CN107271997A - Airborne multichannel CSSAR ground moving object motion parameters estimation methods - Google Patents

Airborne multichannel CSSAR ground moving object motion parameters estimation methods Download PDF

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CN107271997A
CN107271997A CN201710413596.1A CN201710413596A CN107271997A CN 107271997 A CN107271997 A CN 107271997A CN 201710413596 A CN201710413596 A CN 201710413596A CN 107271997 A CN107271997 A CN 107271997A
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CN107271997B (en
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李永康
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Northwestern Polytechnical University
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    • 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
    • G01S13/9088—Circular SAR [CSAR, C-SAR]
    • 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
    • G01S13/9021—SAR image post-processing techniques
    • G01S13/9029—SAR image post-processing techniques specially adapted for moving target detection within a single SAR image or within multiple SAR images taken at the same time
    • 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
    • G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
    • G01S13/50—Systems of measurement based on relative movement of target
    • 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
    • G01S13/9004—SAR image acquisition techniques
    • G01S13/9017—SAR image acquisition techniques with time domain processing of the SAR signals in azimuth

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  • Radar, Positioning & Navigation (AREA)
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  • Computer Networks & Wireless Communication (AREA)
  • General Physics & Mathematics (AREA)
  • Electromagnetism (AREA)
  • Signal Processing (AREA)
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Abstract

The invention provides a kind of airborne multichannel CSSAR ground moving object motion parameters estimation methods, it is related to radar signal processing field.The present invention adjust the distance compression after echo signal enter row distance to Fourier transformation, and carry out base band Doppler center compensation, orientation Fourier transformation is carried out to the echo signal after compensation, estimate the doppler ambiguity number and doppler frequency rate of target, and two-dimensional frequency reference function is constructed, the location parameter for carving target according to the doppler centroid of target, doppler frequency rate and positive side apparent time estimates the kinematic parameter of target.The present invention establishes the exact relationship formula of the coupling between target location and speed, propose to release the coupling between target location and speed using positional information of the doppler frequency rate, doppler centroid and target of target in SAR image, can accurately estimate the kinematic parameter of ground moving object under airborne multichannel CSSAR.

Description

Airborne multichannel CSSAR ground moving object motion parameters estimation methods
Technical field
The present invention relates to radar signal processing field, especially a kind of synthetic aperture radar ground moving object kinematic parameter Method of estimation.
Background technology
Airborne Circular test band synthetic aperture radar (Circular Stripmap Synthetic Aperture Radar, CSSAR) there is the characteristics of wide coverage and periodicity are revisited, thus it is suitable for ground moving object instruction (Ground Moving Target Indication, GMTI).Target moving parameter estimation be GMTI systems basic task it One, it is therefore necessary to ground moving object motion parameters estimation method of the research suitable for airborne CSSAR.
It is existing suitable for conventional linear track synthetic aperture radar (Synthetic Aperture Radar, SAR) Ground moving object motion parameters estimation method is typically the orientation speed that Frequency Estimation target is adjusted according to the orientation of target, root According to target Estimation of Doppler central frequency target distance to speed.However, for airborne CSSAR, its circular motion Track causes to occur in that coupling between the position of target and speed so that the above-mentioned action reference variable for straight path SAR Method cannot be used directly for airborne CSSAR.
The content of the invention
In order to overcome the deficiencies in the prior art, the present invention proposes that one kind can tackle target location for airborne multichannel CSSAR The motion parameters estimation method of coupling between speed.Deposited for the position and speed of ground moving object under airborne CSSAR Coupling cause the problem of existing ground moving object motion parameters estimation method is not used to airborne CSSAR, propose a kind of The motion parameters estimation method of the coupling between target location and speed is coped with, ground under airborne multichannel CSSAR is realized The accurate estimation of moving target kinematic parameter.
The step of the technical solution adopted for the present invention to solve the technical problems is：
Step 1, hypothesis：(1) clutter uses displaced phase center antenna by airborne multichannel CSSAR systems (Displaced Phase Center Antenna, DPCA) method suppresses；(2) echo signal after Range compress is carried Take, and echo signal is located at initial data domain；
The echo signal after compression of adjusting the distance enters row distance to Fourier transformation, by echo signal transform to orientation time domain away from Off-frequency domain, into step 2；
Step 2, step 1 is transformed to orientation time domain apart from frequency domain echo signal carry out base band Doppler center compensation, Comprise the following steps：
A) echo signal s of the orientation time domain apart from frequency domainDPCA(fr,ta) be expressed as
Wherein, Wr() is frequency of distance envelope, wa,1() is the transmitting-receiving round trip antenna radiation pattern of reference channel, fcFor thunder Up to the carrier frequency of transmission signal, frFor frequency of distance, taFor the orientation slow time, c is the light velocity, and λ is wavelength, tbIt is located at reference for target At the time of the positive side-looking direction of passage displaced phase center, RbFor tbMoment target is to the distance of radar, KaFor the Doppler of target Frequency modulation rate, facFor the doppler centroid of target, and facIt is represented by fac=fac,b+ MPRF, wherein fac,bFor target base Band doppler centroid, M is target Doppler fuzzy number, and PRF is the pulse recurrence frequency of radar；
B) target base band doppler centroid f is assumedac,bEstimate beThen according to orientation time domain frequency of distance The expression formula of domain echo signal, base band Doppler center penalty function can be configured to
The echo signal of formula (1) is multiplied with the penalty function of formula (2) compensation of base band Doppler center can be achieved；
Step 3, in step 2 base band Doppler center compensation after echo signal carry out orientation Fourier transformation, will Echo signal changes to two-dimensional frequency；
Step 4, the doppler ambiguity number and doppler frequency rate for estimating target, obtained doppler ambiguity number and Doppler The estimate of frequency modulation rate is respectivelyWith
Step 5, the target Doppler fuzzy number estimated using step 4 and doppler frequency rate construct the reference of two-dimensional frequency Function, the echo signal in step 3 is multiplied with the reference function and carries out target imaging, then carries out two to the signal after multiplication Echo signal is changed to image area by dimension inverse Fourier transform；
Two-dimensional frequency reference function is configured to：
Wherein faFor base band orientation frequency, and satisfaction-PRF/2≤fa≤ PRF/2,WithRespectively target Doppler is adjusted The estimate of frequency and doppler ambiguity number；
Step 6, the location parameter for carving according to location estimation positive side apparent time of the target in image area target, including following step Suddenly：
A) image area echo signal is expressed as：
Wherein, trFor apart from fast time, pr() is Range compress impulse response function, pa() is Azimuth Compression impulse Receptance function；
B) according to the expression formula of the image area echo signal of formula (4), positive side apparent time carves the location parameter R of targetbAnd side Parallactic angle θbDrawn by following formula estimation：
Wherein,WithRespectively RbAnd θbEstimate, ta,imgAnd tr,imgFor the orientation position of target in image area With distance to position, ω is the angular speed of radar motion；
Step 7, the location parameter according to the doppler centroid of target, doppler frequency rate and positive side apparent time quarter target Estimate the kinematic parameter of target, the kinematic parameter of target is estimated using equation below：
In formula (7) and formula (8),
Wherein,For vxEstimate,For vyEstimate, vxAnd vyBe respectively target along x-axis and the speed of y-axis, For facEstimate,For rbEstimate, rbTarget is carved to the distance of the origin of coordinates, r for positive side apparent timeaIt is radar motion rail The radius of mark, h is the height of radar.
The step 4 of the present invention is adjusted using the doppler ambiguity number and Doppler of the method estimation target based on maximum-contrast Frequency, is comprised the following steps that：
A) by the two-dimensional frequency echo signal S (f after base band Doppler center compensation in step 3r,fa) be expressed as
Wherein Wa(fa) it is orientation frequency envelope；
B) doppler ambiguity number and doppler frequency rate of target are estimated using the following method based on maximum-contrast：
In formula (4),
s(tr,ta；ka, m)=IDFT2{S(fr,fa)·H2(fr,fa；ka,m)} (14)
Wherein, IDFT2() represents two-dimentional inverse Fourier transform, E () representation space average operation, Contrast () Represent the contrast of image, kaIt is construction two-dimensional frequency reference function H respectively with m2(fr,fa；ka, how general the target used when m) is Strangle frequency modulation rate and doppler ambiguity number.
The beneficial effects of the present invention are the exact relationship formula of the coupling established between target location and speed, and according to Echo signal model, proposes the position in SAR image using doppler frequency rate, doppler centroid and the target of target Information releases the coupling between target location and speed.The present invention can accurately estimate ground under airborne multichannel CSSAR and transport The kinematic parameter of moving-target, it may also be used for Ground moving target imaging.
Brief description of the drawings
Fig. 1 is the schematic flow sheet of the present invention.
Fig. 2 is Airborne Dual-Channel CSSAR observation geometries, wherein raFor the radius of radar motion track, ω is radar Angular speed, h is radar altitude, vxAnd vyRespectively target is along x-axis and the speed of y-axis, r0And θ0It is former to coordinate for zero moment target The distance of point and the azimuth of target.
Fig. 3 is the imaging simulation result figure of target 1, wherein, Fig. 3 (a) is the target image focused on, and Fig. 3 (b) is orientation To profile, Fig. 3 (c) is distance to profile.
Fig. 4 is the imaging simulation result figure of target 2, wherein, Fig. 4 (a) is the target image focused on, and Fig. 4 (b) is orientation Profile, Fig. 4 (c) is distance to profile.
Fig. 5 is the imaging simulation result figure of target 3, wherein, Fig. 5 (a) is the target image focused on, and Fig. 5 (b) is orientation To profile, Fig. 5 (c) is distance to profile.
Embodiment
The present invention is further described with reference to the accompanying drawings and examples.
Fig. 1 is the schematic flow sheet of the present invention, and of the invention comprises the following steps that：
Step 1, hypothesis：(1) clutter uses displaced phase center antenna by airborne multichannel CSSAR systems (Displaced Phase Center Antenna, DPCA) method suppresses；(2) echo signal after Range compress is carried Take, and echo signal is located at initial data domain；
The echo signal after compression of adjusting the distance enters row distance to Fourier transformation, by echo signal transform to orientation time domain away from Off-frequency domain, into step 2；
Fig. 2 is Airborne Dual-Channel CSSAR observation geometries.The movement locus of radar platform is that a radius is raCircle. The angular speed of radar platform is ω, and flying height is h.Radar beam perpendicular to velocity attitude and points to the outer of movement locus all the time Side.Assuming that target linear uniform motion, and its speed along x-axis and y-axis is respectively vxAnd vy.It is assumed that in ta=0 moment (taFor The orientation slow time), the displaced phase center of radar passage 1 (reference channel) is located at (ra, 0, h), the equivalent phase of radar passage 2 It is centrally located at (ra,-d, h), target are located at (r0cosθ0,r0sinθ0, 0), wherein, r0For ta=0 moment target is to the origin of coordinates Distance, θ0For taThe azimuth of=0 moment target, d is the distance between the adjacent displaced phase center of radar (baseline length).
According to Fig. 2, the instantaneous distance R of the displaced phase center of target to i-th (i=1,2) individual passagei(ta) be represented by
In formula,
vta=vycos(θb)-vxsin(θb) (19)
Wherein, tbAt the time of being located at the positive side-looking direction of reference channel displaced phase center for target, i.e., positive side apparent time is carved, θbFor ta=tbThe azimuth of moment target, rbTarget is carved to the distance of the origin of coordinates, v for positive side apparent timetrMesh is carved for positive side apparent time Mark projection of the speed on radar line of sight direction, vtaProjection of the target velocity on radar motion direction, v are carved for positive side apparent timet For the sum velocity of target.
After carrier frequency de not modulation and Range compress, i-th of channel reception to target echo signal be represented by
Wherein, pr() is Range compress impulse response function, wa,i() is the transmitting-receiving round trip antenna side of i-th of passage Xiang Tu, trFor apart from the fast time, λ is wavelength, c is the light velocity.For simplicity of exposition, the constant amplitude in echo signal have ignored.
It is that target echo signal after reference channel, the registration of passage 2 is represented by with passage 1
In formula,
Wherein wa,1(ta) be reference channel (passage 1) transmitting-receiving round trip antenna radiation pattern, R2() for passage 2 target away from From equation, R2,reg(ta) be the registration of passage 2 after target range equation.
Due to R2,reg(ta) and R1(ta) between difference, i.e. vtrd/(raω) will much smaller than one distance samples unit.Cause This can ignore the difference between them in envelope, so that the echo signal after DPCA clutter recognitions is represented by
Enter row distance to above formula to Fourier transformation, the echo signal in orientation time domain frequency of distance domain can be obtained：
Wherein, frFor frequency of distance, fcFor the carrier frequency of radar emission signal, Wr(fr) it is frequency of distance envelope.
Step 2, step 1 is transformed to orientation time domain apart from frequency domain echo signal carry out base band Doppler center compensation, Comprise the following steps：
A) first by R1(ta) be rewritten into
In formula (26),
Wherein, KaFor the doppler frequency rate of target, RbFor tbMoment target is to the distance of radar, facFor the how general of target Strangle centre frequency, and facIt is represented by fac=fac,b+ MPRF, wherein fac,bFor target base band doppler centroid, M is Target Doppler fuzzy number, PRF is the pulse recurrence frequency of radar.
According to formula (26), echo signal s of the orientation time domain apart from frequency domainDPCA(fr,ta) be expressed as
B) the base band doppler centroid of target can be by conventional average cross correlation coefficient (Average Cross Correlation Coefficient, ACCC) method estimates.It is noted herein that, due to target range migration also It is not corrected, the base band doppler centroid of target need to be estimated using ACCC methods in orientation time domain frequency of distance domain.Assuming that The target base band doppler centroid estimated isThen according to the expression of orientation time domain frequency of distance domain echo signal Formula, base band Doppler center penalty function can be configured to
The echo signal of formula (1) is multiplied with the penalty function of formula (2) compensation of base band Doppler center can be achieved.Root According to sDPCA(fr,ta) and H1(fr) expression formula, base band Doppler center compensation after echo signal be represented by
Step 3, in step 2 base band Doppler center compensation after echo signal carry out orientation Fourier transformation, will Echo signal changes to two-dimensional frequency；
Using guard station phase principle, to S (fr,ta) orientation Fourier transformation is carried out, two-dimensional frequency echo signal can be obtained：
Wherein Wa(fa) it is orientation frequency envelope；
Step 4, the doppler ambiguity number and doppler frequency rate for estimating target, obtained doppler ambiguity number and Doppler The estimate of frequency modulation rate is respectivelyWith
Pass through last three exponential terms in echo signal expression formula in compensation formula (11), so that it may realize that target is focused on. Therefore, the present invention estimates the doppler ambiguity number and doppler frequency rate of target using the following method based on maximum-contrast：
In formula (4),
s(tr,ta；ka, m)=IDFT2{S(fr,fa)·H2(fr,fa；ka,m)} (14)
Wherein, IDFT2() represents two-dimentional inverse Fourier transform, E () representation space average operation, Contrast () The contrast of image is represented,WithRespectively target Doppler frequency modulation rate and the estimate of doppler ambiguity number, kaWith m difference It is construction two-dimensional frequency reference function H2(fr,fa；ka, the target Doppler frequency modulation rate and doppler ambiguity number used when m)；
Step 5, the target Doppler fuzzy number estimated using step 4 and doppler frequency rate construct the reference of two-dimensional frequency Function, the echo signal in step 3 is multiplied with the reference function and carries out target imaging, then carries out two to the signal after multiplication Echo signal is changed to image area by dimension inverse Fourier transform；
According to the expression formula of two-dimensional frequency echo signal in formula (3), two-dimensional frequency reference function is configured to：
According to the reference function and the expression formula of two-dimensional frequency echo signal, image area echo signal is represented by：
Wherein pa() is Azimuth Compression impulse response function.
It can be seen that, there is not the position skew of orientation in image area in target, and this is beneficial to follow-up target Action reference variable.
Step 6, the location parameter of target is carved according to location estimation positive side apparent time of the target in image area.
According to the expression formula of the image area echo signal of formula (4), positive side apparent time carves target to radar apart from RbAnd mesh Target azimuth angle thetabEstimated by following formula：
Wherein,WithRespectively RbAnd θbEstimate, ta,imgAnd tr,imgFor the orientation position of target in image area With distance to position, ω is the angular speed of radar motion.
Step 7, the location parameter of target is carved according to the doppler centroid of target, doppler frequency rate and positive side apparent time Estimate the kinematic parameter of target, the kinematic parameter of target is estimated using equation below：
In formula (7) and formula (8),
Wherein,For vxEstimate,For vyEstimate, vxAnd vyBe respectively target along x-axis and the speed of y-axis, For facEstimate,For rbEstimate, rbTarget is carved to the distance of the origin of coordinates, r for positive side apparent timeaIt is radar motion rail The radius of mark, h is the height of radar.
The effect of the present invention is further illustrated by following emulation experiment：
Airborne Dual-Channel CCSAR systematic parameters are shown in Table 1, simulate three point targets, and parameter is shown in Table 2, the letter of three targets Make an uproar and be set to 30dB than (Signal-to-Noise Ratio, SNR).Target is being estimated using the method based on maximum-contrast Doppler ambiguity number and doppler frequency rate when, parameter M and KaHunting zone be set to [- 1,1] and [563Hz/s, 634Hz/s], this target velocity scope that can be covered is [- 35m/s, 35m/s].Parameter M is integer, is to its step-size in search 1, parameter KaStep-size in search Δ KaIt is set as 1Hz/s, this can guarantee that the broadening of the Azimuth Compression impulse response function of target is less than 2% (because)。
The Airborne Dual-Channel CSSAR systematic parameters of table 1
Radar platform speed 150m/s
Flying radius 2.5km
Radar platform height 10km
Scene center distance 20km
Carrier frequency 10GHz
Transmitted signal bandwidth 75MHz
Sample frequency 100MHz
Pulse recurrence frequency 1500Hz
The synthetic aperture time 0.9s
Adjacent displaced phase center spacing 0.2m
System revisit time 104.72s
SNR 20dB
The target component of table 2
vx(m/s) vy(m/s) r0(km) θ0(rad)
Target 1 -24 13 19.9 0.01
Target 2 -3 -4 20.1 0.04
Target 3 -18 3 19.8 -0.02
The image quality parameter of table 3
The Target moving parameter estimation result of table 4
Fig. 3 is the imaging simulation result figure of target 1, wherein, Fig. 3 (a) is the target image focused on, and Fig. 3 (b) is orientation To profile, Fig. 3 (c) is distance to profile.
Fig. 4 is the imaging simulation result figure of target 2, wherein, Fig. 4 (a) is the target image focused on, and Fig. 4 (b) is orientation Profile, Fig. 4 (c) is distance to profile.
Fig. 5 is the imaging simulation result figure of target 3, wherein, Fig. 5 (a) is the target image focused on, and Fig. 5 (b) is orientation To profile, Fig. 5 (c) is distance to profile.
Fig. 3 is target imaging simulation result, and table 3 gives the image quality parameter of measurement, wide including impulse response Spend (IRW), integration secondary lobe ratio (ISLR), peak sidelobe ratio (PSLR).Wherein PSLRLIt is defined as main lobe and the highest secondary lobe on the left side Height ratio, PSLRRIt is defined as the height ratio of main lobe and the highest secondary lobe on the right.As can be seen from Table 3, the orientation of all targets 2% is respectively less than to IRW broadenings, and distance is zero to IRW broadenings, the imaging results of this explanation present invention very well, also illustrate this Invention has accurately estimated the doppler ambiguity number and doppler frequency rate of target.
Table 4 gives the simulation result of Target moving parameter estimation, it can be seen that Target moving parameter estimation of the invention Precision is higher, and the absolute value of evaluated error is respectively less than 0.4m/s.

Claims (2)

1. a kind of airborne multichannel CSSAR ground moving object motion parameters estimation methods, it is characterised in that comprise the steps：
Step 1, hypothesis：(1) clutter is suppressed by airborne multichannel CSSAR systems using displaced phase center antenna method；(2) Echo signal after Range compress has been extracted, and echo signal is located at initial data domain；
The echo signal after compressing of adjusting the distance enters row distance to Fourier transformation, and echo signal is transformed into orientation time domain distance frequency Domain, into step 2；
Step 2, step 1 is transformed to orientation time domain apart from frequency domain echo signal carry out base band Doppler center compensation, including Following steps：
A) echo signal s of the orientation time domain apart from frequency domainDPCA(fr,ta) be expressed as
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Wherein, Wr() is frequency of distance envelope, wa,1() is the transmitting-receiving round trip antenna radiation pattern of reference channel, fcSent out for radar Penetrate the carrier frequency of signal, frFor frequency of distance, taFor the orientation slow time, c is the light velocity, and λ is wavelength, tbIt is located at reference channel for target At the time of the positive side-looking direction of displaced phase center, RbFor tbMoment target is to the distance of radar, KaFor the Doppler FM of target Rate, facFor the doppler centroid of target, and facIt is represented by fac=fac,b+ MPRF, wherein fac,bIt is many for target base band General Le centre frequency, M is target Doppler fuzzy number, and PRF is the pulse recurrence frequency of radar；
B) target base band doppler centroid f is assumedac,bEstimate beThen according to orientation time domain frequency of distance domain mesh The expression formula of signal is marked, base band Doppler center penalty function can be configured to
<mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> </mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
The echo signal of formula (1) is multiplied with the penalty function of formula (2) compensation of base band Doppler center can be achieved；
Step 3, in step 2 base band Doppler center compensation after echo signal carry out orientation Fourier transformation, by target Signal changes to two-dimensional frequency；
Step 4, the doppler ambiguity number and doppler frequency rate for estimating target, obtained doppler ambiguity number and Doppler FM The estimate of rate is respectivelyWith
Step 5, the target Doppler fuzzy number estimated using step 4 and doppler frequency rate construct the reference letter of two-dimensional frequency Number, the echo signal in step 3 is multiplied with the reference function and carries out target imaging, then carries out two dimension to the signal after multiplication Echo signal is changed to image area by inverse Fourier transform；
Two-dimensional frequency reference function is configured to：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> <mo>,</mo> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> </mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <mfrac> <mrow> <msup> <mover> <mi>M</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <msup> <mi>PRF</mi> <mn>2</mn> </msup> </mrow> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> </mfrac> <mo>}</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mfrac> <mrow> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> </mrow> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
Wherein faFor base band orientation frequency, and satisfaction-PRF/2≤fa≤ PRF/2,WithRespectively target Doppler frequency modulation rate With the estimate of doppler ambiguity number；
Step 6, the location parameter for carving according to location estimation positive side apparent time of the target in image area target, comprise the following steps：
A) image area echo signal is expressed as：
<mrow> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>&amp;lsqb;</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mi>b</mi> </msub> <mo>+</mo> <mfrac> <msub> <mi>f</mi> <mrow> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <msub> <mi>p</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>-</mo> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mi>b</mi> </msub> </mrow> <mi>&amp;lambda;</mi> </mfrac> <mo>+</mo> <msub> <mi>f</mi> <mrow> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Wherein, trFor apart from fast time, pr() is Range compress impulse response function, pa() is Azimuth Compression impulse response Function；
B) according to the expression formula of the image area echo signal of formula (4), positive side apparent time carves target to radar apart from RbWith target Azimuth angle thetabEstimated by following formula：
<mrow> <msub> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>=</mo> <mi>&amp;omega;</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>i</mi> <mi>m</mi> <mi>g</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>=</mo> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> <mi>m</mi> <mi>g</mi> </mrow> </msub> <mo>-</mo> <mfrac> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <msub> <mi>t</mi> <mrow> <mi>a</mi> <mo>,</mo> <mi>i</mi> <mi>m</mi> <mi>g</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
Wherein,WithRespectively RbAnd θbEstimate, ta,imgAnd tr,imgOrientation position and distance for target in image area To position, ω is the angular speed of radar motion；
Step 7, the location parameter estimation according to the doppler centroid of target, doppler frequency rate and positive side apparent time quarter target The kinematic parameter of target, the kinematic parameter of target is estimated using equation below：
<mrow> <msub> <mover> <mi>v</mi> <mo>^</mo> </mover> <mi>x</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>&amp;lambda;</mi> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </msub> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mi>&amp;omega;</mi> <mo>-</mo> <msqrt> <mrow> <msubsup> <mi>r</mi> <mi>a</mi> <mn>2</mn> </msubsup> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mrow> <msup> <mi>&amp;lambda;</mi> <mn>2</mn> </msup> <msubsup> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>&amp;lambda;</mi> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>v</mi> <mo>^</mo> </mover> <mi>y</mi> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>&amp;lambda;</mi> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </msub> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mi>&amp;omega;</mi> <mo>-</mo> <msqrt> <mrow> <msubsup> <mi>r</mi> <mi>a</mi> <mn>2</mn> </msubsup> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <msup> <mi>&amp;omega;</mi> <mn>2</mn> </msup> <mo>-</mo> <mfrac> <mrow> <msup> <mi>&amp;lambda;</mi> <mn>2</mn> </msup> <msubsup> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>-</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> <msub> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </msqrt> <mo>)</mo> </mrow> <mi>cos</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
In formula (7) and formula (8),
<mrow> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>+</mo> <msub> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mover> <mi>r</mi> <mo>^</mo> </mover> <mi>b</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msubsup> <mover> <mi>R</mi> <mo>^</mo> </mover> <mi>b</mi> <mn>2</mn> </msubsup> <mo>-</mo> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>+</mo> <msub> <mi>r</mi> <mi>a</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
Wherein,For vxEstimate,For vyEstimate, vxAnd vyBe respectively target along x-axis and the speed of y-axis,For fac Estimate,For rbEstimate, rbTarget is carved to the distance of the origin of coordinates, r for positive side apparent timeaIt is radar motion track Radius, h is the height of radar.
2. airborne multichannel CSSAR ground moving object motion parameters estimation methods according to claim 1, its feature exists The doppler ambiguity number and doppler frequency rate of target are estimated using the method based on maximum-contrast in step 4, including it is following Step：
A) by the two-dimensional frequency echo signal S (f after base band Doppler center compensation in step 3r,fa) be expressed as
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>W</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>W</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>)</mo> </mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mfrac> <mn>2</mn> <mi>c</mi> </mfrac> <msub> <mi>R</mi> <mi>b</mi> </msub> <mo>+</mo> <mfrac> <msub> <mi>f</mi> <mrow> <mi>a</mi> <mi>c</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;t</mi> <mi>b</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mi>M</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mo>}</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> </mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <mfrac> <mrow> <msup> <mi>M</mi> <mn>2</mn> </msup> <msup> <mi>PRF</mi> <mn>2</mn> </msup> </mrow> <msub> <mi>K</mi> <mi>a</mi> </msub> </mfrac> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mfrac> <mrow> <mi>M</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> </mrow> <msub> <mi>K</mi> <mi>a</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mi>M</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mo>}</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mi>M</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mi>K</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
Wherein Wa(fa) it is orientation frequency envelope；
B) doppler ambiguity number and doppler frequency rate of target are estimated using the following method based on maximum-contrast：
<mrow> <mo>(</mo> <msub> <mover> <mi>K</mi> <mo>^</mo> </mover> <mi>a</mi> </msub> <mo>,</mo> <mover> <mi>M</mi> <mo>^</mo> </mover> <mo>)</mo> <mo>=</mo> <mi>arg</mi> <munder> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mrow> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> </mrow> </munder> <mo>{</mo> <mi>C</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>s</mi> <mi>t</mi> <mo>&amp;lsqb;</mo> <mi>s</mi> <mo>(</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> <mo>&amp;rsqb;</mo> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
In formula (4),
<mrow> <mi>C</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>r</mi> <mi>a</mi> <mi>s</mi> <mi>t</mi> <mo>&amp;lsqb;</mo> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>=</mo> <mfrac> <msqrt> <mrow> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>|</mo> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mi>E</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>}</mo> </mrow> </msqrt> <mrow> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>|</mo> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>t</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>}</mo> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
s(tr,ta；ka, m)=IDFT2{S(fr,fa)·H2(fr,fa；ka,m)} (14)
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>,</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>;</mo> <msub> <mi>k</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> </mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> </mfrac> <mfrac> <mrow> <msup> <mi>m</mi> <mn>2</mn> </msup> <msup> <mi>PRF</mi> <mn>2</mn> </msup> </mrow> <msub> <mi>k</mi> <mi>a</mi> </msub> </mfrac> <mo>}</mo> <mi>exp</mi> <mo>{</mo> <mi>j</mi> <mn>2</mn> <mi>&amp;pi;</mi> <mfrac> <mrow> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> </mrow> <msub> <mi>k</mi> <mi>a</mi> </msub> </mfrac> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>j</mi> <mi>&amp;pi;</mi> <mfrac> <mrow> <msub> <mi>f</mi> <mi>c</mi> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>a</mi> </msub> <mo>+</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>P</mi> <mi>R</mi> <mi>F</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mi>k</mi> <mi>a</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>c</mi> </msub> <mo>+</mo> <msub> <mi>f</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
Wherein, IDFT2() represents two-dimentional inverse Fourier transform, and E () representation space average operation, Contrast () is represented The contrast of image,WithRespectively target Doppler frequency modulation rate and the estimate of doppler ambiguity number, kaIt is structure respectively with m Make two-dimensional frequency reference function H2(fr,fa；ka, the target Doppler frequency modulation rate and doppler ambiguity number used when m).
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