Radar Maneuvering Target Motion Estimation ... - Semantic Scholar

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 12, DECEMBER 2012

Radar Maneuvering Target Motion Estimation Based on Generalized Radon-Fourier Transform Jia Xu, Member, IEEE, Xiang-Gen Xia, Fellow, IEEE, Shi-Bao Peng, Ji Yu, Ying-Ning Peng, Senior Member, IEEE, and Li-Chang Qian

Abstract—The slant range of a radar maneuvering target is usually modeled as a multivariate function in terms of its illumination time and multiple motion parameters. This multivariate range function includes the modulations on both the envelope and the phase of an echo of the coherent radar target and provides the foundation for radar target motion estimation. In this paper, the maximum likelihood estimators (MLE) are derived for motion estimation of a maneuvering target based on joint envelope and phase measurement, phase-only measurement and envelope-only measurement in case of high signal-to-noise ratio (SNR), respectively. It is shown that the proposed MLEs are to search the maximums of the outputs of the proposed generalized Radon-Fourier transform (GRFT), generalized Radon transform (GRT) and generalized Fourier transform (GFT), respectively. Furthermore, by approximating the slant range function by a high-order polynomial, the inherent accuracy limitations, i.e., the Cramer-Rao low bounds (CRLB), and some analysis are given for high order motion parameter estimations in different scenarios. Finally, some numerical experimental results are provided to demonstrate the effectiveness of the proposed methods. Index Terms—Cramer-Rao low bound (CRLB), generalized Radon-Fourier transform (GRFT), maneuvering target, parameter estimation, polynomial phase signal (PPS), root mean-square error (RMSE).

I. INTRODUCTION

H

IGH-PRECISION motion estimation of maneuvering targets attracts much more attentions for modern radar due to the increasing demands in applications [1]–[5]. First,

Manuscript received December 16, 2011; revised April 19, 2012; accepted July 26, 2012. Date of publication September 04, 2012; date of current version November 20, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Visa Koivunen. This work was supported in part by the National Natural Science Foundation of China under Grant 60971087 and 61271391, by Beijing Municipal Natural Science Foundation under Grant 4122038, by the China Ministry Research Foundation under Grants 9140A07021012JW0101, by the Aerospace Innovation Foundation under Grant CASC201104, by the Aerospace Supporting Foundation, by the Tsinghua National Laboratory for Information Science and Technology (TNList) Cross-discipline Foundation, and Xia’s work was supported in part by Air Force Office of Scientific Research (AFOSR) under Grant FA9550-12-10055 and the World Class University (WCU) Program, National Research Foundation, South Korea. J. Xu is with the School of Information and Electronic, Beijing Institute of Technology, Beijing 100081, China, and also with Tsinghua University, Beijing 100084, China (e-mail: [email protected]). X.-G. Xia is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA, and also with the Chonbuk National University, Jeonju, South Korea. S.-B. Peng, J. Yu, and Y.-N. Peng are with Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. L.-C. Qian is with the Radar Academy of Airforce, Wuhan 430010, China. Digital Object Identifier 10.1109/TSP.2012.2217137

more and more high-speed and high-maneuvering targets appear in the radar searching area with the development of advanced technologies. Second, for radars equipped with phased array antennas [1], [2], the digital beam forming (DBF) technique has been widely used to obtain a long illumination time. Thus, the high-order motions, e.g., acceleration and jerk, can not be omitted to realize the long-time integration [1], [2], [21]–[23] even for a slowly moving target in this long observation time. Third, it is essential for high-resolution imaging radars, e.g., synthetic aperture radar (SAR) [3] and inverse synthetic aperture radar (ISAR) [4], [5], to estimate and compensate the high-order non-cooperative motions for obtaining the focused images. Normally, the slant range of a maneuvering radar target can be modeled as a multivariate function in terms of its illumination time and multiple motion parameters, such as velocity and acceleration. In the mean time, the target’s motion may cause the modulations on both the envelope and the phase of an echo of the coherent radar target, from which radar can measure target’s motion accordingly. In this regard, there have been significant studies over the past decades. Bello [6] and Kelly [7] gave some fundamental discussions in the 1960s to jointly estimate target’s range, velocity and acceleration. Nevertheless, the relationships can not be explicitly depicted between the measurement accuracy and the radar parameters, such as waveform duration, bandwidth and transmission power. For the widely used radar linear frequency modulation (LFM) pulse train, Abatzoglou [8], [9] derived the Cramer-Rao low bound (CRLB) in 1990s for velocity and acceleration estimations based on the maximum likelihood estimator (MLE). Unfortunately, the motion estimation was not further discussed for maneuvering targets with motion-order higher than acceleration. On the other hand, many literatures [10]–[20] have discussed motion estimation of a maneuvering target based on polynomial phase signal (PPS) model. However, the envelope modulation information is normally omitted for these works and may not obtain the ultimate accuracy based on joint envelope and phase information in real applications. Apart from the joint envelope and phase measurement of the coherent radar, some other radars can only estimate target motion based on partial information, i.e., envelope-only and phase-only measurements. For the envelope-only applications, there are two typical examples. First, because the magnetron of transmitter generates unknown pulse-by-pulse initial phases, the early incoherent radar is contaminated with the uncertain phase modulation and only the envelope information can be directly used for measuring target’s motion [1], [2]. Second, to overcome target’s unknown translational motion, it is well-known that ISAR normally makes range alignment based on envelopes

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XU et al.: RADAR MANEUVERING TARGET MOTION ESTIMATION

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of high-resolution range profiles (HRRPs) [4], [5]. As for the phase-only radar applications, two typical examples are as follows. First, the continuous wave (CW) radar [1], [2] has been widely used to measure vehicle motions based on Doppler information, i.e., phase-only measurement. Second, for the searching radar with extremely low range resolution, e.g., the over-thehorizon radar (OTHR), the targets will normally not exceed a range unit during the coherent integration time and the echoes’ envelopes can not reflect the target’s motion. Therefore, the phase-only information, i.e., the Doppler frequency, is normally used for OTHR to retrieve the target motion. It is then interesting to ask what the optimal methods and the inherent accuracy limitations are for maneuvering radar target motion estimation in different cases. In this paper, we provide a unified framework for target motion estimation in three scenarios, i.e., the joint envelope and phase measurement, the phase-only measurement and the envelope-only measurement in case of high SNR. Specifically, for a maneuvering target with arbitrary motion, we propose different signal models and the corresponding maximum likelihood estimators (MLE), i.e., generalized Radon-Fourier transform (GRFT) [21]–[23], generalized Radon transform (GRT) and generalized Fourier transform (GFT), respectively. Interestingly, the proposed GRFT has the same form of the likelihood ratio detector (LRT) [21]–[23] for maneuvering targets, which is the statistically optimal detector in a noisy background. Besides, the last two proposed MLEs, i.e., GFT and GRT, are the two special cases of GRFT where phase-only or envelop-only information is used. Then, by using the proposed GRFT the optimal estimator and detector can be unified for radar moving target detection and parameter estimation jointly. Furthermore, by approximating the arbitrary multivariate slant range function of a maneuvering target into a high-order polynomial, the CRLBs are also obtained as well as some performance analysis for high-order motion estimation. The contributions of this paper include four aspects for maneuvering target motion estimation. First, the signal models are proposed in a unified framework for three different radar scenarios. Second, the optimal MLE methods, i.e., GRFT, GRT and GFT, are derived for the above three applications. Third, CRLBs are further obtained by approximating the slant range function as a high-order polynomial. Fourth, some novel analyses are provided with the proposed CRLB. As a result, the inherent accuracy can be analytically obtained for high-order motion estimation via the proposed CRLB and radar equation based on the given system parameters. The remainder of this paper is organized as follows. In Section II, the signal model and MLEs are established for three kinds of radar scenarios. In Section III, CRLBs are derived. In Section IV, numerical experiments are given as well as the performance analysis for the proposed MLEs. In Section V, some conclusions are given. II. THE MAXIMUM LIKELIHOOD ESTIMATORS MANEUVERING RADAR TARGET

A. MLE Based on Joint Envelope and Phase Measurement Suppose radar transmits a linear frequency modulated (LFM) waveform [1], [2], i.e., (1) where is the quick-time, i.e., the intra-pulse sampling time, is the pulse duration, is the frequency modulation rate of LFM and is the signal bandwidth. During the radar illumination time , the target’s two-dimensional (2D) echoes may be represented as

(2) where is the slow-time, i.e., the inter-pulse sampling time, is the complex amplitude of the target, and are light speed and wavelength, respectively, is the time-varying range of the target during the illumination. Normally, these 2D echoes of a moving target mixed with noise will be A/D converted into the discrete form as shown in Fig. 1(a) along the quick-time with sampling frequency and sampling interval . Also, they are sampled along the slow-time with radar pulse repetition frequency (PRF) and pulse repetition interval (PRI) . Then, the th range unit sampling of the th pulse can be rewritten as

(3) where is the number of pulses, is the number of range units in the range gate where holds as shown in Fig. 1(a) to obtain the whole echoes of a moving target. is the stationary band-pass zero-mean complex additive Gaussian-distributed noise with a restricted receiver passing band and variance . According to the radar equation [1], [2], we have the signal-to-noise ratio (SNR) for discrete signal (3) as (4) and are the radar transmission power and antenna where gain, respectively, and are the backscattering radar cross section (RCS) and the initial range of the target, respectively, is the Boltzmann constant, is the system Kelvin temperature, is the noise coefficient, is the system loss. To apply range discrete Fourier transform (DFT) on (3), we have the result as shown in Fig. 1(b) via the principle of stationary phase (POSP) [3] as

FOR

In this section, the signal models will be proposed in a unified framework as well as the MLEs for three radar scenarios, i.e., the joint envelope and phase measurement, the phase-only measurement and the envelope-only measurement, respectively.

(5)

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Fig. 1. Multiple echoes of a moving target and range compression. (a) Raw echoes. (b) Range frequency domain. (c) After range compression.

, , , is the signal’s unit number in frequency domain and is the zero-mean complex noise with variance . Therefore, the SNR after range DFT may be given as

further extract the 2D signal in range frequency domain from (8) as

(6)

where . Because the noise is approximately independently distributed among range-frequency units, we have independent sampling vectors as shown in Fig. 1(b) with SNR in (6) in range frequency domain for the successive parameter estimation. On the other hand, with the given and it is also shown from (11) that the parameter is linearly proportional to as (5) and (11) while the SNR is inversely proportional to as (6) in frequency domain. Therefore, it is an interesting problem whether the motion parameter estimation accuracies will depend on and we will answer this question in Section III via the derived CRLB. For a certain observation of the moving target with non-fluctuating backscattering, the amplitude is independent of the time, is an envelope operator, while the initial phase is uniformly distributed in but it is also fixed during the integration time . If the time-varying range can be modeled as a multivariate function with parameters as , the unknown parameters of interest can be rewritten as -dimensional vector . Notably, the parameters in this Section are generalized representations for motion parameters and they may have different meanings in different cases. For example, to describe a target motion in a direction of slant range, are used to represent motion parameters with different orders as range, velocity, acceleration, ., respectively. This model is widely used in the conventional radar measurement and its CRLB analysis will be discussed in Section III of this paper. Besides, this model can also be generalized for a target in a 2D plane. For a 2D rectilinearly moving target, we have , where , and are the 2D initial location and velocity of the target, respectively. The CRLB derivation for 2D case is omitted in this paper. Besides, for describing the same motion, may have different equivalent combinations as discussed in [21]. Therefore, we will take a generalized motion model with parameters for the following discussions in Section II. Normally, the ultimate

where

. Then, let us define the LFM matched function where in range frequency domain as (7) Multiplying (7) to (5), we have

(8) Apply IDFT on (8) to obtain the range compressed echoes as shown in Fig. 1(c) as

(9) , is also the corwhere related Gaussian-distributed noise with zero-mean and variance . Therefore, the SNR after range compression is (10) It is shown from (9) and (10) that the target’s 2D rangecompressed peaks are approximately distributed along a curve with certain SNR . Because only units in frequency domain contain the target information, we

(11)


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estimation accuracies may be analyzed based on maximizing the likelihood function [8]–[10] in a noisy background. For the proposed parametric motion model (11), the log-likelihood function [8] can be given as shown in (12), at the bottom of the page, where , the MLE of is equivalent to minimizing the negative of the likelihood function over the parameters. Because the parameter appears linearly in the data, it can be solved in terms of the other parameters in the minimization procedure. Then the MLE of the other parameters should follow

verify the effectiveness of (15), let’s define the proposed GRFT [21]–[23], GFT and GRT in a unified discrete form in this paper as follows. Suppose a parameterized dimensional multivariate function and a 2D complex function is given in 2D discrete plane where and , then the discrete GRFT, GRT and GFT can be defined as

(16.a)

(16.b) (13) If the unknown parameter vector is estimated as via (13), then the MLE of can be given as

(14) Furthermore, by exchanging the summations in (14) we can rewrite the MLE of as

(16.c) is a given constant integer, respectively, where and are also constants decided by the system parameters [21]. For example, we have and for the proposed model (9). Compared with (15), it is shown that the MLE of is to search the maximum in the -dimensional output space of the proposed GRFT (16.a) on the range-compressed sampling in (9). Also, the effectiveness of the proposed GRT and GFT will be further demonstrated in the following subsections. B. MLE Based on Phase-Only Measurement For the maneuvering target echoes of low-resolution OTHR radar or CW radar as mentioned in Section I, they may only appear in a single range unit and the signal can be rewritten as

(17)

(15) is the range compressed output of (9) for where the term the th range sampling in the th pulse. Clearly, the proposed RFT in [21] is just the special case of the above GRFT for a rectilinearly moving target where only and appear. To

may be regarded as a special case of the where 1D signal 2D signal (3) with a fixed range index as (16.b). With the similar derivation as (15), the MLE of unknown parameters can be directly given as

(18)

(12)

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It is obvious that the MLE is to search the maximum in the output -dimensional space of the proposed GFT as (16.b) on the 1D signal . From the phase part of (18), it is also shown that the Fourier transform (FT) is a special case of the GFT and the GFT is also a special case of the GRFT with only phase compensation. C. MLE Based on Envelope-Only Measurement For the range-compressed signal envelope as

instantaneous frequency rate (IFR) [17] and high-order phase function (HPF) [18], [19] have been proposed recently. Also, a good tutorial on local polynomial transforms for PPS detection and other related works can be found in [20]. Different from the existing PPS models, the PPS signal introduced in this paper for a maneuvering target in modern radar may be distributed into several range units during the illumination time. Substitute the multivariate range function with -order polynomial as

in (9), we extract its (22) (19)

and are the real and the imagery parts of where as (9), respectively. Also, the signal in range frequency domain after range FT on can be approximated in high-SNR case as

where are the range, velocity, acceleration, jerk and the other high order motion parameters of maneuvering target along slant range at , which all have the real kinetic meanings. Furthermore, by substituting (22) into (9) and (11), we obtain the signal in time and frequency domains, respectively, as follows:

(20) are the complex additive Gaussian distributed samwhere ples with zero mean and variance and the SNR . The proof of (20) can be found in Appendix A. With the similar derivation as (15), the MLE of unknown parameters can also be directly given as

(23)

(24) where

(21) It is shown that the MLE is to search the maximum in the -dimensional output space of the proposed GRT as (16.c) on the signal range-compressed sampling in (9). Also, GRT has the similar expression as (15) but the phase difference is omitted. Notably, (21) is only an approximated MLE for the target motion parameters in high SNR case. III. CRAMER-RAO LOW BOUNDS FOR HIGH-ORDER MOTION PARAMETER ESTIMATION To describe high-order motion of a maneuvering target, a PPS signal [10]–[20] may be a well-used model for coherent radar. It is known that an arbitrary time-varying motion function can be well approximated via a polynomial function as long as the polynomial order is high enough. Besides, the parameter estimation of PPS has drawn much attention in the past decades. Peleg and Porat derived the CRLB for PPS parameter estimation [10]. Normally, the MLE based on high-dimensional search may be computationally expensive. Therefore, many sub-optimal methods, e.g., discrete polynomial transform (DPT) [11]–[13] or the high-order ambiguity function (HAF) [14], [15], polynomial Wigner-Ville distribution (PWVD) [16],

and . Comparing the equivalent models (23) and (24) with the existing PPS models [10]–[20], there are significant differences between them. For example, each target will contribute to a complete PPS component versus sampling time in existing literatures [10]–[20] while in the proposed time domain model (23) a signal component is distributed along a polynomial curve and can be divided into several parts among neighboring range units. That is, the existing model is normally 1D distributed while the proposed (23) can be 2D distributed. Also, for the existing PPS models [10]–[20] each target only contributes to a single component of PPS. In contrast, the proposed frequency domain model (24) has components of PPS for a single target versus frequency units. A. CRLB for Parameter Estimation Based on Envelope and Phase Jointly Let us rewrite the frequency domain signal model (24) as (25) Also, due to

, we have (26)


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For a given observation and , we define the logarithm of the conditional joint probability density function of as

The Fisher information matrix can therefore be rewritten as (32)

(27) The entries of the Fisher information matrix can be given [10] by

where

is the Hankel matrix whose entries are given by (33)

The matrix

can be written as (34)

(28)

and (35)

and

(36) Therefore, the CRLB is given by

(37) The entries (36) of

are given by

Furthermore, the entries of (28) can be further represented as

(38) Numerical experiments reveal that is extremely ill-conditioned [10], even for moderate and . Therefore, the exact result (37) has little practical value. Consequently, assuming that is moderately large compared to the , the approximation can be computed [10] and we may obtain the approximated bound for different parameters as

(39) (29)

is the coherent integration time and the SNR where is in (6) in the frequency domain. B. CRLB for Parameter Estimation Based on Phase-Only Measurement

where (30)

For the phase signal model (17), the phase term can be . With this rewritten as and the same derivations like previous sub-section, the CRLB can also be obtained for estimation of unknown parameters in (17). The specific derivation is omitted in this paper for simplicity and the results are given directly as follows. Let us define a constant

(31)

(40)

and

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Similar to (25)–(38), the CRLB may be given for different order motion parameters in (17) as

(45) (41) Notably, main signals.

in (41) is the original

in (4) in the time-do-

C. CRLB Based on Envelope-Only Measurement For the envelope signal in (20), we define (42) Similar to (25)–(38), the CRLB may be given for different order motion parameters in (24) as

where the approximations are due to the conditions of and in normal radar applications, respectively. It is shown from (44) and (45) that to increase time-bandwidth product of the transmitting waveform will improve the accuracies of motion parameter estimation. Also, to decrease the wavelength may dramatically improve the estimation accuracy via joint envelope and phase measurement while it will not be helpful to improve the accuracy of envelope-only measurement as (45). The most validated method to improve the envelope-only measurement is to increase the waveform bandwidth as shown in (45). Besides, (44) and (45) also answer the question raised in Section II, i.e., the motion parameter estimation accuracies will be approximately unchanged with the sampling gate . Furthermore, to demonstrate the effectiveness of the GRFT, let us define the accuracy gain on the root mean square error (RMSE) in dB as

(43) where is defined in (A.7). Notably, due to the approximation of (20) in high-SNR case, the CRLB (43) only provides an approximate bound for motion parameter estimation and it may be departed more from the true value in a lower SNR case. D. Analysis of the Proposed CRLBs Based on the proposed CRLB in the previous sub-Section for three typical radar scenarios, it is shown that the CRLB of phase-only measurement in (41) is identical to the results in [10]. But for the radar applications, the definition of in (40) tells us that the smaller wavelength is, the higher accuracy may be obtained for phase-only measurement with the other given system parameters. Furthermore, the derived CRLB for joint envelope and phase measurement and envelop-only measurement, i.e., (39) and (43), may have more special properties than the similar results in [10] for parameter estimation. For example, with the given number of pulses and integration time , let us extract the parts of (39) and (43) with varying variables as

(44)

(46) where and are the derived CRLB for parameter as (43) and (39), respectively. It is clear that the accuracy gain in RMSE of the joint envelope and phase measurement may be approximately linearly increased with the ratio of . Notably, although the performance analysis is given based on the LFM pulse train as (1), the basic conclusions may also be applicable for other radar waveforms [1], [2] with the same ratio . IV. NUMERICAL EXPERIMENTS AND PERFORMANCE ANALYSIS In this section, some numerical experiments are given to verify the effectiveness of the proposed ML estimators and the bounds of (39), (41) and (43), respectively. Firstly, an experimental -band searching radar is designed as follows. Radar carrier frequency , transmitting peak power , transmitting LFM waveform with duration and bandwidth , antenna gain , , noise coefficient , pulse repetition frequency , sampling frequency , system loss and system coherent integration time . Also, there is an isolated maneuvering target in the searching beam with RCS and motion parameter vector . Based on the given systematic parameters, it is obvious that the


Fig. 2. Matched filtering of maneuvering target echoes with

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.

target’s range migration will exceed several range units during the radar illumination time. Furthermore, the target’s SNR is approximately according to the radar (4) and the given parameters. Then, the 2D raw echoes of this maneuvering target are generated as Fig. 2(a), which are totally buried in the system noise with . Then the range DFT is implemented to the 2D raw echoes as (5) to generate the 2D signal in 2D range frequency domain as shown in Fig. 2(b). Furthermore, the range compressed result is given in Fig. 2(c) after matched function multiplication and range IDFT as (6) and (7), respectively. Due

Fig. 3. Maneuvering target motion estimation with different methods. (a) The ; (b) the estimated target velocity; estimated range shifted from (c) the estimated target acceleration.

to about 31 dB SNR gain of the range compression, not only the buried echoes of the target can be dramatically compressed but also the output envelopes can be clearly observed as a “brightened curve” on the Fig. 2(c). A. Comparison Between GRFT and the Existing Method The key contribution of this paper is the handling of maneuvering target with motion order higher than acceleration. Therefore, for the maneuvering target with varied jerk the comparison is given based on 100 trails

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Fig. 4. Target motion parameter estimation via different GRFT and GRT. (a) Target range estimation RMSE versus SNR; (b) target velocity estimation RMSE versus SNR; (c) target Acceleration estimation RMSE versus SNR; (d) target Jerk estimation RMSE versus SNR.

between the proposed GRFT and existing 2-order motion estimation method proposed by Abotzoglou [8]. When the target has no jerk component and the Abotzoglou method has the same output of the proposed GRFT with four-dimensional


Fig. 5. Target motion parameter estimation via GFT. (a) Target range estimation RMSE versus SNR; (b) target velocity estimation RMSE versus SNR; (c) target acceleration estimation RMSE versus SNR; (d) target Jerk estimation RMSE versus SNR.

searching space along . However, with the increase of , the order mismatch of the Abotzoglou method becomes obvious and its estimation results will fluctuated around the true values as shown in Fig. 3(a)–Fig. 3(c). This tells us that


the existing method may be invalidated without the consideration of the high-order motion of a maneuvering target higher than acceleration. In Fig. 3(a), because the range estimation errors of both methods are far smaller than the target’s true range , the estimated range shifted from , i.e., , is used for depicting the estimation result where is the estimated range via different methods. B. MLEs for the Proposed GRFT and GRT In this sub-Section the experiments are given to verify the performances of the GRFT and GRT, where the root mean square error (RMSE) is given in dB as and the MSE is the mean square error. The SNR is varied in [ 25 dB, 15 dB] and 100 trails are given for each SNR value. For the different parameter estimations, Fig. 4(a)–Fig. 4(d) give the results of GRFT and GRT, as well as the derived CRLB as (39) and (41), respectively. It is shown that GRFT and GRT may well approach the CRLB. Meanwhile, it is also shown that both CRLB and real results of GRT may be remarkably higher than those of GRFT with accuracy gain in RMSE about 20 dB. This result also verifies the proposed accuracy gain as (46). Generally speaking, the measurement accuracy of GRFT will greatly outperform GRT by jointly using the phase information.

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Remark 3: The high computational complexity [21]–[23] may be a main problem for the proposed MLEs due to the multi-dimensional search. We have also provided two fast implementations for the special case of GRFT, i.e., RFT, in the fast-time frequency domain [23]. They are called frequency bin Radon-Fourier transform (FBRFT) and sub-band Radon-Fourier transform (SBRFT), respectively, which may reduce the complexity of RFT into that of the conventional Doppler filter bank processing. Also, for the high-dimensional search needed by the GRFT, the heuristic search may be a better choice in the case of multiple maneuvering targets. After saying so, the efficient algorithms for the proposed GRFT, GRT and GFT, are of great interests and still under further investigations. Remark 4: The proposed CRLB for the high-order motion parameter estimation may be suitable for high SNR background. In low-SNR case, not only the performance of GRT, but also that of the proposed GRFT and GFT may also deviate from the proposed CRLB [24]–[27]. Some other tight bounds, e.g., Barankin bound (BRB), Hammersley-Chapman-Robbins bounds (HCR) and so on [25]–[27], may be used to predict larger parameter variances than the proposed CRLB. Fortunately, as long as the target can be well detected [1], [2], [21]–[23], the needed detection SNR may be high enough to apply the proposed CRLB in this paper for motion estimation.

C. MLE for GFT of Low-Resolution Radar To verify performance of GFT and avoid the envelope modulation caused by the moving target’s across range unit (ARU) effect [21]–[23], we reduce the radar bandwidth to and the maneuvering target will appear in single range unit during the coherent integration time. Furthermore, the approximate target’s SNR according to the radar equation. Therefore, the 100 trails are implemented versus SNR area [ 20 dB, 30 dB]. For different order parameter estimation, Fig. 5(a)–Fig. 5(d) give the CRLB and the Monte Carlo results for GFT. It is shown that GFT may well approach the proposed CRLB. D. Some Remarks on the Proposed CRLBs and MLEs Based on the proposed MLEs, CRLB and the numerical experiments in this Section, several remarks are provided in this sub-Section. Remark 1: The proposed CRLB in this paper are the bounds for different polynomial order parameter estimations in the approximations. Their accuracies may be further improved with the increase of the number of pulses when the polynomial order is given. The accuracy of the proposed CRLB has the similar properties as discussed in [10], [11]. Normally, they may be good enough for analyzing the accuracies of the proposed MLEs. Remark 2: In the derivations of the proposed MLEs, the backscattering of the target is assumed fixed. That is, the target is assumed as a Swerlling-0 type [1], [2]. In a longer coherent integration time, the varying phase and envelope caused by fluctuated backscattering should be considered, which may cause a larger error for estimation. Therefore, some further research should be extended to the fluctuating targets in future applications.

V. CONCLUSION Based on parametric modeling of a maneuvering target’s motion, we proposed three kinds of MLEs for motion parameter estimations in this paper. They are called GRFT, GRT and GFT proposed for joint envelope and phase measurement, phase-only measurement and envelope-only measurement, respectively. Furthermore, after developing the 2D PPS model both in time and frequency domains, we obtained the CRLB as well as some analysis for high-order motion parameter estimations in the above three scenarios. Then, we can directly obtain the inherent accuracy limitations for high-order motion estimations via radar equation based on given system parameters. Finally, some numerical experiment results were also provided to demonstrate the effectiveness of the proposed methods. APPENDIX A PROOF OF (20) With parametric modeling of time-varying range, let’s rewrite the range-compressed signal (9) as

(A.1) where its envelope as

. We extract

(A.2) and are the real and imagery parts of , respecwhere tively. It is known that (A.2) may be Rician-distributed [28], i.e.,

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(A.3)

(A.4)

shown in (A.3) at the top of the page where is a 0th order modified Bessel function of first kind. It is known that the shape of Rician distribution is decided by the SNR and it may be well approximated with Gaussian-distributed [28] at high SNR like , as shown in (A.4) at the top of the page. Then, (A.2) may be approximated [28] as (A.5) where are the real additive Gaussian distributed samples with variance . Furthermore, the range frequency domain of (A.5) may be approximated as (A.6) are the complex additive Gaussian distributed samwhere ples with mean 0 and variance and the samples have the SNR as (A.7) ACKNOWLEDGMENT The authors would like to thank Prof T. Long, Prof. J. Yang, Prof. X, T Wang, Prof. Y. L. Wang and the anonymous reviewers for their valuable suggestions to improve the quality of this paper. REFERENCES [1] D. K. Barton, Radar System Analysis and Modeling. Beijing, China: Publishing House of Electronics Industry, 2004. [2] M. I. Skolnik, Introduction To Radar System, 3rd ed. New York: McGraw-Hill, 2002. [3] I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation. Norwood, MA: Artech House, 2005. [4] C.-C. Chen and H. C. Andrews, “Target-motion-induced radar imaging,” IEEE Trans. Aerosp. Electron. Syst., vol. 16, no. 1, pp. 2–14, Jan. 1980. [5] S. B. Peng, J. Xu, Y. N. Peng, and J. B. Xiang, “Parametric ISAR maneuvering target motion compensation based on particle swarm optimizer,” Proc. IET Radar, Sonar Navigat., vol. 5, no. 3, pp. 305–314, 2011. [6] P. Bello, “Joint estimation of delay, Doppler and Doppler rate,” IRE Trans. Inf. Theory, vol. 6, no. 3, pp. 330–341, Jun. 1960.

[7] E. J. Kelley, “The radar measurement of range, velocity and acceleration,” IRE Trans. Military Electron., vol. 5, pp. 51–57, Apr. 1961. [8] T. Abotzoglou and G. O. Gheen, “Range, radial velocity, and acceleration MLE using radar LFM pulse train,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 4, pp. 1070–1083, Oct. 1998. [9] T. Abotzoglou, “Fast maximum likelihood joint estimation of frequency and frequency rate,” IEEE Trans. Aerosp. Electron. Syst., vol. 22, no. 6, pp. 708–715, Nov. 1986. [10] S. Peleg and B. Porat, “The Cramer-Rao lower bound for signals with constant amplitude and polynomial phase,” IEEE Trans. Signal Process., vol. 39, no. 3, pp. 749–752, Mar. 1991. [11] S. Peleg and B. Friedlander, “The discrete polynomial phase transform,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 1901–1914, Aug. 1995. [12] S. Golden and B. Friedlander, “A modification of the discrete polynomial transform,” IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1452–1455, May 1998. [13] B. Porat and B. Friedlander, “Asymptotic statistical analysis of the high-order ambiguity function for parameter estimation of polynomial phase signals,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 995–1001, Mar. 1996. [14] S. Barbarossa, A. Scaglione, and G. Giannakis, “Product high-order ambiguity function for multi-component polynomial phase signal modeling,” IEEE Trans. Signal Process., vol. 48, no. 3, pp. 691–708, Mar. 1998. [15] X.-G. Xia, “Dynamic range of the detectable parameters for polynomial phase signals using multiple-lag diversities in high-order ambiguity functions,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1378–1384, May 2001. [16] B. Barkat and B. Boashash, “Design of higher order polynomial Wigner-Ville distributions,” IEEE Trans. Signal Process., vol. 47, no. 9, pp. 2608–2611, Sep. 1999. [17] P. O’Shea, “A new technique for instantaneous frequency rate estimation,” IEEE Signal Process. Lett., vol. 9, no. 8, pp. 251–252, Aug. 2002. [18] P. O’Shea, “A fast algorithm for estimating the parameters of a quadratic FM signals,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 385–393, Feb. 2004. [19] P. Wang, I. Durovic, and J. Yang, “Generalized high-order phase function for parameter estimation of polynomial phase signal,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 3023–3028, Jul. 2008. [20] X. Li, G. Bi, S. Stankovic, and A. Zoubir, “Local polynomial Fourier transform: A review on recent developments and applications,” Signal Process., vol. 91, no. 6, pp. 1370–1393, 2011. [21] 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. [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. [23] J. Yu, J. Xu, Y. N. Peng, and X.-G. Xia, “Radon-Fourier transform (RFT) for radar target detection (III): Optimality and fast implementations,” IEEE Trans. Aerosp. Electron. Syst., vol. 48, no. 2, pp. 991–1004, Apr. 2012.


[24] J. Yu, J. Xu, and Y. N. Peng, “Upper bound of coherent integration loss for symmetrically distributed phase noise,” IEEE Signal Process. Lett., vol. 15, no. 4, pp. 661–664, Apr. 2008. [25] H. L. Van Trees and K. L. Bell, Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking. New York: Wiley-IEEE, 2007. [26] J. S. Abel, “A bound on mean-square-estimate error,” IEEE Trans. Inf. Theory, vol. 39, no. 5, pp. 1675–1680, Sep. 1993. [27] W. Xu, A. B. Baggeroer, and C. D. Richmond, “Bayesian bounds for matched-field parameter estimation,” IEEE Trans. Signal Process., vol. 52, no. 12, pp. 3293–3305, Dec. 2004. [28] S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 24, pp. 46–156, Jan. 1945. Jia Xu (M’05) was born in Anhui Province, P.R. China, in 1974. He received the B.S. and M.S. degree from the Radar Academy of Air Force, Wuhan, China, in 1995 and 1998, and the Ph.D. degree from Navy Engineering University, Wuhan, in 2001. He was an Associated Professor in the Radar Academy of Air Force during 2006–2009 and was an Associated Professor in Tsinghua University during 2009–2012. Currently, he is a Professor in the School of Information and Electronic, Beijing Institute of Technology, China. His current research interests include detection and estimation theory, SAR/ISAR imaging, target recognition, array signal processing, and adaptive signal processing. He has authored or coauthored more than 120 journal and conference papers. Dr. Xu is a senior member of the Chinese Institute of Electronics. He received the Outstanding Post-Doctor Honor of Tsinghua University in 2004.

Xiang-Gen Xia (M’97–S’00–F’09) received the B.S. degree in mathematics from Nanjing Normal University, Nanjing, China, the M.S. degree in mathematics from Nankai University, Tianjin, China, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983, 1986, and 1992, respectively. He was a Senior/Research Staff Member at Hughes Research Laboratories, Malibu, CA, during 1995–1996. In September 1996, he joined the Department of Electrical and Computer Engineering, University of Delaware, Newark, where he is the Charles Black Evans Professor. His current research interests include space-time coding, MIMO and OFDM systems, digital signal processing, and SAR and ISAR imaging. He has more than 240 refereed journal articles published and accepted, seven U.S. patents awarded, and is the author of the book Modulated Coding for Intersymbol Interference Channels (New York: Marcel Dekker, 2000). Dr. Xia received the National Science Foundation (NSF) Faculty Early Career Development (CAREER) Program Award in 1997, the Office of Naval Research (ONR) Young Investigator Award in 1998, and the Outstanding Overseas Young Investigator Award from the National Nature Science Foundation of China in 2001. He also received the Outstanding Junior Faculty Award of the Engineering School of the University of Delaware in 2001. He is currently an Associate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON SIGNAL PROCESSING, Signal Processing (China), and the Journal of Communications and Networks (JCN). He was a Guest Editor of Space-Time Coding and Its Applications in the EURASIP Journal of Applied Signal Processing in 2002. He served as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 1996 to 2003, the IEEE TRANSACTIONS ON MOBILE COMPUTING during 2001 to 2004, IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY during 2005 to 2008, the IEEE SIGNAL PROCESSING LETTERS during 2003 to 2007, Signal Processing (EURASIP) during 2008 to 2011, and the EURASIP Journal of Applied Signal Processing during 2001 to 2004. He served as a Member of the

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Signal Processing for Communications Committee from 2000 to 2005 and a Member of the Sensor Array and Multichannel (SAM) Technical Committee from 2004 to 2009 in the IEEE Signal Processing Society. He serves as IEEE Sensors Council Representative of IEEE Signal Processing Society (from 2002) and served as the Representative of IEEE Signal Processing Society to the Steering Committee for IEEE TRANSACTIONS ON MOBILE COMPUTING during 2005 to 2006. He is the Technical Program Chair of the Signal Processing Symposium, Globecom 2007 in Washington DC, and the General Co-Chair of ICASSP 2005 in Philadelphia.

Shi-Bao Peng was born in JiangXi Province, China, in 1980. He received the B.S., M.S., and Ph.D. degree from Air Force Radar Academy, Wuhan, China, in 2004, 2007, and 2011, respectively. His current research interests are in the areas of detection and estimation theory, SAR/ISAR imaging.

Ji Yu was born in Jiangxi Province, China, in 1982. He received the B.S. degree from Beijing Normal University, Beijing, China, in 2005. He received Ph.D. degree from Tsinghua University, Beijing, in 2010. His current research interests are in the areas of moving target detection and tracking and array signal processing.

Ying-Ning Peng (M’93–SM’97) was born in Sichuan Province, P.R. China, in 1939. He received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1962 and 1965, respectively. Since 1993, he has been with the Department of Electronic Engineering, Tsinghua University, where he is now a Professor and Director of Institute of Signal Detection and Processing. He has worked with real-time signal processing for many years and has published more than 200 papers. His recent research interests include processing, parallel signal processing, and radar polarimetry. Prof. Peng is a fellow of Chinese Institute of Electronics. He has received many awards for his contributions to research and education in China.

Li-Chang Qian was born in Jiangsu Province, China, in 1985. He received the B.S. and M.S. degrees from the Air Force Radar Academy, Wuhan, China, in 2006 and 2009, respectively. He is a Ph.D. candidate in the Air Force Early Warning Academy, Wuhan. His current research interests are in the areas of weak target detection and tracking.