(t_r-\frac{2\·R_n}{c})}^2]\·\exp (-j\·\frac{4\·\mathrm{\π}}{\mathrm{\λ}}\·R_n)---\left(36\right)$

U wherein _r() and u _a() be respectively the distance to the orientation to envelope, t _rAnd t _aBe respectively the distance to the orientation to the time, K _rFor the distance to the frequency modulation rate, c is the light velocity, λ is a wavelength, R _nHistorical for the oblique distance of target, the formula of embodying can use formula (2) to represent.

2) echo is carried out range migration preliminary correction and three phase places removals

The distance to the orientation behind FFT, can obtain the 2-d spectrum expression formula of SAR echo, promptly

$S(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{{2k}_1}{c}(f_r+f_c)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$ (37)

$\·\exp [-j\·2\·\mathrm{\π}\·b(f_a,f_r)]\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_a\right)}\·f_r]\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{K_s(f_a,R)}]\·\exp [j\·{\mathrm{\φ}}_3(f_a,R)\·f_r^3]$

Need do two work in two-dimensional frequency.The first, remove the migration phase place b (f at the RP place that in 2-d spectrum is derived, obtains _a, f _r), to make things convenient for the derivation of subsequent algorithm, the expression formula of penalty function is (38).Need to prove that the range migration of removing is the part of RP range migration here; The second, multiply by a nonlinear frequency modulation function, expression formula is (39), the influence that this function will be removed three phase places on the one hand is used to adjust because residual three phase errors that the adjustment of follow-up frequency modulation rate space-variant shape is introduced on the other hand.

H ₁＝exp[j·2·π·B ₁₀(f _a，f _r)] (38)

$H_2=\exp [j\·\frac{2\·\mathrm{\π}}{3}\·\left(f_a\right)\·f_r^3]---\left(39\right)$

Wherein

$Y\left(f_a\right)=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(\mathrm{\α}-0.5)}{{K_s}^3(f_a,R_0)\·(\mathrm{\α}-1)}-\frac{3}{2\·\mathrm{\π}}\·{\mathrm{\φ}}_3(f_a,R_0)---\left(40\right)$

In (40), the expression formula of α does

$\mathrm{\α}=\frac{M\left(f_{\mathrm{ref}}\right)}{M\left(f_a\right)}---\left(41\right)$

f _RefFor the orientation to reference frequency.

Echo expression formula after treatment does

$S_1(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{{2k}_1}{c}(f_r+f_c)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$ (42)

$\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_a\right)}\·f_r]\·\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{K_s(f_a,R)}]\·\exp [j\·\frac{2\·\mathrm{\π}}{3}\·Y_m\left(f_a\right)\·f_r^3]$

Wherein

$Y_m\left(f_a\right)=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(\mathrm{\α}-0.5)}{K_s^3(f_a,R_0)\·(\mathrm{\α}-1)}---\left(43\right)$

The adjustment distance of space-variant property of space-variant property and frequency modulation rate of 3) echo data being carried out range migration is to IFFT, and the expression formula apart from the Doppler territory that obtains echo does

$S_1(t_r,f_a)=u_r\left\{\frac{K_s(f_a,R)}{k_r}\right[t_r-\frac{2\·R}{c\·M\left(f_a\right)}\left]\right\}\·u_a\left(f_a\right)\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·$

$\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]\·\exp \{j\·\mathrm{\π}\·K_s(f_a,R){[t_r-\frac{2\·R}{c\·M\left(f_a\right)}]}^2\}\·---\left(44\right)$

$\exp \{j\·\frac{2\·\mathrm{\π}}{3}\·Y_m\left(f_a\right)\·K_s(f_a,R){[t_r-\frac{2\·R}{c\·M\left(f_a\right)}]}^3\}$

Carrying out non-linear CS operation apart from the Doppler territory, mainly is the space-variant property of adjustment range migration and the space-variant property of frequency modulation rate.Through after the operation in this step, the space-variant property removal in the scene realizes that range migration can carry out Unified Treatment in two-dimensional frequency.Non-linear CS handling function does

$H_3=\exp \{j\·\mathrm{\π}\·q_2\·{[t_r-\mathrm{\τ}(f_a,R_0)]}^2\}\·\exp \{j\·\frac{2\·\mathrm{\π}}{3}\·q_3\·{[t_r-\mathrm{\τ}(f_a,R_0)]}^3\}---\left(45\right)$

Wherein

q ₂＝q ₂＝k _s(f _a，R ₀)·(α-1) (46)

$q_3=q_3=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(\mathrm{\α}-1)}{2}---\left(47\right)$

4) echo is carried out distance to compression and range migration correction

(45) and after (44) multiply each other, carry out distance again to FFT.The echo data of this moment is in two-dimensional frequency, and this moment, the echo expression formula did

$S_2(f_r,f_a)=u_r\left[\frac{f_r}{K_s(f_a,R_0)\·\mathrm{\α}}\right]\·u_a\left(f_a\right)\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]\·$

$\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\α}\·K_s(f_a,R_0)}]\·\exp \{j\·\frac{2\·\mathrm{\π}}{3}\·\frac{[Y_m\left(f_a\right)\·K_s^3(f_a,R_0)+q_3]}{{[\mathrm{\α}\·K_s(f_a,R_0)]}^3}\·f_r^3\}\·---\left(48\right)$

$\exp \{-j\·\frac{4\·\mathrm{\π}\·R_0}{c}\·[\frac{1}{M\left(f_a\right)}-\frac{1}{M\left(f_{\mathrm{ref}}\right)}]\·f_r\}\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_{\mathrm{ref}}\right)}\·f_r]\·\exp (j\·\mathrm{\π}\·C_0)$

In (48), last exponential term is about C ₀, it is to find the solution Y _m(f _a), q ₂And q ₃In the process under residual, the formula of embodying does

$C_0=K_s(f_a,R)\·{\mathrm{\Δ\τ}}^2\·{(\frac{1}{\mathrm{\α}}-1)}^2+\frac{2}{3}\·Y_m\left(f_a\right)\·K_s^3(f_a,R)\·{\mathrm{\Δ\τ}}^3\·{(\frac{1}{\mathrm{\α}}-1)}^3$ (49)

$+q_2\·{\left(\frac{\mathrm{\Δ\τ}}{\mathrm{\α}}\right)}^2+\frac{2}{3}\·q_3\·{\left(\frac{\mathrm{\Δ\τ}}{\mathrm{\α}}\right)}^3$

Wherein

Δτ＝τ(f _a，R)-τ(f _a，R ₀) (50)

C ₀Not only relevant to frequency with the orientation, and be along distance to changing, therefore the compensation of last exponential term can only carried out apart from the Doppler territory in (48).

Because through the non-linear CS operation in a last step, range migration correction can carry out Unified Treatment by frequency domain in the orientation, and will carry out distance to compression and secondary range compression etc. in two-dimensional frequency.Distance to compression function does

$H_4=\exp [j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\α}\·K_s(f_a,R_0)}]---\left(51\right)$

The secondary range compression function does

$H_5=\exp \{-j\·\frac{2\·\mathrm{\π}}{3}\·\frac{[Y_m\left(f_a\right)\·K_s^3(f_a,R_0)+q_3]}{{[\mathrm{\α}\·K_s(f_a,R_0)]}^3}\·f_r^3\}---\left(52\right)$

The range migration correction function does

$H_6=\exp \{j\·\frac{4\·\mathrm{\π}\·R_0}{c}\·[\frac{1}{M\left(f_a\right)}-\frac{1}{M\left(f_{\mathrm{ref}}\right)}]\·f_r\}---\left(53\right)$

The echo expression formula of distance behind compression, secondary range compression and range migration correction does

$S_3(f_r,f_a)=u_r\left\{\frac{f_r}{K_s(f_a,R_0)\·\mathrm{\α}}\right\}\·u_a\left(f_a\right)\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·$ (54)

$\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_{\mathrm{ref}}\right)}\·f_r]\·\exp (j\·\mathrm{\π}\·C_0)$

5) echo is carried out the orientation to compression

Behind compression and range migration correction, the echo of this moment makes progress in distance, and line focus is good through distance, need carry out be the orientation to compression, the orientation will be upgraded along the different distances door to compression function.At first will carry out distance to IFFT, obtain echo expression formula apart from the Doppler territory at the echo data after two-dimensional frequency is through distance compression and migration treatment for correcting

$S_3(t_r,f_a)=\sin c[t_r-\frac{2\·R}{c\·M\left(f_{\mathrm{ref}}\right)}]\·u_a\left(f_a\right)\·---\left(55\right)$

$\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]\·\exp (j\·\mathrm{\π}\·C_0)$

Carry out the orientation to compression, compression function does

H ₇＝exp[-j·2·π·ф _az(f _a，R)] (56)

And then carry out the removal of residual phase, expression formula does

H ₈＝exp[-j·2·π·ф _RP(R)-j·π·C ₀] (57)

6) final step be the orientation to IFFT, can echo be changed to the two-dimensional time territory, the SAR image that obtains focusing on.

Carry out simulating, verifying below.Here utilize following parameter to carry out simulating, verifying: distance is to bandwidth 18MHz, and SF 20MHz, PRF are 200Hz, pulse width 20us, and the synthetic aperture time is 100s.The result who obtains such as Fig. 2～shown in Figure 6, wherein Fig. 2,3,5,6 is the simulation result of scene marginal point, Fig. 4 is the simulation result of scene center point.The two-dimentional secondary lobe that can find these 5 points is high-visible, does not have the generation of coupling phenomenon.

The above is preferred embodiment of the present invention, and the present invention should not be confined to the disclosed content of this embodiment and accompanying drawing.Everyly do not break away from the equivalence of accomplishing under the disclosed spirit of the present invention or revise, all fall into the scope of the present invention's protection.

Claims (1)

1. improvement NCS imaging algorithm that is applicable to geostationary orbit SAR; It is characterized in that its improvements are two parts: the one, set up the serpentine track signal model and replace the equivalent straight line model in the former NCS imaging algorithm; The 2nd, and on the basis of setting up the serpentine track signal model, obtain the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm, the detailed process of two parts is respectively:

1) process of setting up the serpentine track signal model is:

Definition satellite and the target in each pulse repetition time (PRT) coordinates were

and

satellite and target slant range between the real history is expressed as

$R_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{gn}}\left|\right|---\left(1\right)$

Formula (1) is carried out obtaining the serpentine track model after the Taylor expansion

R _n＝R+k ₁·t _a+k ₂·t _a ²+k ₃·t _a ³+k ₄·t _a ⁴+… (2)

T wherein _aFor the orientation to the time, R, k ₁, k ₂, k ₃And k ₄Be R _nThe Taylor expansion coefficient on 0 to 4 rank, k wherein ₁, k ₂, k ₃And k ₄The formula of embodying is respectively:

$k_1=k_{10}+{\stackrel{.}{k}}_1\·(R-R_0)---\left(3\right)$

$k_2=k_{20}+{\stackrel{.}{k}}_2\·(R-R_0)---\left(4\right)$

$k_3=k_{30}+{\stackrel{.}{k}}_3\·(R-R_0)---\left(5\right)$

$k_4=k_{40}+{\stackrel{.}{k}}_4\·(R-R_0)---\left(6\right)$

In formula (3)～(6), k ₁₀～k ₄₀,

The formula that embodies be respectively:

$k_{10}={\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T/\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|---\left(7\right)$

$k_{20}=\frac{{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}---\left(8\right)$

$k_{30}=\frac{{\stackrel{\→}{b}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·{\stackrel{\→}{a}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T}{6\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}+\frac{{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^3}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^5}$ (9)

$-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}$

$k_{40}=\frac{{\stackrel{\→}{d}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·{\stackrel{\→}{b}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T}{24\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}+\frac{{\left|\right|{\stackrel{\→}{a}}_{s0}\left|\right|}^2}{8\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{k_2^2+2\·k_1\·k_3}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}---\left(10\right)$

${\stackrel{.}{k}}_1=\frac{v_{s0x}}{r_{s0x}}-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{{R_0}^2}---\left(11\right)$

${\stackrel{\·}{k}}_2=\frac{a_{s0x}\·{R_0}^2-r_{s0x}\·({\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2)}{2\·{R_0}^2\·r_{s0x}}-\frac{v_{s0x}\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{{R_0}^2\·r_{s0x}}+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^4}$

(12)

${\stackrel{\·}{k}}_3=\frac{b_{s0x}\·{R_0}^2-r_{s0x}\·[{\stackrel{\→}{b}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·{\stackrel{\→}{a}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T]}{6\·{R_0}^2\·r_{s0x}}+$

$\frac{3\·v_{s0x}\·{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^2-5\·r_{s0x}\·{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^3}{2\·{R_0}^6\·r_{s0x}}---\left(13\right)$

$-\frac{v_{s0x}\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+a_{s0x}\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^2\·r_{s0x}}$

$+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^4}-\frac{v_{s0x}\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{R_0}^2\·r_{s0x}}+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{R_0}^4}$

${\stackrel{.}{k}}_4=\frac{d_{s0x}\·{R_0}^2-r_{s0x}\·[{\stackrel{\→}{d}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+4\·{\stackrel{\→}{b}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T]}{24\·{R_0}^2\·r_{s0x}}-\frac{{\left|\right|{\stackrel{\→}{a}}_{s0}\left|\right|}^2}{8\·{R_0}^2}$ (14)

$+\frac{k_{20}\·{\stackrel{\·}{k}}_2+k_{30}\·{\stackrel{.}{k}}_1+k_{10}\·{\stackrel{.}{k}}_3}{r_{s0x}}+\frac{k_{20}^2+2\·k_{10}\·k_{30}}{2\·{R_0}^2}$

In formula (7)～formula (14),

With

Represent satellite at aperture center position vector constantly, With

Represent respectively satellite aperture center constantly velocity, acceleration, acceleration vector and add acceleration vector, R ₀Expression satellite and reference point target are at aperture center distance constantly, r _S0x, a _S0x, b _S0x, v _S0xAnd d _S0xBe respectively

With

Distance under the scene coordinate system is to component;

2), obtain the two-dimensional analysis frequency spectrum table 2-d spectrum that is applicable to the NCS imaging algorithm and do based on serpentine track

$S(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{2\·k_1}{c}\·\left(f_r{+f}_c\right)]\·$

$\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$

$\·\exp [-j\·2\·\mathrm{\π}\·b(f_a,f_r)]\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_a\right)}\·f_r]\·---\left(15\right)$

$\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{K_s(f_a,R)}]\·\exp [j\·{\mathrm{\φ}}_3(f_a,R)\·f_r^3]$

Formula (15) is further described as follows:

2.1 the ф on formula (15) equal sign right side _Az(f _a, R) be the orientation to modulating function, the formula of embodying does

${\mathrm{\φ}}_{\mathrm{az}}(f_a,R)=[\frac{k_1}{2\·k_2}+\frac{3\·k_1^2\·k_3}{8\·k_2^3}+\frac{k_1^3\·(9\·k_3^2-4\·k_2\·k_4)}{16\·k_2^5}]\·f_a+$

$[\frac{\mathrm{\λ}}{8\·k_2}+\frac{3\·\mathrm{\λ}\·k_1\·k_3}{16\·k_2^3}+\frac{3\·\mathrm{\λ}\·k_1^2\·(9\·k_3^2-4\·k_2\·k_4)}{64\·k_2^5}]\·f_a^2+---\left(16\right)$

$[\frac{{\mathrm{\λ}}^2\·k_3}{32\·k_2^3}+\frac{\mathrm{\λ}\·k_1\·(9\·k_3^2-4\·k_2\·k_4)}{64\·k_2^5}]\·f_a^3+\frac{{\mathrm{\λ}}^3\·(9\·k_3^2-4\·k_2\·k_4)}{512\·k_2^5}\·f_a^4$

2.2 the ф on formula (15) equal sign right side _RP(R) be excess phase after the accurate 2-d spectrum Taylor expansion, expression formula does

${\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)=\frac{k_1^2}{2\·\mathrm{\λ}\·k_2}+\frac{k_1^3\·k_3}{4\·\mathrm{\λ}\·k_2^3}+\frac{k_1^4\·(9\·k_3^2-4\·k_2\·k_4)}{32\·\mathrm{\λ}\·k_2^5}-\frac{2\·R}{\mathrm{\λ}}---\left(17\right)$

2.3 the b (f on formula (15) equal sign right side _a, f _r) the RP place migration phase place that obtains when launching for 2-d spectrum, its expression formula does

$b(f_a,f_r)=-[\frac{k_{10}^2}{2\·k_{20}\·c}+\frac{k_{10}^3\·k_{30}}{4\·k_{20}^3\·c}+\frac{k_{10}\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·c\·k_{20}^5}]\·f_r+$

$[\frac{\mathrm{\λ}}{8\·k_{20}\·f_c}+\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{16\·k_{20}^3\·f_c}+\frac{3\·\mathrm{\λ}{k}_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{64\·f_c\·k_{20}^5}]\·f_a^2\·f_r+$ (18)

$[\frac{{\mathrm{\λ}}^2\·k_{30}}{16\·k_{20}^3\·f_c}+\frac{{\mathrm{\λ}}^2\·k_{10}\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·f_c\·k_{20}^5}]\·f_a^3\·f_r+$

$\frac{3\·{\mathrm{\λ}}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{512\·f_c\·k_{20}^5}{f}_a^4\·f_r+(\frac{2\·R_0}{c}-{\stackrel{\·}{B}}_1\·R_0)\·f_r$

Wherein

${\stackrel{\·}{B}}_1=-\frac{2\·k_{10}\·k_{20}\·{\stackrel{.}{k}}_1-k_{10}^2\·{\stackrel{.}{k}}_2}{2\·c\·k_{20}^2}-\frac{(3\·k_{10}^2\·k_{30}\·{\stackrel{\·}{k}}_1+k_{10}^3\·{\stackrel{\·}{k}}_3)\·k_{20}-3\·k_{10}^3\·k_{30}\·{\stackrel{\·}{k}}_2}{4\·c\·k_{20}^4}-\frac{A_3\·{\stackrel{.}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3}{32\·c}+$

$\left\{\begin{array}{c}-\frac{\mathrm{\λ}\·{\stackrel{\·}{k}}_2}{8\·f_c\·k_{20}^2}+\frac{3\·\mathrm{\λ}\·[(k_{10}\·{\stackrel{\·}{k}}_3+k_{30}\·{\stackrel{\·}{k}}_1)\·k_{20}-3\·k_{10}\·k_{30}\·{\stackrel{\·}{k}}_2]}{16\·f_c\·k_{20}^4}\\ +\frac{3\·\mathrm{\λ}\·(2\·k_{10}\·A_3\·{\stackrel{.}{k}}_1-k_{10}^2\·{\stackrel{\·}{A}}_3)}{64\·f_c}\end{array}\right\}\·f_a^2$

$+[\frac{{\mathrm{\λ}}^2\·(k_{20}\·{\stackrel{\·}{k}}_3-3\·k_{30}\·{\stackrel{\·}{k}}_2)}{16\·f_c\·k_{20}^4}+\frac{{\mathrm{\λ}}^2\·(A_3\·{\stackrel{\·}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3)}{32\·f_c}]\·f_a^3+\frac{3\·{\mathrm{\λ}}^2\·{\stackrel{.}{A}}_3}{512\·f_c}\·f_a^4+\frac{2}{c}$

(19)

$A_3=\frac{9\·k_{30}^2-4\·k_{20}\·k_{40}}{k_{20}^5}---\left(20\right)$

${\stackrel{.}{A}}_3=\frac{[18\·k_{30}\·{\stackrel{.}{k}}_3-4\·(k_{20}\·{\stackrel{\·}{k}}_4+k_{40}\·{\stackrel{\·}{k}}_2)]k_{20}-5\·(9\·k_{30}^2-4\·k_{20}\·k_{40})\·{\stackrel{.}{k}}_2}{k_{20}^6}---\left(21\right)$

2.4 the 4th exponential term on formula (15) equal sign right side is

In,

Be range migration, M (f _a) be the migration factor and

$M\left(f_a\right)=\frac{1}{{\stackrel{.}{B}}_1\·c}---\left(22\right)$

2.5 the 5th exponential term on formula (15) equal sign right side is

In

For distance to modulation item, K wherein _s(f _a, R) be new distance to frequency modulation factor, and

K _s(f _a，R)＝K _s(f _a，R ₀)+Δk _s(f _a)·[τ(f _a，R)-τ(f _a，R ₀)] (23)

Wherein

$\frac{1}{K_s(f_a,R_0)}=-[\frac{\mathrm{\λ}}{4\·k_{20}\·f_c^2}+\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{8\·k_{20}^3\·f_c^2}+\frac{3\·\mathrm{\λ}\·k_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·k_{20}^5\·f_c^2}]\·f_a^2-$

$[\frac{3\·{\mathrm{\λ}}^2\·k_{30}}{16\·k_{20}^3\·f_c^2}+\frac{3\·{\mathrm{\λ}}^2\·k_{10}\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{32\·k_{20}^5\·f_c^2}]\·f_a^3---\left(24\right)$

$-\frac{3\·{\mathrm{\λ}}^3\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{128\·k_{20}^5\·f_c^2}\·f_a^4+\frac{1}{K_r}$

${\mathrm{\Δk}}_s\left(f_a\right)=\frac{K_s^2(f_a,R_0)\·c\·M\left(f_a\right)}{2}.$

$\left\{\begin{array}{c}-\frac{\mathrm{\λ}\·{\stackrel{\·}{k}}_2\·f_a^2}{4\·f_c^2\·k_{20}^2}+\frac{3\·\mathrm{\λ}\·[(k_{10}\·{\stackrel{\·}{k}}_3+k_{30}\·{\stackrel{\·}{k}}_1)\·k_{20}-3\·k_{10}\·k_{30}\·{\stackrel{\·}{k}}_2]}{8\·f_c^2\·k_{20}^4}\·f_a^2+\\ \frac{3\·\mathrm{\λ}\·(2\·k_{10}\·A_3\·{\stackrel{.}{k}}_1-k_{10}^2\·{\stackrel{\·}{A}}_3)}{32\·f_c^2}\·f_a^2+\frac{3\·{\mathrm{\λ}}^2\·(k_{20}\·{\stackrel{\·}{k}}_3-3\·k_{30}\·{\stackrel{\·}{k}}_2)}{16\·f_c^2\·k_{20}^4}\·f_a^3\\ +\frac{3\·{\mathrm{\λ}}^2\·(A_3\·{\stackrel{\·}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3)}{32\·f_c^2}\·f_a^3+\frac{3\·{\mathrm{\λ}}^3\·{\stackrel{\·}{A}}_3}{128\·f_c^2}{f}_a^4\end{array}\right\}---\left(25\right)$

$\mathrm{\τ}(f_a,R)=\frac{2\·R}{c\·M\left(f_a\right)}---\left(26\right)$

$\mathrm{\τ}(f_a,R_0)=\frac{2\·R_0}{c\·M\left(f_a\right)}---\left(27\right)$

According to formula (23)～(27), obtain the operations factor Y that in the NCS algorithm, uses _m(f _a), q ₂And q ₃, its expression formula is respectively (28)～(30), wherein Y _m(f _a) be used to eliminate the influence of three phase places and be used to adjust residual three phase errors of introducing owing to the adjustment of follow-up frequency modulation rate space-variant shape, q ₂And q ₃Be used to adjust the space-variant property of range migration and the space-variant property of frequency modulation rate.

$Y_m\left(f_a\right)=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-0.5)}{K_s^3(f_a,R_0)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-1)}---\left(28\right)$

q ₂＝K _s(f _a，R ₀)·(M(f _ref)/M(f _a)-1) (29)

$q_3=\frac{{\mathrm{\Δk}}_s\left(f_a\right)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-1)}{2}---\left(30\right)$

2.6 the 6th exponential term on formula (15) equal sign right side is In For what obtain when the 2-d spectrum decoupling zero, relevant with distance to the cube of frequency, adopt RP R here ₀The value at place, promptly

${\mathrm{\φ}}_3(f_a,R)={\mathrm{\φ}}_3(f_a,R_0)=$

$2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{\mathrm{\λ}\·f_a^2}{8\·k_{20}\·f_c^3}-\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{16\·k_{20}^3\·f_c^3}\·f_a^2-\frac{3\·\mathrm{\λ}\·k_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{64\·k_{20}^5\·f_c^3}\·f_a^2\\ -\frac{{\mathrm{\λ}}^2\·k_{30}\·f_a^3}{8\·k_{20}^3\·f_c^3}-\frac{{\mathrm{\λ}}^2\·k_1\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{16\·k_{20}^5\·f_c^3}\·f_a^3-\frac{5\·{\mathrm{\λ}}^3\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{256\·k_{20}^5\·f_c^3}\·f_a^4\end{array}\right]$

(31)

Images