CN107102330A - A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR - Google Patents

A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR Download PDF

Info

Publication number
CN107102330A
CN107102330A CN201710270564.0A CN201710270564A CN107102330A CN 107102330 A CN107102330 A CN 107102330A CN 201710270564 A CN201710270564 A CN 201710270564A CN 107102330 A CN107102330 A CN 107102330A
Authority
CN
China
Prior art keywords
mrow
msub
msup
mfrac
mtr
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN201710270564.0A
Other languages
Chinese (zh)
Inventor
王岩
杨健
李景文
孙兵
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Tsinghua University
Original Assignee
Tsinghua University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Tsinghua University filed Critical Tsinghua University
Priority to CN201710270564.0A priority Critical patent/CN107102330A/en
Publication of CN107102330A publication Critical patent/CN107102330A/en
Pending legal-status Critical Current

Links

Classifications

    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/904—SAR modes
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques
    • G01S13/904—SAR modes
    • G01S13/9041—Squint mode

Landscapes

  • Engineering & Computer Science (AREA)
  • Remote Sensing (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Physics & Mathematics (AREA)
  • Electromagnetism (AREA)
  • Computer Networks & Wireless Communication (AREA)
  • General Physics & Mathematics (AREA)
  • Other Investigation Or Analysis Of Materials By Electrical Means (AREA)

Abstract

It is applied to airborne geo Reference Strip synthetic aperture radar (Synthetic Aperture Radar the invention discloses one kind, abbreviation SAR) wide cut Non-linear chirp scaling image processing method (Wide Nonlinear Chirp Scaling, abbreviation WNLCS), WNLCS methods mainly include six steps：Step 1: distance is compensated to quadratic phase, Step 2: non-linear consistent Range migration correct and distance are to Signal Compression, Step 3: the remaining Range migration correct based on time domain interpolation, Step 4: orientation frequency disturbance is handled, Step 5: the compensation of orientation quadratic phase and orientation Signal Compression, Step 6: focusedimage Geometry rectification.WNLCS image processing methods proposed by the present invention have the data-handling capacity of full aperture, and data redundancy amount is low, are a kind of efficient, accurate image processing methods suitable for airborne geo Reference Strip SAR earth observation demands.

Description

A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR
Technical field
The present invention relates to synthetic aperture radar (Synthetic Aperture Radar, abbreviation SAR) technical field, specifically It is to say, refers to a kind of wide cut Non-linear chirp scaling (Wide Nonlinear suitable for airborne geo Reference Strip SAR Chirp Scaling, abbreviation WNLCS) image processing method.
Background technology
Satellite-borne SAR is a kind of round-the-clock, round-the-clock microwave imaging instrument, it is adaptable to the earth's surface prison under IFR conditions Survey.Traditional satellite-borne SAR typically operates in nearly polar orbit, to make satellite have the observing capacity to high latitude area.By Do not possess continuous distance to beam scanning capabilities in traditional satellite-borne SAR, their earth's surface beam coverage area approximately along " north-south " Directional Extension.But observe and applying for specific earth's surface, such as earthquake fault band is monitored, and target observation region is walked Larger angle is there may be between " north-south " direction.Now still use the beam coverage area moved towards approximately along " north-south " The reduction of observed efficiency will be caused.This is due to that, to ensure the integrality of data acquisition, actual earth's surface beam coverage area is needed Target observation region is at least external in, this inevitably results in the increase of required beam coverage area width, data redundancy amount Rise and the decline of data-handling efficiency.Under extreme conditions, be likely less than must for the width of spaceborne stripmap SAR beam coverage area The width of the beam coverage area needed, now can not data acquisition merely with single track observation.Many rail repeated measures are One solution route, but this can cause the significantly extension of data acquisition cycle；The scan pattern of time-division is another solution route, But this diminution that can cause azimuth accumulation angle and the decline of azimuthal observation resolution ratio.Therefore, above two solution route is equal It is undesirable.
Airborne geo Reference Strip SAR can solve the problems, such as above-mentioned data acquisition well.It is carried out by adjusting wave beam Continuous distance carries out Proactive traceback to scanning to target area, is covered so as to generate the wave beam consistent with target observation regional strike Cover fillet band.The pattern at least contribute to the benefit of three aspects：Less total amount of data, shorter data acquisition cycle and Geng Gao Azimuth resolution.But at the same time, the pattern will bring both sides difficult to echo imaging.First, airborne geo Reference Strip SAR big strabismus data acquisition configuration can make echo-signal have serious two-dimentional coupled characteristic；Secondly, spacebornely Reason Reference Strip SAR distance can make echo-signal have strong space-variant in azimuth to continuous beam scanning.At present, still without appoint A kind of what image processing method can realize efficient, accurate imaging to airborne geo Reference Strip SAR echo data.
The content of the invention
The present invention is for the characteristics of the coupling of airborne geo Reference Strip SAR echo signal is serious, space-variant is strong, it is proposed that one Plant efficient, the accurate WNLCS image processing methods suitable for the pattern.WNLCS algorithms are made up of following six key step：
Step 1: distance is compensated to quadratic phase.
Step 2: non-linear consistent Range migration correct and distance are to Signal Compression.
Step 3: the remaining Range migration correct based on time domain interpolation.
Step 4: orientation frequency disturbance is handled.
Step 5: the compensation of orientation quadratic phase and orientation Signal Compression.
Step 6: focusedimage Geometry rectification.
The advantage of the invention is that：
(1) a kind of new method of nonlinear consistent Range migration correct is proposed, this method can be corrected along earth's surface in batches The range migration of the target of beam trajectory distribution；
(2) a kind of new method of the remaining Range migration correct based on time domain interpolation is proposed, this method can be corrected in batches The remaining range migration of all targets in scene；
(3) a kind of new method of the orientation frequency disturbance of amendment is proposed, this method can eliminate doppler frequency rate Azimuthal dependence, is convenient for follow-up orientation signal focus processing.
Brief description of the drawings
Fig. 1 is the schematic diagram of WNLCS method image processing steps in the present invention；
Fig. 2 is the airborne geo Reference Strip SAR pattern diagrams that signal acquisition is used in the present invention；
Fig. 3 is the effect diagram of WNLCS methods progress Range migration correct in the present invention；
Fig. 4 is the effect diagram of WNLCS methods progress orientation frequency disturbance in the present invention；
Fig. 5 is the flow chart of WNLCS method signal transactings in the present invention；
Fig. 6 is the distribution map of simulation objectives in the embodiment of the present invention；
Fig. 7 is remaining Range migration correct design sketch in the embodiment of the present invention；
Fig. 8 is focusedimage Geometry rectification design sketch in the embodiment of the present invention；
Fig. 9 is the profile of Scene of embodiment of the present invention edge destination and scene center target.
Embodiment
Below in conjunction with drawings and examples, the present invention is described in further detail.
The present invention is a kind of WNLCS image processing methods suitable for airborne geo Reference Strip SAR.This method is by six Process step is constituted, as shown in Figure 1.
Step 1: distance is compensated to quadratic phase.Specially：
If airborne geo Reference Strip SAR data acquisition configuration is as shown in Figure 2.Cartesian coordinate system in figure is by as follows Method is set up：Origin O and observation scene center superposition, surface projection direction of the X-axis along beam center, Y-axis is along wave cover band Trend, Z axis is away from the earth's core direction and perpendicular to earth's surface.In follow-up discussion, X-direction is referred to as distance to Y-axis side by us To referred to as orientation.V is satellite velocities, and H is satellite altitude, and θ is defined as geographical oblique angle, represents wave cover band trend and satellite The angle of velocity, RcWith RrefThe center oblique distance and its surface projection at t=0 moment are represented respectively, and S is in wave cover band Any one coordinate is (xs, ys, 0) and target, tsAt the time of target inswept for beam center, RsFor target S instantaneous oblique distance, βsWith φsTarget S center downwards angle of visibility and center angle of squint is represented respectively.
WNLCS methods that the present invention is introduced based on the fact that：Airborne geo Reference Strip SAR beam position hangs down all the time Directly in wave cover band, as shown in Figure 2.Now target S instantaneous oblique distance RsWith being expressed as below：
Wherein RgAnd tsThere is one-to-one relation between the coordinate of target
If using traditional linear FM signal as transmission signal, the echo-signal from target S is demodulated into fundamental frequency, The echo-signal is represented by after demodulation
Wherein c is the light velocity, and t and τ represent the orientation slow time and apart from fast time, f respectivelycFor signal carrier frequency, γ is signal Frequency modulation rate.Because the amplitude of signal does not influence imaging, formula (3) eliminates the amplitude of signal.Different from traditional satellite-borne SAR, star Can be up to by carrying the oblique distance change of Geographic Reference stripmap SAR by tens kilometers, and tens thousand of Range resolution units of correspondence can not now be kept away The problem of transmitting is blocked can be run into exempting from.Therefore, airborne geo Reference Strip SAR employs a kind of adopting for multi-emitting pulse spacing Sample technology, has the echo-signal from target S and is expressed as below：
Wherein τdelayThe fast time delay relied on for the orientation introduced by multi-emitting pulse spacing Sampling techniques.To formula (4) Enter row distance to Fourier transformation, obtain
Wherein fτRepresent frequency of distance.In formula (5), Section 1 is that, apart from unrelated doppler phase, Section 2 is represented should The distance removed is to quadratic phase, and Section 3 represents range migration dependent phase.Therefore, second of formula (5) can be based on Phase term, generates distance to quadratic phase compensating filter FRC
Formula (6) is multiplied with formula (5), obtained signal distance frequency spectrum is
Step 2: non-linear consistent Range migration correct and distance are to Signal Compression.Specially：
In formula (7), second phase term represents the phase term caused by range migration, should be given before orientation processing To remove.WNLCS algorithms assume that the range migration of target in scene only exists linear component in the present invention.WNLCS in the present invention Image processing method is first by introducing a nonlinear wave filter FBRMC, carry out consistent Range migration correct.The filtering Device is represented by:
Formula (8) is multiplied with formula (7), obtaining signal distance frequency spectrum is
Wherein
It can be obtained by (10), for the arbitrfary point (point that i.e. abscissa is zero) in Y-axis in Fig. 2, their Rs1Linear portion Divide the synthetic aperture central instant (t=t in targets) be always zero, i.e.,
Therefore, for all targets being distributed along Y-axis, their range migration is all completely removed.Line-spacing is entered to formula (9) Descriscent inverse Fourier transform, can be in the signal after time domain obtains Range compress
Wherein BrSignal bandwidth is represented, function sinc (), which has, to be defined as below
Fig. 3 further demonstrates the principle that WNLCS methods carry out Range migration correct.Numbering is given in Fig. 3 (a) from A To nine of the I targets for being located at diverse location in scene, Fig. 3 (b) give carry out non-linear consistent Range migration correct it Before, the range migration curve of 9 targets.Fig. 3 (c) gives complete non-linear consistent Range migration correct after, 9 targets Range migration curve.It can be found that, although the range migration for the target being now distributed along Y-axis is completely removed, and deviates Y-axis point The target of cloth still has range migration, it is necessary to further be removed.Notice that the range migration curve shown in Fig. 3 (b) is to formula (7) enter row distance to what is obtained after inverse Fourier transform, be not the intermediate result of WNLCS methods.
Step 3: the remaining Range migration correct based on time domain interpolation.Specially：
Target for deviateing Y-axis distribution, relation shown in formula (11) is no longer set up.WNLCS methods are employed in the present invention A kind of remaining range migration in new time domain interpolation processing correction-type (10).The key idea of the time domain interpolation method be by All targets with identical X-coordinate are reoriented in identical range cell.The use of range cell sequence during t=0 is work The reference of interpolation.If Rc1And Rc2Respectively represent interpolation before and after target S center oblique distance, they are represented by
Wherein τsWith τ 'sThe target S corresponding fast time is represented before and after interpolation respectively
Based on formula (14), the X-coordinate x for obtaining target S before and after interpolation can be calculated1And x2
The purpose of time domain interpolation processing is to realize following mapping
x1(S)→x2(S) (17)
It should be noted that although above-mentioned derivation is carried out for target S, the time domain interpolation in actual process is rectified Just it is being that there is global universality.Therefore when doing the expression of time domain interpolation mapping, ts, τsWith τ 'sShould respectively by t, τ and τ ' replacements, wherein τ ' are new distance to the fast time.By formula (14), formula (16) substitutes into formula (17), can obtain time domain interpolation and reflect The expression formula penetrated
Formula (18) is substituted into formula (12), time-domain signal E after interpolation processing can be obtaineds2Expression formula be
The function δ () of Section 1 has and is defined as below in formula (19)
After time domain interpolation processing shown in perfect (18), target S range migration curve can be tried to achieve by below equation
δ(τ′,t；S)=0 (21)
By solving formula (21), it can obtain
Wherein RnewRepresent the new range migration of target S after time domain interpolation
Now, formula (19) can be rewritten as following form
Easily checking, in the oblique distance shown in formula (23), following formula perseverance is set up
As shown in formula (24), now the range migration of all targets is completely removed in scene.For 9 in Fig. 3 (a) Individual target, is completed after remaining Range migration correct, range migration curve such as Fig. 3 (d) of 9 targets is shown.
Step 4: orientation frequency disturbance is handled.Specially：
The Section 2 of formula (24) represents doppler phase.Utilize R in formula (10)s1Definition, can be by target S Doppler Centre frequency fdWith doppler frequency rate frIt is expressed as below
As shown in formula (26), frDependent on the position (R of target in the scenegAnd ts).Therefore positioned at same range cell but Target with different azimuth position can not unanimously be compressed along orientation.Fig. 4 (a) illustrate complete Range migration correct it Afterwards, signal relationship between frequency and time figure of each target along orientation in Fig. 3 (a).
By to doppler centroid and doppler frequency rate heart moment t in the targetsTaylor expansion is carried out, can be to formula (26) approximate processing is carried out.The WNLCS image processing methods of the present invention are thought to doppler centroid fdSecond order Taylor's exhibition Open and to doppler frequency rate frFirst order Taylor expansion have enough approximation qualities.Therefore, formula (26) can be rewritten as
Each term coefficient wherein in formula (27) has following expression
Notice that coefficients all in formula (28) only depend on the X-coordinate of target, they can facilitate in processing procedure Ground is updated by range cell.The WNLCS image processing methods of the present invention employ a kind of orientation frequency disturbance of amendment Operation, the operation can be pointed to same range cell but the realization of goal doppler frequency rate of different azimuth position (Y-coordinate) It is unified, while parameter l in removable (27)0The component of the doppler centroid unrelated with target bearing position represented.It is first First, if the orientation frequency disturbance wave filter F of the amendmentMAFPWith following form
FMAFP(t, τ ')=exp { j π (μ (τ ') t+ α (τ ') t3)} (29)
Wherein μ and α is two undetermined coefficients related to target X-coordinate.Formula (29) is multiplied with formula (24), believed after filtering Number it is
In formula (30), second phase term is the doppler centroid with azimuthal dependence, its coefficient τ '= τ′sShi Yingwei zero；3rd phase term is the doppler frequency rate with azimuthal dependence, and its coefficient is in τ '=τ 'sWhen also should be Zero.Therefore it can obtain
Due to target S arbitrariness, the τ ' in formula (31)sShould be with τ ' replacements.Now, formula (31) can be rewritten as
Wherein, l0With q1For the result after expression formula in formula (28) is rewritten, it is represented by
The orientation frequency that formula (33) is substituted into the amendment that WNLCS image processing methods are used in formula (29), the present invention is disturbed Dynamic wave filter is represented by
Formula (32) is substituted into formula (30), Es3Can be by further abbreviation
After orientation frequency disturbance operation by amendment, orientation time-frequency characteristic such as Fig. 4 (b) institutes of each target in Fig. 3 (a) Show.Convolution (35) and Fig. 4 (b) are can be found that, although many positioned at same range cell but with different azimuth Place object General Le frequency modulation rate is unified, and their doppler centroid still has space-variant in azimuth, corresponding to the 3rd in formula (35) Phase term.The residual phase will introduce orientation geometric warping after focusing in image, the distortion will be corrected in step 6.
Step 5: the compensation of orientation quadratic phase and orientation Signal Compression.Specially：
Orientation Fourier transformation is carried out to signal shown in formula (35), ignores amplitude and constant phase, obtains signal Range Doppler frequency spectrum is
Wherein ftRepresent Doppler frequency.Second phase term in formula (36) is orientation quadratic phase compensating filter The phase that should be compensated.The orientation quadratic phase compensating filter that WNLCS image processing methods are used in the present invention is represented by
Wherein q0For the result after expression formula in formula (28) is rewritten, it is represented by
Formula (37) is multiplied with formula (36), filtered signal is obtained
To signal Es3a1Orientation inverse Fourier transform is carried out, obtaining the signal expression after orientation compression is
Wherein BaFor target S doppler bandwidth.As can be seen that target S orientation reconstructed positions are from formula (40)
Step 6: focusedimage Geometry rectification.Specially：
Target S is not reconstituted in by t it can be seen from formula (41)sThe correct position of orientation represented.In other words, focus on There is the geometric warping of orientation in image afterwards, need to be corrected.Geometric warping shown in consideration formula (41) is space-variant, this hair Bright middle WNLCS image processing methods are focused the Geometry rectification of image using the method for time domain interpolation.The time domain interpolation can table It is shown as
Wherein t ' is new orientation time, l1With l2For the result after expression formula in formula (28) is rewritten, it is represented by
If final products are slant-range image, the final magnitude image E after Geometry rectifications51It is represented by
If final products are distance image, it would be desirable to extra distance is carried out once to time domain interpolation, by oblique distance figure As being converted to distance image.The distance is represented by time domain interpolation
Wherein τ " is the new Distance Time after interpolation.Now, the final magnitude image E after Geometry rectifications52It is represented by
The signal processing flow figure for the WNLCS image processing methods that the present invention is introduced is as shown in Figure 5.
Embodiment
In an embodiment of the present invention, if the size of airborne geo Reference Strip SAR observation areas be 40km × 50km (away from From × orientation), 25 targets to be observed are uniformly distributed among this region, as shown in Figure 6.The main performance ginseng of the emulation experiment Number, systematic parameter and spatial parameter are as shown in table 1.
The airborne geo Reference Strip SAR simulation parameter lists of table 1
Step 1: based on simulation objectives distribution in parameter, Fig. 6 in table 1 and the target echo expression formula shown in formula (4), entering Planet carries the emulation of Geographic Reference stripmap SAR echo data, obtains scene echoes data Es.Calculated according to parameter in table 1 with formula (6) Distance is to quadratic phase compensating filter FRC, to echo data EsCarry out orientation carry out Fourier transformation, obtain signal away from Off-frequency composes Esr, by quadratic phase compensating filter FRCWith EsrBe multiplied, obtain filtered signal apart from frequency spectrum Esr1。
Step 2: generating nonlinear linear range migration correcting filter F with formula (8) based on parameter in table 1BRMC, will FBRMCWith signal apart from frequency spectrum Esr1Be multiplied, obtain filtered signal apart from frequency spectrum Esr2.To Esr2In enter row distance to inverse Fu Leaf transformation, obtains distance to the time-domain signal E after compressions1, such as shown in Fig. 7 (a).
Step 3: based on parameter in table 1 and formula (18), being handled by time domain interpolation and carrying out remaining Range migration correct, obtained Signal E after to remaining Range migration corrects2, such as shown in Fig. 7 (b).
Step 4: based on parameter, formula (33) in table 1 and formula (34), generating the orientation frequency disturbance wave filter of amendment FMAFP.By FMAFPWith signal Es2In time domain multiplication, the time-domain signal E after obtaining the orientation frequency disturbance processing by amendments3。
Step 5: based on parameter, formula (37) in table 1 and formula (38), generation orientation quadratic phase compensating filter FAC.It is right Signal Es3Orientation Fourier transformation is carried out, the range Doppler spectrum E of signal is obtaineds3a.By FACWith Es3aIt is multiplied, is filtered The range Doppler of signal composes E afterwardss3a1.To signal Es3a1Orientation inverse Fourier transform is carried out, the figure after orientation compression is obtained As Es4, such as shown in Fig. 8 (a).
Step 6: based on parameter, formula (42) in table 1 and formula (43), being corrected by orientation time domain interpolation in focusedimage Along the geometric distortion of orientation.Based on parameter in table 1 and formula (45), realize slant-range image to distance to time domain interpolation by distance The conversion of image, the image product E finally gatheredS52, such as shown in Fig. 8 (b).Fig. 9 gives scene edge destination T11、T13、 T15、T31、T35、T51And scene center target T33Profile, it is seen that in figure institute a little by well focussed.
Invention describes a kind of WNLCS image processing methods suitable for airborne geo Reference Strip SAR, this method Handling process does not introduce extra redundant data, is provided simultaneously with the energy to carrying out full aperture imaging compared with wide-scene echo data Power, therefore the WNLCS methods have higher imaging efficiency, are that one kind is applied to airborne geo Reference Strip SAR wide cuts Efficient, the accurate image processing method of earth observation demand.

Claims (1)

1. it is a kind of towards airborne geo Reference Strip synthetic aperture radar (Synthetic Aperture Radar, abbreviation SAR) Wide cut Non-linear chirp scaling (Wide Nonlinear Chirp Scaling, abbreviation WNLCS) image processing method, including Following six step：
Step 1: distance is compensated to quadratic phase.Specially：
In airborne geo Reference Strip SAR data acquisition, the cartesian coordinate system set up as follows：Origin O With observation scene center superposition, surface projection direction of the X-axis along beam center, Y-axis is moved towards along wave cover band, and Z axis is away from ground Heart direction and perpendicular to earth's surface.In follow-up discussion, X-direction is referred to as distance to Y direction is referred to as orientation by us.V For satellite velocities, H is satellite altitude, and θ is defined as geographical oblique angle, represents the folder of wave cover band trend and satellite velocity vector Angle, RcWith RrefThe center oblique distance and its surface projection at t=0 moment are represented respectively, and S is any one coordinate in wave cover band For (xs, ys, 0) and target, tsAt the time of target inswept for beam center, RsFor target S instantaneous oblique distance, βsAnd φsRepresent respectively Target S center downwards angle of visibility and center angle of squint.
WNLCS methods that the present invention is introduced based on the fact that：Airborne geo Reference Strip SAR beam position all the time perpendicular to Wave cover band.Now target S instantaneous oblique distance RsWith being expressed as below：
<mrow> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>(</mo> <mi>S</mi> <mo>)</mo> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <mi>&amp;theta;</mi> <msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein RgAnd tsThere is one-to-one relation between the coordinate of target
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>y</mi> <mi>s</mi> </msub> <mrow> <mi>V</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
If using traditional linear FM signal as transmission signal, the echo-signal from target S is demodulated into fundamental frequency, demodulation The echo-signal is represented by afterwards
<mrow> <msub> <mi>E</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>&amp;tau;</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mi>&amp;gamma;</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>&amp;tau;</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
Wherein c is the light velocity, and t and τ represent the orientation slow time and apart from fast time, f respectivelycFor signal carrier frequency, γ is signal frequency modulation Rate.Because the amplitude of signal does not influence imaging, formula (3) eliminates the amplitude of signal.Different from traditional satellite-borne SAR, spacebornely Reason Reference Strip SAR oblique distance change can be up to tens kilometers, correspond to tens thousand of Range resolution units, now inevitably The problem of transmitting is blocked can be run into.Therefore, airborne geo Reference Strip SAR employs a kind of sampling skill in multi-emitting pulse spacing Art, has the echo-signal from target S and is expressed as below：
<mrow> <msub> <mi>E</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>&amp;tau;</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mi>&amp;gamma;</mi> <msup> <mrow> <mo>(</mo> <mrow> <mi>&amp;tau;</mi> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mi>c</mi> </mfrac> <mo>-</mo> <msub> <mi>&amp;tau;</mi> <mrow> <mi>d</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Wherein τdelayFor the fast time delay that orientation is relied on caused by multi-emitting pulse spacing Sampling techniques.Formula (4) is carried out Distance is obtained to Fourier transformation
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> <mo>}</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <msubsup> <mi>&amp;pi;f</mi> <mi>&amp;tau;</mi> <mn>2</mn> </msubsup> </mrow> <mi>&amp;gamma;</mi> </mfrac> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>&amp;tau;</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msub> <mi>&amp;tau;</mi> <mrow> <mi>d</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
Wherein fτRepresent frequency of distance.In formula (5), Section 1 is that Section 2 represents should give apart from unrelated doppler phase The distance removed is to quadratic phase, and Section 3 represents range migration dependent phase.Therefore, second phase term of formula (5) can be based on, Distance is generated to quadratic phase compensating filter FRC
<mrow> <msub> <mi>F</mi> <mrow> <mi>R</mi> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mfrac> <mrow> <msubsup> <mi>&amp;pi;f</mi> <mi>&amp;tau;</mi> <mn>2</mn> </msubsup> </mrow> <mi>&amp;gamma;</mi> </mfrac> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
Formula (6) is multiplied with formula (5), obtained signal distance frequency spectrum is
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mi>r</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>}</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>&amp;tau;</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> <mo>+</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msub> <mi>&amp;tau;</mi> <mrow> <mi>d</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>y</mi> </mrow> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Step 2: non-linear consistent Range migration correct and distance are to Signal Compression.Specially：
In formula (7), second phase term represents the phase term caused by range migration, should be gone before orientation processing Remove.WNLCS algorithms assume that the range migration of target in scene only exists linear component in the present invention.WNLCS is imaged in the present invention Processing method is first by introducing a nonlinear wave filter FBRMC, carry out consistent Range migration correct.The wave filter can It is expressed as:
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mrow> <mi>B</mi> <mi>R</mi> <mi>M</mi> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>&amp;tau;</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;theta;</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msub> <mi>&amp;tau;</mi> <mrow> <mi>d</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>y</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;theta;</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
Formula (8) is multiplied with formula (7), obtaining signal distance frequency spectrum is
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mi>r</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>&amp;tau;</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Wherein
<mrow> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
It can be obtained by (10), to the arbitrfary point (point that i.e. abscissa is zero) in Y-axis, their Rs1Linear segment target conjunction Into aperture central instant (t=ts) be always zero, i.e.,
<mrow> <mfrac> <mrow> <msub> <mi>dR</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Therefore, for all targets being distributed along Y-axis, their range migration is all completely removed.Formula (9) is entered row distance to Inverse Fourier transform, can be in the signal after time domain obtains Range compress
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>r</mi> </msub> <mo>(</mo> <mrow> <mi>&amp;tau;</mi> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
Wherein BrSignal bandwidth is represented, function sinc (), which has, to be defined as below
<mrow> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>&amp;pi;</mi> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>&amp;pi;</mi> <mi>x</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
Step 3: the remaining Range migration correct based on time domain interpolation.Specially：
Target for deviateing Y-axis distribution, relation shown in formula (11) is no longer set up.WNLCS methods employ one kind in the present invention Remaining range migration in new time domain interpolation processing correction-type (10).The key idea of the time domain interpolation method is will be all Target with identical X-coordinate is reoriented in identical range cell.The use of range cell sequence during t=0 is to insert The reference of value.If Rc1And Rc2Respectively represent interpolation before and after target S center oblique distance, they are represented by
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msub> <mi>&amp;tau;</mi> <mi>s</mi> </msub> <mo>+</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>R</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msubsup> <mi>&amp;tau;</mi> <mi>s</mi> <mo>&amp;prime;</mo> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
Wherein τsWith τ 'sThe target S corresponding fast time is represented before and after interpolation respectively
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;tau;</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mi>c</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;tau;</mi> <mi>s</mi> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mi>c</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
Based on formula (14), the X-coordinate x for obtaining target S before and after interpolation can be calculated1And x2
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>c</mi> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
The purpose of time domain interpolation processing is to realize following mapping
x1(S)→x2(S) (17)
It should be noted that although above-mentioned derivation is carried out for target S, time domain interpolation correction is tool in actual process There is global universality.Therefore when doing the expression of time domain interpolation mapping, ts, τsWith τ 'sShould be respectively by t, τ and τ ' replacement, its Middle τ ' is new distance to the fast time.By formula (14), formula (16) is substituted into formula (17), can obtain the expression formula of time domain interpolation mapping For
<mrow> <mi>&amp;tau;</mi> <mo>&amp;RightArrow;</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
Formula (18) is substituted into formula (12), time-domain signal E after interpolation processing can be obtaineds2Expression formula be
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>r</mi> </msub> <mi>&amp;delta;</mi> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>,</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>
The function δ () of Section 1 has and is defined as below in formula (19)
<mrow> <mi>&amp;delta;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>,</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <mrow> <mo>(</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <msub> <mi>R</mi> <mi>s</mi> </msub> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>
After time domain interpolation processing shown in perfect (18), target S range migration curve can be tried to achieve by below equation
δ(τ′,t；S)=0 (21)
By solving formula (21), it can obtain
<mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>
Wherein RnewRepresent the new range migration of target S after time domain interpolation
<mrow> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>g</mi> </msub> <mo>(</mo> <mi>S</mi> <mo>)</mo> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <mi>&amp;theta;</mi> <msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>t</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>
Now, formula (19) can be rewritten as following form
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>r</mi> </msub> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>4</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>
Easily checking, in the oblique distance shown in formula (23), following formula perseverance is set up
<mrow> <mfrac> <mrow> <msub> <mi>dR</mi> <mrow> <mi>n</mi> <mi>e</mi> <mi>w</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>
As shown in formula (24), now the range migration of all targets is completely removed in scene.
Step 4: orientation frequency disturbance is handled.Specially：
The Section 2 of formula (24) represents doppler phase.Utilize R in formula (10)s1Definition, can be by target S Doppler center Frequency fdWith doppler frequency rate frIt is expressed as below
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>-</mo> <mi>V</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <mi>V</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;theta;</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <mi>&amp;theta;</mi> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>
As shown in formula (26), frDependent on the position (R of target in the scenegAnd ts).Therefore it is located at same range cell but has The target of different azimuth position can not unanimously be compressed along orientation.
By to doppler centroid and doppler frequency rate heart moment t in the targetsTaylor expansion is carried out, can be to formula (26) Carry out approximate processing.The WNLCS image processing methods of the present invention are thought to doppler centroid fdThe second Taylor series and To doppler frequency rate frFirst order Taylor expansion have enough approximation qualities.Therefore, formula (26) can be rewritten as
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&amp;ap;</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>f</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&amp;ap;</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <mo>(</mo> <mi>S</mi> <mo>)</mo> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>S</mi> <mo>)</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>
Each term coefficient wherein in formula (27) has following expression
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>l</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mo>-</mo> <mfrac> <mrow> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>V</mi> <mn>3</mn> </msup> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>V</mi> <mn>3</mn> </msup> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msubsup> <mi>&amp;theta;R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>V</mi> <mn>3</mn> </msup> <msub> <mi>sin&amp;theta;R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msubsup> <mi>&amp;theta;R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>V</mi> <mn>3</mn> </msup> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>
Notice that coefficients all in formula (28) only depend on the X-coordinate of target, they in processing procedure can easily by Range cell is updated.The WNLCS image processing methods of the present invention employ a kind of orientation frequency disturbance operation of amendment, The operation can be pointed to the unification of same range cell but the realization of goal doppler frequency rate of different azimuth position (Y-coordinate), While parameter l in removable (27)0The component of the doppler centroid unrelated with target bearing position represented.First, if The orientation frequency disturbance wave filter F of the amendmentMAFPWith following form
FMAFP(t, τ ')=exp { j π (μ (τ ') t+ α (τ ') t3)} (29)
Wherein μ and α is two undetermined coefficients related to target X-coordinate.Formula (29) is multiplied with formula (24), filtered signal is
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>l</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>&amp;alpha;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>3</mn> <mi>&amp;alpha;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>
In formula (30), second phase term is the doppler centroid with azimuthal dependence, and its coefficient is in τ '=τ 'sWhen It should be zero；3rd phase term is the doppler frequency rate with azimuthal dependence, and its coefficient is in τ '=τ 'sWhen also should be zero. Therefore it can obtain
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;tau;</mi> <mi>s</mi> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>l</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;alpha;</mi> <mrow> <mo>(</mo> <msubsup> <mi>&amp;tau;</mi> <mi>s</mi> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mrow> <mn>3</mn> <mi>c</mi> </mrow> </mfrac> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow> 5
Due to target S arbitrariness, the τ ' in formula (31)sShould be with τ ' replacements.Now, formula (31) can be rewritten as
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>&amp;alpha;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mrow> <mn>3</mn> <mi>c</mi> </mrow> </mfrac> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>
Wherein, l0With q1For the result after expression formula in formula (28) is rewritten, it is represented by
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>l</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>V</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <msqrt> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>-</mo> <mn>4</mn> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mrow> <msup> <mi>c&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <mi>V</mi> <mi> </mi> <msub> <mi>sin&amp;theta;R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> <msqrt> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <msup> <mi>V</mi> <mn>3</mn> </msup> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <msqrt> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>-</mo> <mn>4</mn> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mfrac> <mrow> <mo>(</mo> <mn>12</mn> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <msup> <mi>c</mi> <mn>5</mn> </msup> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>V</mi> <mn>3</mn> </msup> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>
Formula (33) is substituted into the orientation frequency disturbance filter for the amendment that WNLCS image processing methods are used in formula (29), the present invention Ripple device is represented by
<mrow> <msub> <mi>F</mi> <mrow> <mi>M</mi> <mi>A</mi> <mi>F</mi> <mi>P</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mrow> <mn>4</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>l</mi> <mn>0</mn> </msub> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <mi>t</mi> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> </mrow> <mrow> <mn>3</mn> <mi>c</mi> </mrow> </mfrac> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mi>t</mi> <mn>3</mn> </msup> <mo>)</mo> </mrow> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>
Formula (32) is substituted into formula (30), Es3Can be by further abbreviation
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mfrac> <mrow> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mfrac> <mrow> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>c</mi> </msub> </mrow> <mi>c</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>
After orientation frequency disturbance operation by amendment, although positioned at same range cell but with different azimuth Place object Doppler frequency rate unified, their doppler centroid still has space-variant in azimuth, corresponding to the in formula (35) Three phase terms.The residual phase will after focusing in image introduce orientation geometric warping, the distortion will in step 6 quilt Correction.
Step 5: the compensation of orientation quadratic phase and orientation Signal Compression.Specially：
Orientation Fourier transformation is carried out to signal shown in formula (35), ignores amplitude and constant phase, obtains the distance of signal Doppler frequency spectrum is
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>3</mn> <mi>a</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>t</mi> </msub> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mi>c</mi> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <msubsup> <mi>f</mi> <mi>t</mi> <mn>2</mn> </msubsup> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mi>t</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>
Wherein ftRepresent Doppler frequency.Second phase term in formula (36) should be mended for orientation quadratic phase compensating filter The phase repaid.The orientation quadratic phase compensating filter that WNLCS image processing methods are used in the present invention is represented by
<mrow> <msub> <mi>F</mi> <mrow> <mi>A</mi> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>t</mi> </msub> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mi>c</mi> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mi>c</mi> </msub> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <msubsup> <mi>f</mi> <mi>t</mi> <mn>2</mn> </msubsup> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>
Wherein q0For the result after expression formula in formula (28) is rewritten, it is represented by
<mrow> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>c&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow>
Formula (37) is multiplied with formula (36), filtered signal is obtained
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>3</mn> <mi>a</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>t</mi> </msub> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mi>exp</mi> <mo>{</mo> <mrow> <mi>j</mi> <mi>&amp;pi;</mi> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msub> <mi>f</mi> <mi>t</mi> </msub> </mrow> <mo>}</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>39</mn> <mo>)</mo> </mrow> </mrow>
To signal Es3a1Orientation inverse Fourier transform is carried out, obtaining the signal expression after orientation compression is
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow>
Wherein BaFor target S doppler bandwidth.As can be seen that target S orientation reconstructed positions are from formula (40)
<mrow> <msub> <mover> <mi>t</mi> <mo>^</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>41</mn> <mo>)</mo> </mrow> </mrow>
Step 6: focusedimage Geometry rectification.Specially：
Target S is not reconstituted in by t it can be seen from formula (41)sIn the correct position of orientation represented.In other words, the figure after focusing There is the geometric warping of orientation as in, need to be corrected.Geometric warping shown in consideration formula (41) is space-variant, in the present invention WNLCS image processing methods are focused the Geometry rectification of image using the method for time domain interpolation.The time domain interpolation is represented by
<mrow> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msubsup> <mi>t</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>q</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;RightArrow;</mo> <msup> <mi>t</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>42</mn> <mo>)</mo> </mrow> </mrow>
Wherein t ' is new orientation time, l1With l2For the result after expression formula in formula (28) is rewritten, it is represented by
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mn>8</mn> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mi>c</mi> <mn>3</mn> </msup> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> <msup> <mi>sin</mi> <mn>2</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <msup> <mi>V</mi> <mn>3</mn> </msup> <msup> <mi>sin</mi> <mn>3</mn> </msup> <msup> <mi>&amp;theta;H</mi> <mn>2</mn> </msup> </mrow> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>-</mo> <mfrac> <mrow> <mn>16</mn> <msqrt> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>-</mo> <mn>4</mn> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mrow> <msup> <mi>c</mi> <mn>5</mn> </msup> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>R</mi> <mrow> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow> <mn>5</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>43</mn> <mo>)</mo> </mrow> </mrow>
If final products are slant-range image, the final magnitude image E after Geometry rectifications51It is represented by
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>51</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>t</mi> <mo>&amp;prime;</mo> </msup> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>r</mi> </msub> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>a</mi> </msub> <mo>(</mo> <mrow> <msup> <mi>t</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>44</mn> <mo>)</mo> </mrow> </mrow>
If final products are distance image, it would be desirable to carry out once extra distance to time domain interpolation, slant-range image is turned It is changed to distance image.The distance is represented by time domain interpolation
<mrow> <msup> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msqrt> <mrow> <msubsup> <mi>R</mi> <mi>g</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>H</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>&amp;RightArrow;</mo> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mo>&amp;prime;</mo> </mrow> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>45</mn> <mo>)</mo> </mrow> </mrow>
Wherein τ " is the new Distance Time after interpolation.Now, the final magnitude image E after Geometry rectifications52It is represented by
<mrow> <msub> <mi>E</mi> <mrow> <mi>s</mi> <mn>52</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>t</mi> <mo>&amp;prime;</mo> </msup> <mo>,</mo> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mo>&amp;prime;</mo> </mrow> </msup> <mo>;</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>r</mi> </msub> <mo>(</mo> <mrow> <msup> <mi>&amp;tau;</mi> <mrow> <mo>&amp;prime;</mo> <mo>&amp;prime;</mo> </mrow> </msup> <mo>-</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mi>g</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>sin</mi> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>B</mi> <mi>a</mi> </msub> <mo>(</mo> <mrow> <msup> <mi>t</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <msub> <mi>t</mi> <mi>s</mi> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>46</mn> <mo>)</mo> </mrow> </mrow> 8
CN201710270564.0A 2017-04-24 2017-04-24 A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR Pending CN107102330A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201710270564.0A CN107102330A (en) 2017-04-24 2017-04-24 A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201710270564.0A CN107102330A (en) 2017-04-24 2017-04-24 A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR

Publications (1)

Publication Number Publication Date
CN107102330A true CN107102330A (en) 2017-08-29

Family

ID=59658159

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201710270564.0A Pending CN107102330A (en) 2017-04-24 2017-04-24 A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR

Country Status (1)

Country Link
CN (1) CN107102330A (en)

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111650586A (en) * 2020-04-03 2020-09-11 中国科学院电子学研究所苏州研究院 Target positioning method based on satellite-borne SAR azimuth NLCS imaging
CN111751821A (en) * 2020-06-08 2020-10-09 北京理工大学 MWNLCS imaging method suitable for high-resolution satellite-borne scene matching SAR

Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101369019A (en) * 2008-10-10 2009-02-18 清华大学 Polarization interference synthetic aperture radar three-dimensional imaging method based on polarization data amalgamation
CN102288961A (en) * 2011-07-07 2011-12-21 电子科技大学 Imaging method for synthetic aperture radar nonlinear frequency modulation label change
CN102331577A (en) * 2011-09-21 2012-01-25 北京理工大学 Improved NCS (Nonlinear Chirp Scaling) imaging algorithm suitable for geosynchronous orbit (GEO) SAR (Synthetic Aperture Radar)
CN103412309A (en) * 2013-08-26 2013-11-27 电子科技大学 Shift-invariant double-foundation foresight synthetic aperture radar NLCS imaging method
CN103901428A (en) * 2014-03-28 2014-07-02 西安电子科技大学 Missile-borne SAR sub-aperture forward squint high-order nonlinear chirp scaling imaging method
CN104035093A (en) * 2014-04-10 2014-09-10 西安电子科技大学 Chirp scaling algorithm based satellite-bone SAR (Synthetic Aperture Radar) ground accelerated movement object imaging method
CN105259552A (en) * 2015-09-17 2016-01-20 中国科学院电子学研究所 Synthetic aperture radar imaging method and device based on non-linear frequency-modulated signals
CN105572648A (en) * 2016-02-01 2016-05-11 中国科学院电子学研究所 Synthetic aperture radar echo data range cell migration correction method and device
CN106291489A (en) * 2016-08-23 2017-01-04 中国人民解放军国防科学技术大学 It is applicable to the synthetic aperture radar echo simulation method of multiple transmitting signal waveform

Patent Citations (9)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101369019A (en) * 2008-10-10 2009-02-18 清华大学 Polarization interference synthetic aperture radar three-dimensional imaging method based on polarization data amalgamation
CN102288961A (en) * 2011-07-07 2011-12-21 电子科技大学 Imaging method for synthetic aperture radar nonlinear frequency modulation label change
CN102331577A (en) * 2011-09-21 2012-01-25 北京理工大学 Improved NCS (Nonlinear Chirp Scaling) imaging algorithm suitable for geosynchronous orbit (GEO) SAR (Synthetic Aperture Radar)
CN103412309A (en) * 2013-08-26 2013-11-27 电子科技大学 Shift-invariant double-foundation foresight synthetic aperture radar NLCS imaging method
CN103901428A (en) * 2014-03-28 2014-07-02 西安电子科技大学 Missile-borne SAR sub-aperture forward squint high-order nonlinear chirp scaling imaging method
CN104035093A (en) * 2014-04-10 2014-09-10 西安电子科技大学 Chirp scaling algorithm based satellite-bone SAR (Synthetic Aperture Radar) ground accelerated movement object imaging method
CN105259552A (en) * 2015-09-17 2016-01-20 中国科学院电子学研究所 Synthetic aperture radar imaging method and device based on non-linear frequency-modulated signals
CN105572648A (en) * 2016-02-01 2016-05-11 中国科学院电子学研究所 Synthetic aperture radar echo data range cell migration correction method and device
CN106291489A (en) * 2016-08-23 2017-01-04 中国人民解放军国防科学技术大学 It is applicable to the synthetic aperture radar echo simulation method of multiple transmitting signal waveform

Non-Patent Citations (4)

* Cited by examiner, † Cited by third party
Title
ALBERTO MOREIRA等: "Extended Chirp Scaling Algorithm for Air- and Spaceborne SAR Data Processing in Stripmap and ScanSAR Imaging Modes", 《IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING》 *
F.W.WONG等: "New applications of nonlinear chirp scaling in SAR data processing", 《IEEE TRANS.GEOSCI.REMOTE SENS.》 *
LEPING CHEN等: "A NLCS focusing approach for low frequency UWB one-stationary bistatic SAR", 《2015 16TH INTERNATIONAL RADAR SYMPOSIUM》 *
安道祥等: "基于去调频技术的斜视聚束SAR成像方法", 《中国科学》 *

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111650586A (en) * 2020-04-03 2020-09-11 中国科学院电子学研究所苏州研究院 Target positioning method based on satellite-borne SAR azimuth NLCS imaging
CN111650586B (en) * 2020-04-03 2022-03-22 中国科学院电子学研究所苏州研究院 Target positioning method based on satellite-borne SAR azimuth NLCS imaging
CN111751821A (en) * 2020-06-08 2020-10-09 北京理工大学 MWNLCS imaging method suitable for high-resolution satellite-borne scene matching SAR
CN111751821B (en) * 2020-06-08 2022-05-20 北京理工大学 MWNLCS imaging method suitable for high-resolution satellite-borne scene matching SAR

Similar Documents

Publication Publication Date Title
CN103543453B (en) Elevation inversion method for geosynchronous orbit synthetic aperture radar interference
CN108490441B (en) Dive section large squint SAR sub-aperture imaging space-variant correction method based on two-stage filtering
EP2650695B1 (en) Imaging method for synthetic aperture radar in high squint mode
CN102721948A (en) Large-scene SAR deception jamming implementation method
CN106249237A (en) Big Squint SAR frequency domain imaging method under a kind of curvilinear path
CN109270528B (en) One-station fixed type double-station SAR imaging method based on full-analytic distance model
CN113589285B (en) SAR real-time imaging method for aircraft
CN107918124A (en) Airborne big strabismus High Resolution SAR imaging method with the correction of orientation space-variant
CN112748431A (en) Ground moving target imaging method of medium-orbit spaceborne SAR
CN105759263A (en) High resolution satellite-borne squint SAR imaging method in large-scale scene
CN110596701B (en) Non-level-flight double-station SAR frequency domain FENLCS imaging method based on quadratic ellipse model
CN106950565A (en) Space-borne SAR Imaging jitter compensation method, imaging method
Zuo et al. Unified coordinate system algorithm for terahertz video-SAR image formation
Deng et al. A space-variant phase filtering imaging algorithm for missile-borne BiSAR with arbitrary configuration and curved track
CN107102330A (en) A kind of WNLCS imaging methods towards airborne geo Reference Strip SAR
CN105180852A (en) GB-SAR deformation monitoring method based on triple stepping
CN111208515B (en) SAR motion compensation method based on two-dimensional nonlinear mapping
CN108469612B (en) Bistatic time-varying acceleration foresight SAR imaging method based on equivalent slope distance
CN106054152A (en) Non-ideal locus SAR echo obtaining method based on inverse extended Omega-K algorithm
CN103837874A (en) Two-dimensional non-linear frequency conversion and modulation method for SAR imaging of geosynchronous orbit
CN103630904B (en) Array 3-D SAR formation method is looked under airborne
CN111751821B (en) MWNLCS imaging method suitable for high-resolution satellite-borne scene matching SAR
CN112578383A (en) High maneuvering platform TOPS SAR imaging method based on expanded wave number spectrum reconstruction
CN101833094B (en) Imaging processing device of satellite-borne TOPSAR (Terrain Observation by Progressive Scans Synthetic Aperture Radar) data and processing method thereof
KR102289433B1 (en) Image decoding apparatus based on squint sar and method of decoding image using the same

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
WD01 Invention patent application deemed withdrawn after publication
WD01 Invention patent application deemed withdrawn after publication

Application publication date: 20170829