CN103278819B - Onboard high-resolution strabismus bunching synthetic aperture radar (SAR) imaging method based on sliding receiving window - Google Patents

Abstract

The invention discloses an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window, which includes the following steps: Step 1: read in the original echo data and related imaging parameters; Frequency modulation processing; step three: azimuth Fourier transform processing; step four: azimuth de-chirp residual phase error compensation processing; step five: range to Fourier transform processing; step six: consistent compression processing; step seven: Si Tolt (stolt) interpolation processing; step eight: azimuth to Fourier inverse transform processing; step nine: geometric correction processing; step ten: distance to Fourier transform processing; the present invention proposes a sliding receiving window based The airborne high-resolution squint spotlight SAR imaging method solves the current situation that there is no imaging method for the original echo data of airborne high-resolution squint spotlight SAR based on sliding receiving window technology.

Description

Airborne High Resolution Squint Spotlight SAR Imaging Method Based on Sliding Receiving Window

technical field

The invention belongs to the field of signal processing, in particular to an airborne high-resolution squint spotlight synthetic aperture radar synthetic aperture radar (Synthetic Aperture Radar, SAR) imaging method based on a sliding receiving window.

Background technique

SAR is an active remote sensor that works in the microwave frequency band. It overcomes the limitations of optical imaging due to weather and light conditions. Camouflage, etc., greatly improved the information capture capability of the radar. Therefore, SAR has become a popular research field of radar technology, and more and more countries pay attention to it. Compared with the traditional side-looking SAR system, airborne squint SAR imaging has high flexibility and maneuverability in practical applications. By adjusting the antenna beam pointing, the SAR system can freely and flexibly select the observation area, and can quickly revisit Sensitive areas, greatly improving the observation capabilities of SAR. In addition, with the improvement of SAR resolution, the ability of SAR target reconnaissance and recognition has been significantly improved. Therefore, high-resolution squint imaging has become an important development direction in recent years.

However, high-resolution high-squint SAR imaging also brings new technical difficulties. On the one hand, as the squint angle increases, the azimuth and range coupling of the original SAR echo becomes more serious, making high-precision imaging more difficult; The geometric progression increases, leading to a significant increase in the amount of SAR original echo data, increasing the difficulty of data storage and real-time imaging. The sliding receiving window technology refers to eliminating the one-time item in the distance migration by changing the opening time of the echo window, that is, the distance walking. The sliding receiving window technology can reduce the distance migration of SAR original echoes, thereby reducing the amount of SAR original echo data. However, the sliding receiving window technology adjusts the start time of echo data acquisition, and the Doppler history of each target in the observation area also changes with the azimuth, making the traditional imaging method no longer applicable.

Contents of the invention

The purpose of the present invention is to solve the above problems. Based on the technical characteristics of the sliding receiving window and combined with the traditional wavenumber domain imaging method, an airborne high-resolution squint spotlight SAR imaging method based on the sliding receiving window is proposed.

An airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window, comprising the following steps:

Step 1: Read in the original echo data and related imaging parameters;

Read in the airborne high-resolution squint spotlight SAR two-dimensional original echo simulation complex data S _start and the corresponding imaging parameters based on the sliding receiving window, including: the number of sampling points in azimuth N _a , the number of sampling points in range N _r , the signal Sampling rate f _s , signal bandwidth Bw, pulse width τ, frequency modulation slope b, pulse repetition frequency PRF, reference slope distance R _ref , Doppler center frequency fd ₀ , Doppler modulation frequency f _r0 , satellite velocity P _v , etc. Effective oblique angle Signal wavelength λ, signal carrier frequency f ₀ , signal propagation speed c;

Step 2: azimuth dechirp processing;

The two-dimensional echo simulation complex data S _start is subjected to azimuth dechirp processing: firstly, the complex data S _start is multiplied by the factor H ₁ to obtain the complex data S _{1_1} ; secondly, the complex data S _{1_1} is subjected to Fourier transform in azimuth direction, That is, fast Fourier transform (FFT) is performed along each distance direction (by column) to obtain complex data S _{1_2} ; finally, the complex data S _{1_2} is multiplied by the factor H ₂ to obtain the complex data after de-chirping the final azimuth direction S ₁ ;

Step 3: Azimuth to Fourier transform processing;

Perform fast Fourier transform (FFT) on the complex data S ₁ obtained in step 2 along each distance direction (by column) to obtain complex data S ₂ in the azimuth direction spectrum;

Step 4: Azimuth de-chirp residual phase error compensation processing;

Multiply the complex data S ₂ obtained in step 3 with the azimuth dechirp residual phase error compensation factor Ω ₁ at the corresponding azimuth moment to obtain the compensated complex data S ₃ ;

Step five: distance to Fourier transform processing;

Perform fast Fourier transform (FFT) on the complex data S ₃ obtained in step 4 along each azimuth moment (by row) to obtain two-dimensional spectrum complex data S ₄ ;

Step 6: consistent compression processing;

Multiply the complex data S ₄ obtained in step 5 with the corresponding consistent compression factor Ω ₂ to obtain the coarse-focus complex data S ₅ ;

Step 7: Stolt (stolt) interpolation processing;

For the complex data S ₅ obtained in step 6, Singer (sinc) interpolation method is used to perform stolt interpolation processing to obtain complex data S ₆ mapped from the two-dimensional frequency domain to the two-dimensional wavenumber domain;

Step 8: Azimuth inverse Fourier transform processing;

Perform inverse fast Fourier transform (IFFT) on the complex data S ₆ obtained in step 7 along each distance direction (column), to obtain the complex data S ₇ in the azimuth time domain distance wavenumber domain;

Step 9: Geometric correction processing;

Multiply the complex data S ₇ obtained in step 8 with the geometric correction factor Ω ₃ to obtain complex data S ₈ after geometric correction;

Step ten: distance to Fourier transform processing;

Perform inverse fast Fourier transform (IFFT) on the complex data S ₉ obtained in step 9 along each azimuth moment (by row) to obtain the final imaging result S _end ;

The present invention has the advantage that:

(1) The present invention proposes an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window, which solves the problem that the original echo data of airborne high-resolution squint spotlight SAR based on sliding receiving window technology is not imaged The current state of the method.

(2) The present invention proposes an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window, which has the characteristics of high-precision focused imaging. Since the imaging method proposed in the present invention is an improved wavenumber domain imaging method, the advantage of the wavenumber domain imaging method is that it can achieve high-precision focusing as long as the condition of constant speed is satisfied (airborne SAR just meets this condition). . Therefore, the present invention can realize high-precision focused imaging of the scene.

(3) The present invention proposes an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window, which has the characteristics of strong applicability. On the one hand, since the imaging method proposed by the present invention is an improved wavenumber domain imaging method, and the wavenumber domain imaging method is not limited by the angle of the squint, the present invention can also realize the image of the scene when the angle of the squint is large. Precise focus to get high-quality SAR images. On the other hand, the present invention can realize 0.1m ultra-high resolution imaging and obtain precise focus, therefore, the present invention is applicable to current imaging requirements of various resolutions.

Description of drawings

Fig. 1 is a kind of airborne high-resolution squint spotlight SAR imaging method flow chart based on the sliding receiving window proposed by the present invention;

Fig. 2 is a schematic diagram of an embodiment simulation scene;

Fig. 3 is embodiment imaging result;

Fig. 4 is the upper left point target sectional view of the embodiment;

Fig. 5 is the cross-sectional view of the middle point target of the embodiment;

Fig. 6 is a sectional view of the lower right point target of the embodiment.

Detailed ways

The present invention will be further described in detail with reference to the accompanying drawings and embodiments.

The invention proposes an airborne resolution squint spotlight SAR imaging method based on a sliding receiving window. The processing object is the original echo data of the airborne high resolution squint spotlight SAR based on a sliding receiving window, and the obtained result is a High resolution squint images.

The sliding receiving window means that when the SAR receives the echo, the opening time of the echo receiving window changes with the azimuth time, thereby reducing the range migration of the target, or eliminating the range walking of the target (the linear part in the range migration). The opening time T'(i) of the echo window at each azimuth time of the SAR system using the sliding receiving window is:

${\mathrm{<span\; class="notranslate">\; T</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; fd</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Among them, T ₀ is the opening time of the traditional SAR system fixed echo window under the same conditions, fd ₀ refers to the Doppler center frequency, λ refers to the signal wavelength, c refers to the signal propagation speed, t(i) refers to the azimuth time, and , i=0, 1, 2, . . . , N _a -1. The sliding receiving window technology reduces the distance migration of the echo, thereby reducing the amount of echo data and relieving the pressure of data storage and real-time imaging.

The present invention is an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window. The specific process is shown in Figure 1, comprising the following steps:

Step 1: Read in the airborne high-resolution squint spotlight SAR two-dimensional original echo complex data S _start and the corresponding imaging parameters based on the sliding receiving window. Among them, S _start is a two-dimensional complex array with a size of N _a ×N _r , and the imaging parameters specifically include: the number of sampling points in the azimuth direction N _a , the number of sampling points in the range direction N _r , the signal sampling rate f _s , the signal bandwidth Bw, and the frequency modulation Slope b, pulse repetition frequency PRF, reference slant distance R _ref , Doppler center frequency fd ₀ , Doppler modulation frequency f _r0 , satellite velocity P _v , equivalent slant angle , signal wavelength λ, signal carrier frequency f ₀ , signal propagation speed c;

Step 2: Perform azimuth dechirp processing on the two-dimensional original echo data S _start , which can be specifically divided into the following steps:

(a) Construct two one-dimensional sequences i,j, where i represents the azimuth sequence (row), and j represents the distance sequence (column);

i=[1,2,...,N _a ] (2)

j=[1,2,…,N _r ]

(b) Obtain the azimuth time t(i) corresponding to each row of the two-dimensional original echo data S _start ;

$\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; PRF</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

(c) Multiply the two-dimensional original echo complex data S _start with the derotation factor H ₁ to obtain the data S _{1_1} (i,j), where the factor H ₁ (i) is a one-dimensional complex number with a size of N _a ×1 group, the formula is:

H ₁ (i)=exp{jπ(fr ₀ t ² (i)+2 fd ₀ t(i))} (4)

Then the two-dimensional complex array S _{1_1} can be obtained by the following formula:

S _{1_1} (i, j) = S _start (i, j) · H ₁ (i) (5)

(d) Perform fast Fourier transform (FFT) on complex data S _{1_1} along each distance direction (by column) to obtain complex data S _{1_2} ;

S _{1_2} (:, j) = FFT(S _{1_1} (:, j)) (6)

Among them, S _{1_2} (:, j) represents the nth column of S _{1_2} , S _{1_1} (:, j) represents the nth column of S _{1_1} , and FFT(·) represents the fast Fourier transform of a one-dimensional array.

(e) Obtain the equivalent pulse repetition frequency PRF' after derotation;

${\mathrm{<span\; class="notranslate">\; PRF</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; fr</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{\mathrm{<span\; class="notranslate">\; PRF</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

(g) Combining formula (7) to obtain the azimuth time t ₁ (i) corresponding to each row of the two-dimensional complex data;

${\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; PRF</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

(h) Multiply the complex data S _{1_2} by the factor H ₂ to obtain the final azimuth dechirped data S ₁ . The factor H ₂ (i) is a one-dimensional complex array with a size of N _a ×1, and its formula is:

${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; j\&pi;</span>}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; fr</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Then the two-dimensional complex array S ₁ can be obtained by the following formula:

S ₁ (i, j) = S _{1_2} (i, j) · H ₂ (i) (10)

Step 3: Perform fast Fourier transform (FFT) on the complex data S ₁ (i,j) obtained in Step 2 along each distance direction (column), and obtain the complex data S ₂ (i,j) of the azimuth spectrum;

S ₂ (:,j)=FFT(S ₁ (:,j)) (11)

Among them, S ₂ (:, j) represents the jth column of S ₂ , S ₁ (:, j) represents the jth column of S ₁ , and FFT(·) represents the fast Fourier transform of a one-dimensional array.

Step 4: Multiply the complex data S ₂ (i,j) obtained in Step 3 with the azimuth dechirp residual phase error compensation factor Ω ₁ (i) at the corresponding azimuth moment to obtain the compensated complex data S ₃ (i ,j);

(a) Combining formula (7) to obtain the azimuth frequency f _a (i) corresponding to each row of the two-dimensional azimuth frequency domain distance time domain complex data S ₂ (i,j);

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; PRF</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

(b) Combining formula (12) to obtain a one-dimensional compensation factor Ω ₁ (i) whose size is N _a ×1;

${\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; j\&pi;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>\frac{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; fr</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

(c) Obtain the compensated two-dimensional complex data S ₃ (i,j);

S ₃ (i, j) = S ₂ (i, j)·Ω ₁ (i) (14)

Step 5: Perform fast Fourier transform (FFT) on the complex data S ₃ (i,j) obtained in Step 4 along each azimuth time (by row) to obtain two-dimensional spectrum complex data S ₄ (i,j);

S ₄ (i,:) = FFT(S ₃ (i,:)) (15)

Wherein, S ₃ (i,:) represents the i-th row of S ₃ , and S ₄ (i,:) represents the i-th row of S ₄ .

Step 6: Multiply the complex data S ₄ (i,j) obtained in Step 5 by the corresponding consistent compression factor Ω ₂ (i,j) to obtain coarse-focus complex data S ₅ (i,j).

(a) Obtain the shortest slant distance R _min according to the reference slant distance R _ref ;

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; ref</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; c</span>}}{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>\frac{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$

(b) Obtain the distance frequency domain f _τ (j) corresponding to each column of the two-dimensional frequency domain complex data S ₄ (i,j);

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; Nr</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

(c) Combining Equation (12) and Equation (17) to obtain the two-dimensional frequency domain complex data S ₄ (i,j), the azimuth wavenumber k _x (i) corresponding to each column and the range wavenumber k _rc ( j);

${\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}}$ $<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}$

(d) Combining formulas (16) to (18) to obtain a two-dimensional consistent compression factor Ω ₂ (i,j) with a size of N _a ×N _r ;

(e) Obtain consistent compressed two-dimensional complex data S ₅ (i,j) by combining formula (19);

S ₅ (i, j) = S ₄ (i, j)·Ω ₂ (i, j) (20)

Step 7: For the complex data S ₅ (i, j) obtained in Step 6, use the sinc interpolation method to perform stolt interpolation processing, and obtain the complex data S ₆ (i, j) mapped from the two-dimensional frequency domain to the two-dimensional wavenumber domain;

(a) According to the mapping relationship from the distance frequency domain to the distance wavenumber domain, combined with formula (18) to obtain the wavenumber k' _rc (i,j) in the distance wavenumber domain;

(b) Traverse to obtain the maximum value k' _{rc ,max} and the minimum value k' _rc,max of the wavenumber k' rc (i,j) in the distance wavenumber domain, and obtain the equal interval Δk' _rc of the wavenumber in the distance wavenumber domain;

$\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; max</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; min</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; two</span>}<span\; class="notranslate">\; )</span>$

(c) Obtain the uniform distance wavenumber domain wavenumber of the two-dimensional wavenumber domain data

${\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}\mathrm{<span\; class="notranslate">\; min</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&Delta;k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$

(d) Obtain the uniform distance wavenumber domain wavenumber of the two-dimensional wavenumber domain complex data Position p(i,j) in uneven k' _rc (i,:) corresponding to each row.

The method is to perform the following operations by row: take the position p(1,1) of the first row and first column as an example, first obtain the absolute difference n=[1,2,…,N _r ], to obtain the smallest absolute difference Δk _min and n at the corresponding position, if $k_{\mathrm{rc}}^e\left(1\right)<k_{\mathrm{rc}}^{\&prime;}(1,\mathrm{no}),$ p(1,1)=n-1, if $k_{\mathrm{rc}}^e\left(1\right)\&Greater\; Equal;k_{\mathrm{rc}}^{\&prime;}(1,\mathrm{no}),$ p(1,1)=n,

By analogy, each position p(i,j) is obtained.

(e) Combining the position p(i,j) obtained in the previous step, obtain the sampling point interval q(i,j,n) required for sinc interpolation;

$q(i,j,\mathrm{no})=\frac{k_{\mathrm{rc}}^e\left(j\right)-k_{\mathrm{rc}}^{\&prime;}(i,(p(i,j)+\mathrm{no}))}{\frac{4\mathrm{\&pi;}{f}_{\mathrm{the\; s}}}{c}\&Center\; Dot;\frac{1}{N_r}},$ n=[-N/2,-N/2+1,...,N/2-1] (24)

₍ f) Combining formula (24) with the sinc interpolation method to obtain the two-dimensional data S ₆ after stolt interpolation, since the two-dimensional data is complex data, the real part S _{6_re} ( i, j) and the imaginary part S _{6_im} (i, j) are obtained by sinc interpolation method respectively.

Among them, N is the interpolation kernel length, sinc( ) refers to the interpolation function S _{5_re} (i,j) refers to the real part of the i-th row and j-th column of the two-dimensional data S _{5_re} , and S _{5_im} (i,j) refers to the imaginary part of the i-th row and j-th column of the two-dimensional data S _{5_im} .

Step 8: Perform Inverse Fast Fourier Transform (IFFT) on the complex data S ₆ (i,j) obtained in Step 7 along each distance direction (column), and obtain complex data S ₇ (i ,j);

S ₇ (:, j) = IFFT (S ₆ (:, j)) (27)

Among them, S ₆ (:, j) represents the j-th column of S ₆ , S ₇ (:, j) represents the j-th column of S ₇ , and IFFT (·) represents the inverse fast Fourier transform of a one-dimensional array.

Step 9: Multiply the complex data S ₇ (i,j) obtained in Step 8 by the geometric correction factor Ω ₃ (i,j) to obtain complex data S ₈ (i,j) after geometric correction;

(a) Combine formula (8) and formula (12) to obtain geometric correction factor Ω ₄ (i,j);

${\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>\frac{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; fd</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 28</span>}<span\; class="notranslate">\; )</span>$

(b) Use formula (30) to obtain geometrically corrected complex data S ₈ (i,j);

S ₈ (i, j) = S ₇ (i, j)·Ω ₃ (i, j) (29)

Step 10: Perform Inverse Fast Fourier Transform (IFFT) on the complex data S ₈ (i,j) obtained in Step 9 along each azimuth moment (by row) to obtain the final imaging result S _end (i,j);

S _end (i,:)＝IFFT(S ₈ (i,:)) (30)

Wherein, S ₈ (i,:) represents the i-th row of S ₈ , and S _end (i,:) represents the i-th row of S _end .

Example:

This embodiment proposes an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window. The simulation scene is a 3×3 dot matrix, and the interval between points is 100m. Finally, the upper left and middle of the simulation scene are respectively , and the lower right three points were evaluated. The specific simulation scene is shown in Figure 2, and the imaging parameters involved in the imaging process are shown in Table 1.

Table 1 embodiment parameters

This embodiment specifically includes the following steps:

Step 1: Read in the airborne high-resolution squint spotlight SAR two-dimensional original echo complex data S _start and the corresponding imaging parameters based on the sliding receiving window. Among them, S _start is a two-dimensional complex array with a size of 65536×16384, and the specific imaging parameters are shown in Table 1;

Step 2: Perform azimuth dechirp processing on the two-dimensional original echo data S _start . The specific operation steps are:

(a) Construct a one-dimensional sequence, as shown in formula (2), i=[1,2,…,65536], j=[1,2,…,16384];

(b) Obtain the azimuth time t(i) corresponding to each line of the two-dimensional original echo data S _start , and the specific process is carried out according to formula (3);

(c) Multiply the data S _start by the derotation factor H ₁ (i) to obtain the data S _{1_1} (i,j). The factor H ₁ is a one-dimensional complex array with a size of 65536×1. The specific acquisition process is carried out according to formula (4), while the two-dimensional complex array S _{1_1} (i,j) is obtained according to formula (5);

(d) Perform fast Fourier transformation (FFT) on the complex data S _{1_1} (i,j) according to formula (6) along each distance direction to obtain the complex data S _{1_2} (i,j);

(e) Obtain the equivalent pulse repetition frequency PRF' after derotation, and the acquisition process is carried out according to formula (7);

(g) Combining formula (7) to obtain the azimuth time t ₁ (i) corresponding to each row of two-dimensional complex data, the acquisition process is carried out according to formula (8);

(h) Multiply the data S _{1_2} (i,j) with the factor H ₂ (i) to obtain the final azimuth dechirped data S ₁ (i,j). Among them, the factor H ₂ (i) is a one-dimensional complex array with a size of 65536×1, and its acquisition process is carried out according to formula (9), while the acquisition process of the two-dimensional complex array S ₁ is carried out according to formula (10);

Step 3: Perform fast Fourier transform (FFT) on the complex data S ₁ (i,j) obtained in Step 2 along each distance direction (column), and obtain the complex data S ₂ (i,j) in the azimuth direction. The specific operation process is carried out according to formula (11);

Step 4: Multiply the complex data S ₂ (i,j) obtained in Step 3 with the azimuth dechirp residual phase error compensation factor Ω ₁ (i) at the corresponding azimuth moment to obtain the compensated complex data S ₃ (i ,j);

(a) Obtain the azimuth frequency f _a corresponding to each row of the two-dimensional azimuth, frequency domain, distance and time domain complex data S ₂ (i,j), and the specific acquisition process is carried out according to formula (12);

(b) Combining formula (7) to obtain a one-dimensional compensation factor Ω ₁ (i) with a size of 65536×1, the specific acquisition process is carried out according to formula (13);

(c) Obtain the compensated two-dimensional complex data S ₃ (i,j), the specific acquisition process is carried out according to formula (14);

Step 5: Perform fast Fourier transform (FFT) on the complex data S ₃ (i,j) obtained in Step 4 along each azimuth moment (by row) to obtain the two-dimensional spectrum complex data S ₄ (i,j), The specific operation process is carried out according to formula (15);

Step 6: Multiply the complex data S ₄ (i,j) obtained in Step 5 by the corresponding consistent compression factor Ω ₂ (i,j) to obtain coarse-focus complex data S ₅ (i,j).

(a) Obtain the shortest slant distance R _min according to the reference slant distance R _ref =19.31km, and the specific acquisition process is carried out according to formula (16);

(b) Obtain the distance frequency domain f _τ (j) corresponding to each column of the two-dimensional frequency domain complex data S ₄ (i,j), and the specific acquisition process is carried out according to formula (17);

(c) Combining formula (7) and formula (11) to obtain the two-dimensional frequency domain complex data S ₄ (i,j) corresponding to the azimuth wave number k _x (i) of each column and the range wave number k _rc ( j), the specific acquisition process is carried out according to formula (18);

(d) Obtain the two-dimensional uniform compression factor Ω ₂ (i, j) with a size of 65536×16384 by combining formulas (10) to (12), and the specific acquisition process is carried out according to formula (19);

(e) Obtain consistent compressed two-dimensional complex data S ₅ (i,j) in combination with formula (13), and the specific acquisition process is carried out according to formula (20);

Step 7: For the complex data S ₅ (i, j) obtained in Step 6, use the sinc interpolation method to perform stolt interpolation processing, and obtain the complex data S ₆ (i, j) mapped from the two-dimensional frequency domain to the two-dimensional wavenumber domain;

(a) According to the mapping relationship from the distance frequency domain to the distance wavenumber domain, the wavenumber k' _rc (i, j) in the distance wavenumber domain is obtained by combining formula (18), and the specific acquisition process is carried out according to formula (21);

(b) Traverse to obtain the maximum value k' _rc,max =69.12rad/s and the minimum value k' _rc,max =-105.06rad/s of the wavenumber k' _rc (i,j) in the distance wavenumber domain, and obtain the distance wavenumber domain The wave number equal interval Δk' _rc , the specific operation is carried out according to formula (22);

(c) Obtain the uniform distance wavenumber domain wavenumber of complex data in the two-dimensional wavenumber domain The specific operation is carried out according to formula (23);

(d) Obtain the uniform distance wavenumber domain wavenumber of the two-dimensional wavenumber domain complex data Position p(i,j) in uneven k' _rc (i,:) corresponding to each row. The method is to perform the following operations by row: first obtain the position p(1,1) of the first row and first column, and obtain the absolute difference n=[1,2,…,16384], to obtain the smallest absolute difference Δk _min and n of the corresponding position, if $k_{\mathrm{rc}}^e\left(1\right)<k_{\mathrm{rc}}^{\&prime;}(1,\mathrm{no}),$ p(1,1)=n-1, if $k_{\mathrm{rc}}^e\left(1\right)\&Greater\; Equal;k_{\mathrm{rc}}^{\&prime;}(1,\mathrm{no}),$ p(1,1)=n,

By analogy, each position p(i,j) is obtained.

(e) Combining the position p(i,j) obtained in the previous step, obtain the sampling point interval q(i,j,n) required for sinc interpolation, and the specific operation is carried out according to formula (24);

(f) Combining formula (18) with the sinc interpolation method, select the sinc interpolation kernel length as N=8, and obtain the two-dimensional complex data S ₆ (i,j) after stolt interpolation. Since the two-dimensional data is complex data, The real part S _{6_re} (i,j) and the imaginary part S _{6_im} (i,j) of S ₆ (i,j) need to be obtained by sinc interpolation respectively, and the specific operation is carried out according to formula (25) (26) .

Step 8: Perform Inverse Fast Fourier Transform (IFFT) on the complex data S ₆ (i,j) obtained in Step 7 along each distance direction (column), and obtain complex data S ₇ (i ,j), the specific operation is carried out according to formula (27);

Step 9: Multiply the complex data S ₇ (i,j) obtained in Step 8 by the geometric correction factor Ω ₃ (i,j) to obtain complex data S ₈ (i,j) after geometric correction;

(a) Combine formula (8) and formula (12) to obtain the geometric correction factor Ω ₃ (i,j), and the specific operation is carried out according to formula (28);

(b) Use formula (28) to obtain geometrically corrected complex data S ₈ (i, j), and the specific operation is carried out according to formula (29);

Step 10: Perform Inverse Fast Fourier Transform (IFFT) on the complex data S ₈ (i,j) obtained in Step 9 along each azimuth moment (by row) to obtain the final imaging result S _end (i,j), The specific operation is carried out according to formula (30);

After the imaging processing of the above steps, the final scene imaging result is obtained as shown in FIG. 3 . Table 2 shows the imaging evaluation results of the three point targets in the upper left, middle, and lower right of the scene. Figure 4, Figure 5, and Figure 6 respectively show the cross-sectional views of the three point targets in the upper left, middle, and lower right of the scene.

Table 2 Imaging evaluation results

According to the evaluation results in Table 2 and the cross-sectional diagrams shown in Figures 4 to 6, it can be concluded that: on the one hand, the imaging method can still focus accurately when the squint angle is 70 degrees, which shows that the method proposed by the present invention is not limited by the squint angle On the other hand, the imaging method can still accurately focus for 0.1m ultra-high resolution, indicating that the method proposed by the present invention can achieve accurate focus for various resolutions at present. Therefore, the method proposed in the present invention can realize accurate imaging of airborne high-resolution oblique-angle spotlight SAR based on a sliding receiving window, and obtain high-precision imaging results.

Claims (1)

1. an airborne high-resolution squint spotlight SAR imaging method based on a sliding receiving window, comprising the following steps: Step 1: read in the airborne high-resolution squint spotlight SAR two-dimensional original echo complex data S _start and the corresponding imaging parameters based on the sliding receiving window; S _start is a two-dimensional complex array with a size of N _a ×N _r . The imaging parameters include: sampling points N _a in azimuth direction, sampling points N _r in range direction, signal sampling rate f _s , signal bandwidth Bw, frequency modulation slope b, pulse Repetition frequency PRF, reference slant distance R _ref , Doppler center frequency fd ₀ , Doppler modulation frequency f _r0 , satellite velocity P _v , equivalent slant angle , signal wavelength λ, signal carrier frequency f ₀ , signal propagation speed c; Step 2: Perform azimuth dechirp processing on the two-dimensional original echo data S _start , which specifically includes the following steps: (a) Construct two one-dimensional sequences i, j, where i represents the azimuth sequence, and j represents the distance sequence; i=[1,2,…,N _a ] (2) j=[1,2,…,N _r ] (b) Obtain the azimuth time t(i) corresponding to each row of the two-dimensional original echo data S _start ; $\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; PRF</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ (c) Multiply the two-dimensional original echo complex data S _start with the derotation factor H ₁ to obtain the data S _{1_1} (i,j), where the factor H ₁ (i) is a one-dimensional complex number with a size of N _a ×1 group, the formula is: H ₁ (i)=exp{jπ(fr ₀ t ² (i)+2 fd ₀ t(i))} (4) Then the two-dimensional complex array S _{1_1} is obtained by the following formula: S _{1_1} (i,j)=S _start (i,j)·H ₁ (i) (5) (d) carry out fast Fourier transformation to complex data S _{1_1} along each distance direction, obtain complex data S _{1_2} ; S _{1_2} (:, j) = FFT(S _{1_1} (:, j)) (6) Wherein, S _{1_2} (:, j) represents the jth column of S _{1_2} , S _{1_1} (:, j) represents the jth column of S _{1_1} , and FFT( ) represents performing fast Fourier transform on a one-dimensional array; (e) obtaining the equivalent pulse repetition frequency PRF' after derotation; $\mathrm{<span\; class="notranslate">\; PR</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; fr</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{\mathrm{<span\; class="notranslate">\; PRF</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ (g) Combining formula (7) to obtain the azimuth time t ₁ (i) corresponding to each row of the two-dimensional complex data; ${\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; PRF</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ (h) Multiply the complex data S _{1_2} with the factor H ₂ to obtain the final azimuth dechirped data S ₁ ; where the factor H ₂ (i) is a one-dimensional complex array with a size of N _a ×1, its formula for: ${\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; j\&pi;</span>}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; fr</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$ Then the two-dimensional complex array S ₁ is obtained by the following formula: S ₁ (i,j)=S _{1_2} (i,j)·H ₂ (i) (10) Step 3: Perform fast Fourier transform on the complex data S ₁ (i,j) obtained in Step 2 along each distance direction to obtain complex data S ₂ (i,j) in the azimuth direction spectrum; S ₂ (:,j)=FFT(S ₁ (:,j)) (11) Among them, S ₂ (:, j) represents the j-th column of S ₂ , S ₁ (:, j) represents the j-th column of S ₁ , and FFT ( ) represents performing fast Fourier transform on a one-dimensional array; Step 4: Multiply the complex data S ₂ (i,j) obtained in Step 3 with the azimuth dechirp residual phase error compensation factor Ω ₁ (i) at the corresponding azimuth moment to obtain the compensated complex data S ₃ (i ,j); (a) Obtain the azimuth _frequency f a (i) corresponding to each row of the two-dimensional azimuth, frequency domain, distance and time domain complex data S ₂ (i,j) by combining formula (7); ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; PRF</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$ (b) Combining formula (12) to obtain a one-dimensional compensation factor Ω ₁ (i) whose size is N _a ×1; ${\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; j\&pi;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>\frac{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; fr</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$ (c) Obtain the compensated two-dimensional complex data S ₃ (i,j); S ₃ (i,j)=S ₂ (i,j)·Ω ₁ (i) (14) Step 5: Perform fast Fourier transform on the complex data S ₃ (i,j) obtained in Step 4 along each azimuth time to obtain two-dimensional spectrum complex data S ₄ (i,j); S ₄ (i,:) = FFT(S ₃ (i,:)) (15) Among them, S ₃ (i,:) represents the i-th row of S ₃ , and S ₄ (i,:) represents the i-th row of S ₄ ; Step 6: Multiply the complex data S ₄ (i,j) obtained in Step 5 by the corresponding consistent compression factor Ω ₂ (i,j) to obtain coarse-focus complex data S ₅ (i,j); (a) Obtain the shortest slant distance R _min according to the reference slant distance R _ref ; ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; ref</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; c</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; \&Center\; Dot;</span>\frac{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$ (b) Obtain the distance frequency domain f _τ (j) corresponding to each column of the two-dimensional frequency domain complex data S ₄ (i,j); ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; Nr</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$ (c) Combining formula (12) and formula (17) to obtain the two-dimensional frequency-domain complex data S ₄ (i, j) corresponding to the azimuth wave number k _x (i) of each row and the range wave number k _rc ( j); $\begin{array}{c}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; v</span>}}}\\ {\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$ (d) Obtain the two-dimensional uniform compression factor Ω ₂ (i,j) with the size of N _a ×N _r by combining formulas (16) to (18); (e) Obtain consistent compressed two-dimensional complex data S ₅ (i,j) by combining formula (19); S ₅ (i,j)=S ₄ (i,j)·Ω ₂ (i,j) (20) Step 7: For the complex data S ₅ (i, j) obtained in Step 6, use the sinc interpolation method to perform stolt interpolation processing, and obtain the complex data S ₆ (i, j) mapped from the two-dimensional frequency domain to the two-dimensional wavenumber domain; (a) According to the mapping relationship from the distance frequency domain to the distance wavenumber domain, the wavenumber k' _rc (i, j) in the distance wavenumber domain is obtained by combining formula (18); (b) traverse to obtain the maximum value k' _{rc ,max} and the minimum value k' _rc,max of the wave number k' rc (i,j) in the distance wave number domain, and obtain the equal interval Δk' _rc of the wave number in the distance wave number domain; ${\mathrm{<span\; class="notranslate">\; \&Delta;k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; max</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; min</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; two</span>}<span\; class="notranslate">\; )</span>$ (c) Obtain the uniform distance wavenumber domain wavenumber of the two-dimensional wavenumber domain data ${\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}\mathrm{<span\; class="notranslate">\; min</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; \&Delta;k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$ (d) Obtain the uniform distance wavenumber domain wavenumber of the two-dimensional wavenumber domain complex data position p(i,j) in the uneven k' _rc (i,:) corresponding to each row; Specifically: take the position p(1,1) of the first row and first column as an example, first obtain the absolute difference n=[1,2,…,N _r ], get the smallest absolute difference Δk _min and n of the corresponding position, if p(1,1)=n-1, if p(1,1)=n, similarly, and so on, get each position p(i,j); (e) Combining the position p(i,j) obtained in the previous step, obtain the sampling point interval q(i,j,n) required for sinc interpolation; $\mathrm{<span\; class="notranslate">\; q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; rc</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}{\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; four</span>}<span\; class="notranslate">\; )</span>$ (f) Combining formula (24) with the sinc interpolation method to obtain the two-dimensional data S ₆ after stolt interpolation, since the two-dimensional data is complex data, the real part S _{6_re (} i, j) and the imaginary part S _{6_im} (i, j) are obtained by sinc interpolation method respectively; Among them, N is the interpolation kernel length, sinc( ) refers to the interpolation function S _{5_re} (i, j) refers to the real part of the i-th row and j-column of the two-dimensional data S _{5_re} , and S _{5_im} (i, j) refers to the imaginary part of the i-th row and j-column of the two-dimensional data S _{5_im} ; Step 8: Perform inverse fast Fourier transform on the complex data S ₆ (i,j) obtained in Step 7 along each distance direction to obtain complex data S ₇ (i,j) in the azimuth time domain and distance wavenumber domain; S ₇ (:, j) = IFFT (S ₆ (:, j)) (27) Among them, S ₆ (:, j) represents the j-th column of S ₆ , S ₇ (:, j) represents the j-th column of S ₇ , and IFFT ( ) represents performing an inverse fast Fourier transform on a one-dimensional array; Step 9: Multiply the complex data S ₇ (i,j) obtained in Step 8 by the geometric correction factor Ω ₃ (i,j) to obtain complex data S ₈ (i,j) after geometric correction; (a) Combine formula (8) and formula (12) to obtain geometric correction factor Ω ₄ (i,j); ${\mathrm{<span\; class="notranslate">\; \&Omega;</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j</span>}\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>\frac{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; fd</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; c</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 28</span>}<span\; class="notranslate">\; )</span>$ (b) Use formula (30) to obtain geometrically corrected complex data S ₈ (i,j); S ₈ (i,j)=S ₇ (i,j)·Ω ₃ (i,j) (29) Step 10: Perform inverse Fast Fourier Transform on the complex data S ₈ (i,j) obtained in Step 9 along each azimuth moment to obtain the final imaging result S _end (i,j); S _end (i,:)＝IFFT(S ₈ (i,:)) (30) Wherein, S ₈ (i,:) represents the i-th row of S ₈ , and S _end (i,:) represents the i-th row of S _end .