CN115453536A - Forward-looking azimuth pitching two-dimensional super-resolution imaging method for motion platform - Google Patents
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Abstract
The invention discloses a method for two-dimensional super-resolution imaging of a forward-looking direction and a pitching direction of a motion platform. The method solves the problems that the existing real-aperture radar is low in azimuth-elevation two-dimensional resolution and is not suitable for a moving platform, and compared with the existing two-dimensional scanning radar super-resolution method, the method not only can be used for two-dimensional super-resolution imaging of the moving platform, but also has excellent target resolution capability and lower calculation complexity and is suitable for engineering realization.
Description
Technical Field
The invention belongs to the technical field of radar imaging, and particularly relates to a forward-looking azimuth pitching two-dimensional super-resolution imaging method for a motion platform.
Background
The Real Aperture Radar (RAR) has forward-looking target detection capability which cannot be realized by a Synthetic Aperture Radar (SAR) and a Doppler Beam Sharpening (DBS) in the aspect of an imaging mechanism, and is widely applied to civil fields such as topographic mapping, autonomous landing, material airdrop and the like. Existing RAR super-resolution methods can improve azimuth imaging resolution, but the result is typically an azimuth-distance two-dimensional image with a fixed pitch angle, regardless of the scanning modulation of the pitch beam antenna pattern.
To obtain high azimuthal resolution matching range resolution, the literature "Zhang Y, li J, li M, et al, online space Reconstruction for Scanning Radar Using Beam-Updating q-spice, ieee Geoscience and Remote Sensing Letters,2021, 19: 1-5' provides a generalized spatial-based estimation (SPICE) method based on beam updating, the method utilizes the sparsity of a target scene, compared with an iterative adaptive method, the method further improves the azimuth resolution, but the processing sequence of the method to the azimuth and the pitch direction can seriously deteriorate the imaging result, and the method can not be directly used for two-dimensional super-resolution imaging; the literature "Tuo X, xia Y, zhang Y, et al. Super-Resolution Imaging for Real Aperture radio by Two-Dimensional resolution.2021 IEEE International Geoscience and Remote Sensing Symposium IGARSS. IEEE,2021:6630-6633 "proposes a two-dimensional super-resolution imaging Method based on sparse Alternating Direction multiplier (ADMM), which uses regularized 1 norm constraint and uses ADMM Method to solve, improving the two-dimensional resolution of RAR, but the Method is only effective for fixed platform, but is difficult to realize airborne moving platform radar azimuth pitching two-dimensional resolution.
Disclosure of Invention
In order to solve the technical problem, the invention provides a two-dimensional super-resolution imaging method for pitching a forward-looking direction of a motion platform.
The technical scheme adopted by the invention is as follows: a forward-looking azimuth pitching two-dimensional super-resolution imaging method for a motion platform comprises the following specific steps:
s1, an orientation-pitching two-dimensional scanning echo model of an airborne radar platform;
if the airborne radar platform is located at the point A (0, gamma) when t =0 and moves at a constant speed along the y-axis direction at the speed v, gamma represents the height of the airborne platform, and if the airborne platform moves to the point B at the moment t, AB = vt can be obtained, and if a point P (x) in the scanning area is set P ,y P ,z P ) The projection on the xOy plane is the Q point, R P The distance between the radar platform and the target at the initial moment,
is the pitch angle, θ, of the target relative to the radar platform P Is the azimuth angle of the target with respect to the y-axis direction. According to the geometrical relationship, the distance R (t) between the airborne platform and the target at the time t can be represented as follows:
according to the following geometrical relationship:
substituting the formula (2) into the formula (1) to obtain:
taylor unfolding of R (t) gives:
wherein, O (t) 2 ) Denotes t 2 In the coherent accumulation time (CPI) of an arbitrary point target, satisfies vt < R P Thus, the quadratic and higher order terms of the distance history are negligible, and equation (4) can be approximated as:
setting a scanning radar to transmit a chirp signal:
where τ is the distance time variable, T p For the pulse width, rect (-) represents a rectangular window function, f c Denotes the carrier frequency, K r Indicating the chirp rate. According to the platform motion, beam scanning, signal transmitting and receiving processes, the echo signal of the target P point can be expressed as:
where t is the platform motion time variable (slow time), τ is the distance time variable (fast time), σ P The scattering coefficient of the target P is represented by k, a constant containing various gains and losses, c is the propagation speed of the electromagnetic wave, and h (t) is the antenna pattern modulation function. In distance, high distance resolution is achieved through pulse compression, and after carrier frequency removal and pulse compression processing, echoes can be written as:
wherein, B r Represents the signal bandwidth and satisfies B r ＝K r T p And sinc (·) represents a distance signal envelope after pulse compression, and is particularly marked as sinc (γ) = sin (π γ)/π γ, the last term exp { -j4 π R (t)/λ } represents a Doppler phase term caused by stage motion, and λ represents wavelength.
The method for scanning the target airspace omega is as follows: for the first pitch dimension, scanning along the azimuth direction, jumping to the second pitch dimension, scanning along the azimuth direction, and so on, and scanning densely in a Z shape line by line, wherein the sum of delta theta and delta theta
Respectively representing the azimuth and elevation wave position jump, phi θ And
respectively representing azimuth and elevation sweep ranges, N θ And
respectively representing the number of azimuth and pitch samples. In the scanning process of the airborne radar platform, the fast time tau and the slow time t respectively meet the following conditions:
wherein PRF denotes the pulse repetition frequency, θ and
representing the azimuth pitch angle variable and R the range variable.
S2, echo pretreatment;
substituting the formula (9) and the formula (10) into the echo formula (8) can obtain a distance-azimuth-pitch three-dimensional echo expression:
wherein,
representing a two-dimensional antenna pattern, in order to eliminate the range walk created by the relative motion of the platform and target, the range walk amount can be expressed as:
constructing a phase compensation factor in the distance frequency domain:
wherein f is R And (3) representing a distance frequency variable, performing Fast Fourier Transform (FFT) on the distance direction of the formula (11), multiplying the distance direction by the formula (13), and performing Inverse Fast Fourier Transform (IFFT) on the distance direction to obtain an echo after walking correction:
wherein, the fourth term in the formula
A constant term related to the initial slope distance is expressed, the azimuth and elevation two-dimensional processing is not influenced and can be approximately ignored, and a fifth term expresses Doppler phase modulation generated in the relative motion process. When the target is located in the forward looking range of the airborne platform,
in this case, the doppler history of each target in the forward-looking region is approximately equal, so equation (14) can be approximated as:
is located in an airspace omega
Has a target backscattering coefficient of
R 0 ,θ 0 ,
Respectively represent objects
The total echo of the detection airspace can be written as:
for a fixed range unit, only azimuth and pitch changes are considered. The azimuth-elevation two-dimensional scanning of the detection area is equivalent to the two-dimensional convolution of a two-dimensional antenna directional diagram and each scattering point of the detection area by a target scattering coefficient and an azimuth elevation two-dimensional antenna directional diagram. Thus, equation (16) can be simplified to an echo model for a single range bin as:
wherein,
is equivalent to a two-dimensional convolution kernel,
representing the target azimuth pitch scatter distribution for the corresponding range bin.
Considering additive noise, there are:
wherein, denotes a two-dimensional convolution,
is equivalent to a two-dimensional convolution kernel,
representing additive gaussian noise.
For simplicity of presentation, the two-dimensional convolution model (18) can be discretized as:
Y＝AXB T +N (19)
wherein,
an echo matrix representing a slice at a certain distance,
is a backscattering coefficient matrix of the target, K 1 、K 2 Respectively representing the number of azimuth and elevation sampling points of the backscatter coefficient matrix,
in order to be an additive noise, the noise is,
representing a real space, T represents the transpose of a matrix,
and
the azimuth and elevation antenna pattern modulation matrices are represented separately, written specifically as:
wherein,
representing the azimuth dimension antenna pattern sampling points,
representing the antenna pattern sample point, L, in elevation 1 And L 2 Respectively representing the sampling points of the directional diagram and the pitching-dimensional antenna directional diagram, and the sampling points M, N and K 1 ，K 2 ，L 1 ，L 2 Satisfies the relationship:
s3, calculating an autocorrelation matrix R;
scattering result x from the target (q+1) Can be written as:
x (q+1) ＝P (q) F H (R (q) ) -1 y (23)
wherein q represents the number of iterations, R (q) ＝F H P (q) F is P by the q-th iteration (q) Is obtained by calculation, P (q) ＝diag(x (q) ) Diag (·) denotes the construction of a diagonal matrix from a vector, H denotes the conjugate transpose operation, and Y = vec (Y) is the one-dimensional vectorized version of the echo Y. For simplicity of representation, the number of iterations of R is omitted from equation (24) and onward. Furthermore, due to the particular structure of the scan matrix F, it can be seen that R is actually a singular matrix, which results in R -1 Is wrong. In order to ensure the accuracy of the angle estimation and prevent the noise amplification problem, R is diagonally loaded:
R＝FPF H +μI (24)
wherein μ represents a regularization parameterA number, which balances noise and resolution performance, I denotes a unit matrix. Because the dimension of the matrix R is overlarge, R is directly solved -1 The computational complexity is too high, and R is first reduced by CG algorithm -1 The computational complexity of (2).
S4, iteratively updating U;
defining variables
According to the Fletcher-Reeves conjugate gradient algorithm, the variable u can be updated by the following formula:
u l+1 ＝u l +α l d l (25)
wherein l represents the number of CG iterations, d l Denotes the direction of conjugation, α, with respect to R l Represents a step size, satisfies
ρ l Representing an intermediate variable, a weight vector w l Satisfies the following conditions:
w l ＝Rd l+1 (26)
equation (25) is written in two-dimensional form:
wherein, U l 、D l Represents u l And d l In two-dimensional form of (1), satisfy u l ＝vec(U l )，d l ＝vec(D l ) Thus, there are:
U l+1 ＝U l +α l D l (28)
wherein,
wherein an indicates a conjugate operation, an indicates a Hadamard product of the matrix,
and
respectively represent vectors d l+1 And w l The (i) th element of (a),
represent
The full 1 vector of row 1 and column,
represents N θ All 1 vectors, W, of row 1 column l Satisfy w l ＝vec(W l ) And can be derived by the following steps.
S5, iteratively updating W;
from the Kronecker product property, equation (26) is written as a two-dimensional form:
wherein, sigma is the target scattering power, and satisfies the condition of sigma ij ＝|X ij | 2 The relation P = diag { vec (Σ) }, Σ is satisfied with the matrix P ij I row j column element representing the target scattering power matrix Σ, i =1 1 ，j＝1,...,K 2 ，X ij Representing the ith row and jth column of matrix object X. Since P is a diagonal array, P is converted into sigma-and (A) H D l+1 B * ) Hadamard product relationship of (a), thus:
W l ＝A(Σ⊙(A H D l+1 B * ))B T +μD l+1 (31)
s6, iteratively updating D;
d l the one-dimensional update expression can be written as:
d l+1 ＝g l +β l d l (32)
wherein, g l Representing gradient vectors, intermediate variable beta l+1 ＝ρ l+1 /ρ l 。
Writing equation (32) in two dimensions:
wherein G is l Is represented by g l In two-dimensional form of (1), satisfies g l ＝vec(G l ) Thus, there are:
D l+1 ＝G l +β l D l (35)
wherein, beta l+1 The update method of (2) is the same as that of (32), and
||·|| F representing the matrix Frobenius norm.
S7, iteratively updating G;
in the formula (32), the gradient g l The update formula of (c) is as follows:
g l+1 ＝g l -α l w l (36)
converting the above formula to a two-dimensional form:
thus, the following results are obtained:
G l+1 ＝G l -α l W l (38)
when it is satisfied with
Or stopping CG iteration after reaching the specified iteration times, wherein epsilon represents a small positive number, and outputting a two-dimensional CG iteration nodeFruit
S8, iterating a two-dimensional form of an estimation result;
according to equation (23), the two-dimensional representation of the scattering of the target can be written as:
similar to formula (30), wherein P is used (q) ＝diag{vec(Σ (q) ) And obtaining a two-dimensional target scattering iteration formula:
X (q+1) ＝Σ (q) ⊙(A H U (q) B * ) (40)
wherein, U (q) Satisfy vec (U) (q) )＝R -1 y. At the start of the target scatter iteration, the variable Σ is initialized to: sigma (0) ＝A H YB H . At the start of a CG iteration, variables are initialized as: u shape 0 ＝0，D 0 ＝0，β 0 ＝0，G 0 ＝Y，
By repeated iterations of two-dimensional target scatter (equation (40)) and CG (equations (28), (29), (31), (35), and (38)), X in model equation (19) can be solved directly.
The invention has the beneficial effects that: the method comprises the steps of establishing an orientation-pitching two-dimensional scanning echo model of the airborne radar platform, then carrying out distance walking correction on the orientation-pitching-distance echo, enabling the echo model to be approximate to a two-dimensional convolution model, and finally utilizing a conjugate gradient method and the Kronecker product property of a matrix to derive the scattering intensity of an orientation-pitching two-dimensional target. The method solves the problems that the existing real aperture radar is low in azimuth-elevation two-dimensional resolution and is not suitable for a motion platform, and compared with the existing two-dimensional scanning radar super-resolution method, the method not only can be used for two-dimensional super-resolution imaging of the motion platform, but also has excellent target resolution capability and lower calculation complexity, and is suitable for engineering implementation.
Drawings
FIG. 1 is a flow chart of a forward-looking azimuth elevation two-dimensional super-resolution imaging method of a motion platform according to the invention.
Fig. 2 is a geometric model of the movement of the airborne radar platform in the embodiment of the invention.
FIG. 3 is a schematic diagram of a spatial domain target two-dimensional scanning mode according to an embodiment of the present invention.
Fig. 4 is an azimuthally tilted two-dimensional antenna pattern in an embodiment of the present invention.
FIG. 5 is a diagram of original scene three-dimensional point target layout in an embodiment of the present invention.
FIG. 6 is a diagram of the setting of an azimuth-elevation two-dimensional point target in an embodiment of the invention.
FIG. 7 is a diagram of the results of the pre-processed azimuth-elevation two-dimensional real beam echo in the embodiment of the present invention.
FIG. 8 is a diagram of super-resolution results of two-dimensional wiener inverse filtering in the embodiment of the present invention.
FIG. 9 is a graph of super-resolution results according to an embodiment of the present invention.
Detailed Description
The method of the present invention is further illustrated by the following examples in conjunction with the accompanying drawings.
In the embodiment, the effectiveness of the proposed method for the two-dimensional super-resolution imaging of the pitching of the forward-looking direction of the motion platform is verified through a simulation experiment. The steps and results in this embodiment are verified on a MATLAB 2018b simulation platform.
As shown in fig. 1, a flow chart of a two-dimensional super-resolution imaging method for pitching a forward-looking azimuth of a motion platform of the present invention includes the following specific steps:
s1, establishing a signal geometric model;
as shown in table 1, simulation parameters of the radar motion platform are listed. The sampling rate satisfies the nyquist sampling law.
TABLE 1
The airborne radar platform is located at a point A (0, Γ) when t =0, Γ =1000m, and moves at a uniform velocity in the y-axis direction with a velocity v =90 m/s. And when the airborne platform moves to the point B at the moment t, AB = vt can be obtained. Let a point P (x) in the scanning area P ,y P ,z P ) The projection on the xOy plane is the Q point. R is P =30000m is the distance between the radar platform and the target at the initial moment,
is the pitch angle, theta, of the target relative to the radar platform P For the azimuth angle of the target relative to the y-axis direction, as shown in fig. 2, according to the geometric relationship, the distance R (t) between the airborne platform and the target at time t can be written as:
according to fig. 2, there are the following geometrical relations:
substituting formula (42) for formula (41) yields:
taylor expansion of R (t) and neglecting the quadratic and higher terms of the distance history, we can:
according to the platform motion, beam scanning, signal transmitting and receiving processes, the echo signal of the target P point can be expressed as:
wherein tau is a distance time variable, rect (-) represents a rectangular window function, K r Expressing the chirp slope, t is the platform motion time variable (slow time), τ is the distance time variable (fast time), σ P The scattering coefficient of the target P is represented by k, a constant containing various gains and losses, c is the propagation speed of the electromagnetic wave, and h (t) is the antenna pattern modulation function.
After carrier frequency removal and after pulse compression processing, the echo can be written as:
wherein, sinc (·) represents the distance direction signal envelope after pulse compression, and is specifically denoted as sinc (γ) = sin (π γ)/π γ, the last term exp { -j4 π R (t)/λ } represents the Doppler phase term caused by stage motion, and λ represents the wavelength.
As shown in fig. 3, the target airspace Ω is scanned in the following manner: and for the first pitch dimension, scanning along the azimuth direction, jumping to the second pitch dimension, scanning along the azimuth direction, and so on, and performing Z-shaped dense scanning line by line. Wherein N is θ And
respectively representing the number of azimuth and pitch samples. In the scanning process of the airborne radar platform, the fast time tau and the slow time t respectively meet the following requirements:
wherein PRF denotes the pulse repetition frequency, θ and
representing the azimuth pitch angle variable and R the range variable.
S2, preprocessing an echo;
as can be seen from equations (46), (47) and (48), the range-azimuth-pitch three-dimensional echo expression can be written as:
wherein,
representing a two-dimensional antenna pattern function with an energy distribution as shown in figure 4. The phase compensation factors are constructed in the distance frequency domain as follows:
wherein, f R And (3) representing a distance frequency variable, performing Fast Fourier Transform (FFT) on the distance direction of the formula (49), multiplying the distance direction by the formula (50), and performing Inverse Fast Fourier Transform (IFFT) on the distance direction to obtain an echo after walking correction:
when the target is located in the direct forward-looking range of the airborne platform,
the doppler history of each target in the forward-looking region is approximately equal, so equation (51) can be approximated as:
is located in an airspace omega
Has a target backscattering coefficient of
R 0 ,θ 0 ,
Respectively representing objects
The target setting is shown in fig. 5 and 6. The total echo in the probe space domain can be written as:
according to the above equation, the two-dimensional scanning convolution model can be discretized as:
Y＝AXB T +N (54)
wherein,
an echo matrix representing a slice at a certain distance,
is a backscattering coefficient matrix of the target, K 1 、K 2 Respectively representing the number of azimuth and elevation sampling points of the backscatter coefficient matrix,
in order to be additive to the noise,
representing a real space, T represents the transpose of a matrix,
and
representing azimuth and elevation antenna pattern modulation matrices, respectively, in particularWrite as:
wherein,
the azimuth dimension antenna pattern sampling points are represented,
represents the elevation dimension antenna pattern sample point, L 1 And L 2 Respectively representing the sampling point number of the antenna directional diagram in the azimuth dimension and the elevation dimension.
S3, iteratively updating U;
defining variables
The variable u is updated by the following formula:
u l+1 ＝u l +α l d l (57)
wherein l represents the number of CG iterations, d l Denotes the conjugation direction, α, with respect to R l Represents a step size, satisfies
ρ l Represents an intermediate variable, w l Representing a weight vector.
Writing equation (58) in two-dimensional form:
wherein, U l 、D l Denotes u l And d l In two-dimensional form of (1), satisfy u l ＝vec(U l )，d l ＝vec(D l ) Thus, there are:
U l+1 ＝U l +α l D l (59)
wherein,
wherein, indicates a conjugate operation, indicates a Hadamard product,
and
respectively represent vectors d l+1 And w l The (i) th element of (2),
to represent
The full 1 vector of row 1 and column,
represents N θ All 1 vectors, W, of row 1 column l Satisfy w l ＝vec(W l )。
S4, iteratively updating W;
in equation (57):
w l ＝Rd l+1 (61)
equation (61) is written in two dimensions:
wherein, the sigma is the target scattering power, and the requirement of sigma is satisfied ij ＝|X ij | 2 The relation P = diag { vec (Σ) }, Σ is satisfied with the matrix P ij I row j column element representing the target scattering power matrix Σ, i =1 1 ，j＝1,...,K 2 ，X ij Representing the ith row and the jth column of matrix object X. Since P is a diagonal array, P is converted into sigma-and (A) H D l+1 B * ) Hadamard product relationship of (a), thus:
W l ＝A(Σ⊙(A H D l+1 B * ))B T +μD l+1 (63)
s5, iteratively updating D;
d l+1 and d l Is updated by:
d l+1 ＝g l +β l d l (64)
wherein, g l Representing gradient vectors, intermediate variable beta l+1 ＝ρ l+1 /ρ l 。
Writing equation (64) in two dimensions:
wherein, G l Is represented by g l In two-dimensional form of (b), satisfies g l ＝vec(G l ) Thus, there are:
D l+1 ＝G l +β l D l (67)
wherein beta is l+1 The update method of (1) is the same as formula (64), and
||·|| F representing the Frobenius norm of the matrix.
S6, iteratively updating G;
in equation (64), the gradient g may be updated by the equation l ：
g l+1 ＝g l -α l w l (68)
Converting the above formula to a two-dimensional form:
thus, the following results were obtained:
G l+1 ＝G l -α l W l (70)
when it is satisfied with
Or terminate the CG iteration after up to 10 iterations, e =10 -3 Representing a small positive number and outputting a two-dimensional CG iteration result
S7, iterating a two-dimensional form of an estimation result;
after the CG iteration is terminated, outputting a CG iteration result u (q) ＝u l Substituting the formula (23) to obtain the q-th iteration result of the target scattering:
wherein u is (q) R in the q-th target scattering iteration (23) derived by CG iteration -1 y。
According to equation (23), the two-dimensional iterative representation of the target scatter can be written as:
therefore, there are:
X (q+1) ＝Σ (q) ⊙(A H U (q) B * ) (73)
wherein, U (q) Satisfy vec (U) (q) )＝R -1 y. At the start of the target scatter iteration, the variable Σ is initialized to: sigma-shaped (0) ＝A H YB H . At the start of a CG iterationWhen, the variables are initialized to: u shape 0 ＝0，D 0 ＝0，β 0 ＝0，G 0 ＝Y，
By repeated iterations of two-dimensional target scatter (equation (73)) and CG (equations (59), (60), (63), (67), and (70)), X in model equation (54) can be solved directly. On one hand, the method does not need matrix inversion, on the other hand, the method does not need to construct and calculate multiplication of high-dimensional matrix and vector, and can greatly reduce the calculation complexity of the target scattering iteration method.
The complexity of the method is specifically analyzed: in the CG iteration, four matrices, U respectively, need to be stored l ，W l ，D l And G l . Each CG iteration requires 4 matrix-matrix multiplications (A (Sigma |, A) H D l+1 B * ))B T ) 2 matrix-matrix Hadamard multiplication (one-time as in equation (63) & Sigma & (A) H D l+1 B * ) In another case in formula (60)
) 4 matrix-scalar multiplication (U) l ,D l ,G l Update of (2) and μ D in equation (63) l+1 ) And 1 scalar multiplication
If L iterations are required, the computational complexity of the two-dimensional CG algorithm is
Two matrix multiplications (sigma) are required for each iteration of the target scatter (q) ⊙(A H U (q) B * ) And a scalar multiplication (∑ y) (q) ＝X (q) ⊙X (q) ). Assuming that Q target scattering iterations are performed, the overall computational complexity of the algorithm is:
fig. 7, 8 and 9 show the point target super-resolution results of different methods. Limited by the internal memory of the platform, the number of azimuth and pitch sampling points is reduced, wherein,
fig. 7 shows the real beam results, and it can be seen that 4 targets are completely indistinguishable. Fig. 8 is a processing result of the two-dimensional wiener inverse filtering method. The method has high processing speed. It can be seen that although the four targets are substantially separated, the method has poor resolution performance and also has strong side lobe effects near the 0 ° azimuth and 0 ° pitch dimensions. Both methods can be implemented quickly in the frequency domain by FFT, but with limited resolution. Fig. 9 is a processing result of the method proposed by the present invention, and it can be seen that the resolution performance is significantly better than the real beam result and the two-dimensional wiener filtering method.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Various modifications and alterations to this invention will become apparent to those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.
Claims (1)
1. A method for two-dimensional super-resolution imaging of pitching of a forward-looking azimuth of a motion platform comprises the following specific steps:
s1, an orientation-pitching two-dimensional scanning echo model of an airborne radar platform;
assuming that the airborne radar platform is located at a point A (0, gamma) when t =0 and moves at a constant speed along the y-axis direction at a speed v, gamma represents the height of the airborne platform, and the airborne platform moves to a point B at the time t, AB = vt can be obtained, and a point P (x) in a scanning area is assumed P ,y P ,z P ) The projection on the xOy plane is the Q point, R P The distance between the radar platform and the target at the initial moment,
is the pitch angle, θ, of the target relative to the radar platform P According to the geometrical relationship, the distance R (t) between the airborne platform and the target at the time t is expressed as follows:
according to the following geometrical relationship:
substituting formula (2) into formula (1) to obtain:
taylor unfolding R (t) gives:
wherein, O (t) 2 ) Denotes t 2 Satisfies vt & lt R within the coherent accumulation time (CPI) of an arbitrary point target P Equation (4) is approximated as:
setting a scanning radar to transmit a chirp signal:
where τ is the distance time variable, T p For pulse width, rect (·)Representing a rectangular window function, f c Denotes the carrier frequency, K r Representing the chirp slope, the echo signal at target P point is represented as:
where t is the platform motion time variable (slow time), τ is the distance time variable (fast time), σ P For the target scattering coefficient of the target P, κ represents a constant containing various gains and losses, c is the electromagnetic wave propagation speed, h (t) is the antenna pattern modulation function, high range resolution is achieved by pulse compression over distance, and after carrier frequency removal and pulse compression processing, the echo is written as:
wherein, B r Represents the signal bandwidth and satisfies B r ＝K r T p Sine (·) represents the distance-wise signal envelope after pulse compression, and is specifically denoted as sine (γ) = sin (π γ)/π γ, and the last term exp { -j4 π R (t)/λ } represents the Doppler phase term due to stage motion, and λ represents wavelength;
scanning a target space domain omega, wherein delta theta and
respectively representing the azimuth and elevation wave position jump, phi θ And
respectively representing azimuth and elevation sweep ranges, N θ And
respectively representing the number of azimuth sampling points and the number of pitching sampling points, wherein in the scanning process of the airborne radar platform, the fast time tau and the slow time t respectively meet the following requirements:
wherein PRF denotes the pulse repetition frequency, θ and
representing an azimuth pitch angle variable, and R representing a distance variable;
s2, echo pretreatment;
substituting the formula (9) and the formula (10) into the echo formula (8) can obtain a distance-azimuth-pitch three-dimensional echo expression:
wherein,
representing a two-dimensional antenna pattern, the distance walk is represented as:
constructing a phase compensation factor in the distance frequency domain:
wherein f is R And (3) representing a distance frequency variable, performing Fast Fourier Transform (FFT) on the distance direction of the formula (11), multiplying the distance direction by the formula (13), and performing Inverse Fast Fourier Transform (IFFT) on the distance direction to obtain an echo after walking correction:
wherein, the fourth term in the formula
A constant term related to the initial slope distance is shown, a fifth term represents Doppler phase modulation generated by the relative motion process, when the target is positioned in the forward looking range of the airborne platform,
the approximation is:
locate in omega of airspace
Has a target backscattering coefficient of
R 0 ,θ 0 ,
Respectively representing objects
The total echo of the detection airspace is written as:
equation (16) reduces to the echo model for a single range bin:
wherein,
is equivalent to a two-dimensional convolution kernel,
representing the target azimuth pitch scattering distribution of the corresponding range unit;
considering additive noise, there are:
wherein, denotes a two-dimensional convolution,
is equivalent to a two-dimensional convolution kernel,
representing additive gaussian noise;
discretization of the two-dimensional convolution model (18) is:
Y＝AXB T +N (19)
wherein,
an echo matrix representing a slice at a certain distance,
is a backscattering coefficient matrix of the target, K 1 、K 2 Respectively representing the position of the backscattering coefficient matrix and the number of pitching sampling points,
in order to be additive to the noise,
representing a real space, T represents the transpose of a matrix,
and
the azimuth and elevation antenna pattern modulation matrices are represented separately, written specifically as:
wherein, [ a (θ) ] L1 ),a(θ L1-1 ),…,a(θ 1 )]The azimuth dimension antenna pattern sampling points are represented,
representing the antenna pattern sample point, L, in elevation 1 And L 2 Respectively representing the number of sampling points of an azimuth dimension antenna directional diagram and a pitching dimension antenna directional diagram, and the number of the sampling points M, N and K 1 ，K 2 ，L 1 ，L 2 Satisfies the relationship:
s3, calculating an autocorrelation matrix R;
scattering result x from the target (q+1) Writing as follows:
x (q+1) ＝P (q) F H (R (q) ) -1 y (23)
whereinQ denotes the number of iterations, R (q) ＝F H P (q) F is P by the q-th iteration (q) Is obtained by calculation, P (q) ＝diag(x (q) ) Diag (·) denotes constructing a diagonal matrix from a certain vector, H denotes a conjugate transpose operation, Y = vec (Y) is a one-dimensional vectorization form of the echo Y, and R is diagonally loaded:
R＝FPF H +μI (24)
where μ represents a regularization parameter, I represents a unit matrix, and R is reduced by the CG algorithm -1 The computational complexity of (2);
s4, updating U in an iterative manner;
defining variables
The variable u is updated by the following formula:
u l+1 ＝u l +α l d l (25)
wherein l represents the number of CG iterations, d l Denotes the direction of conjugation, α, with respect to R l Represents a step size, satisfies
ρ l Representing an intermediate variable, a weight vector w l Satisfies the following conditions:
w l ＝Rd l+1 (26)
equation (25) is written in two-dimensional form:
wherein, U l 、D l Represents u l And d l In two-dimensional form of (1), satisfy u l ＝vec(U l )，d l ＝vec(D l ) Thus, there are:
U l+1 ＝U l +α l D l (28)
wherein,
wherein an indicates a conjugate operation, an indicates a Hadamard product of the matrix,
and
respectively represent vectors d l+1 And w l The (i) th element of (a),
to represent
The full 1 vector of row 1 and column,
represents N θ All 1 vectors, w, of row 1 column l Satisfy w l ＝vec(W l )；
S5, iteratively updating W;
equation (26) is written in two dimensions:
wherein, the sigma is the target scattering power, and the requirement of sigma is satisfied ij ＝|X ij | 2 The relation P = diag { vec (Σ) }, Σ is satisfied with the matrix P ij I row j column element representing the target scattering power matrix Σ, i =1 1 ，j＝1,...,K 2 ，X ij The ith row and the jth column of the matrix target X are represented, P is a diagonal matrix, and P is converted into sigma-sum (A) H D l+1 B * ) Hadamard product relationship of (a), thus:
W l ＝A(Σ⊙(A H D l+1 B * ))B T +μD l+1 (31)
s6, iteratively updating D;
d l the one-dimensional update expression is written as:
d l+1 ＝g l +β l d l (32)
wherein, g l Representing gradient vectors, intermediate variable beta l+1 ＝ρ l+1 /ρ l ；
Writing equation (32) in two dimensions:
wherein, G l Denotes g l In two-dimensional form of (b), satisfies g l ＝vec(G l ) Thus, there are:
D l+1 ＝G l +β l D l (35)
wherein, beta l+1 The update method of (2) is the same as formula (32), and
||·|| F representing a matrix Frobenius norm;
s7, iteratively updating G;
in the formula (32), the gradient g l The update formula of (2) is as follows:
g l+1 ＝g l -α l w l (36)
converting the above formula to a two-dimensional form:
obtaining:
G l+1 ＝G l -α l W l (38)
when it satisfies
Or stopping CG iteration after reaching the specified iteration times, wherein epsilon represents a small positive number, and outputting a two-dimensional CG iteration result
S8, iterating a two-dimensional form of an estimation result;
according to equation (23), the two-dimensional representation of the scattering of the target is written as:
obtaining a two-dimensional target scattering iterative formula:
X (q+1) ＝Σ (q) ⊙(A H U (q) B * ) (40)
wherein, U (q) Satisfy vec (U) (q) )＝R -1 y, at the start of the target scatter iteration, the variable Σ is initialized to: sigma-shaped (0) ＝A H YB H At the start of a CG iteration, variables are initialized to: u shape 0 ＝0，D 0 ＝0，β 0 ＝0，G 0 ＝Y，
By repeated iterations of two-dimensional object scatter (equation (40)) and CG (equations (28), (29), (31), (35), and (38)), X in model equation (19) can be solved directly.
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