CN118011342A - Error calibration method for space target imaging system of foundation step frequency radar - Google Patents

Abstract

The invention relates to a method for calibrating errors of a space target imaging system by a ground-based step frequency radar, and belongs to the technical field of radar imaging. Firstly, a complex coupling error echo signal model of a system error and a motion error is established according to the characteristics of a stepping frequency chirp radar and high-speed motion. The errors are then decoupled into sub-band distance imaging errors, sub-band azimuth imaging errors, and full sub-band composite imaging errors, depending on the effect of the error pattern on the different processing steps. Then solving the gradient solution of the image entropy to the error after the sub-band distance imaging, the sub-band azimuth imaging and the full-sub-band synthetic imaging, and respectively solving the error by using an adaptive moment estimation method. And finally, carrying out cyclic processing on the steps, iterating the calibration error for a plurality of times, and obtaining a high-quality image.

Description

A method for error calibration of ground-based stepped-frequency radar imaging system for space targets

Technical Field

The invention relates to a method for calibrating errors of a ground-based stepped-frequency radar on a space target imaging system, and belongs to the technical field of radar imaging.

Background technique

Ground-based radar is an effective means of detecting and imaging air and space targets. Improving resolution can obtain more target details, which is convenient for classification and identification. To achieve high resolution, the radar needs to be able to transmit large-bandwidth signals. If an instantaneous large-bandwidth signal is transmitted, according to the Nyquist sampling theorem, a higher sampling rate must be achieved to match it, which is a huge challenge for system hardware such as analog-to-digital conversion and direct digital synthesis. Frequency-stepped signals have the advantages of instantaneous narrowband and synthetic broadband, and have been widely used in the radar field.

The frequency-stepped chirp signal has the advantages of both frequency-stepped and chirp signals. On the one hand, the difficulty of the system transmitting a large-bandwidth signal is reduced through the form of carrier frequency stepping. On the other hand, each sub-pulse has a large time-bandwidth product, which balances the contradiction between the effective range and the range resolution.

However, the fuzzy function property of the frequency-modulated stepped frequency signal determines that grating lobes are very likely to be generated when obtaining high-resolution range images. There are two main ways to suppress grating lobes: first, design appropriate waveform parameters to avoid information loss and redundancy in the synthetic broadband; second, correct various errors in the signal to achieve perfect coherent synthesis of multiple sub-band signals. The second point is more important for data processing.

Stepped frequency radar needs to use different local oscillators and sub-links to transmit different sub-pulses, and it is difficult to achieve strict consistency in time and phase, so there will be systematic errors. The applicability of the method based on internal calibration is limited, and the observation of space targets usually does not have isolated strong points, so it is necessary to study self-calibration technology based on radar data.

At the same time, the stepped frequency chirp radar synthesizes a large bandwidth signal through multiple sub-pulses, that is, the repetition frequency of the burst is smaller than the repetition frequency of the sub-pulses. For high-speed space targets, it is necessary to consider not only the motion error between bursts, but also the motion error between sub-pulses and within sub-pulses caused by high-speed motion. However, there is little research on the coupling error of system error and target motion error in practical applications.

In summary, there is currently no algorithm that can calibrate this complex coupling error to obtain better images. It is necessary to study a technology for high-speed space target imaging and coupling error self-calibration based on minimum entropy stepped frequency chirp radar.

Summary of the invention

In view of this, the present invention proposes a method for calibrating the error of a ground-based stepped-frequency radar for a space target imaging system, which can obtain high-precision space target imaging results.

The technical solution of the present invention is:

A method for calibrating the error of a ground-based stepped frequency radar imaging system for a space target, the method comprising the steps of:

The first step is to establish a coupling error echo signal model. The coupling error echo signal model includes four errors, namely, system phase error, initial sampling distance error, quadratic phase error within the pulse, and first and zeroth order phase errors caused by motion.

In the second step, the four errors in the coupled error echo signal model established in the first step are decoupled into three errors, namely, sub-band range imaging error, sub-band azimuth imaging error and full sub-band synthetic imaging error;

The third step is to perform sub-band range imaging on the space target echo data collected by the ground-based stepped frequency radar, and obtain the gradient of the image entropy after sub-band range imaging to the sub-band range imaging error obtained in the second step, and then use the adaptive moment estimation method to solve the sub-band range imaging error according to the obtained gradient;

The fourth step is to perform sub-band azimuth imaging on the data after sub-band range imaging, and obtain the gradient of the image entropy after sub-band azimuth imaging to the sub-band azimuth imaging error obtained in the second step, and then use the adaptive moment estimation method to solve the sub-band azimuth imaging error according to the obtained gradient;

The fifth step is to perform full sub-band synthetic imaging on the data after sub-band azimuth imaging, and obtain the gradient of the image entropy after full sub-band synthetic imaging to the full sub-band synthetic imaging error obtained in the second step, and then use the adaptive moment estimation method to solve the full sub-band synthetic imaging error according to the obtained gradient;

Step 6: Repeat steps 3 to 5 until the image is focused and the final image is obtained.

In the first step, the coupling error echo signal model established is:

Where _Aq (·) is the amplitude error, is the system phase error of sampling number n,/> is the starting sampling distance error of sampling number n,/> is the quadratic phase error in the pulse with sampling number n,/> The first and zeroth order phase errors caused by the motion with sampling number n;/> is the fast frequency, _tn is the slow time with sampling number n, n = 1, 2, 3, ..., N, N is the azimuth sampling number;

In the second step, the sub-band range imaging error is:

in, is the amplitude error,/> For / > The relevant polynomial coefficients;

The sub-band azimuth imaging error is:

Wherein, _fq is the carrier frequency of the qth sub-pulse; q=1, 2, 3, ..., Q, Q is the total number of sub-bands; p=1, 2, 3, ..., P, P is the polynomial order of the error; are the polynomial coefficients related to t _n ;

The full sub-band synthetic imaging error is:

in, is a constant amplitude error, p = 0 or 1,/> For / > The relevant polynomial coefficients;

In the third step, the formula for obtaining the gradient is:

Among them, X _1,q is the image after sub-band range imaging, En(X _1,q ) is the image entropy of the image after sub-band range imaging, is the error parameter, /> is the error parameter, S(X _1,q ) is the total energy of the image after sub-band range imaging, t _q is the azimuth sampling time corresponding to the qth sub-pulse, /> is the fast time, x _1,q is the sub-element of the image after sub-band range imaging; is the inverse Fourier transform, /> is the data before sub-band range imaging;

In the fourth step, the formula for obtaining the gradient is:

Among them, X _2,q is the image after sub-band azimuth imaging, En(X _2,q ) is the image entropy of the image after sub-band azimuth imaging, S(X _2,q ) is the total energy of the image after sub-band azimuth imaging, is the azimuth slow frequency, x2 _,q is the sub-element of the image after sub-band azimuth imaging, /> is the Fourier transform, /> It is the data before sub-band azimuth imaging;

In the fifth step, the formula for obtaining the gradient is:

Among them, X _3,q is the image after full subband synthesis imaging, En(X _3,q ) is the image entropy after full subband synthesis imaging, S(X _3,q ) is the total energy of the image after full subband synthesis imaging, This is the data before full sub-band synthesis imaging;

The update formula of the adaptive moment estimation method is:

Where e is the parameter to be estimated, H(e) is the gradient, m ₁ and m ₂ are the first-order and second-order moment estimates, respectively, k is the number of iterations, β ₁ , β ₂ and α are adjustable parameters, and ε>0.

Beneficial Effects

The present invention accurately models the coupling error between the stepped frequency radar system and the high-speed moving target, and can realize high-resolution imaging of high-speed space targets. The present invention decomposes the coupling error into sub-band range imaging error, sub-band azimuth imaging error and full sub-band synthetic imaging error, obtains the gradient of the entropy for the error in each step in a step-by-step solution, combines the adaptive moment estimation method to obtain the error and compensate, and realizes the self-calibration of the coupling error. The frequency step signal has the advantages of instantaneous narrowband and synthetic broadband, can image space targets with high resolution, and has been widely used in the radar field. But the frequency step frequency observation. However, the echo in the observation of the space target by the frequency step radar contains the error formed by the coupling of the system error and the target high-speed motion error, and the traditional method is difficult to image. Therefore, the present invention proposes a self-calibration technology for high-speed space target imaging and coupling error of the stepped frequency chirp radar based on minimum entropy. First, according to the characteristics of the stepped frequency chirp radar and high-speed motion, a complex coupling error echo signal model of the system error and motion error is established. Then, according to the influence of the error form on different processing steps, the error is decoupled into sub-band range imaging error, sub-band azimuth imaging error and full sub-band synthetic imaging error. Then, the gradient of the image entropy error after sub-band range imaging, sub-band azimuth imaging and full sub-band synthetic imaging is solved, and the error is solved respectively using the adaptive moment estimation method. Finally, the above steps are processed in a loop, and the error is calibrated iteratively for multiple times to obtain high-quality images. The proposed method aims to provide a robust high-resolution imaging algorithm for high-speed space targets using stepped-frequency chirp radar, which is expected to be applied to fields such as ground-based radar observation of space targets.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a frequency domain representation of the error;

Figure 2 is a time domain representation of the error;

FIG3 is a flow chart of a coupling error self-calibration algorithm;

FIG4 is a satellite dot array simulation diagram, wherein FIG4(a) is a simulated point target, FIG4(b) is an imaging result without error correction, FIG4(c) is an imaging result with only average range correction, FIG4(d) is an imaging result without corrected sub-band errors, and FIG4(e) is an imaging result of this method;

FIG5 is a diagram showing the change of entropy along the processing flow.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

Example

Each frame of the frequency-stepped chirp signal consists of a group of Q sub-pulses with linear carrier frequency jumps. Assuming that the starting carrier frequency is f ₀ and the step amount is Δf, the carrier frequency of the qth sub-pulse is f _q ＝f ₀ +qΔf, q＝0,1,...,Q-1. The expression of the transmitted signal with f _q as the carrier frequency is:

here For quick time, /> is the chirp baseband signal, T _pulse is the pulse width, γ is the frequency modulation slope, rect(·) is the rectangular window function, and the sub-pulse bandwidth/> Since the spectra of different sub-pulses overlap, the maximum synthesized bandwidth that can be obtained for a frame of signal is B=B ₀ +(Q-1)Δf.

If the target delay corresponding to the n=gQ+qth sub-pulse in azimuth is is the frame index, t _n = _nTPRT , _TPRT is the repetition time of the sub-pulse. Since f _n = f _q , the received echo is

σ is the complex scattering intensity of the target. After matched filtering, its spectrum is

Here (·) ^* is a conjugate operator. It can be seen that after the matched filtering, only the first-order phase and the zero-order phase remain. The coupling error between the target motion and the system error is explained below.

Error modeling: In actual radar systems, the amplitude-frequency and phase-frequency characteristics of radar systems are distorted due to the unsatisfactory characteristics of components such as amplifiers and filters. This error can be modeled as a polynomial. At the same time, since the system works in frequency hopping mode, the local oscillator and link used when transmitting sub-pulses of different carrier frequencies are different, so its channel characteristics are related to the carrier frequency index q. At the same time, there will be an error between the actual value and the nominal value of the frequency source of the radar system's transmission and trigger signals, so there will be an error between the design value and the actual starting sampling time, which is a first-order and constant phase error.

For motion error, due to high-speed moving targets, motion will introduce phase into the pulse. Set the direction away from the radar as positive. Since the target motion is usually continuously changing, the echo form becomes:

Here is the scaling factor caused by high-speed motion, /> The reason for the approximation is that the intra-pulse changes caused by target motion are small during the radar observation time, so the same value can be used for intra-pulse phase compensation. The spectrum after matched filtering is

Since the speed is changing, both ρ and the delay τ here are related to the slow time t _n . High-speed motion will cause spectral shift and amplitude scaling, but they are usually negligible. The secondary phase of will cause HRRP waveform distortion and affect the quality of two-dimensional imaging. Therefore, it is the main compensation quantity for the intra-pulse error. The second and third terms are both related to _fq . The first and zeroth phase errors will not cause waveform distortion.

After the above analysis, the system error of the step frequency and the motion error of the target will form a complex coupling error, which is expressed as

Where _Aq (·) is the amplitude error, They represent the system phase error, the initial sampling distance error, the quadratic phase error within the pulse, and the first-order and zero-order phase errors caused by motion.

Figure 1 shows the phase-frequency and amplitude-frequency diagrams of broadband synthesis using two sub-pulses. In Figure 1(a), the ideal phase splicing result should be continuous and have the same slope, without second-order and higher-order phase errors. The first sub-pulse is affected by second-order and higher-order phase errors, and its phase is nonlinear. The second sub-pulse contains first-order and zero-order phase errors, and the phase slope changes. At the same time, a phase jump occurs during splicing. In Figure 1(b), the amplitude spectrum of the first sub-pulse is ideal, while the second sub-pulse has a polynomial amplitude error, so broadband synthesis cannot be achieved.

Figure 2 shows the ideal one-dimensional range image and the one-dimensional range image with errors. In Figure 2(a), the dotted line and the solid line represent the one-dimensional range image of a low-resolution single subband without errors and the one-dimensional range image of the perfect synthesis of two subbands, respectively. It can be seen that the main lobe of the dotted line is narrower, indicating that broadband synthesis can improve the resolution. Figure 2(b) is the one-dimensional range image of the first subband with errors. Even-order phase errors of the second order and above will cause symmetrical sidelobe lift, and odd-order phases will cause sidelobe asymmetry. Figure 2(c) is the one-dimensional range image of the second subband with errors. The amplitude error causes symmetrical distortion of the sidelobes, and the first-order phase error causes the peak position to move. Figure 2(d) is the one-dimensional range image obtained by synthesizing the two subband spectra of Figures 2(b) and 2(c). Due to the existence of multiple errors, it is obviously far from the high-resolution one-dimensional range image in Figure 2(a). At the same time, it can be seen that regular grating lobes will appear at specific positions, which is determined by the fuzzy function properties of the stepped frequency chirp signal. Therefore, the key to obtaining ideal imaging results with stepped frequency chirp signals is to correct various errors.

Error decoupling and self-calibration: According to the previous description, the relationship between the qth subband echo signal with error and the two-dimensional image is modeled as

_Yq ＝ _Eq⊙AXqB + _N

Where A and B are the range Fourier transform and azimuth inverse Fourier transform matrices, _Yq and _Xq are the echo and image, ⊙ is the Hadamard product operator, N is the noise, and _Eq is the coupling error, which can be expressed as:

It is worth noting that _Xq has a higher signal-to-noise ratio after two-dimensional accumulation. The use of stepped frequency chirp signals to achieve high-resolution in the range direction can be considered as obtaining a larger bandwidth through splicing of multiple sub-band spectra to achieve high-resolution imaging, that is, Here (·) ^H is the Hermitian operator.

Image entropy is an important indicator for image quality assessment. The smaller its value, the better the image quality. It is defined as

The total energy of the image is is a constant. The imaging problem can be modeled as

The computational cost of the search-based method has an exponential relationship with the number of parameters. Therefore, it is necessary to decouple the errors and simplify the amount of parameter calculation each time.

Sub-band range imaging error: within each sub-band The secondary and above phase errors and the primary and above amplitude errors will affect the waveform distortion after pulse compression. These errors do not change with t _n . All the data of each sub-band can be used for error estimation and correction, that is, the optimization problem is:

Sub-band azimuth imaging error

Both target motion and system errors can cause This coupling error is difficult to solve directly. Although there are many similar envelope alignment methods commonly used in ISAR, on the one hand, these methods are simpler than the coupling error model considered in this paper. On the other hand, the compensation accuracy of envelope alignment is difficult to achieve the phase accuracy required for synthesizing high-resolution images of stepped frequency chirp radar. Two-dimensional images have higher signal-to-noise ratio and better entropy performance. In addition, each sub-band has more precise error calibration requirements in two-dimensional imaging. Therefore, the objective function can be modeled as

Here includes the error that changes with t _n , that is,

Full subband synthesis imaging error: still exists after the first two steps of error correction The non-time-varying first-order and zero-order phases of the directly synthesized high-resolution image will affect the accumulated gain, causing the appearance of grating lobes and image degradation. This error is called synthetic imaging error. Therefore, the optimization problem can be modeled as

Here Since the time-varying error has been corrected before, only the time-invariant error remains. The first-order and zero-order phases can therefore be simplified. The error estimated in this step is expressed as

It is worth mentioning that It can be regarded as a time-invariant error, that is, it is contained in/> Inside.

After three steps of error decoupling, the final error estimation result can be obtained as

The previously modeled motion errors and system errors are included and decoupled in three different processing steps. The image after the three-step correction is represented as

In order to obtain the decoupled error and compensate it, the most direct method is to perform parameter traversal search. Although the proposed decoupling method can reduce the parameter dimension, it still requires a large amount of calculation. Obviously, it is not realistic to directly solve the analytical solution. Therefore, this paper uses the Adam algorithm with a faster convergence rate to solve it. The gradient is expressed as follows

For each of the three steps, the solution can be completed using adaptive moment estimation (Adam). The update formula of Adam is:

Where e is the parameter to be estimated, m ₁ and m ₂ are the first-order and second-order moment estimates respectively, k is the number of iterations, β ₁ , β ₂ and α are adjustable parameters, ε>0, Adam applies the momentum method and can adaptively adjust the gradient descent rate.

Figure 3 shows the corresponding relationship between the error after decoupling and the correction algorithm, and also provides an intuitive graphical display.

1) Sub-band pulse pressure error estimation: This step utilizes the influence of the second and above phase-frequency response and the first and above amplitude response on pulse compression, performs unified processing on each sub-band, and corrects the consistent non-time-varying error within the sub-band. After this step, the distortion-free range image of all sub-bands is obtained;

2) Sub-band 2D imaging error estimation: This step utilizes the influence of time-varying motion error on each sub-band data, and simultaneously corrects the time-varying envelope and phase errors shared by all sub-bands through sub-band 2D imaging. After this step, a distortion-free low-resolution 2D image of each sub-band is obtained;

3) Synthesis error correction: This step uses the effect of residual error on broadband synthesis, that is, coherent accumulation. Entropy, as an image evaluation criterion, can also measure the accumulation effect. A smaller entropy value indicates that multiple sub-band images have achieved better accumulation. After this step, a high-resolution two-dimensional image is obtained.

It is worth noting that the proposed technology is also applicable to stepped frequency chirp radar data with only a few frames. Even if the target motion cannot form a synthetic aperture, entropy can still measure the coherent accumulation effect, thereby achieving self-calibration of coupling errors. At the same time, the error model is also applicable to other forms, such as sinusoidal errors, which can also be solved using the technology in this article.

Example

A satellite electron with 64 points is set up by computer simulation as shown in Figure 4(a). The motion parameters are set so that the rotation can form a synthetic aperture and two-dimensional imaging can be performed. At the same time, additive Gaussian white noise is added to the image to keep the signal-to-noise ratio at 0dB. The simulated radar parameters are shown in Table 1.

Table 1 Simulation experiment parameters

Radar parameters Numeric unit Starting center frequency 3.5 GHz bandwidth 250 MHz Frequency step 200 MHz Number of subbands 4 indivual Pulse repetition frequency 800 Hz Pulse Width 6.25 us Target distance 482 Km Target speed 1400 m/s Acceleration 100 m/s ²

Table 2 Evaluation of simulation experiment test results

Figure No. Figure 4(c) Figure 4(c) Figure 4(d) Figure 4(e) entropy 5.3103 5.6782 5.7091 4.5182

Figure 4(b) does not correct the error within the sub-band, and the range grating lobe is obvious. Figure 4(c) only performs a simple average range image correction, and its motion compensation accuracy cannot meet the requirements of spectrum synthesis, resulting in image quality degradation. Figure 4(d) does not calibrate the error between sub-bands, and the image range produces a widening of the main lobe. Figure 4(e) is the algorithm proposed in this paper. It can be seen that each point target is well focused. At the same time, in Table 2, the entropy value of Figure 4(e) is the smallest.

The algorithm proposed in this paper is a step-by-step error decoupling, so the entropy value of the processing process of Figure 4(e) is calculated, and the curve is depicted in Figure 5. As shown in the figure, the pulse compression error calibration is first performed, and the entropy value of the sub-band range image after pulse compression is reduced. Then two-dimensional imaging is performed. As the motion error is calibrated in the iteration, the entropy value becomes smaller and smaller. It can be seen that the motion error usually causes a large entropy change. Then the image synthesis is performed, and the error is calibrated in the synthesis, and the entropy value is further reduced. After three steps of error estimation and calibration based on minimum entropy, a better imaging result is finally obtained.

In summary, the above are only preferred embodiments of the present invention and are not intended to limit the protection scope of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims (9)

1. A method for calibrating errors of a space target imaging system by a ground-based step frequency radar is characterized by comprising the following steps:

the method comprises the steps of firstly, establishing a coupling error echo signal model, wherein the coupling error echo signal model comprises four errors, and the four errors are respectively a system phase error, an initial sampling distance error, a secondary phase error in a pulse and a primary and zero phase error caused by motion;

Step two, the four errors in the coupling error echo signal model established in the step one are decoupled into three errors, wherein the three errors are respectively a sub-band distance imaging error, a sub-band azimuth imaging error and a full sub-band synthetic imaging error;

Thirdly, carrying out sub-band distance imaging on the space target echo data acquired by the foundation step frequency radar, solving the gradient of the sub-band distance imaging error obtained in the second step by the image entropy after the sub-band distance imaging, and then solving the sub-band distance imaging error according to the solved gradient by using an adaptive moment estimation method;

fourth, carrying out sub-band azimuth imaging on the data after sub-band distance imaging, solving the gradient of the image entropy after sub-band azimuth imaging on the sub-band azimuth imaging error obtained in the second step, and then solving the sub-band azimuth imaging error according to the solved gradient by using an adaptive moment estimation method;

fifthly, carrying out full-subband synthesis imaging on the data after subband azimuth imaging, solving the gradient of the image entropy after full-subband synthesis imaging on the full-subband synthesis imaging error obtained in the second step, and then solving the full-subband synthesis imaging error according to the solved gradient by using a self-adaptive moment estimation method;

And sixthly, repeating the third step to the fifth step until the image is focused, and obtaining a final image.

2. The method for calibrating the error of a space target imaging system by using the ground-based step-frequency radar according to claim 1, wherein the method comprises the following steps:

in the first step, the established coupling error echo signal model is:

wherein A _q (·) is the amplitude error, For a systematic phase error with sample number n,/>For a starting sampling distance error with a sampling number n,/>Is the secondary phase error in the pulse with the sampling sequence number of n,/>A primary phase error and a zero phase error brought by the motion with the sampling sequence number of n; /(I)For fast frequency, t _n is the slow time with sample number N, n=1, 2,3, …, N is the azimuth sample number.

3. A method for calibrating error of a space target imaging system by using a ground-based step-frequency radar according to claim 1 or 2, wherein:

In the second step, the subband distance imaging error is:

Wherein, For amplitude error,/>Is AND/>The associated polynomial coefficients.

4. A method for calibrating an error of a space target imaging system by a ground-based step-frequency radar according to claim 3, wherein:

The subband azimuth imaging error is:

wherein f _q is the carrier frequency of the q-th sub-pulse; q=1, 2,3, …, Q is the total number of subbands; p=1, 2,3, …, P is the polynomial order of the error; Is the polynomial coefficient associated with t _n.

5. The method for calibrating the error of a space target imaging system by using the ground-based step-frequency radar according to claim 4, wherein the method comprises the following steps:

The full subband synthesis imaging error is:

Wherein, For constant amplitude error, p=0 or 1,/>Is AND/>The associated polynomial coefficients.

6. The method for calibrating the error of a space target imaging system by using the ground-based step-frequency radar according to claim 1, wherein the method comprises the following steps:

in the third step, the formula for obtaining the gradient is as follows:

Wherein X _1,q is the sub-band distance imaged image, en (X _1,q) is the image entropy of the sub-band distance imaged image, Is an error parameter,/>S (X _1,q) is the total energy of the image after sub-band distance imaging, t _q is the azimuth sampling time corresponding to the qth sub-pulse, and is used as an error parameterFor fast time, x _1,q is the subelement of the image after sub-band distance imaging; is inverse Fourier transform,/> Data before imaging for the subband distance.

7. The method for calibrating the error of a space target imaging system by using the ground-based step-frequency radar according to claim 6, wherein the method comprises the following steps:

in the fourth step, the formula for obtaining the gradient is:

Wherein X _2,q is the image after sub-band azimuth imaging, en (X _2,q) is the image entropy of the image after sub-band azimuth imaging, S (X _2,q) is the total energy of the image after sub-band azimuth imaging, For slow azimuth frequencies, x _2,q is a sub-element of the image after sub-band azimuth imaging,Is Fourier transform,/>Data prior to imaging for the subband azimuth.

8. The method for calibrating the error of a space target imaging system by using the ground-based step-frequency radar according to claim 7, wherein the method comprises the following steps:

in the fifth step, the formula for obtaining the gradient is:

Wherein X _3,q is the full-subband synthesized imaged image, en (X _3,q) is the image entropy of the full-subband synthesized imaged image, S (X _3,q) is the total energy of the full-subband synthesized imaged image, The pre-imaging data is synthesized for the full sub-band.

9. The method for calibrating the error of a space target imaging system by using the ground-based step-frequency radar according to claim 1, wherein the method comprises the following steps:

The updating formula of the self-adaptive moment estimation method is as follows:

Wherein e is a parameter to be estimated, H (e) is a gradient, m ₁ and m ₂ are first-order and second-order moment estimates, k is the number of iterations, β ₁、β₂ and α are adjustable parameters, ε >0.