Coherent Integration Algorithm for a Maneuvering ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 63, NO. 17, SEPTEMBER 1, 2015

Coherent Integration Algorithm for a Maneuvering Target With High-Order Range Migration Lingjiang Kong, Xiaolong Li, Guolong Cui, Wei Yi, and Yichuan Yang

Abstract—This paper considers the coherent integration problem for a maneuvering target with complex motions, where the velocity, acceleration, and jerk result in respectively the first-order range migration (FRM), second-order range migration (SRM), and third-order range migration (TRM) within the coherent pulse interval. A new coherent integration algorithm based on generalized keystone transform (KT) and second-order dechirp process is proposed, which employs the third-order KT, six-order KT, second-order KT, and fold factor searching to correct the TRM, SRM, and FRM, respectively. The range migration change during each step and computational complexity are also theoretically analyzed. Compared with the generalized Radon Fourier transform (GRFT) algorithm, the presented method can avoid the blind speed sidelobe (BSSL) and acquire close integration performance but with much lower computational cost. Simulations are provided to demonstrate the effectiveness. Finally, a generalized method, named generalized KT and generalized dechirp process (GKTGDP), is also introduced for the maneuvering target with arbitrary high-order range migration. Index Terms—Maneuvering target, coherent integration, thirdorder range migration, generalized keystone transform, range migration correction.

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I. INTRODUCTION

ITH the development of science technology, especially the highly maneuvering target and high-resolution imaging technique, there is a growing need for detection of maneuvering targets with complex motions in radar society [1]–[4]. The performance of target detection affects the imaging quality, target tracking and identification significantly [5]–[7]. It is known that the coherent integration can increase the signal-to-noise-ratio (SNR) and thus improve the radar detection ability, via compensating the target’s phase fluctuation among multiple radar pulse samplings [8]–[10]. Unfortunately, the complex motions of maneuvering targets, e.g., high velocity, acceleration and jerk, involves the first-order range migration (FRM), second-order range migration (SRM) and third-order

Manuscript received October 18, 2014; revised January 08, 2015 and May 05, 2015; accepted May 06, 2015. Date of publication May 26, 2015; date of current version July 21, 2015. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ana Perez-Neira. This work was supported by the National Natural Science Foundation of China under Grants 61201276, 61178068, and 61301266, the Fundamental Research Funds of Central Universities under Grants ZYGX2012Z001, ZYGX2013J012, ZYGX2014J013 and ZYGX2014Z005, the Chinese Postdoctoral Science Foundation under Grant 2014M550465, and by the Program for New Century Excellent Talents in University under Grant A1098524023901001063. (Corresponding author: Xiaolong Li.) The authors are with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu City 611731, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2015.2437844

range migration (TRM) within the coherent integration time, which result in serious performance loss for the coherent integration processing. The problem of correcting range migration (RM) and achieving the coherent integration for maneuvering targets has received considerable attention recently. These works can be classified into three categories according to the target maneuver. The first category addresses the target with a constant velocity which causes only FRM (also called as range walk). The first-order keystone transform (KT), via rescaling the time axis for each frequency, has been employed to remove the FRM and achieve the coherent integration for high speed moving target [11]–[14]. Besides, a novel Radon Fourier transform (RFT) was proposed to realize the coherent accumulation for the moving target with FRM [15]–[17]. Nevertheless, the SRM induced by the target’s acceleration cannot be mitigated by the first-order KT or RFT and thus it will suffer integration performance loss in case of the SFM. The second category considers the target with a constant acceleration which brings about SRM. In [18], [19], the secondorder KT was applied to eliminate the SRM and achieve the coherent integration. However, they [18], [19] do not consider the Doppler ambiguity. Due to the high speed of target and low radar pulse repetition frequency, Doppler ambiguity would occur and then the second-order KT may become invalid. A method combining second-order KT and modified fractional Radon transform (MFrRT), i.e., SKT-MFrRT, was then introduced to estimate the Doppler ambiguity number and realize the coherent accumulation [20]. Unfortunately, the radial jerk of target is not taken into consideration. For the maneuvering targets with complex motions, it is highly possible that the radial jerk exists and then the algorithms aforementioned [18]–[20] will not be appropriate. The third category focuses on a maneuvering target with jerk motion, which is a challenging task for coherent integration due to the TRM. The generalized Radon Fourier transform (GRFT) was then presented to remove the TRM and conduct the coherent integration processing [15], [21]. Although GRFT can obtain coherent accumulation via jointly searching in target’s motion parameter space, it has two limitations. Firstly, because of discrete pulse sampling, finite range resolution and limited integration time, the blind speed sidelobe (BSSL) may inevitably appear in the GRFT output, which leads to serious false alarm and detection performance deterioration [22]. A BSSL suppression method, based on the minimum operator of two weighted GRFT outputs, was studied in [22]. Nevertheless, it requires to bisect the coherent integration period, thus the integration performance would be decreased by about 3 dB. Secondly, the GRFT method is often computationally prohibitive, since it involves the solution of a four-dimensional searching. An im-

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KONG et al.: COHERENT INTEGRATION ALGORITHM FOR A MANEUVERING TARGET WITH HIGH-ORDER RANGE MIGRATION

proved particle swarm optimization (PSO) was then introduced to reduce the computational burden of GRFT [23]. However, the PSO may sensitive to the initial parameters set and get trapped in the local optimization for the scenario of multiple targets. This paper considers the coherent integration problem for a maneuvering target with jerk motion, involving TRM, SRM and FRM within the coherent pulse interval. A new coherent integration method based on generalized KT and second-order dechirp process (SDP) is proposed, where the third-order KT, six-order KT, second-order KT and fold factor searching are respectively employed to remove the TRM, SRM and FRM. After that, cross range Fourier transform (FT) is carried out to achieve the coherent accumulation of target energy. We also theoretically analyze the RM change and computational complexity. Compared with the GRFT algorithm [15], [21], the presented method can acquire close coherent integration performance and avoid the BSSL but with much lower computational cost. Finally, based on the proposed algorithm, a generalized method, i.e., generalized KT and generalized dechirp process (GKTGDP), is also introduced for a maneuvering target with arbitrary high-order RM. The remainder of this paper is organized as follows. In Section II, we present the coherent integration problem for a maneuvering target with jerk motion. In Section III, the coherent integration algorithm via generalized KT and SDP is proposed. In Section IV, the RM change and computational complexity are analyzed. In Section V, we evaluate the performance of the proposed method via several numerical experiments. Section VI introduces the GKTGDP method. Finally, in Section VII, we provide some concluding remarks and possible future research tracks. II. PROBLEM FORMULATION Suppose that the radar transmits a linear frequency modulated (LFM) signal, i.e.,

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where denotes the signal bandwidth, is the range frequency corresponding to the fast time is the Doppler frequency and is the target’s radial velocity. The instantaneous slant range between radar and the maneuvering target with jerk motion satisfies [21], [24], [25] (4) is the initial slant range from the radar platform to the where target, and denote respectively the radial acceleration and radial jerk of the target. Substituting (4) into (3) yields

(5) Due to the high speed of target and low pulse repetition frequency , undersampling would occur. Therefore, the velocity of target can be expressed as [7] (6) where

is the blind velocity, is the fold factor, and is the unambiguous velocity which satisfies

. Substituting (6) into (5) yields

(1) where

is the pulsewidth,

de-

notes the carrier frequency, is the FM rate, is the fast time, is the number of radar pulses, is the pulse repetition interval, and denotes the slow time. The received signal of a moving target after down conversion can be stated as [15], [21]

(7) where (8) (9)

(2) is the backscattering coefficient, denotes the where is the wavelength, and is the instanlight speed, taneous slant range between radar and the target. After range compression, the compressed signal in the slow time-range frequency domain can be expressed as

(3)

denotes the fold factor term caused by underand sampling. Suppose that and ignoring the effect of on can be simplified into [2], [7]. Thus, (7) and (8) can be rewritten as

(10)

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Fig. 1. Block diagram for the proposed method.

(11) It is worth pointing out that (11) is obtained by substituting [2], [5] into (11). In (10), it is shown that the cubic term, the quadratic term and the first-order term of slow time are all coupled with the range frequency , which would result in respectively TRM, SRM and FRM within the coherent integration time. Additionally, the quadratic term and cubic term of will bring about Doppler frequency migration (DFM) [9], [26] which may make the signal energy defocused. Both RM (including TRM, SRM and FRM) and DFM effects will create difficulties during the coherent accumulation of target energy. III. COHERENT INTEGRATION METHOD GENERALIZED KT AND SDP

From (12), it is observed that the phase in the cross range dimension is a cubic chirp signal. Motivated by the dechirp process technique [5], [27], which is an effective method to estimate the quadratic phase term of a chirp signal, we propose a method named SDP to estimate the cubic phase term. Denote the phase compensation function (14) where and represent the searching radial acceleration and searching radial jerk, respectively. Multiplying (12) with (14), we have

VIA

In this section, a method based on generalized KT and SDP is proposed to eliminate the RM and achieve the coherent integration for the maneuvering target with jerk motion, where the block diagram of the presented method is given in Fig. 1. A. TRM Correction We borrow the idea of the first-order KT [11] and secondorder KT [18], apply the third-order KT, which performs scaling in the domain, to correct the TRM. Substituting the scaling formula into (10) yields

(15) Equation (15) shows that if (independent of range frequency ) and (coupled with range frequency ), the signal becomes a complex sinusoid signal over with a certain and reaches the best focusing. Therefore, can be estimated as (16) where

(12)

denotes the FT over

the process in (14)–(16) is similar to the dechirp process [5], [27], so we call it as SDP. Because the target energy is still not located in the same range cell during the integration time after third-order KT, we need to search for the jerk and acceleration values in the range frequency cells whose frequencies are within the range of signal bandwidth. Then, those estimated values should be averaged to obtain the estimation of target jerk

where (13) By (12), the TRM has been removed. However, part of the SRM and FRM are still present.

dimension. Interestingly,

, where

denotes the number of those range frequency cells. Using the estimated jerk to construct the phase function , and compensating for (12) with , we have

B. Cubic Phase Term Compensation Prior to the correction of the remaining SRM, the cubic phase term, which is caused by the target’s radial jerk, should be estimated and compensated. Without this process, the phase of different echo pulses cannot be aligned completely and the signal energy would be defocused.

(17)

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C. Remaining SRM Correction In order to correct the remaining SRM, the six-order KT, which performs in the domain, is employed. Substituting the scaling formula into (17) yields

By (21), the residual FRM induced by unambiguous velocity is removed. Secondly, define the fold factor term compensation function as (23) where denotes the searching fold factor. Multiplying (21) with (23), we have

(18) where

(24) If

, then (24) can be written as

(19)

(25)

By (18), it can be seen that the residual SRM has been eliminated. Nevertheless, part of the FRM is still existed.

Applying inverse Fourier transform (IFT) to (25) with variable , we have

D. Quadratic Phase Term Compensation The quadratic phase term, which is induced by the target’s radial acceleration, should be estimated and compensated before the correction of the residual FRM. It can be seen from (18) that the phase in the cross range dimension is a chirp signal. Hence, the dechirp process (DP) [5], [27] can be applied for estimating the radial acceleration. With the estimated acceleration , we can construct the quadratic phase compensation function . Then, compensating for (18) with , we have

(20)

(26) Equation (26) shows that when the searching fold factor equals to , the signal becomes a complex sinusoid signal over and reaches the best focusing. Therefore, the fold factor of target can be estimated by (27) where

denotes the IFT over

dimension.

With the estimated fold factor , we can compensate the fold factor term and obtain the signal (26). It can be seen from (26) that the signal energy is concentrated in the same range cell, i.e., the RM is corrected. F. Integration via FT Applying cross range FT to (26) with respect to

, we have

E. Residual FRM Correction From (20), it can be seen that the residual FRM including ,

(28)

which is caused by the half of the unambiguous velocity. The second part is the fold factor term that induced by integral multiples of blind velocity. We will remove this two remaining FRM one by one in the following. Firstly, the second-order KT, which performs scaling

denotes the coherent integration time. where From (28), it is observed that the coherent integration of target energy is achieved. The target is detected if the peak value of (28) is larger than a given threshold. Detailed flowchart of the presented method is shown in Fig. 2. Furthermore, we can obtain the estimations of target’s initial slant range and unambiguous velocity based on the peak location of (28). Then, with the estimated fold factor and the unambiguous velocity , the radial velocity of the target can be estimated as follows (29)

two parts. The first part is the term

in the domain, is applied. Substituting the scaling formula into (20) yields

(21) where (22)

IV. RM CHANGE AND COMPUTATIONAL COMPLEXITY In this section, we first analyze the RM change and computational complexity of the proposed method, then, some important remarks are provided.

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Second, the signal after third-order KT and cubic phase term compensation is shown in (17). By taking first-order Taylor series expansion over (17) can be written as

and

, respectively,

(32) [7]. As a reNote that sult, the TRM caused by jerk is removed and the SRM is reduced to one-third of its original value. Besides, the FRM induced by blind velocity is reduced to two-thirds of its original value. Hence, the RM change in this step is (33) Third, after six-order KT and quadratic phase term compensation, the signal is shown in (20). Taking the first-order Taylor series expansion over

, (20) can be recast as

(34) Fig. 2. Detailed flowchart of the proposed method.

From (34), it can be seen that the residual SRM is eliminated and the FRM caused by blind velocity is reduced to half of the original value. Therefore, the RM change during this step is

A. RM Change Note that the proposed method is composed of four main steps: range compression and range FT, third-order KT and cubic phase term compensation, six-order KT and quadratic phase term compensation, second-order KT and fold factor term compensation, as shown in Fig. 3. In the following, we will discuss the RM change of the presented method step by step. First, the raw signal after range compression and range FT can be rewritten as follows

(35) Finally, after second-order KT and fold factor term compensation with the estimated fold factor, the signal can be rewritten as (36) By (36), the residual FRM induced by the half of the unambiguous velocity and integral multiples of blind velocity has been removed. Thus, the RM correction in this step is (37) The detailed RM change in different steps of the proposed method is shown in Table I. In addition, we verify the quality of the above analysis for RM change in the numerical simulation section (see Figs. 4–7).

(30) It can be seen from (30) that the total RM (including TRM, SRM and FRM) caused by target’s motions, i.e., jerk, acceleration and velocity, during the coherent integration period is (31) where and denote respectively the initial value of TRM, SRM and FRM.

B. Computational Complexity In what follows, the computational complexity of major steps in the proposed method will be analyzed in terms of the number of operations, i.e., complex multiplications (Mc) and additions (Ac). Denote the number of range cells, echo pulses, searching velocity, searching acceleration, searching jerk and searching fold factor by , respectively. For the TRM correction, Mc and Ac are needed. As to radial jerk estimation and


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Fig. 3. Four main steps of the presented method.

Fig. 4. Result after range compression (a) Echoes in the

domain. (b) Echoes of 1st and 200th pulses.

Fig. 5. Result after third-order KT and cubic phase term compensation (a) Echoes in the


TABLE I RM CHANGE DURING DIFFERENT STEPS OF THE PROPOSED METHOD

cubic phase term compensation, Mc and Ac are needed. Besides, for the reMc and Ac are maining SRM correction, needed. As to estimation of acceleration and quadratic phase term compensation, Mc and Ac are needed. Moreover, for the second-order KT and estimation of fold factor, Mc and Ac are needed. As to the residual FRM compensation and coherent integration via FT, Mc and Mc and Ac are needed. On the other hand, Ac are needed for the GRFT algorithm [21].

The detailed computational complexity of the presented method and the GRFT algorithm is given in Table II. Suppose that , then the computational complexity of proposed method is whereas the computational burden of GRFT is . C. Some Remarks on the Proposed Method 1) Remark 1: In the above analysis, the presented method is derived in the scenario with single target. However, it can also be extended to the scenario with multiple targets. Moreover, if the scattering intensities of different targets differ significantly, the CLEAN technique [28]–[30] could be applied to eliminate

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Fig. 6. Result after six-order KT and quadratic phase term compensation (a) Echoes in the

Fig. 7. Result after second-order KT and fold factor term compensation (a) Echoes in the


domain. (b) Echoes of the 1st and 200th pulses.

TABLE II COMPUTATIONAL COMPLEXITY OF THE PROPOSED METHOD AND GRFT

the effect of the strong target. In this way, the coherent integration of strong moving target and weak ones can be achieved iteratively. 2) Remark 2: The avoidance of BSSL for the proposed method is conditional with the maximum operators. This is because when the integration time or the signal bandwidth is reduced, the peaks of BSSL may be as high as the correct peaks. Therefore, an appropriate increase in the coherent integration time or signal bandwidth is still very necessary for real applications. Besides, when multiple targets have the similar power distribution, the presented method may be also difficult extended, although the “CLEAN” technique is used. Fortunately, due to the high speed or mobility of targets, the trajectories are quite different with different motion parameters [3]. Hence, it is not common in real applications and we can separate the multiple targets according to their different motion parameters.

3) Remark 3: For realization of the proposed generalized KT (i.e., third-order KT, six-order KT and second-order KT), the interpolation processing is normally employed for the keystone transformation [31]. Also, there are some realizations of keystone transform without interpolation, such as chirp transform [12], [32], scaled fast FT [33] and Chirp-Z transform [34]. Then the computational burden can be reduced. However, they are at the cost of some performance loss. As a result, we should make a tradeoff between the computational burden and precision. In our paper, the sinc-interpolation is used because of its robustness and desirable performance [5], [35], [36]. V. NUMERICAL RESULTS In this section, several numerical simulations are presented to demonstrate the effectiveness of the proposed coherent integration algorithm for a maneuvering target with jerk motion,


TABLE III SIMULATION PARAMETERS OF RADAR

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well-focused thanks to its ability to deal with TRM, SRM and FRM, which is helpful to target detection. C. Coherent Integration for Multiple Targets

where the parameters of the radar are listed in Table III. Comparisons with other popular coherent integration methods, i.e., moving target detection (MTD), SKT-MFrRT [20], GRFT [15], [21] and weighted GRFT [22] are also given. A. RM Correction Performance We first analyze the RM correction performance of the presented method in Figs. 4–7, where the motion parameters of a maneuvering target are km, m/s, m/s and m/s . The result after range compression is domain and the shown by Fig. 4, where the echoes in the result of the 1st and 200th pulses are shown in Fig. 4(a) and (b), respectively. It is observed that serious RM occurs and the total RM value is 276. Then the third-order KT and cubic phase term compensation are performed to Fig. 4, and the results are shown in Fig. 5(a) and (b). It can be seen from Fig. 5(b) that the total RM value is reduced to 247, i.e., 29 range cells less than that of Fig. 4(b). Theoretically, after third-order KT and cubic phase term compensation, the number of cross-range cells is reduced by , where denote the range sampling rate. Besides, the results after six-order KT and quadratic phase term compensation are shown in Fig. 6(a) and (b). It can be seen from Fig. 6(b) that the total RM value is reduced to 240, which indicates 7 range cells less than that of Fig. 5(b). Theoretically, after six-order KT and quadratic phase compensation, the number of cross range cells is further reduced by . Furthermore, the results after second-order KT and fold factor term compensation are shown in Fig. 7(a) and (b). It can be seen that the RM is reduced to 0, i.e., the RM is removed. Consequently, the simulation results are consistent with the theoretical calculation, which demonstrates the correctness of the derivation in Section IV.A. B. Coherent Integration for a Single Target In the second simulation, we evaluate the coherent integration performance for a single target via MTD, SKT-MFrRT and the proposed method, where complex white Gaussian noise is added to the echoes and the SNR of the target is dB. The motion parameters of the maneuvering target are km, m/s, m/s and m/s . Fig. 8(a) shows the result after range compression in the domain, in which the target trajectory is blurry. Fig. 8(b) shows the MTD processing result. From this figure, we can see that the energy spread and parameters of the target are difficult to estimate, which indicates that MTD becomes ineffective due to the RM. Fig. 8(c) shows the integration result of SKT-MFrRT method, where the target energy is defocused. This is because SKTMFrRT is only appropriate for the target with a constant acceleration and would become invalid in the presence of TRM. Furthermore, the coherent integration result of the proposed method is shown in Fig. 8(d). It can be seen that the target’s energy is

We also analyse the coherent integration performance for multiple targets of the presented method in Fig. 9. The motion parameters of two maneuvering targets are listed in Table IV and the SNRs of the two targets are both dB before pulse compression. Fig. 9(a) shows the result after pulse compression and it can be seen that serious RM occurs. Fig. 9(b) shows the result of second-order dechirp process after third-order KT, where the peak values indicate the estimated jerks of target A and target B. The coherent integration results of target A and target B via the proposed method are shown in Fig. 9(c) and (d), respectively. We can see that there are two peaks clearly located in the outputs of the presented method, which is an advantage for target detection. In addition, Fig. 9(e)–(h) show respectively the coherent integration results of target A and target B via GRFT and weighted GRFT. Although GRFT and weighted GRFT are also able to achieve the coherent accumulation via four-dimensional searching, the BSSL with large peak value appears in the GRFT outputs and weighted GRFT outputs, which may lead to serious false alarm and make it hard to obtain the number of targets. D. Integration Performance and Computational Cost 1) The coherent integration performance of the proposed method is further investigated by Monte Carlo trials, where the simulation parameters of the moving target are the same as those in Fig. 8. Fig. 10 shows the coherent integration performance curves of GRFT, weighted GRFT and the proposed method under different SNR levels (defined before pulse compression). In each case, 500 times of Monte Carlo simulations are done. The theoretical outputs peak value after range compression and long-time coherent integration can be given as (38) where denotes the time bandwidth product of the transmitting signal. The curves show that presented method can achieve close integration performance as the GRFT algorithm. It is worth pointing out that the proposed method suffers a slight integration performance loss in comparison with GRFT, which is caused by the interpolation of the keystone transformation. The plots also illustrate that the integration performance of the presented method is superior to that of weighted GRFT method. This is because the weighted GRFT needs to bisect the coherent integration period and then the integration performance would be decreased by about 3 dB. 2) The computational complexity and time cost of the proposed method and GRFT are also evaluated. Specifically, we first compare the computation complexities of the presented method and GRFT in Fig. 11, where the computational complexity curves for different pulse numbers are plotted. It is evident that the computational cost of GRFT is much larger than the proposed method. For , the computational complexity of the presented method is about of the GRFT. Furthermore, Table V illustrates the time cost of GRFT and the proposed method. We

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Fig. 8. Coherent integration for a single target via MTD, SKT-MFrRT and the proposed method. (a) Result after range compression in result of MTD. (c) Integration result of SKT-MFrRT. (d) Integration result of the proposed method. TABLE IV MOTION PARAMETERS OF TARGETS

can see that the GRFT costs more time than the presented method due to the four-dimensional searching. E. Detection Performance Analysis In this subsection, the detection performances of MTD, SKT-MFrRT, GRFT and the proposed method are investigated by Monte Carlo trials. We combine the constant false alarm (CFAR) detector and the four methods as corresponding detectors. The Gaussian noises are added to the target echoes and the false alarm ratio is set as . Fig. 12 shows the detection probability of the four detectors versus different SNR levels and in each case, 1000 times of Monte Carlo simulations are done. The simulation results show that the presented method can obtain close detection performance as the GRFT. In addition, the probability of the detector based on the proposed method is superior to MTD and SKT-MFrRT thanks to its ability to deal with the high-order RM. From the simulation results above (Figs. 8–12 and Table V), we can conclude that the proposed method is able to achieve close integration and detection performance and avoid the BSSL effect but with much lower computational cost, in comparison with the GRFT algorithm. Besides, the coherent

domain. (b) Integration

integration ability of the presented method is superior to MTD, SKT-MFrRT and weighted GRFT for the maneuvering target with jerk motion. VI. EXTENSIONS TO ARBITRARY HIGH-ORDER RM Based on the method presented in Section III, we further introduce a more generalized coherent integration method via generalized keystone transform and generalized dechirp process (GKTGDP), for a maneuvering target with arbitrary high-order RM. The detailed discussion is given in the following. A. Definitions of Generalized KT and Generalized DP 1) The th-order KT is defined as (39) where and denote respectively the slow time before and after -order KT . Obviously, the first-order KT [11] and the second-KT [18] are respectively two special cases of (39) when and . Hence, we call (39) as generalized keystone transform (GKT). 2) Suppose that is a polynomial phase signal (PPS) given by [37] (40) where and

is the order of the PPS, is the amplitude denote the phase parameters.


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Fig. 9. Coherent integration for multiple targets via the proposed method, GRFT and weighted GRFT. (a) Result after pulse compression. (b) Result of secondorder dechirp process. (c) Integration result of target A via the proposed method. (d) Integration result of target B via the proposed method. (e) Integration result of target A via GRFT. (f) Integration result of target B via GRFT. (g) Integration result of target A via weighted GRFT. (h) Integration result of target B via weighted GRFT.

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Fig. 10. Coherent integration performance of GRFT, weighted GRFT and the proposed method under different SNR background.

Fig. 12. Detection probability of MTD, SKT-MFrRT, GRFT and the proposed method.

where are used to denote the target motion parameters with different orders as acceleration, jerk, , respectively. Similarly to (7), the received signal after range compression in the slow time-range frequency domain can be expressed as

(44)

Fig. 11. Computational complexity versus integration pulse number.

Then, the th-order DP is defined as (41) denotes the th-order searching where parameters. From (41), we can see that when reaches its maximum value. Hence, the parameters, i.e., , can be estimated by (42) It is worth pointing out that the DP is a special cases of (41)–(42). Therefore, we call (41)–(42) as generalized dechirp process (GDP). B. GKTGDP Method Suppose that the slant range between the radar and a maneuvering target with th-order RM is as follows [21] (43)

Then, the coherent integration method for a maneuvering target with th-order RM based on GKTGDP, can be described as follows. Step 1) Initialization: . Step 2) Perform th-order KT to correct the th-order RM. Step 3) Apply the th-order DP to estimate and then compensate the th-order phase term; and . Step 4) perform th-order KT to remove the remaining th-order RM. Step 5) If , go to steps 3–4. Step 6) If , obtain the estimation of via fold factor searching. Step 7) Compensate the fold factor phase term using and do IFT with respect to range frequency . Step 8) Perform cross range FT and achieve the coherent integration of target energy. If the peak value is higher than the predetermined threshold, then the target is detected. The flowchart of the coherent integration algorithm based on GKTGDP is shown in Fig. 13. VII. CONCLUSION In this paper, we have addressed the coherent integration problem for a maneuvering target with jerk motion, involving


TABLE V TIME COST OF GRFT AND THE PROPOSED METHOD

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[25]. Finally, it might be of interest to extend the framework to SAR imaging or inverse SAR imaging [6], [38]. REFERENCES

Fig. 13. Flowchart of the coherent integration algorithm for a maneuvering target with high-order RM based on GKTGDP.

TRM, SRM, and FRM within the coherent integration period. Summarizing: • We proposed a method based on GKT and second-order DP to realize the coherent integration for the maneuvering target with TRM. This method can effectively correct the TRM, SRM and FRM one by one via third-order KT, sixorder KT, second-order KT and fold factor searching. After that, cross range FT is carried out to achieve the coherent accumulation of target energy. • We also theoretically analyzed the RM change and computational complexity during each step of the presented method. The step-by-step complexity analysis shows that the proposed method can reduce the complexity order from to in comparison with the GRFT algorithm. • We have evaluated the performance of the presented method by several numerical simulations. Compared with GRFT algorithm, the results highlighted that, the proposed method can avoid the BSSL and obtain close integration performance but with much lower computational cost. In addition, the presented method is superior to MTD and SKT-MFrRT in terms of coherent integration ability thanks to its capacity to deal with TRM, SKM and FRM. • Furthermore, based on the proposed algorithm, we introduced a more generalized coherent integration method, i.e., GKTGDP, for the maneuvering target with arbitrary high-order RM. A possible future work might concern the study of the coherent integration for targets with motion order higher than jerk

[1] S. Q. Zhu, G. S. Liao, D. Yang, and H. H. Tao, “A new method for radar high-speed maneuvering weak target detection and imaging,” IEEE Geosci. Remote Sens. Lett., vol. 11, no. 7, pp. 1175–1179, Jul. 2014. [2] M. D. Xing, J. H. Su, G. Y. Wang, and Z. Bao, “New parameter estimation and detection algorithm for high speed small target,” IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 1, pp. 214–224, Jan. 2011. [3] X. L. Chen, J. Guan, N. B. Liu, and Y. He, “Maneuvering target detection via radon-fractional fourier transform-based long-time coherent integration,” IEEE Trans. Signal Process., vol. 62, no. 4, pp. 939–953, Feb. 2014. [4] X. L. Li, G. L. Cui, L. J. Kong, W. Yi, X. B. Yang, and J. J. Wu, “High speed maneuvering target detection based on joint keystone transform and CP function,” in Proc. IEEE Radar Conf., May 2014, pp. 0436–0440. [5] J. Su, M. D. Xing, G. Wang, and Z. Bao, “High-speed multi-target detection with narrowband radar,” IET Radar Sonar Navig., vol. 4, no. 4, pp. 595–603, Aug. 2010. [6] Y. C. Li, M. D. Xing, Y. H. Quan, and Z. Bao, “A new algorithm of ISAR imaging for maneuvering targets with low SNR,” IEEE Trans. Aerosp. Electron. Syst., vol. 49, no. 1, pp. 543–557, Jan. 2013. [7] J. Tian, W. Cui, and S. Wu, “A novel method for parameter estimation of space moving targets,” IEEE Geosci. Remote Sens. Lett., vol. 11, no. 2, pp. 389–393, Feb. 2014. [8] M. A. Richards, “Coherent integration loss due to white Gaussian phase noise,” IEEE Signal Process. Lett., vol. 10, no. 7, pp. 208–210, July 2003. [9] R. Tao, N. Zhang, and Y. Wang, “Analysing and compensating the effects of range and Doppler frequency migrations in linear frequency modulation pulse compression radar,” IET Radar Sonar and Navig., vol. 5, no. 1, pp. 12–22, Jan. 2011. [10] X. L. Li, G. L. Cui, W. Yi, and L. J. Kong, “Coherent integration for maneuvering target detection based on Radon-LV’s distribution,” IEEE Signal Process. Lett., vol. 22, no. 9, pp. 1467–1471, Sep. 2015. [11] R. P. Perry, R. C. Dipietro, and R. L. Fante, “SAR imaging of moving targets,” IEEE Trans. Aerosp. Electron. Syst., vol. 35, no. 1, pp. 188–200, Jan. 1999. [12] D. Y. Zhu, Y. Li, and Z. D. Zhu, “A keystone transform without interpolation for SAR ground moving-target imaging,” IEEE Geosci. Remote Sens. Lett., vol. 4, no. 1, pp. 18–22, Jan. 2007. [13] G. Li, X. G. Xia, and Y. N. Peng, “Doppler keystone transform: An approach suitable for parallel implementation of SAR moving target imaging,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 4, pp. 573–577, Oct. 2008. [14] S. J. Yuan, T. Wu, M. Mao, G. J. Mei, and X. Wei, “Application research of keystone transform in weak high-speed target detection in low-PRF narrowband chirp radar,” in Proc. 9th Int. Conf. Signal Process., Beijing, China, Oct. 2008, pp. 2452–2456. [15] J. Xu, J. Yu, Y. N. Peng, and X. G. Xia, “Radon-fourier transform (RFT) for radar target detection (I): Generalized doppler filter bank processing,” IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 2, pp. 1186–1202, Apr. 2011. [16] L. C. Qian, J. Xiu, X. G. Xia, W. F. Sun, T. Long, and Y. N. Peng, “Wideband-scaled Radon-Fourier transform for high-speed radar target detection,” IET Radar Sonar Navig., vol. 8, no. 5, pp. 501–512, June 2014. [17] J. Xu, J. Yu, Y. N. Peng, X. G. Xia, and T. Long, “Space-time Radon-Fourier transform and applications in radar target detection,” IET Radar Sonar Navig., vol. 6, no. 9, pp. 846–857, Dec. 2012. [18] D. Kirkland, “Imaging moving targets using the second-order keystone transform,” IET Radar Sonar Navig., vol. 5, no. 8, pp. 902–910, Oct. 2011. [19] Y. Jungang, H. Xiaotao, J. Thompson, J. Tian, and Z. Zhimin, “Low-frequency ultra-wideband synthetic aperture radar ground moving target imaging,” IET Radar Sonar Navig., vol. 5, no. 9, pp. 994–1001, Dec. 2011. [20] G. Sun, M. D. Xing, Y. Wang, F. Zhou, Y. Wu, and Z. Bao, “Improved ambiguity estimation using a modified fractional radon transform,” IET Radar Sonar Navig., vol. 5, no. 4, pp. 489–495, Apr. 2011. [21] J. Xu, X. G. Xia, S. B. Peng, J. Yu, Y. N. Peng, and L. C. Qian, “Radar maneuvering target motion estimation based on generalized Radon-Fourier transform,” IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6190–6201, Dec. 2012. [22] J. Xu, J. Yu, Y. N. Peng, and X. G. Xia, “Radon-Fourier transform (RFT) for radar target detection (II): Blind speed sidelobe suppression,” IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 4, pp. 2473–2489, Oct. 2011.

4486


[23] L. C. Qian, J. Xu, X. G. Xia, W. F. Sun, T. Long, and Y. N. Peng, “Fast implementation of generalized Radon-Fourier transform for manoeuvring radar target detection,” Electron. Lett., vol. 48, no. 22, pp. 1427–1428, Oct. 2012. [24] Y. Wang and Y. C. Jiang, “Inverse synthetic aperture radar imaging of maneuvering target based on the product generalized cubic phase function,” IEEE Geosci. Remote Sens. Lett., vol. 8, no. 5, pp. 958–962, Sept. 2011. [25] Y. Wang and Y. C. Jiang, “ISAR imaging of maneuvering target based on the L-class of fourth-order complex-lag PWVD,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 3, pp. 1518–1527, Mar. 2010. [26] X. L. Li, G. L. Cui, W. Yi, and L. J. Kong, “A fast maneuvering target motion parameters estimation algorithm based on ACCF,” IEEE Signal Process. Lett., vol. 22, no. 3, pp. 270–274, Mar. 2015. [27] W. P. Li, “Wigner distribution method equivalent to dechirp method for detecting a chirp signal,” IEEE Trans. Acoust., Speech, Signal Process., vol. 35, no. 8, pp. 1210–1211, Aug. 1987. [28] J. Tsao and B. D. Steinberg, “Reduction of sidelobe and speckle artifacts in microwave imaging: The CLEAN technique,” IEEE Trans. Antennas Propag., vol. 36, no. 4, pp. 543–556, Apr. 1998. [29] H. Deng, “Effective CLEAN algorithms for performance-enhanced detection of binary coding radar signals,” IEEE Trans. Signal Process., vol. 52, no. 1, pp. 72–78, Jan. 2004. [30] Q. Wang, M. D. Xing, G. Y. Lu, and Z. Bao, “SRMF-CLEAN imaging algorithm for space debris,” IEEE Trans. Antennas Propag., vol. 55, no. 12, pp. 3524–3533, Dec. 2007. [31] K. Huo, Y. X. Liu, J. M. Hu, W. D. Jiang, and X. Li, “A novel imaging method for fast rotating targets based on the segmental pseudo keystone transform,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 4, pp. 1464–1472, Apr. 2011. [32] G. C. Sun, M. D. Xing, X. G. Xia, Y. R. Wu, and Z. Bao, “Robust ground moving-target imaging using deramp-keystone processing,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 2, pp. 966–982, Feb. 2013. [33] K. Natroshvili, O. Loffeld, H. Nies, A. M. Ortiz, and S. Knedlik, “Focusing of general bistatic SAR configuration data with 2-D inverse scaled FFT,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 10, pp. 2718–2727, Oct. 2006. [34] R. Lanari, “A new method for the compensation of the SAR range cell migration based on the chirp Z-transform,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 5, pp. 1296–1299, Sep. 1995. [35] T. D. Deng and C. S. Jiang, “Evaluations of keystone transforms using several interpolation methods,” in Proc. IEEE CIE Int. Conf. Radar, Chengdu, China, Oct. 2011, vol. 2, pp. 1876–1878. [36] S. S. Zhang, T. Zeng, T. Long, and H. P. Yang, “Dim target detection based on keystone transform,” in Proc. IEEE Int. Radar Conf., May 2005, pp. 889–894. [37] S. Peleg and B. Friedlander, “The discrete polynomial-phase transform,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 1901–1904, Aug. 1995. [38] A. De Maio, G. Fornaro, and A. Pauciullo, “Detection of single scatterers in multidimensional SAR imaging,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 7, pp. 2284–2297, Jul. 2009. Lingjiang Kong was born in 1974. He received the B.S., M.S., and Ph.D. degrees from the University of Electronic Science and Technology of China (UESTC) in 1997, 2000 and 2003, respectively. From September 2009 to March 2010, he was a visiting researcher with the University of Florida. He is currently a professor with the School of Electronic Engineering, University of Electronic Science and Technology of China (UESTC). His research interests include multiple-input multiple-output (MIMO) radar, through the wall radar, and statistical signal processing.

Xiaolong Li was born in Jiangxi, China. He received the B.S. degree in University of Electronic Science and Technology of China, in 2011. Currently, he is working toward the Ph.D. degree at the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China. His research interests include signal detection, parameter estimation and inverse synthetic aperture radar (ISAR) signal processing (particular emphasis on coherent integration technique for maneuvering target detection).

Guolong Cui received the B.S., M.S., and Ph.D. degrees from University of Electronic Science and Technology of China (UESTC), Chengdu, in 2005, 2008, and 2012, respectively. From January 2011 to April 2011, he was a visiting researcher with University of Naples Federico II, Naples, Italy. From June 2012 to August 2013, he was a postdoctoral researcher in the Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ. Since September 2013, he has been an Associate Professor in University of Electronic Science and Technology of China (UESTC). His current research interests include statistical signal processing in the field of statistical signal processing with emphasis on radars, waveform optimization, and passive sensing.

Wei Yi received the B.E. and the Ph.D. degrees in 2006 and 2012, respectively, both in electronic engineering from the University of Electronic Science and Technology of China, Chengdu. From March 2010 to February 2012, he was a visiting student in the Melbourne Systems Laboratory, University of Melbourne, Australia. Since 2013 he has been a Research Fellow at the School of Electronic Engineering, University of Electronic Technology and Science of China. His research interests include particle filtering and target tracking (particular emphasis on multiple target tracking and track-before-detect techniques).

Yichuan Yang was born in 1988. He received the B.S. degree in University of Electronic Science and Technology of China, in 2011. Currently, he is working toward the Ph.D. degree at the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China. His research interests include signal detection, parameter estimation and multiple-input multiple-output (MIMO) radar signal processing.