Radar high-speed maneuvering target detection based on joint second-order keystone transform and modified integrated cubic phase function

Abstract. Long-time coherent integration is an effective means to improve the radar detection ability of high-speed maneuvering targets with jerk motion. However, the range migration (RM) and Doppler frequency migration (DFM) have a great impact on the integration performance. To overcome these problems, a unique method, called the second-order keystone transform modified integrated cubic phase function (SKT-MICPF), is proposed. In this method, the velocity compensation and SKT are jointly employed to correct the RM. After the RM correction, the azimuth echoes of a range cell where a target is located can be modeled as a cubic phase signal (CPS), whose chirp rate (CR) and quadratic CR are related to the target’s radial acceleration and jerk, respectively. Thereafter, an effective parameters’ estimation algorithm for CPS, called MICPF, is proposed and applied to compensate the DFM. After that, coherent integration and target detection are accomplished via the fast Fourier transform and constant false alarm rate technique, successively. Compared with the improved axis rotation discrete chirp Fourier transform, the SKT-MICPF achieves close detection performance, but greatly reduces the computational complexity. The results of simulation and real radar data demonstrate the validity of the proposed algorithm.

Radar high-speed maneuvering target detection based on joint second-order keystone transform and modified integrated cubic phase function

Introduction
Modern national defenses have higher requirements for the detection of high-speed maneuvering targets. 1,2 Prolonging the integration time is an effective means to improve the detection performance. However, with the increasing of the integration time, the integration performance will degrade due to the range migration (RM) and Doppler frequency migration (DFM).
To address these problems, various successful algorithms [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have been developed. The Radon Fourier transform (RFT), 2 the keystone transform (KT), 3 the axis rotation moving target detection, 4 and the scaled inverse Fourier transform-based method 5 can be applied to correct the RM and accomplish coherent integration for a target with a uniform radial velocity. The Radon fractional FT, 6 the Radon Lv's distribution (RLVD), 7 the KT Lv's transform, 8 the two-dimensional (2-D) match filter-based method, 9 and the KT-RFT 10 are the effective methods for a high-speed maneuvering target with a uniform radial acceleration. However, for a highly maneuvering target with a jerky motion, the aforementioned algorithms 2-10 will suffer from integration loss due to the RM and DFM induced by the target's jerky motion.
For a high-speed maneuvering target with a uniform jerk, an algorithm based on the generalized KT and second-order dechirp process is proposed in Ref. 11, which employs the range frequency domain form of the compressed echoes, i.e., Eq. (5), to accomplish the target's motion parameters estimation. However, because the target's energy spreads in the range frequency domain, this method suffers from great signal-to-noise ratio (SNR) loss. The cubic phase function (CPF)-based methods 12,13 and the adjacent correlation function LVT 14 also face similar problems. An improved axis rotation discrete chirp Fourier transform (IAR-DCFT) method is presented in Ref. 15, which can correct the RM by the IAR, and achieve coherent integration via the DCFT. Nevertheless, the implementation of the DCFT is computationally demanding. 16,17 After the RM correction, the azimuth echoes of a range cell where a target is located can be considered as a cubic phase signal (CPS), whose chirp rate (CR) and quadratic CR (QCR) are related to the target's radial acceleration and jerk, respectively. Therefore, the estimation algorithms for the CPS can be applied to estimate the target's radial acceleration and jerk to further compensate the DFM. The scaled FT (SCFT)-based method, 18 the keystone time chirp rate distribution, 19 and the modified chirp rate quadratic chirp rate distribution 20 are the successful algorithms for CPS. However, due to the fourth-order kernel functions that are applied, the antinoise performances of those methods are still low. The integrated cubic phase function (ICPF) 21 which is developed from the CPF 22,23 can obtain a higher antinoise performance than the aforementioned approaches [18][19][20] due to its bilinearity. In Ref. 24, a space switching-based ICPF (SS-ICPF) algorithm is proposed for CPS. Nevertheless, the space switch method which includes the circular shifting operation and map operation would burden the complexity of the estimator and increase the computational cost.
Motivated by the previous work, an algorithm, known as second-order keystone transform modified integrated cubic phase function (SKT-MICPF), is presented for the high-speed maneuvering targets with jerky motions. First, the velocity compensation and SKT are jointly employed to correct the RM. Second, an estimation method for CPS, named MICPF, is proposed and applied to compensate the DFM. Finally, the fast Fourier transform (FFT) and constant false alarm rate (CFAR) technique are employed to accomplish coherent integration and target detection, successively. Comparisons with the IAR-DCFT show that the SKT-MICPF can reduce the computational cost greatly with little detection performance loss.
The rest of this paper is organized as follows. In Sec. 2, the signal model for high-speed maneuvering targets with jerky motion is built. The SKT-based RM correction method is presented in Sec. 3. In Sec. 4, an MICPF method is proposed for CPS and its performances are also analyzed. A detection approach for high-speed maneuvering targets with jerky motion is illustrated in Sec. 5. In Sec. 6, the experiments with synthetic data and real data are performed to demonstrate the effectiveness of the proposed algorithm. Section 7 gives the conclusions.

Signal Model for High-Speed Maneuvering Targets
Suppose that the radar transmits a linear frequency modulation signal, which can be expressed as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 6 ; 2 9 7 s t ðtÞ ¼ rectðt∕T r Þ exp½jπð2f c t þ γt 2 Þ; (1) , T r is the pulse duration, and t, f c , and γ denote the fast time, the carrier frequency, and the modulation rate, respectively. Assume that there are P high-speed maneuvering targets with a constant radial jerk in the scene, thus the instantaneous slant range between the radar and the p'th target satisfies E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 6 ; 2 0 3 where t m represents the slow time, and r p ; v p ; a p , and k p denote the initial slant range, the radial velocity, acceleration, and jerk of the p'th target, respectively. The received baseband echoes can be expressed as 2 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 6 ; 1 3 3 S r ðt m ; tÞ ¼ where A p denotes the p'th target reflectivity, and c and T represent the speed of light and the integration time, respectively. Nðt m ; tÞ is the complex white Gaussian noise. After the pulse compression, the echoes in the slow time-fast time domain can be stated as Applying the inverse FFT (IFFT) on Eq. (10) along the f r -axis, we get E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 1 1 6 ; 6 5 1 S 3 ðt n ; tÞ ≈ From Eq. (11), it can be easily seen that the RMs induced by the targets' radial accelerations and part of the RMs caused by the targets' radial jerks have been corrected. Assume that the radial velocity of the p'th target is matched by the searching velocity, i.e., jv Tp − v s j ≤ Δv s ∕2, where Δv s denotes the interval of the searching velocity. Therefore, the residual RM of the p'th target satisfies E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 1 1 6 ; The RMs induced by the targets' radial jerks are usually small for a narrowband radar. 12,15 We limit the residual RMs induced by the targets' radial jerks to be half a range resolution in this paper. If the interval of searching velocity satisfies E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 1 1 6 ; 4 2 8 Δv s ≤ 2ΔR∕T; (13) where ΔR ¼ c∕ð2BÞ is the range resolution, the residual RM of the p'th target will be less than a range resolution, which indicates that the RM effect can be ignored. 2 Thus, the interval of searching velocity can be selected in terms of Eq. (13).
With the searching procedure, all the targets can be corrected successively. However, the DFMs induced by the targets' radial accelerations and jerks still remain, which have a great impact on the integration performance.

Parameters Estimation of CPS Based on Modified Integrated Cubic Phase Function
In order to compensate the DFM, a modified ICPF is proposed for CPS and applied to compensate the DFM.

Modified Integrated Cubic Phase Function
Consider a monocomponent CPS expressed as follows: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 1 1 6 ; 1 9 6 sðt m Þ ¼ ρ exp j2π where ρ, a 1 , a 2 , and a 3 denote the amplitude, the centroid frequency (CF), the CR, and the QCR, respectively. T represents the integration time.
The instantaneous frequency rate (IFR) 22,23 of sðt m Þ can be stated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 5 ; 1 1 6 ; 1 1 5 where Φðt m Þ denotes the phase of sðt m Þ.
The CPF 22,23 is defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 1 1 6 ; 7 2 3 CPFðt m ; fÞ ¼ where τ m is a lag-time variable with respect to t m , f denotes the IFR domain, and δð·Þ is the Dirac delta function. Rðt m ; τ m Þ is the kernel function defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 7 ; 1 1 6 ; 6 2 5 After the Fourier transform, the signal energy peaks along the inclined line f − a 2 − a 3 t m ¼ 0 in the time-IFR domain. The FT performed on Eq. (16) is with respect to τ 2 m , which indicates that the unambiguous range of f is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 8 ; 1 1 6 ; 5 2 6 jfj ≤ f 2 s ∕2; where f s denotes the sampling frequency.
In realistic applications, the IFR of a CPS is usually much smaller than f 2 s ∕2. [18][19][20][22][23][24] To obtain a higher estimation accuracy, we limit f to be jfj ≤ f s ∕2, and introduce a zoom factor to balance the resolution and range of the IFR. Thus, a modified CPF is defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 9 ; 1 1 6 ; 4 4 6 MCPFðt m ; where η is the introduced zoom factor. The selection criterion will be discussed in Sec. 4.2.
Because the sampling corresponds to τ 2 m is nonuniform, the Fourier transform performed on Eq. (19) cannot be implemented by the FFT. Fortunately, the nonuniform FFT (NUFFT) can be applied to speed up the FT with the computational cost of OðM 2 log 2 MÞ, 24,25 where M is the number of integration pulses.
After the NUFFT, the signal energy peaks along the inclined line E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 0 ; 1 1 6 ; 2 8 4 f τ − a 2 ∕η − a 3 t m ∕η ¼ 0: (20) Due to the coupling between t m and τ m in the second exponential phase term of Rðt m ; τ m Þ, the inclined line is not parallel to the t m -axis. Due to the first exponential phase term of Rðt m ; τ m Þ, the SCFT 18,26 cannot remove this coupling. It is observed from Eq. (19) that, after the NUFFT operation, the CR and QCR have been packaged into the inclined line, i.e., Eq. (20). The Radon transform, 27 the Hough transform, 28 and the modified Hough transform 29 are the effective methods to accomplish energy accumulation along a straight line. Nevertheless, a 2-D searching procedure is inevitable which will burden the system complexity and increase the computational cost.
To realize energy accumulation along the inclined line without brute-force searching, the IFFT is performed on jMCPFðt m ; f τ Þj 2 with respect to f τ E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 1 ; 1 1 6 ; 1 3 3 where τ 0 m denotes the lag-time variable with respect to f τ . IFFT f τ ½· denotes the IFFT operation along the f τ -axis. The first exponential phase term of Rðt m ; τ m Þ does not appear in R 1 ðt m ; τ 0 m Þ, which indicates that the SCFT becomes applicable to remove the coupling between t m and τ 0 m . Applying the SCFT on Eq. (21) along the t m -axis, we obtain E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 2 ; 1 1 6 ; 7 1 1 where f t is the frequency domain with respect to τ 0 m t m , SCFT ðξτ 0 m t m Þ ½· denotes the SCFT operation along the t m -axis, ξ is a zoom factor which is introduced to balance the resolution and range of the QCR, and its selection criterion will be analyzed in Sec. 4.2.
After the SCFT, the signal energy distributes along the beeline f t − a 3 ∕ηξ ¼ 0 in the τ 0 m − f t domain. Thereafter, applying the FFT along the τ 0 m -axis to achieve further energy accumulation, we obtain E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 3 ; 1 1 6 ; 5 9 5 MICPFðf t ; f 0 where FFT τ 0 m ½· denotes the FFT operation along the τ 0 m -axis. A sole peak has been accumulated at the point ½a 2 ∕η; a 3 ∕ðηξÞ in the f 0 τ − f t domain. Thereafter, a peak detection technique [18][19][20]24 is employed to estimate the CR and QCR of the CPS.
From the above discussion, the abbreviated expression of the proposed MICPF can be expressed as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 4 ; 1 1 6 ; 4 7 8 where KF½· denotes the kernel function defined in Eq. (17).

Selection Criteria of Zoom Factors
Equation (23) indicates that the CR and QCR of the CPS can be estimated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 5 ; 1 1 6 ; 3 8 4 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 6 ; 1 1 6 ; 3 4 8 where ðf t ;f 0 τ Þ denotes the peak location. Furthermore, the estimated ranges of the CR and QCR are E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 7 ; 1 1 6 ; 2 9 6 ja 2 j ≤ ηf s ∕2; E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 8 ; 1 1 6 ; 2 6 3 To avoid spectrum aliasing, it is assumed that ; t e m p : i n t r a l i n k -; e 0 2 9 ; 1 1 6 ; 2 2 5 ja 2 þ a 3 t m j ≤ 13f s ∕ð4TÞ: In order to avoid energy loss, Eq. (19) indicates that η should satisfy E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 0 ; 1 1 6 ; 1 8 2 Therefore, η should be E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 1 ; 1 1 6 ; 1 3 9 η ≥ 6.5∕T: Equations (27) and (31) indicate that the estimated range of the CR is larger than its unambiguous range, so it is suitable for the CR estimation. Similarly, the estimated range of the QCR should be larger than its unambiguous range, which indicates that ξ should be ξ ≥ 9∕ðηT 2 Þ: However, a bigger zoom factor will result in resolution degradation. Therefore, we suggest η ¼ 6.5∕T, ξ ¼ 18∕ð13TÞ for the MICPF.
In realistic applications, the CR and QCR of a CPS may be limited to a smaller range. Under this situation, the zoom factors can be selected in terms of the real ranges of the parameters.

Performance Analyses
The cross-term suppression performance, the computational complexity, and the antinoise performance are analyzed to evaluate the performance of the proposed MICPF. The SCFT-based method, 18 the SS-ICPF, 24 and the method in Ref. 30 are also presented for comparison. These are chosen due to the following: (1) these three approaches are the typical methods for the parameters estimation of the CPS, (2) no or few redundant data is necessary, and (3) the CR and QCR are estimated simultaneously, which can suppress the error propagation.

Cross-term suppression performance
The bilinearity of the MICPF indicates that cross-terms will appear for multi-CPS. After the NUFFT, the cross-terms in the time-IFR domain take the cosine oscillation structure. 24 Thus, the cross-terms cannot be accumulated as the self-terms, because: (1) MICPF can achieve energy accumulation along the inclined line with where signal energy peaks, and (2) the cosine oscillation structure will spread the cross-terms energy. This conclusion can be proven by the experiment performed in Sec. 6.1.
However, if the amplitudes of different CPSs differ significantly, the weak CPSs may be submerged by the cross-terms generated by the strong CPSs. Under this situation, similar to other estimation methods, 18-20 the CLEAN technique 31 can be employed to separate the weak and strong CPSs, then the strong CPSs and weak CPSs can be estimated iteratively.  Table 1.

Computational complexity
It can be seen from Table 1 that the computational complexity of the proposed method is lower than that of the SS-ICPF. Moreover, for the SS-ICPF, the space switch method which includes a circular shifting operation and map operation will burden the complexity of the

Antinoise performance
First-order perturbation analysis as used in Ref. 21 is an effective method to analyze the antinoise performance of estimation algorithms. However, due to the NUFFT operation, the modulus square operation, the IFFT operation, the SCFT operation, and the FFT operation that are performed on Rðt m ; τ m Þ, the first-order perturbation analysis becomes extremely difficult. Similar to Refs. 18-20, the mean square error (MSE) is adopted to analyze the antinoise performance of MICPF in this section. Example: Consider a mono-CPS contaminated with a zero-mean complex white Gaussian noise, where the sampling frequency and sampling points of the signal are 128 Hz and 256, respectively. The CF, CR, and QCR of the CPS are set as 40 Hz, 60 Hz∕s, and 50 Hz∕s 2 , respectively, and the test input SNRs are [−10∶1∶0] (dB). 100 iterations of Monte-Carlo experiments are performed for each input SNR value. The zoom factors η and ξ are set as 2 and 0.5, respectively. It is worthwhile noting that the zoom factors are set in terms of the parameters' actual values. However, it is not in conflict with the selection criteria described in Sec. 4.2. Moreover, in Sec. 6, the zoom factors are also set in terms of the real ranges of the targets' motion parameters.
The simulation results and the corresponding Cramer-Rao bounds (CRBs) are shown in Fig. 1, and the expressions of the CRBs can be found in Ref. 32. In Fig. 1, the threshold SNR of the MICPF is −8 dB, because: (1) the bilinear kernel function is applied in the MICPF and (2) signal energy is well accumulated by the proposed nonsearching method. The SS-ICPF has the same antinoise performance as the MICPF, but with a greater computational cost. However, the threshold SNR of the SCFT-based method is −3 dB due to the fourth-order nonlinear kernel function being applied. Compared with the SCFT-based method, the method in Ref. 30 introduces a constant delay to suppress noise, and achieves a 1 dB antinoise performance gain. Nevertheless, it still has a 4 dB loss compared with the MICPF.
From the above analyses and experiments, we can conclude that the proposed method is more suitable for CPS than the SS-ICPF, the SCFT-based method, and the method in Ref. 30.

Target Detection Based on Second-Order Keystone Transform Modified Integrated Cubic Phase Function
In this section, by employing the SKT-based RM correction method and the MICPF, a coherent detection algorithm, known as SKT-MICPF, is presented to realize the detection of a high-speed maneuvering target with a constant radial jerk.

Target Detection Based on Second-Order Keystone Transform Modified Integrated Cubic Phase Function
For simplicity, a single target is considered in the scene. After the RM correction with the proposed method in Sec. 3, the azimuth echoes of a range cell where a target is located can be stated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 3 ; 1 1 6 ; 6 6 7 Performing the MICPF on sðt n Þ, we get E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 4 ; 1 1 6 ; 6 1 1 Target's radial acceleration and jerk can be estimated via peak detection technique. Thus, a compensation function is constructed to compensate the DFM, which takes the form E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 5 ; 1 1 6 ; 5 4 3 whereâ andk denote the estimated acceleration and jerk, respectively. After the DFM compensation, the FFT is applied to achieve coherent integration E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 6 ; 1 1 6 ; 4 7 4 The processing procedures of the proposed method, i.e., velocity compensation, SKT, and DFM compensation, will not change the distributions. Therefore, the CFAR technique is utilized to target detection, and the threshold can be set as 33 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 7 ; 1 1 6 ; 3 8 1T where N w , P FA , andβ 2 denote the length of the reference windows, the probability of false alarm, and the estimated noise power, respectively. It is worth noting that the strong target may be detected even when the RM has not been completely corrected, i.e., different search velocities yield positive detections. Under this situation, we confirm that the target's radial velocity has been matched by the searching velocity, if both of the following two criteria are satisfied: (1) the searching velocity satisfies jVj ≤ Δv s ∕2, whereV denotes the estimated value of V; and (2) the searching velocity corresponds to the best integration result (the maximum integrated peak).
By the criterion 1, the range of searching velocity has been confined to v sn ¼ v s0 þ nv a ðn ¼ 0; AE1; AE2; : : : Þ, where v s0 denotes the searching velocity which matches the target's radial velocity v T , i.e., jv T − v s0 j ≤ Δv s ∕2. Thereafter, the criterion 2 is introduced to select v s0 among v sn ¼ v s0 þ nv a ðn ¼ 0; AE1; AE2; : : : Þ. On one hand, the RM can be completely corrected when the searching velocity is v s0 , then optimal integration can be obtained. On the other hand, due to the long-time integration, v a T > ΔR can be easily satisfied, which indicates that the RM cannot be completely removed when the searching velocities are v sn ¼ v s0 þ nv a ðn ¼ AE1; AE2; : : : Þ, and will result in integration loss. Therefore, criterion 2 implies that the final matched velocity can be selected from the set of v sn .
Furthermore, the target's radial velocity can be estimated as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 8 ; 1 1 6 ; 1 2 8v Although a single target is considered in the above analysis, SKT-MICPF has a good detection performance for multitargets since it can distinguish multitargets from the initial range, the radial velocity, acceleration, and jerk.

Procedure of Second-Order Keystone Transform Modified Integrated Cubic Phase Function
The flowchart of the SKT-MICPF is shown in Fig. 2, which is mainly composed of seven steps. I and N denote the number of searching velocities and range cells, respectively.
Step 1: Complete the downconversion, pulse compression, and range FFT. Initialize the range of the searching velocity, let i ¼ 1.
Step 2: Construct the i'th compensation function H 1 ðt m ; f r ; v si Þ as Eq. (6), and multiply H 1 ðt m ; f r ; v si Þ with S c ðt m ; f r Þ. Thereafter, the SKT and range IFFT are performed successively. Initialize the range of searching range cell, let n ¼ 1.
Step 3: Get the data sðt n Þ of the n'th range cell, and perform the MICPF on sðt n Þ. Carry out the peak detection in the f t − f 0 τ domain. If a peak has been detected, estimate the acceleration and jerk, and then go to step 4. Otherwise, go to step 5.
Step 4: Construct a compensation function H 2 ðâ;k; t n Þ as Eq. (35), and multiply H 2 ðâ;k; t n Þ with sðt n Þ. Accomplish coherent integration and target detection via the FFT and CFAR detection, successively. If a target has been detected, estimate V, and note down the coordinate, then go to step 5. Otherwise, go directly to step 5.
Step 5: If n < N, set n ¼ n þ 1, and repeat step 3 and step 4 until n ¼ N.
Step 6: If i < I, set i ¼ i þ 1, and repeat step 2 to step 5 until i ¼ I.
Step 7: After all the searching velocities and range cells have been detected, estimate targets' radial velocities by the two criteria in Sec. 5.1.

Analysis of Computational Complexity
Denote the number of searching velocities, integration pulses, and range cells by I, M, and N, respectively. Assume that the number of searching angles of IAR-DCFT is I. The main procedures of the SKT-MICPF include: I times chirp-z-based SKT [OðMNI log 2 MÞ], MI times N points IFFT [OðMNI log 2 NÞ], and NI times MICPF [OðM 2 NI log 2 MÞ]. Therefore, the computational cost of the SKT-MICPF is OðM 2 NI log 2 MÞ. On the other hand, the computational cost of the IAR-DCFT is OðM 3 NI log 2 MÞ. 15 From the above analysis, we can see that the computational complexity of the proposed method is much less than that of the IAR-DCFT. In real applications, the initial slant range and radial velocity of a target are limited to some fixed region via prior knowledge, 6 and the computational cost of the proposed method can be further decreased.
Remark: Although the SKT-MICPF is proposed for high-speed maneuvering targets' detection, it can also be applied to inverse synthetic aperture radar (ISAR) imaging. For ISAR imaging of a target with complex motion, after the range alignment and the phase adjustment, the azimuth echoes of a range cell can be modeled as multi-CPS. [17][18][19][20]24,30 If the parameters of those CPSs can be estimated, high-resolution imaging can be obtained. Moreover, the performance of parameters' estimation has a great impact on the imaging quality. The proposed MICPF can achieve high-precision estimation for multi-CPS. Comparisons with the SS-ICPF, the SCFTbased method, and the method in Ref. 30 show that the proposed MICPF outperforms those methods, which indicates that the proposed MICPF can also be applied to ISAR imaging, and may achieve a better imaging quality than those methods.

Verification of the Proposed Algorithm
We are to validate the effectiveness of the proposed detection algorithm with synthetic radar data and real radar data.

Verification with Synthetic Radar Data
Two high-speed maneuvering targets with a constant radial jerk are contained in the radar echoes. The radar parameters and targets' motion parameters are listed in Tables 2 and 3, respectively. The echoes are contaminated with a zero-mean complex white Gaussian noise.
Assume the ranges of the targets' radial acceleration and jerk are limited to ½−50;50 m∕s 2 and ½−40;40 m∕s 3 , respectively. The zoom factor of the MICPF can be set as η ¼ 1.2, ξ ¼ 0.5 in terms of the radar parameters and the ranges of the target's motion parameters. According to the above parameters, the estimated resolutions of acceleration and jerk are 0.09 m∕s 2 and 0.045 m∕s 3 , respectively. The SNRs (after the pulse compression) of the two targets are SNR A ¼ −3 dB and SNR B ¼ −6 dB. The interval of the searching velocity is set as 15 m∕s for the RM correction in terms of Eq. (13). Figure 3 shows the simulation results of the SKT-MICPF. The result after the pulse compression is given in Fig. 3(a). Because of the low SNR, the two targets are buried in the noise.   If there is another target (target C) with a radial velocity, e.g., 2000 m∕s, it is also located in the 200th range cell. Assume that the searching velocities are set as ½1800∶15∶2400 m∕s. When the searching velocity is 1995 m∕s, the RM of target C can be completely compensated, whereas serious RMs still remain for targets A and B. Therefore, target C can be detected first. When the searching velocity is 2100 m∕s, the RMs of targets A and B can be eliminated further to parameters estimation and targets detection.
The integration results of the RFT, 2 RLVD, 7 IAR-DCFT, 15 and the proposed SKT-MICPF are further performed for comparison. Consider target A in the scene, where the SNR after the pulse compression is set as −13 dB.   Fig. 4(b). The RM induced by the target's radial velocity and acceleration and the DFM caused by the target's radial acceleration can be eliminated by the RLVD. However, the DFM induced by the target's radial jerk affects the integration performance. The integration results of the IAR-DCFT and SKT-MICPF are shown in Figs. 4(c) and 4(d), respectively. The target is well-focused via the two approaches, which indicates that the IAR-DCFT and SKT-MICPF are suitable for the detection of a high-speed maneuvering target with a jerky motion. However, the SKT-MICPF has a lower computational complexity, which has been analyzed in Sec. 5.3. The time costs of the SKT-MICPF and IAR-DCFT are also given in Table 4.
The detection performances of the RFT, RLVD, IAR-DCFT, and SKT-MICPF are further evaluated by Monte-Carlo experiments, and target A is considered in the scene. The false alarm rate is set to be 10 −6 , the test input SNRs are [−25∶1∶20] (dB, after the pulse compression), and 100 Monte-Carlo experiments are performed for each input SNR value. The simulation results are shown in Fig. 5.
It is observed from Fig. 5 that, due to its ability to deal with the DFM effect induced by the target's radial jerk, the detection performance of the proposed method is superior to those of the RFT and RLVD. The IAR-DCFT has a better detection performance (2 dB) than the SKT-MICPF. This is because: (1) the DCFT has the optimal antinoise performance 16,17 and (2) the MICPF suffers from SNR loss due to its bilinearity. However, the computational cost of the IAR-DCFT is much greater than that of the SKT-MICPF.  From the above experiments, we can see that the SKT-MICPF offers a much better performance than the RFT and RLVD, while offering a much lower computational cost than the higher performing IAR-DCFT. This makes the proposed approach a stronger candidate for practical adoptions.

Verification with Real Radar Data
The real radar data were collected in February 2010 by the National University of Defense Technology, Hunan, China. The parameters of the radar system are listed in Table 5. The 2000 integration pulses and 200 range cells are extracted from the collected data.
According to Eq. (13), the interval of the searching velocity is set as 30 m∕s for the RM correction. The processing results are shown in Fig. 6.
Figures 6(a) and 6(b) show the original radar echoes and the result after the pulse compression, respectively. It is obvious from Fig. 6(b) that the target envelope has been shifted away from its original position due to the RM. Figure 6(c) gives the result of RM correction via the proposed method (the searching velocity is −180 m∕s). After the velocity compensation and SKT, the RM has been corrected. The result after the MICPF performed on the 121th range cell is shown in Fig. 6(d). We can see that a peak has been accumulated, and then the target's radial acceleration and jerk can be estimated by the peak detection technique. Figure 6(e) gives the integration result via the MTD, and the nearby area of the integrated peak is also zoomed-in to evaluate the integration performance. Due to the RM and DFM, target energy distributes in both the range and Doppler directions. The integration result of the proposed method is shown in Fig. 6(f). Because the proposed method can compensate the RM and DFM, the integrated peak of the proposed SKT-MICPF is larger than that of the MTD.
Although the maneuverability and velocity of the passenger plane are low, the RM and DFM affect the integration performance greatly. The advantages of the SKT-MICPF will be more obvious for a target moving at a higher speed and maneuverability.

Conclusions
In this paper, an SKT-based RM correction method is first proposed then an MICPF is presented for the parameters' estimation of CPS. Thereafter, by using the proposed RM correction method and MICPF, a coherent target detection algorithm, known as SKT-MICPF, is proposed for a high-speed maneuvering target with a constant jerkiness. The advantages of the proposed method include the following: (1) the estimations of a target's radial acceleration and jerk are not searched which greatly reducing the computational cost; (2) the RM correction and DFM compensation can be accomplished in a low SNR scene, which indicates the proposed method can get a high antinoise performance; and 3) it has a good detection performance for multitargets since it can distinguish multitargets from the initial slant range, radial velocity, radial acceleration, and radial jerk. Comparisons with several methods show that the SKT-MICPF offers a much better performance than the RFT and RLVD, while offering a much lower computational cost than the higher performing IAR-DCFT. This makes the proposed approach a stronger candidate in realistic applications.