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Noiserobust range alignment method for inverse synthetic aperture radar based on aperture segmentation and average range profile correlation
EURASIP Journal on Advances in Signal Processing volume 2021, Article number: 4 (2021)
Abstract
Range alignment is an essential procedure in the translation motion compensation of inverse synthetic aperture radar imaging. Global optimization or maximumcorrelationbased algorithms have been used to realize range alignment. However, it is still challenging to achieve range alignment in low signaltonoise ratio scenarios, which are common in inverse synthetic aperture radar imaging. In this paper, a novel antinoise range alignment approach is proposed. In this new method, the target motion is modeled as a uniformly accelerated motion during a short subaperture time. Minimum entropy optimization is implemented to estimate the motion parameters in each subaperture. These estimated parameters can be used to align the profiles of the current subaperture. Once the range profiles of each subaperture are aligned, the noncoherent accumulation gain is obtained by averaging all profiles in each subaperture, which can be used as valuable information. The accumulation and correlation method is applied to align the average range profiles of each subaperture because the former step focuses mainly on alignment within the subapertures. Experimental results based on simulated and real measured data demonstrate the effectiveness of the proposed algorithm in low signaltonoise ratio scenarios.
Introduction
Inverse synthetic aperture radar (ISAR), which is used to obtain images of noncooperative and moving targets, has been widely applied in many civil and military domains in the last few decades [1–5]. ISAR achieves high resolutions in both the range and azimuth directions by exploiting the wideband characteristics and angular diversity during the coherent processing interval. During the process of imaging, a target’s movement can be divided into two parts: translational and rotational motion. It is well known that only the rotational motion contributes to imaging, while translational motion can cause blurring of the ISAR images and therefore must be compensated for. Range alignment and autofocusing constitute the basic workflow of translational motion compensation. The range alignment procedure is the prerequisite for the subsequent finetuned phase compensation, which ensures that the amount of energy of the same scatters are in the same range bins. Wellaligned range profiles are the foundation of a focused ISAR image [6–9]. This article focuses mainly on a novel and effective method for range alignment.
The traditional range alignment methods can be sorted into three categories. The first is maximumcorrelationbased methods, which maximize the correlation function between adjacent profiles or between the current profile and a template one. A representative one was discussed in detail in [1]. To improve the performance of these methods in low SNR scenarios, [10] proposed that the average of aligned profiles can be used as a reference. Xue et al. [11] demonstrated a highorder symmetric accumulated crosscorrelation method (HSACM) to achieve a high realtime computation speed. In extremely low SNR scenarios, however, these methods can be impacted by noise, and misalignment errors accumulate during the coherent processing interval (CPI). The second category of methods is based on the dominant scatters of targets [12, 13]. Although these methods have high computational efficiency, they are sensitive to angular glint and noise, and they have difficulty in tracking strong scatters in real situations. The third category of methods is optimizationbased and uses several global metrics, such as entropy ([14, 15]) and contrast ([16, 17]), as references to iteratively compensate for misalignment. However, as mentioned in [18], the synthetic profiles in the global methods are equal to the noncoherent accumulation, and under a low SNR, the SNR gain from the noncoherent integrant is not sufficient to overcome the strong noise, and as a result, the metrics cannot fully indicate the quality of the range alignment. In short, the current range alignment methods suffer performance reductions in low SNR situations, and it is still challenging to propose an antinoise range alignment method.
By referring to [19] and [20], it is likely that ISAR signals encounter interference from strong noises because of the long observation distance and low transmitting energy. For example, when the target of interest is a highorbit satellite, the SNR of echoes can be very low. It has been concluded that if the required SNR decreases by 1 dB, the surveillance range will increase by 8%. Therefore, it is essential to implement algorithms with high robustness under low SNRs. Methods such as those by [18] and [19] model the total translation motion as a highorder polynomial and use coordinate descent or a particle swarm algorithm to carry out the optimization. According to [21] and [22], the plain coordinate descent used in [18] can easily be trapped in a local optimum when the cost function is nonconvex. The particle swarm algorithm is heuristic, which means a relatively complex parameter setting scheme is required. In addition, the above algorithms jointly solve range alignment and phase compensation, but none of them is specifically aimed at range alignment. Furthermore, joint compensation demands higher accuracy (range, 1/4 ~1/8 range unit; phase, wavelength/8) than that of the separated methods, and joint compensation requires a highorder model, which represents a high computational cost.
To improve the performance of the range alignment methods in low SNR scenarios and overcome the shortcomings of the above algorithms, the proposed method is designed with the following three steps. First, the target motion information is considered. In a relatively short observation time, it is reasonable to model the target motion as uniform acceleration. However, in a real situation, the CPI is always long. Therefore, we split the full aperture into several subapertures to achieve a short observation time. Extra information is of benefit to suppress strong noise. In each subaperture, the minimum entropy principle is implemented to estimate the motion parameters and obtain the envelope deviations of all profiles in the current subaperture. As an improvement to [18], the coordinate descent algorithm (CDA) with proximal point updating is used to obtain better convergent properties. Second, after optimization and compensation in each subaperture, noncoherent accumulation gain can be realized through the average of the profiles in the subaperture, and we then use a maximumcorrelationbased algorithm to align the average range profiles of every subaperture and obtain the envelope deviation between every two subapertures. This step is necessary because optimization in each subaperture mainly focuses on the alignment of the current subaperture. After the former two steps, the envelope deviations of all profiles in one subaperture and the deviations of the average range profiles of every subaperture, i.e., the error within and between subapertures, are obtained. Finally, locally weighted regression is applied to fit the total envelope deviations to eliminate the step change between every two subapertures. In general, the reason that the proposed method has enhanced performance in low SNR scenarios is that the target motion information and global optimization are used, and in addition, the noncoherent accumulation gain is adopted.
The main contributions of this article can be summarized as follows:

1)
A novel range alignment approach is proposed with consideration of the target motion, and the global optimization and maximumcorrelationbased methods are combined to achieve high performance in low SNR scenarios.

2)
The CDA with a proximal update scheme is used to achieve stability. Furthermore, in each iteration of the CDA, the LevenbergMarquardt (LM) method is implemented as the solver. The LM method, which blends the advantages of the gradient descent and Newton’s method, can achieve a better robustness and a faster rate of convergence.
The remainder of this paper is organized as follows. In Section 2, the signal model is introduced, and related works, such as the maximumcorrelationbased algorithm (MCA) and minimization of the entropy of the average range profile (MEARP) [7], are briefly summarized for comparison. In Section 3, the threestep procedure of the proposed algorithm is discussed in detail. Section 4 presents experimental results based on simulated and real measured data to verify the effectiveness of the proposed algorithm. Some conclusions are presented in Section 5.
Signal model and related works
In this section, the models of transmitted and received signals obtained using the dechirping technique are briefly deduced, and the formula of envelope deviation compensation is demonstrated. For comparison, a barebones introduction to the related works, including the MCA [1] and MEARP [7], is presented.
Signal model
In ISAR imaging, the linear frequency modulation (LFM) signal is often transmitted, which can be expressed as
where rect(·) is the rectangular function \(\left (rect\left (\tau \right) = \left \{ \begin {array}{l} 1,\tau  \le \frac {1}{2}\\ 0,\tau  > \frac {1}{2} \end {array} \right.\right)\). T_{p},f_{c},γ, and \(\hat {t}\) represent the pulse duration, carrier frequency, chirp rate, and fast time, respectively.
After applying the wellestablished dechirping technique, including residual video phase (RVP) term elimination, and the fast Fourier transform (FFT) operations to the fast time, highresolution range profiles (HRRPs) can be obtained:
where A, f, and ΔR denote the amplitude modulation, frequency variable, and radial range difference between the scatter and reference point, respectively. t_{s} is the slow time with 0≤t_{s}≤T_{a}, where T_{a} is the CPI [18], and the variation in t_{s} is reflected by ΔR:
where R_{0} is the radial distance between the scatter and the radar, R_{ref} is that between the reference point and the radar, d is the distance between the scatter and the reference point, and Ω represents the angular velocity of the target.
The range misalignment, due to the translation motion of the target, can be reflected by the envelopes of the HRRPs under the dechirping model. The peak of the sinc function in (2), which is located at f=−2γΔR/c, denotes the range information of the scatter. However, because of the variation in ΔR during the CPI, the peaks exist in different range bins among all sweeps of the radar, resulting in the defocusing of the scatter energy in the final ISAR image.
To achieve range alignment, the misaligned envelopes of the HRRPs should be shifted along the direction of fast time. According to the shift property of the Fourier transform, alignment can be accomplished in the time domain as follows:
where \(\tilde {s}\left ({{t_{s}},\hat {t}} \right)\) is the compensated signal echo and Φ(t_{s}) represents the envelope deviations of different sweeps.
Related works
The two main approaches to achieving range alignment are the MCA and MEARP. In this subsection, both are briefly introduced.
One of the widely used maximumcorrelationbased algorithms is the accumulation and correlation method (ACM), which uses the average of already aligned envelopes as the template (reference envelope) and maximizes the correlation function between the template and the next unaligned envelope. Its formula is as follows:
where x_{i} and x_{m}(m=1,2,…,Q) represent the ith unaligned and first Q aligned envelopes, respectively. R(·) denotes the correlation function, which is maximized with respect to Δ_{τ}, i.e., the shift value of the envelope. τ_{s} is the final result used to calibrate the current envelope. The approaches for averaging the already aligned envelopes in (5) can be varied for the sake of accuracy and stability [10, 23].
For the MEARP, it is iterated with the entropy minimization principle as follows [7]:
and
In (6) and (7), r and m represent the radial range and pulse index, respectively. k denotes the iteration number, and p(r,m) is the envelope of the rangecompressed ISAR signal. The other mathematical symbols are the same as those in (5).
The two algorithms above are representative of the maximumcorrelationbased and globaloptimizationcriterionbased methods, respectively. They are used for comparison with the proposed algorithm.
Methods
In this section, the workflow of the proposed range alignment method is discussed in detail. In general, as mentioned in Section 1, this novel algorithm contains three steps:

1
Split the full aperture and estimate the target motion parameters in each subaperture based on the CDA and the LevenbergMarquardt (LM) optimization method

2
Align the envelopes of the average range profiles (ARPs) of every subaperture using the MCA

3
Smooth the estimated deviations by means of locally weighted regression (LOESS)
Estimating the motion parameters in the subapertures
Traditional range alignment methods often focus on the similarity among range profiles without fully considering the target motion information. As a result, these algorithms are vulnerable in low signaltonoise ratio (SNR) scenarios.
In a relatively short observation time, the target translation motion can be regarded as stable movement, i.e., uniformly accelerated motion; thus, it is rational to model the envelope shift as a secondorder polynomial with respect to the slow time. With the adoption of the motion information, the algorithm robustness under a low SNR can be considerably improved. However, in real ISAR imaging, hundreds or even thousands of echoes are accumulated, which leads to a long observation time. To use the motion information, we split the full aperture into a certain number of subapertures to obtain a short slowtime span. In each subaperture, optimization based on the minimum entropy principle is performed to estimate the velocity and acceleration of the target, which is discussed at length in the following.
Assume every M consecutive echoes of the full aperture are viewed as a subaperture and that the echo in the kth subaperture is \({x_{k}}\left ({{t_{m}},\hat {t}} \right)\), where t_{m} is the slow time and t_{n} is the fast time (discrete). As mentioned above, the envelope shift in one subaperture can be modeled as a secondorder polynomial in t_{m} with two unknown parameters, i.e., acceleration a and velocity v:
where Φ(t_{m}) represents the envelope shift error.
As (4) indicates, once the optimal values of the unknown parameters are obtained, modulation in the time domain can be carried out to compensate for the range shift:
where \({\tilde {x}_{k}}\) denotes the compensated echo.
To estimate a and v, the minimum entropy method is introduced. By referring to [24], compared with contrastbased methods, this method can attain a good compromise among all kinds of scatters contained in the echo and result in a globally highquality image. The entropy of the ARP of the current subaperture is chosen as the metric, and the unknown parameters are estimated by minimizing the entropy.
The HRRP of one subaperture is denoted by
where fft{·} represents the FFT operation on t_{n}.
Based on (9) and (10), the ARP of the subaperture can be expressed as:
where \({f_{k}}\left ({v,a} \right){\mathrm { = }}fft\left \{ {{x_{k}}\left ({{t_{m}},t_{n}} \right) \cdot \exp \left [ {  j2\pi \hat {t}\left ({v{t_{m}} + a{t_{m}}^{2}} \right)} \right ]} \right \}\) and m is the index of each echo in the subaperture. Equation (11) is apparently a onedimensional real function with a length equal to the number of fast time sampling points N.
According to [18], the entropy of the ARP can be written as
where n is the index of the fasttime sampling points and S_{arp} is the intensity of the ARP, namely,
Combining (11) and (12), it can be seen that the entropy E is a function of the unknown parameters a and v. Therefore, the problem of estimating these parameters can be abstracted to the following form:
where \(\hat {a}\) and \(\hat {v}\) are the estimated values that minimize the entropy.
Equation (14) is a twodimensional optimization. In the proposed algorithm, the CDA is implemented as the optimization solver, which is an iterative method with outer and inner iterations. The inner ones, with the same number of unknown parameters, are accomplished by minimizing the objective along a certain dimension while fixing the remaining components of the vector of the parameters at their current values. The outer one is not to be terminated until the criteria on the tolerance of the change in the cost function or the preset maximum loop times are met [21].
According to [25], by using a proximal point update technique, the CDA can achieve better robustness in solving onedimensional subproblems. Suppose the vector of the unknown parameters is θ and that the CDA procedure is in the pth outer loop and implemented to update the i_{p}th parameter. The CDA updating scheme with the proximal point update can be written as
where \({\frac {1}{{2a_{{i_{p}}}^{p  1}}}\left \left {\theta _{{i_{p}}}}  \theta _{{i_{p}}}^{p  1}\right \right _{2}^{2}}\) is the socalled quadratic proximal term and \(a_{{i_{p}}}^{p  1}\) serves as a step size and can be any bounded positive number. The addition of the quadratic proximal term makes the function of each subproblem dominate the original objective around the current iteration and therefore produces increased stability and better convergence properties, especially in the case of nonsmooth optimization [25].
For the onedimensional search in the CDA, the LM algorithm, which has been the de facto standard for most optimization problems [26], is utilized. In the LM method, the cost function in the neighborhood of the current iteration θ_{i} can be approximated as
where Δ is the update value and L(Δ) represents the approximation of E(θ_{i}+Δ).
With two parameters, i.e., the damping parameter λ and the division factor γ, the LM procedure can be summarized as in Algorithm 1, where the initial values of λ and γ are empirically obtained.
It can be seen from Algorithm 1 that the first and second derivatives of the cost functions are needed to complete the LM method, as illustrated in the Appendix.
Assume that the number of subapertures is SN; therefore, there are M·SN echoes in total. The envelope deviations obtained by parameter estimation can be written as
where v_{i} and a_{i} (i=1,2,…,SN) denote the estimated parameters and t_{sm} (m=1,2,…,M) represents the slowtime vector with length M of each subaperture. Therefore, the total length of vector Δ_{sub} is M·SN.
Aligning the ARPs of the subapertures
After the parameter estimation based on the method mentioned in the previous subsection and compensation according to the estimated results, the envelopes in each subaperture have been aligned. However, due to the following two reasons, some finetuned techniques are required to achieve better alignment.
On the one hand, the parameter estimation method focuses mainly on the alignment in the subapertures, and as a result, there exist envelope fluctuations among different subapertures; on the other hand, after the misalignment compensation in each subaperture, the processing gain of noncoherent integration can be obtained by means of averaging all envelopes in a subaperture, which provides useful information for performance enhancement in low SNR scenarios.
To make full use of the noncoherent integration gain and improve the effect of alignment between subapertures, we implement the ACM (5) on the ARPs of all subapertures. The framework of this finetuned technique is demonstrated below.
Suppose the estimated motion parameters of the kth subaperture are \(\hat {v}\) and \(\hat {a}\); therefore, the compensated echoes of this subaperture can be written as
where x_{ck}(t_{m},t_{n}) denotes the aligned echoes of the kth subaperture. The ARP of the kth subaperture can be expressed as
Again, M is the number of echoes in one subaperture, and fft{·} represents application of the FFT along the fasttime direction.
Assume that the number of subapertures is SN. After the ACM, the SN values of the envelope deviations are obtained, i.e.,
where Δ_{i}(i=1,2,…,SN) represents the envelope deviation of each subaperture’s ARP and Δ_{ave} denotes the vector of all deviations.
The length of vector Δ_{ave} should be extended to M when carrying out compensation for each echo. The longer version of Δ_{ave} can be expressed as
Total error fitting using locally weighted regression
In the previous two subsections, the envelope deviations in each subaperture and between every two subapertures were obtained through optimization and the ACM, respectively. With the combination of (17) and (21), the total envelope deviations can be expressed as
where Δ_{total} denotes the total deviations of the envelopes.
After aligning the ARPs of all subapertures, in general, the misaligned envelopes can be calibrated well. To achieve a higher performance, some fine tuning is still required. Figure 1 shows an estimation result for a full aperture’s envelope misalignment error, where relatively accurate error estimation could be achieved; however, according to the enlarged error estimation curve, there exist step changes between the two subapertures, which can undermine the imaging quality. Because each subaperture is aligned as a whole by the method proposed in the previous subsection, the step changes are inevitable.
These step changes can be easily smoothed and eliminated by some curvefitting techniques. By referring to [27] and [11], we use locally weighted regression (LOESS) to smooth the step changes between every two adjacent subapertures. The procedure of LOESS is briefly introduced in the following:

1
Suppose that there are N points to be fitted, i.e., [x_{1},…,x_{N}]^{T}. For the ith point x_{i}, put N·f_{r}(0<f_{r}≤1) points into its neighborhood Ω_{i}.

2
Determine the weight w_{k}(x_{i}),k=1,2,…,N·f_{r} for the weighted least squares (WLS) in the neighborhood of x_{i} using tricube functions.

3
Because of the short length of the neighborhood, it is reasonable to model the points in the neighborhood as quadratic. After conducting WLS, the ith fitted value can be expressed as \({y_{i}} = {\beta _{0}} + {\beta _{1}}{x_{i}} + {\beta _{2}}x_{i}^{2}\), where β_{i},i=0,1,2 are the estimated coefficients of the quadratic polynomial.

4
Repeat steps 1–3 for all N points to obtain N fitted values.
After LOESS, accurate envelope deviations can be obtained, and good alignment can be achieved. The whole framework is shown in Fig. 2.
Optimal selection and computational complexity
This subsection discusses how to choose the number of subapertures and the computational complexity of the proposed algorithm.
By referring to [28] and [29], we develop the following adaptive selection method:

1
Initialize SN. The principle of initializing SN is that the envelope deviation in one subaperture should not exceed half the range unit, i.e., c/4F_{s}, where c is the velocity of light and F_{s} is the sampling rate.

2
Implement the minimum entropy optimization. The estimated error of the pth subaperture is ΔR_{p}(t_{m}), where t_{m} represents the slow time of the current aperture.

3
Double SN and implement the minimum entropy optimization. The pth aperture in (2) is split into two equal subapertures, and the estimated envelope errors in each one can be expressed as ΔR_{p1}(t_{m}) and ΔR_{p2}(t_{m}). They can be denoted as ΔR_{pNew}(t_{m})=[ΔR_{p1}(t_{m}),ΔR_{p2}(t_{m})], and the length of t_{m} is equal to that in (2).

4
If the following condition is satisfied, the initialized SN can be used; otherwise, return to (3) and repeat.
$$ \begin{aligned} &\max \left(\Delta {R_{p}}({t_{m}})  \Delta {R_{pNew}}({t_{m}})\right) \\& \min \left(\Delta {R_{p}}({t_{m}})  \Delta {R_{pNew}}({t_{m}})\right) \le c/4{F_{s}} \end{aligned} $$
In addition to the selection scheme above, in real cases, it is also important to jointly consider prior information, such as the motion parameters, the expected gain through accumulated echoes, and the computational burden (a larger number of apertures means a higher computational complexity). Moreover, when the accumulated echoes are insufficient, a certain number of echoes can be reused by two different subapertures. In general, the selection of the number of subapertures requires thorough consideration.
In the following part of this subsection, the computational complexity is briefly numerically analyzed, with the detailed derivation shown in the Appendix. As mentioned above, the proposed algorithm contains three parts. In the first part, we use the CDA to solve the optimization in each subaperture. In each loop, the computational burden is devoted mainly to obtaining the entropy and its first and second derivatives. The computational complexity of these operations is denoted as
where \(N_{mul}^{OP}\) and \(N_{add}^{OP}\) represent the numbers of multiplications and additions of the proposed algorithm’s optimization procedure, respectively, and M and N denote the numbers of Doppler and range cells of each subaperture, respectively. It also can be seen that each subaperture is independent; thus, parallel programming can be used to execute all optimizations concurrently.
The second step is the ACM in each loop, of which the computational burden arises mainly from obtaining the correlation function between the current profile and the reference. The computational complexity of this step is
where \(N_{mul}^{ACM}\) and \(N_{add}^{ACM}\) represent the numbers of multiplications and additions of the proposed algorithm’s ACM procedure, respectively, N denotes the number of range cells of each subaperture, and SN is the number of subapertures.
With reference to [30], LOESS also decomposes the problem into independent pieces, with all operations being completely parallel. Therefore, with welldesigned concurrent programming, the computational complexity of the LOESS step is equal to that of a weighted least squares operation with few points.
Results and discussion
In this section, experiments based on simulated and real measured data are reported to verify the effectiveness of the antilowSNR characteristics of the proposed algorithm. Entropy and contrast were chosen to evaluate the quality of the range profiles and ISAR images. The ACM (5) and MEARP (6) were also used for comparison. The computation platform was based on the Windows 10 64bit operating system, an Intel Core i59300H@2.40 GHz CPU, 8 GB of memory, and MATLAB version 2017b.
Simulated experiment
In this subsection, a simple plane model, as shown in Fig. 3a, is used as the target of the simulated experiments. The radar system has a 15GHz carrier frequency and 2GHz bandwidth. The numbers of fasttime sampling points and accumulated echoes are both 512. Figure 3b is the original ISAR image of this plane model. Figure 3c and d demonstrate the original (without misalignment) and the misaligned HRRPs.
To preliminarily verify the effectiveness of the proposed algorithm using this simulated data, we added extra white Gaussian noise to change the SNR to − 10 and − 13 dB.
First, the alignment results, the final ISAR images and their evaluations, and the estimated error curve under − 10 dB SNR are given. For the purpose of comparison, these results obtained by the proposed algorithm, ACM, and MEARP are all displayed.
Figure 4 gives the aligned HRRPs under the − 10 dB SNR obtained through the three methods above. It can be seen that due to the strong background noise, the maximumcorrelationbased algorithm has the worst alignment result and that a few profiles at the beginning of the slow time cannot be identified. The proposed and MEARP algorithms utilize the global information so that their range profiles have higher sharpness than those of the ACM.
The theoretical and estimated envelope deviation curves are given in Fig. 5 to compare the accuracies of the three methods. In Fig. 5, the dotted lines are the estimated curves, and the full lines are the theoretical ones. It can be seen from the first subfigure that the estimated curve is almost consistent with the theoretical one, indicating the high accuracy of the proposed algorithm under low SNR situations. However, there exist several abrupt changes in the estimation of the ACM, which is the result of the noise interference. The results of the MEARP, with fewer abrupt changes than those of the ACM, reveal an overall shift of the envelopes and are still not as good as those of the proposed algorithm.
As shown in Fig. 4, it is difficult to distinguish the pictures of the range profiles under strong noise. Moreover, under very low SNRs, the entropy of the range profiles cannot fully reflect the degree of focus due to the noise disturbance. Therefore, the final image is obtained, and the image quality is used as another reference for evaluation. After the range alignment, by using the autofocusing method proposed in [31], we can obtain the final ISAR image. Figure 6 demonstrates the ISAR imaging results of the three methods. The qualities of the images evaluated by the entropy and contrast are shown in Table 1. Through Fig. 6 and Table 1, it can be seen that the proposed algorithm can obtain the best degree of focus, namely, the lowest entropy and highest contrast. It should also be noted that the ACM performs worse than the two other algorithms, with some scatter points overwhelmed by the ground noise.
In more challenging situations (− 15 dB SNR), the proposed algorithm can still achieve high performance. Figure 7 shows an ISAR image under − 15 dB for the three methods. Their evaluations, conducted using the entropy and contrast, are listed in Table 2. As is the case in the − 10 dB SNR situation, the proposed algorithm achieves the highest performance. It can be seen in Fig. 7 that the ACM cannot image at all and that some scatters drowned in noise in the result of the MEARP can hardly be identified. However, the imaging result of the proposed algorithm is almost the same as the original one. Through Table 2, it can be seen that compared with the qualities of the results of the ACM and MEARP, the proposed algorithm achieves the highest contrast and lowest entropy, indicating that the proposed algorithm can produce the mostfocused images.
The estimated error curves obtained through the three methods under − 15 dB are also demonstrated. As in Fig. 5, in Fig. 8, the proposed algorithm realizes the highest accuracy.
Experiment on real measured data
Real measured Yak42 data were adopted to validate the performance of the proposed algorithm. The radar recording the dataset was a Cband ISAR experimental system with a 400MHz bandwidth, 5.52GHz carrier frequency, 100Hz PRF, and 25.6 μs pulse width. The numbers of fasttime sampling points and accumulated echoes were both 256. The SNR of the original echoes (before range compression) was − 15 dB. To test the robustness of the proposed algorithm in low SNR scenarios, extra white Gaussian noise was added, making the SNR of echoes − 20, − 22, and − 25 dB. Monte Carlo experiments were performed under these low SNRs.
The threeorder envelope deviation error, which can be expressed as (27), was added:
where R_{error} is the added error and t_{s} represents the slow time.
The original ISAR image, the range profiles of 256 echoes, the added error, and the contaminated range profiles are demonstrated in Fig. 9. In the following part of this subsection, the performances of the proposed algorithm under − 20, − 22, and − 25 dB SNR are evaluated. The imaging results and estimated error curves of the three algorithms are demonstrated. To verify the robustness of the proposed algorithm in low SNR scenarios, Monte Carlo experiments were conducted.
Figure 10 demonstrates the ISAR image entropy and MSE of the error estimation achieved after the 150 Monte Carlo experiments. In Fig. 10, the full lines, dotted lines, and dashed lines indicate the results of the proposed algorithm, ACM, and MEARP, respectively. It can be seen in Fig. 10a that the entropy of the ISAR images obtained from the three methods indicates that the proposed algorithm achieved lower entropy than did the ACM and MEARP under − 15 to approximately − 25 dB SNRs. Furthermore, according to Fig. 10b, the proposed algorithm had the lowest MSE of error estimation, which further verifies its robustness in low SNR scenarios. Through Fig. 10a and b, it can be seen that the MEARP performed better than the ACM under − 20 to approximately − 25 dB SNRs, which is consistent with the conclusion in [7].
Figures 11, 12, and 13 demonstrate the ISAR imaging results of the three methods under − 20, − 22, and − 25 dB SNRs, respectively. The first subfigures in these figures show the results of the proposed algorithm, and the second and third subfigures show the results of the ACM and MEARP, respectively. It can be seen from Figs. 11, 12, and 13 that for extremely low SNRs, especially − 22 and − 25 dB, severe defocus occurred in the images yielded by the ACM and MEARP, and the outlines of the planes could not be distinguished. However, the imaging results of the proposed algorithm remained clear and well focused as the background noise increased. Therefore, through Figs. 11, 12, and 13, it can be concluded that in low SNR scenarios, the proposed algorithm can realize the mostfocused ISAR images compared with the ACM and MEARP.
In Figs. 14, 15, and 16, the error estimation curves of the three algorithms under − 20, − 22, and − 25 dB SNRs are demonstrated. The full lines represent the theoretical results, and the dotted lines represent the estimations. It can be seen that compared with the ACM and MEARP, the proposed algorithm is the most robust and has the highest estimation accuracy. The estimated error curve obtained by the proposed algorithm is approximately consistent with the theoretical one; however, there exists a very large estimation error in the curves generated by the ACM and MEARP, which will cause severely defocused images. Therefore, through Figs. 14, 15, and 16, the superiority of the proposed algorithm in low SNR scenarios is further confirmed.
Conclusions
In low SNR scenarios, it is challenging to realize range alignment in ISAR translation motion compensation. Illaligned range profiles can cause imprecise phase compensation results. A novel range alignment method was proposed in this article. First, the target motion information was adopted as useful information, and the full aperture was split into several subapertures. In each subaperture, the minimum entropy principle was applied as a metric, and the CDA with a proximal point updating scheme was implemented as the solver to estimate the motion parameters (i.e., velocity and acceleration) in each subaperture. Second, after parameter estimation in every subaperture, the noncoherent accumulation gain was obtained by averaging the profiles in each subaperture, which could be used as further information to improve the antinoise characteristics. The first step focused mainly on alignment in each subaperture; therefore, we used the ACM to align the ARPs of each subaperture. Third, to eliminate the step change in the total estimated error curve, LOESS was applied to smooth the discontinuity. Experiments based on real measured Yak42 data were conducted, and the classic ACM and MEARP were used for comparison. The results show that the proposed algorithm achieved the best performance in low SNR scenarios. Further work will be focused on parallelization of the proposed algorithm to use it in realtime ISAR scenarios.
Appendix
Proof of the first and second derivatives of the cost function
In this section, the first and second derivatives of (12) are given.
The ARP is a onedimensional function of the unknown parameters v and a. A vector containing an unknown is denoted as θ=[v,a]^{T}. Combined with (11) and (13), the first derivative of (12) can be written as
where θ_{i}(i=1,2) is the ith element of θ.
Therefore,
It can be seen from (28) and (29) that ∂f_{k}(θ)/∂θ_{i} should first be deduced to obtain ∂E(ARP)/∂θ_{i}.
Consider that
and
where f_{k}^{∗} represents the conjugate of f_{k}.
Taking (29) ~(32) in (28), we can obtain an expression for the first derivative of (12).
The second derivative of (12) can be written as
and
It can be seen from (33) and (34) that \({\partial ^{2}}{f_{k}}/\partial \theta _{i}^{2}\) should first be deduced to obtain \({\partial ^{2}}E\left (ARP \right)/\partial \theta _{i}^{2}\).
Consider that
and
Taking (34) ~(37) in (33), we can obtain an expression for the second derivative of (12).
Numerical computational complexity analysis
As (11) shows, f_{k} contains M FFTs in the fasttime direction. Therefore, the numbers of multiplications and additions can be approximately expressed as
According to (30) and (31), the first derivative of f_{k} also contains M FFT operations, whose computational complexity can be written as
Because the ARP, S_{ARP}, and ∂ARP^{2}/∂θ_{i} are all functions of f_{k} or ∂f_{k}/∂θ_{i}, once f_{k} and ∂f_{k}/∂θ_{i} have been obtained, (28) can be obtained. In other words, the computational complexity of (28) is the sum of the complexity of computing f_{k},∂f_{k}/∂θ_{i}, and other multiplications and additions in (28). The final result is
Once (28) is obtained, in (33), only \({\partial ^{2}}AR{P^{2}}/\partial \theta _{i}^{2}\) must be recomputed. According to (34) ~(37), the numbers of multiplications and additions of (34) can be written as
The total numbers of multiplications and additions of (33) are equal to the sum of the multiplications and additions of (34) and other necessary ones contained in (33). Therefore, the computational complexity of (33) can be expressed as
Combining (42), (43), (46), and (47), it can be seen that (23) and (24) have been proved.
For the computational burden of the ACM, the focus is on the correlation function in each loop. To obtain the correlation between the current profile and the template through an FFT, three Npoint FFT/IFFT operations and one Npoint conjugate multiplication must be performed, and the average of the aligned profiles must be taken. An Npoint FFT contains Nlog_{2}N/2 multiplications and Nlog_{2}N additions, and Npoint conjugate multiplication has N multiplications. Therefore, the total computational burden of the ACM can be approximately written as
Availability of data and materials
The data of the Yak42 plane is presented in the additional supporting files.
Abbreviations
 ISAR:

Inverse synthetic aperture radar
 SNR:

Signaltonoise ratio
 HSACM:

Highorder symmetric accumulated crosscorrelation method
 CPI:

Coherent processing interval
 CDA:

Coordinate descent algorithm
 LM:

LevenbergMarquardt
 MEARP:

Minimization of the entropy of the average range profile
 MCA:

Maximumcorrelationbased algorithm
 LFM:

Linear frequency modulation
 RVP:

Residual video phase
 FFT:

Fast Fourier transform
 HRRP:

Highresolution range profile
 ACM:

Accumulation and correlation method
 ARP:

Average range profile
 LOESS:

Locally weighted regression
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Acknowledgements
The authors thank the anonymous reviewers for their enlightening comments and careful review.
Funding
This work was supported in part by the National Natural Science Foundation of China under Grant 61571449 and Grant 61501471.
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YL was a major contributor in writing the manuscript and carried out the simulation work. JY designed the major workflow of the proposed algorithm. YZ helped to improve the source code and modified some details of this algorithm. SX and the authors read and approved the final manuscript.
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Additional file 1
The file containing the data of the Yak42 plane can be opened by MATLAB.
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Lu, Y., Yang, J., Zhang, Y. et al. Noiserobust range alignment method for inverse synthetic aperture radar based on aperture segmentation and average range profile correlation. EURASIP J. Adv. Signal Process. 2021, 4 (2021). https://doi.org/10.1186/s1363402000709z
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Keywords
 Aperture segmentation
 Entropy minimization
 Inverse synthetic aperture radar
 Range alignment