CN102331577B - 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 imaging algorithm suitable for geosynchronous orbit SAR, belonging to the technical field of synthetic aperture radar (SAR) imaging. The improvement of the present invention to the NCS imaging algorithm lies in two parts: one is to establish a curved trajectory signal model to replace the equivalent straight line model in the original NCS imaging algorithm; 2D analytical spectral representation of the NCS imaging algorithm. Compared with the prior art, the present invention has the advantages that a new curved trajectory model suitable for GEO SAR is obtained through the method of high-order Taylor expansion, and the trajectory model can solve the problem that the error of the GEOSAR perigee equivalent straight line model is relatively large. The apogee equivalent straight line model cannot be applied at all; at the same time, based on the equivalent straight line model, a binary spectrum suitable for the NCS algorithm is obtained. Using this spectrum, each compensation function of the NCS algorithm can be obtained, and the GEO SAR is realized. Large scene imaging requirements.

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 synthetic-aperture radar (SAR) technical field of imaging.

Background technology

Present SAR satellite is low orbit satellite, and orbit altitude is no more than 1,000km, to the coverage cycle of particular locality being generally 3 to 5 days, 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 the SAR satellite that operates on 36, the 000km height geostationary orbit; This geostationary orbit is not geostationary orbit, and it has certain angle of inclination, and its sub-satellite track is ' 8 ' font, can obtain thus the relative motion with terrain object, realizes the two-dimensional SAR imaging.

Present SAR imaging algorithm all is based on low rail (LEO) situation and sets up, defective is: LEO SAR is general, and the synthetic aperture time 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 relies on was because 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.For 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 is achieved by the following technical solutions.

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 equivalent straight line model in the former NCS imaging algorithm, the 2nd, and obtain the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm on the basis of setting up the serpentine track signal model, the detailed process of two parts is respectively:

1) traditional NCS algorithm is based on equivalent straight line model, but the problems such as the large even inefficacy of error ratio appear in equivalent straight line model in GEO SAR, therefore need to 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 coordinate of target in each pulse-recurrence time (PRT) are respectively

With

True oblique distance history lists between satellite and the target is shown

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

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 ₄Expression 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,

Expression 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 is 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 proposes 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 is

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

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

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 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), through after deriving, draw GEO SAR 2-d spectrum and be:

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

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

Be further described as follows to formula (17):

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

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

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

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

Because ф _Az(f _a, R) only relevant to frequency and target location with the orientation, with distance to frequency-independent, therefore can compensate at range-Dopler domain;

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 is

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

This and orientation are to frequency with apart to frequency-independent, and be relevant to the position with the distance of target, can be compensated at range-Dopler domain;

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

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

$[\frac{\mathrm{\λ}}{8\·k_{20}\·f_c}+\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{16\·k_{20}^3\·f_c}+\frac{3\·\mathrm{\λ}{k}_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{64\·f_c\·k_{20}^5}]\·f_a^2\·f_r+$ (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 reference point place migration phase place, does not have this in traditional CS based on equivalent straight line model, NCS algorithm, 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, 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 Expression formula shown in (21);

Can find that this is inconsistent to the position migration for different distances, mainly be to multiply by a Chirp signal at range-Dopler domain to adjust this space-variant in the CS algorithm, needs equally to adjust this space-variant in the NCS 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, in the CS algorithm, do not consider its space-variant, it is the part of space-variant modulation in the NCS algorithm, but it is difficult to directly use, and needs to 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 except the impact of three phase places, be used on the other hand adjusting because residual three phase errors that the adjustment of follow-up frequency modulation rate space-variant shape is introduced on the one hand; q ₂And q ₃Be mainly used in adjusting the space-variant of range migration and the space-variant 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 the 6th exponential term on formula (17) equal sign right side In

For what obtain when the 2-d spectrum decoupling zero, relevant to the cube of frequency with distance, 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, 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 as a result k at place ₁₀～k ₄₀, therefore, ф ₃(f _a, R ₀) expression formula be

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

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

Beneficial effect

The present invention compares with respect to prior art, it is advantageous that: the method by the high-order Taylor expansion has obtained a kind of new serpentine track model that is applicable to GEO SAR, it is larger 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 the 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 after the present invention improves;

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

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

Fig. 6 is the simulation results figure of the point target (30km, 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.The echo that obtains is carried out imaging processing, propose a kind of improved NCS algorithm of GEO SAR that is applicable to, its concrete steps comprise as shown in Figure 1:

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 wavelength, R _nHistorical for the oblique distance of target, expression can represent with formula (2).

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, namely

$S(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{{2k}_1}{c}(f_r+f_c)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$ (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 to do two work in two-dimensional frequency.The first, remove the migration phase place b (f at the reference point 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 reference point range migration here; The second, multiply by a nonlinear frequency modulation function, expression formula is (39), the impact that this function will be removed three phase places on the one hand is used for adjusting 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 is

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

f _RefFor the orientation to reference frequency.

Echo expression formula after treatment is

$S_1(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{{2k}_1}{c}(f_r+f_c)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$ (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 of 3) echo data being carried out the space-variant of range migration and frequency modulation rate is to IFFT, and the expression formula that obtains the range-Dopler domain of echo is

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

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

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

Carrying out non-linear CS operation at range-Dopler domain, mainly is to adjust the space-variant of range migration and the space-variant of frequency modulation rate.Through after the operation in this step, the space-variant removal in the scene, the realization range migration can be unified to process in two-dimensional frequency.Non-linear CS handling function is

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

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 again distance to FFT.The echo data of this moment is in two-dimensional frequency, and this moment, the echo expression formula was

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

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

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

In (48), last exponential term is about C ₀, it is to find the solution Y _m(f _a), q ₂And q ₃Under residual in the process, expression is

$C_0=K_s(f_a,R)\·{\mathrm{\Δ\τ}}^2\·{(\frac{1}{\mathrm{\α}}-1)}^2+\frac{2}{3}\·Y_m\left(f_a\right)\·K_s^3(f_a,R)\·{\mathrm{\Δ\τ}}^3\·{(\frac{1}{\mathrm{\α}}-1)}^3$ (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 that therefore the compensation of last exponential term can only be carried out at range-Dopler domain in (48) along distance to changing.

Because through the non-linear CS operation of previous step, range migration correction can be unified to process at the orientation frequency domain, and will carry out distance to compression and secondary range compression etc. in two-dimensional frequency.Distance to compression function is

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

The secondary range compression function is

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

The range migration correction function is

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

Apart from the echo expression formula behind compression, secondary range compression and range migration correction be

$S_3(f_r,f_a)=u_r\left\{\frac{f_r}{K_s(f_a,R_0)\·\mathrm{\α}}\right\}\·u_a\left(f_a\right)\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·$ (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 to carry out be the orientation to compression, the orientation will be upgraded along different range gate to compression function.At first the echo data after two-dimensional frequency is through Range compress and migration correction processing is carried out distance to IFFT, obtain the echo expression formula of range-Dopler domain

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

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

Carry out the orientation to compression, compression function is

H ₇＝exp[-j·2·π·ф _az(f _a，R)] (56)

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

H ₈＝exp[-j·2·π·ф _RP(R)-j·π·C ₀] (57)

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

The below carries out simulating, verifying.Here utilize following parameter to carry out simulating, verifying: distance is to bandwidth 18MHz, and sample frequency 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 finishing under the spirit disclosed in this invention or revise, all fall into the scope of protection of the invention.

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 equivalent straight line model in the former NCS imaging algorithm, the 2nd, and obtain the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm on the basis of setting up the serpentine track signal model; Detailed process is respectively:

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

$S(t_r,t_a)=u_r(t_r-\frac{2\·R_n}{c})\·u_a\left(t_a\right)\·\exp [j\·\mathrm{\π}\·K_r\·{(t_r-\frac{2\·R_n}{c})}^2]\·\exp (-j\·\frac{4\·\mathrm{\π}}{\mathrm{\λ}}\·R_n)---\left(1\right)$

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 wavelength, R _nHistorical for the oblique distance of target, also be the serpentine track signal model, the process of specifically setting up the serpentine track signal model is:

Definition satellite and the coordinate of target in each pulse-recurrence time (PRT) are respectively With , the true oblique distance history lists between satellite and the target is shown

$R_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{gn}}\left|\right|---\left(2\right)$

Formula (2) is carried out obtaining the serpentine track model after the Taylor expansion

$R_n=R+k_1\·t_a+k_2\·{t_a}^2+k_3\·{t_a}^3+k_4\·{t_a}^4+\·\·\·\left(3\right)$

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 ₄Expression is respectively:

$k_1=k_{10}+{\stackrel{\·}{k}}_1\·(R-R_0)---\left(4\right)$

$k_2=k_{20}+{\stackrel{\·}{k}}_2\·(R-R_0)---\left(5\right)$

$k_3=k_{30}+{\stackrel{\·}{k}}_3\·(R-R_0)---\left(6\right)$

$k_4=k_{40}+{\stackrel{\·}{k}}_4\·(R-R_0)---\left(7\right)$

In formula (4) ~ (7), k ₁₀~ K ₄₀,

~

Expression be respectively:

$k_{10}={\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T/\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|---\left(8\right)$

$k_{20}=\frac{{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{{[{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}---\left(9\right)$

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

$-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}---\left(10\right)$

$k_{40}=\frac{{\stackrel{\→}{d}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+4\·{\stackrel{\→}{b}}_{s0}\·{{\stackrel{\→}{v}}_{s0}}^T}{24\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}+\frac{{\left|\right|{\stackrel{\→}{a}}_{s0}\left|\right|}^2}{8\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{k_2^2+2\·k_1\·k_3}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}---\left(11\right)$

${\stackrel{\·}{k}}_1=\frac{v_{s0x}}{r_{s0x}}-\frac{{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{{R_0}^2}---\left(12\right)$

${\stackrel{\·}{k}}_2=\frac{a_{s0x}\·{R_0}^2-r_{s0x}\·({\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2)}{2\·{R_0}^2\·r_{s0x}}-\frac{v_{s0x}\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{{R_0}^2\·r_{s0x}}+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^4}---\left(13\right)$

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

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

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

$+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\stackrel{\→}{a}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{2\·{R_0}^4}-\frac{v_{s0x}\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{R_0}^2\·r_{s0x}}+\frac{3\·{\stackrel{\→}{v}}_{s0}\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·{\left|\right|{\stackrel{\→}{v}}_{s0}\left|\right|}^2}{2\·{R_0}^4}---\left(14\right)$

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

$+\frac{k_{20}\·{\stackrel{\·}{k}}_2+k_{30}\·{\stackrel{\·}{k}}_1+k_{10}\·{\stackrel{\·}{k}}_3}{r_{s0x}}+\frac{k_{20}^2+2\·k_{10}\·k_{30}}{2\·{R_0}^2}---\left(15\right)$

In formula (8) ~ formula (15),

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, obtain the two-dimensional analysis spectrum expression formula that is applicable to the NCS imaging algorithm and be

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

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

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

$\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{K_s(f_a,R)}]\·\exp [j\·{\mathrm{\φ}}_3(f_a,R)\·f_r^2]---\left(16\right)$

Be further described as follows to formula (16):

1. the φ on formula (16) equal sign right side _Az(f _a, R) be the orientation to modulating function, expression is

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

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

$[\frac{{\mathrm{\λ}}^2\·k_3}{32\·k_2^3}+\frac{\mathrm{\λ}\·k_1\·(9\·k_3^2-4{\·k}_2\·k_4)}{64\·k_2^5}]\·f_a^3+\frac{{\mathrm{\λ}}^3\·(9\·k_3^2-4\·k_2\·k_4)}{512\·k_2^5}\·f_a^4---\left(17\right)$

2. the φ on formula (16) equal sign right side _RP(R) be excess phase after the accurate 2-d spectrum Taylor expansion, expression formula is

${\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)=\frac{k_1^2}{2\·\mathrm{\λ}\·k_2}+\frac{k_1^3\·k_3}{4\·\mathrm{\λ}\·k_2^3}+\frac{k_1^4\·(9\·k_3^2-4\·k_2\·k_4)}{32\·\mathrm{\λ}\·k_2^5}-\frac{2\·R}{\mathrm{\λ}}---\left(18\right)$

3. b (the f on formula (16) equal sign right side _a, f _r) the reference point place migration phase place that obtains when launching for 2-d spectrum, its expression formula is

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

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

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

$\frac{3\·{\mathrm{\λ}}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{512\·f_c\·k_{20}^5}{f}_a^4\·f_r+(\frac{2\·R_0}{c}-{\stackrel{\·}{B}}_1\·R_0)\·f_r---\left(19\right)$

Wherein

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

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

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

$A_3=\frac{9\·k_{30}^2-4\·k_{20}\·k_{40}}{k_{20}^5}---\left(21\right)$

${\stackrel{\·}{A}}_3=\frac{[18\·k_{30}\·{\stackrel{\·}{k}}_3-4\·(k_{20}\·{\stackrel{\·}{k}}_4+k_{40}\·{\stackrel{\·}{k}}_2)]\·k_{20}-5\·(9\·k_{30}^2-4\·k_{20}\·k_{40})\·{\stackrel{\·}{k}}_2}{k_{20}^6}---\left(22\right)$

4. the 4th exponential term on formula (16) equal sign right side is

In,

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

$M\left(f_a\right)=\frac{2}{{\stackrel{\·}{B}}_1\·c}---\left(23\right)$

5. the 5th exponential term on formula (16) 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)] (24)

Wherein

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

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

$-\frac{3\·{\mathrm{\λ}}^3\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{128\·k_{20}^5\·f_c^2}\·f_a^4+\frac{1}{K_r}---\left(25\right)$

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

$\left\{\begin{array}{c}-\frac{\mathrm{\λ}\·{\stackrel{\·}{k}}_2\·f_a^2}{4\·f_c^2\·k_{20}^2}+\frac{3\·\mathrm{\λ}\·[(k_{10}\·{\stackrel{\·}{k}}_3+k_{30}\·{\stackrel{\·}{k}}_1)\·k_{20}-3\·k_{10}\·k_{30}\·{\stackrel{\·}{k}}_2}{8\·f_c^2\·k_{20}^4}\·f_a^2+\\ \frac{3\·\mathrm{\λ}\·(2\·k_{10}\·A_3\·{\stackrel{\·}{k}}_1-k_{10}^2\·{\stackrel{\·}{A}}_3)}{32\·f_c^2}\·f_a^2+\frac{3\·{\mathrm{\λ}}^2\·(k_{20}\·{\stackrel{\·}{k}}_3-3\·k_{30}\·{\stackrel{\·}{k}}_2)}{16\·f_c^2\·k_{20}^4}\·f_a^3\\ +\frac{3\·{\mathrm{\λ}}^2\·(A_3\·{\stackrel{\·}{k}}_1-k_{10}\·{\stackrel{\·}{A}}_3)}{32\·f_c^2}\·f_a^3+\frac{3\·{\mathrm{\λ}}^3\·{\stackrel{\·}{A}}_3}{128\·f_c^2}{f}_a^4\end{array}\right\}---\left(26\right)$

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

$\mathrm{\τ}(f_a,R_0)=\frac{2\·R_0}{c\·M\left(f_a\right)}---\left(28\right)$

According to formula (24) ~ (28), obtain the operations factor Y that in the NCS algorithm, uses _m(f _a), q ₂And q ₃, its expression formula is respectively (29) ~ (31), wherein Y _m(f _a) be used for eliminating the impact of three phase places and be used for adjusting residual three phase errors of introducing owing to the adjustment of follow-up frequency modulation rate space-variant shape, q ₂And q ₃Be used for adjusting the space-variant of range migration and the space-variant of frequency modulation rate;

$Y_m\left(f_a\right)=\frac{\mathrm{\Δ}{k}_s\left(f_a\right)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-0.5)}{K_s^3(f_a,R_0)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-1)}---\left(29\right)$

q ₂=K _s(f _a,R ₀)·(M(f _ref)/M(f _a)-1) (30)

$q_3=\frac{\mathrm{\Δ}{k}_s\left(f_a\right)\·(M\left(f_{\mathrm{ref}}\right)/M\left(f_a\right)-1)}{2}---\left(31\right)$

6. the 6th exponential term on formula (16) equal sign right side is In

For what obtain when the 2-d spectrum decoupling zero, relevant to the cube of frequency with distance, adopt reference point R here ₀The value at place, namely

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

$2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{\mathrm{\λ}\·f_a^2}{8\·k_{20}\·f_c^2}-\frac{3\·\mathrm{\λ}\·k_{10}\·k_{30}}{16\·k_{20}^3\·f_c^3}\·f_a^2-\frac{3\·\mathrm{\λ}\·k_{10}^2\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{64\·k_{20}^5\·f_c^3}\·f_a^2\\ -\frac{{\mathrm{\λ}}^2\·k_{30}\·f_a^3}{8\·k_{20}^3\·f_c^3}-\frac{{\mathrm{\λ}}^2\·k_1\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{16\·k_{20}^5\·f_c^3}\·f_a^3-\frac{5\·{\mathrm{\λ}}^3\·(9\·k_{30}^2-4\·k_{20}\·k_{40})}{256\·k_{20}^5\·f_c^3}\·f_a^4\end{array}\right]---\left(32\right)$

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

Need to do two work in two-dimensional frequency; The first, remove the migration phase place b (f at the reference point 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 (33); Need to prove that the range migration of removing is the part of reference point range migration here; The second, multiply by a nonlinear frequency modulation function, expression formula is (34), the impact that this function will be removed three phase places on the one hand is used for adjusting 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)](33)

$H_2=\exp [j\·\·\frac{2\·\mathrm{\π}}{3}\·Y\left(f_a\right)\·f_r^3]---\left(34\right)$

Wherein

$Y\left(f_a\right)=\frac{\mathrm{\Δ}{k}_s\left(f_a\right)\·(\mathrm{\α}-0.5)}{{K_s}^3(f_a,R_0)\·(\mathrm{\α}-1)}-\frac{3}{2\·\mathrm{\π}}\·{\mathrm{\φ}}_3(f_a,R_0)---\left(35\right)$

In (35), the expression formula of α is

$\mathrm{\α}=\frac{M\left(f_{\mathrm{ref}}\right)}{M\left(f_a\right)}---\left(36\right)$

f _RefFor the orientation to reference frequency;

Echo expression formula after treatment is

$S_1(f_r,f_a)=u_r\left(\frac{f_r}{K_r}\right)\·u_a[f_a+\frac{2k_1}{c}(f_r+f_c)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]$

$\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_a\right)}\·f_r]\·\exp [-j\·\mathrm{\π}\·\frac{f_r^2}{K_s(f_a,R)}]\·\exp [j\·\frac{2\·\mathrm{\π}}{3}\·Y_m\left(f_a\right)\·f_r^3]---\left(37\right)$

$Y_m\left(f_a\right)=\frac{\mathrm{\Δ}{k}_s\left(f_a\right)\·(\mathrm{\α}-0.5)}{{K_s}^3(f_a,R_0)\·(\mathrm{\α}-1)}---\left(38\right)$

4) echo data is carried out the adjustment of the space-variant of the space-variant of range migration and frequency modulation rate

Distance is to IFFT, and the expression formula that obtains the range-Dopler domain of echo is

Carrying out non-linear CS operation at range-Dopler domain, mainly is to adjust the space-variant of range migration and the space-variant of frequency modulation rate; Through after the operation in this step, the space-variant removal in the scene, the realization range migration can be unified to process in two-dimensional frequency; Non-linear CS handling function is

$H_3=\exp \{j\·\mathrm{\π}\·q_2\·{[t_r-\mathrm{\τ}(f_a,R_0)]}^2\}\·\exp \{j\·\frac{2\·\mathrm{\π}}{3}\·q_3\·{[t_r-\mathrm{\τ}(f_a,R_0)]}^3\}---\left(40\right)$

Wherein

q ₂=q ₂=K _s(f _a,R ₀)·(α-1)(41)

$q_3=q_3=\frac{\mathrm{\Δ}{k}_s\left(f_a\right)\·(\mathrm{\α}-1)}{2}---\left(42\right)$

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

(40) and after (39) multiply each other, carry out again distance to FFT; The echo data of this moment is in two-dimensional frequency, and this moment, the echo expression formula was

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

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

$\exp \{-j\·\frac{4\·\mathrm{\π}\·R_0}{c}\·[\frac{1}{M\left(f_a\right)}-\frac{1}{M\left(f_{\mathrm{ref}}\right)}]\·f_r\}\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_{\mathrm{ref}}\right)}\·f_r]\·\exp (j\·\mathrm{\π}\·C_0)---\left(43\right)$

In (43), last exponential term is about C ₀, it is to find the solution Y _m(f _a), q ₂And q ₃Under residual in the process, expression is

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

$+q_2\·{\left(\frac{\mathrm{\Δ\τ}}{\mathrm{\α}}\right)}^2+\frac{2}{3}\·q_3\·{\left(\frac{\mathrm{\Δ\τ}}{\mathrm{\α}}\right)}^3---\left(44\right)$

Wherein

Δτ=τ(f _a,R)-τ(f _a,R ₀)(45)

C ₀Not only relevant to frequency with the orientation, and be that therefore the compensation of last exponential term can only be carried out at range-Dopler domain in (43) along distance to changing;

Because through the non-linear CS operation of previous step, range migration correction can be unified to process at the orientation frequency domain, and will carry out distance to compression and secondary range compression etc. in two-dimensional frequency; Distance to compression function is

$H_4=\exp [j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\α}\·K_s(f_a,R_0)}]---\left(46\right)$

The secondary range compression function is

$H_5=\exp \{-j\·\frac{2\·\mathrm{\π}}{3}\·\frac{{[Y}_m\left(f_a\right)\·{K_s}^3(f_a,R_0)+q_3}{{[\mathrm{\α}\·K_s(f_a,R_0)]}^3}\·f_r^3\}---\left(47\right)$

The range migration correction function is

$H_6=\exp \{j\·\frac{4\·\mathrm{\π}\·R_0}{c}\·[\frac{1}{M\left(f_a\right)}-\frac{1}{M\left(f_{\mathrm{ref}}\right)}]\·f_r\}---\left(48\right)$

Apart from the echo expression formula behind compression, secondary range compression and range migration correction be

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

$\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]\·\exp [-j\·\frac{4\·\mathrm{\π}\·R}{c\·M\left(f_{\mathrm{ref}}\right)}\·f_r]\·\exp (j\·\mathrm{\π}\·C_0)---\left(49\right)$

6) 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 to carry out be the orientation to compression, the orientation will be upgraded along different range gate to compression function; At first the echo data after two-dimensional frequency is through Range compress and migration correction processing is carried out distance to IFFT, obtain the echo expression formula of range-Dopler domain

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

$\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{az}}(f_a,R)]\·\exp [j\·2\·\mathrm{\π}\·{\mathrm{\φ}}_{\mathrm{RP}}\left(R\right)]\·\exp (j\·\mathrm{\π}\·C_0)---\left(50\right)$

Carry out the orientation to compression, compression function is

H ₇=exp[-j·2·π·φ _az(f _a,R)](51)

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

H ₈=exp[-j·2·π·φ _RP(R)-j·π·C ₀](52)

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

Images