Parameter Estimation of Micro-Motion Targets for High-Resolution-Range Radar Using Online Measured Reference

Micro-motion dynamics produce Micro-range (m-R) signatures which are important features for target classification and recognition, provided that the range resolution of radar signal is high enough. However, dechirping the echo with reference measured by narrow bandwidth radar would generate the residual translational motion, which exhibits as random shifts of envelopes of range profiles. The residual translational motion would destroy the periodicity of m-R signatures and make a challenge to estimate rotational parameter. In this work, we proposed an efficient high-resolution range profile (HRRP)-based method to estimate rotational parameter, in which online measured reference distances are used to dechirp the radar raw echo. Firstly, the deduction for the modified first conditional comment of range profiles (MFCMRP) is introduced in detail, and the MFCMRP contain periodic and random components when dechirped by measured reference, corresponding to the rotational motion and the reference measured errors compared with actual reference. Secondly, the Wavelet Transform (WT) is utilized to separate the measured errors from the MFCMRP. The estimations of measured errors are used to compensate the MFCMRP, and then autocorrelation is performed on the estimated periodic component to obtain the estimation of rotational period. Lastly, the rotational amplitudes and phases are achieved by inverse Radon transform (IRT) of the compensated HRRP. The effectiveness of the proposed method in this paper is verified by synthetic data and measured radar data.


Introduction
Micro-motion [1] refers to a minor motion part in a whole target, and its characteristics of periodic motion could be utilized to estimate the parameters of micro-motion for target classification and recognition [2,3]. Micro-motion mainly consists of three types, such as rotation, procession, and vibration [4]. Rotation is a popular micro-motion widely seen in rotational blades of helicopters [5], rotational antenna of the navy, and so on. Because of noncooperation, moving targets contain translational motion and rotational motion. Translational motion degenerates the parameter estimation via destroying the periodicity of micro-Range (m-R) and micro-Doppler (m-D) signatures. Thus, estimation and compensation of the translational motion is essential before the parameter estimation of rotational motion.
The conventional method for translational motion compensation in a micro-motion target mainly utilizes the m-D signatures of rotational [5], processional [6], and vibrating motion [4]. An effective orthogonal matching pursuit based [5] method was proposed to estimate translational motion by selecting the optimal velocity atom among the velocity dictionary. However, the order of translational

Micro-Range Signature
Without loss of generality, the point-scatter model is usually constructed to describe the radar signal scattered by a target. In this paper, the linear-frequency modulated (LFM) signal is utilized to acquire the m-R signature. Dechirping the echo of target with a reference signal and compensating the residual video phase, we can obtain HRRPs of the rotational target [6].
where (r, t m ) represent range-slow time domain. K is the number of scattering centers. σ k (t m ), r k (t m ) refer to the backscattering coefficient of the kth scattering center and its instantaneous range to radar. T p , B, r re f , f c are the pulse width, bandwidth, reference distance, and carrier frequency of the radar signal, respectively. Taking the translational motion of target into consideration, the motions of scattering centers on a target can be expressed as follows: r k (t m ) = r T (t m ) + r 0 + R k (t m ), k = 1, 2, · · · , K, R k (t m ) = A k sin(2π f M t m + ϕ k ), k = 1, 2, · · · , K (2) where r T (t m ) is the range induced by translational motion, which is the same for all scatters. r 0 is the initial relative range between radar and rotating center; R k (t m ) is the range induced by rotational motion of the kth scattering center and A k , ϕ k , f M are the amplitude, initial phase, and frequency of rotational motion, respectively.
Due to the reference distance measured by narrowband radar, thus the reference distance satisfied where v(t m ) denotes the ranging errors between actual reference and measured reference. Due to the attitude of target changes all the time, the ranging errors have randomness. In Equation (2), the translational motion r T (t m ) is unnecessary and would be removed in m-R signatures by measured reference.

Estimate Method
In this section, we propose a HRRP-based method to compensate the error of measured reference distance, and estimate the rotational parameters including period, amplitude, and phase. It should be noted that the proposed method is applicable to estimate the parameters that are produced by any micro-motion (not limited to the rotational motion). The proposed method is composed of two stages: (i) calculation for the modified first conditional moment of the range profiles (MFCMRP) and (ii) parameter estimation and image reconstruction.

Calculation for MFCMRP
Compared with function sin c(·) used in [6], sin c 2 (·) has smaller sidelobes, so that it is closer to impulse function δ(·). On the other hand, the second-order sin c function is more robust than the higher-order ones with respect to the presence of noise. Therefore, the MFCMRP is defined and simplified as follows, and the detailed deduction can be seen in Appendix A. According to Equation (1), Equation (4) would be simplified as where σ(t m ) is dependent on the attitude of the target. Since the periodicity of rotational motion dynamics makes the attitude of target change periodically, σ k (t m ) has the same period with rotational motion with respect to slow-time.
where T M is the period of rotational motion. The fourth term in Equation (6) is a periodic function because the all items in it are periodical, that is, From Equations (4)- (7), it is obvious that the MFCMRP can be divided into four terms: translational motion, the initial distance between rotating center and radar, the reference distance, and the periodic term.

Parameters Estimation and Image Reconstruction
When the reference distance is measured by narrowband radar in real-time, as a result of ranging errors, there exist random shifts (residual translational motion) of envelopes of range profile between adjacent pulses. In this situation, the measured reference distance is modeled as a sum of translational motion, random ranging errors, and the the initial relative range between radar and rotating center, as depicted in Equation (3). Moreover, according to Equations (5) and (7), the MFCMRP in (4) can be rewritten as There are two items in the MFCMRP as shown in Equation (8), the former is a periodic item, and the latter is a random item corresponding to ranging errors, also called residual translational motion. For a periodical signal with random noise, the WT is a popular method in stable signal denoising [14,15]. Hence, WT is utilized to separate the periodic item from random item in this paper. Performing autocorrelation on the estimated periodic item would obtain the period estimation. What's more, the random item is corresponding to the shift range of interfered HRRPs in Equation (1). Compensating the contaminated RPs so as to achieve the rotational m-R signatures, and then utilizing the IRT result of compensated signatures, we can obtain the estimation of rotational motion parameter. The flow chart of the whole procedures is shown in Figure 1, and the specific solution procedures are divided into three steps: • According to the HRRPs by dechirping echo with online reference, the MFCMRP can be obtained through Equations (4) and (8).

•
Performing WT on the MFCMRP can get the random item, the residual translational motion, compensating the contaminated HRRPs with the residual translational motion can get the rotational m-R signatures. Autocorrelation operation is a popular method to estimate the period in periodic function and is utilized for the rotational period estimation.

•
Performing inverse radon transform on the compensated HRRPs would get the estimation of rotational parameters, including the amplitude and the phase of rotational motion. • Performing inverse radon transform on the compensated HRRPs would get the estimation of rotational parameters, including the amplitude and the phase of rotational motion.

Experiment 1
A rotating target without translational motion in our experiment is demonstrated in Figure 2. The target contains two rotating four-side corner reflectors driven by a motor, and the rotational angular is set to 21.4 rpm, the rotational period is 2.8 s, and their rotational amplitudes are 16 and 24 cm, the differential phase of them is 2 rad π , respectively. The experiment is carried out in an absorbing chamber, for the transmitted signal, the central frequency 220 GHz , the pulse repetition frequency 1000 Hz s f = and the power is 1.2 mW. The experiment setup is same as [6], the radar echo viewed as a sum of micro-motion modulation part and translational part. The actual RPs can be seen in Figure 2b, and the IRT of Figure 2b is shown in Figure 2c, where two bright points can be clearly observed. The experimental environment can be seen in [4] except for vibrating interference.

Experiment 1
A rotating target without translational motion in our experiment is demonstrated in Figure 2. The target contains two rotating four-side corner reflectors driven by a motor, and the rotational angular is set to 21.4 rpm, the rotational period is 2.8 s, and their rotational amplitudes are 16 and 24 cm, the differential phase of them is π/2rad, respectively. The experiment is carried out in an absorbing chamber, for the transmitted signal, the central frequency f c = 220 GHz, bandwidth B = 12.8 GHz, the pulse repetition frequency f s = 1000 Hz and the power is 1.2 mW. The experiment setup is same as [6], the radar echo viewed as a sum of micro-motion modulation part and translational part. The actual RPs can be seen in Figure 2b, and the IRT of Figure 2b is shown in Figure 2c, where two bright points can be clearly observed. The experimental environment can be seen in [4] except for vibrating interference.

•
Performing inverse radon transform on the compensated HRRPs would get the estimation of rotational parameters, including the amplitude and the phase of rotational motion.

Experiment 1
A rotating target without translational motion in our experiment is demonstrated in Figure 2. The target contains two rotating four-side corner reflectors driven by a motor, and the rotational angular is set to 21.4 rpm, the rotational period is 2.8 s, and their rotational amplitudes are 16 and 24 cm, the differential phase of them is 2 rad π , respectively. The experiment is carried out in an absorbing chamber, for the transmitted signal, the central frequency 220 GHz , the pulse repetition frequency 1000 Hz s f = and the power is 1.2 mW. The experiment setup is same as [6], the radar echo viewed as a sum of micro-motion modulation part and translational part. The actual RPs can be seen in Figure 2b, and the IRT of Figure 2b is shown in Figure 2c, where two bright points can be clearly observed. The experimental environment can be seen in [4] except for vibrating interference.          The simulation target is the corner reflector Q1 in the Figure 2c. As shown in Figure 4, the estimation errors decrease as SNR increase and the rotational amplitude error plays a more important part in the effect of SNR than the initial phase and the period. Due to the benefits of the autocorrelation of the MFCMRP, the rotational period can be estimated well in low SNR. Above all, the method proposed in this paper has a good anti-noise ability, it can also be noted that the estimation error rate for every parameter is no more than 2% when the SNR is greater than −16 dB.

Experiment 2
This experiment demonstrates the result of a method based on high-order difference sequence [6] for comparison. The experimental data is the same with experiment 1. The experimental HRRPs by dechirping on radar echo with the measured reference distance is shown in Figure 3a, and there is not any sinusoidal character compared with Figure 2b because of ranging errors in reference distance (residual translational motion). Motivated by the method based on entropy minimization, the results are different with respect to the different lengths of smooth window. Figure 5a-c denote the range profiles after range alignment corresponding the length of smooth window 5, 10, and 20 slow-time sampling units. It is noted that the optimal length of smooth window would make the right m-R signatures. Thus, the determination for the length of smooth window is vital for range alignment. Selecting 10 slow-time sampling units as the length of smooth window, aligned HRRPs seen in Figure 5b are utilized to estimate the micro-motion period, the procedures of which are depicted in Figure 6. The instantaneous range of the scattering center with largest energy extracted by Viterbi algorithm is plotted in Figure 6a, but that isn't a sum of sinusoidal signal and polynomial signal described in [6]. The reason is that the relative position of scattering points in rotational motion is front-behind variable. In contrast, the largest energy of scattering center in the processional missile is always the top of missile. The high-order difference sequence of Figure 6a is depicted in Figure 6b, it is hard to recognize sinusoidal character in that. The spectrum of high-order difference sequence is shown in Figure 6c, the estimation of the rotational period is 0.324 s, which disagrees with the actual period. The simulation target is the corner reflector Q1 in the Figure 2c. As shown in Figure 4, the estimation errors decrease as SNR increase and the rotational amplitude error plays a more important part in the effect of SNR than the initial phase and the period. Due to the benefits of the autocorrelation of the MFCMRP, the rotational period can be estimated well in low SNR. Above all, the method proposed in this paper has a good anti-noise ability, it can also be noted that the estimation error rate for every parameter is no more than 2% when the SNR is greater than −16 dB.

Experiment 2
This experiment demonstrates the result of a method based on high-order difference sequence [6] for comparison. The experimental data is the same with experiment 1. The experimental HRRPs by dechirping on radar echo with the measured reference distance is shown in Figure 3a, and there is not any sinusoidal character compared with Figure 2b because of ranging errors in reference distance (residual translational motion). Motivated by the method based on entropy minimization, the results are different with respect to the different lengths of smooth window. Figure 5a-c denote the range profiles after range alignment corresponding the length of smooth window 5, 10, and 20 slow-time sampling units. It is noted that the optimal length of smooth window would make the right m-R signatures. Thus, the determination for the length of smooth window is vital for range alignment. Selecting 10 slow-time sampling units as the length of smooth window, aligned HRRPs seen in Figure 5b are utilized to estimate the micro-motion period, the procedures of which are depicted in Figure 6. The instantaneous range of the scattering center with largest energy extracted by Viterbi algorithm is plotted in Figure 6a, but that isn't a sum of sinusoidal signal and polynomial signal described in [6]. The reason is that the relative position of scattering points in rotational motion is front-behind variable. In contrast, the largest energy of scattering center in the processional missile is always the top of missile. The high-order difference sequence of Figure 6a is depicted in Figure 6b, it is hard to recognize sinusoidal character in that. The spectrum of high-order difference sequence is shown in Figure 6c, the estimation of the rotational period is 0.324 s, which disagrees with the actual period.
From the above results and analysis, the following conclusion would be made: (i) for a micro-motion target, the aligned HRRPs would be affected by the length of smooth window; (ii) if the scattering center of target is a front-behind variable, the instantaneous range of the scattering center with the largest energy isn't a sum of sinusoidal signal and polynomial signal. Therefore, their applications to estimate micro-motion parameters interfered with translational motion are limited. From the above results and analysis, the following conclusion would be made: (i) for a micro-motion target, the aligned HRRPs would be affected by the length of smooth window; (ii) if the scattering center of target is a front-behind variable, the instantaneous range of the scattering center with the largest energy isn't a sum of sinusoidal signal and polynomial signal. Therefore, their applications to estimate micro-motion parameters interfered with translational motion are limited.

Conclusions
This work proposes a novel HRRP-based method to estimate rotational parameter under the interference of translational motion. The MFCMRP is utilized to describe the rotational dynamics by function approximation, and the relationship between the MFCMRP and translational motion is established. When the reference distance is measured by narrowband radar signal, the MFCMRP contains a periodic item associated with rotational motion and reference distance measured error item.
Taking advantage of WT transform, the ranging error of the reference could be separated to compensate the contaminated HRRPs. Then, performing autocorrelation operation on the periodic item would obtain the rotational period; compensated HRRPs could be utilized to estimate rotational amplitude and initial phase of rotational target by IRT. The results of synthetic and measured radar data shown that the proposed method is effective to estimate the period, amplitude, and phase of micro-motion with high accuracy, and it is convenient for signal processing compared to the method based on high-order difference sequence. In the method motivated by high-order difference sequence, the results of HRRPs range alignment is important for later processing, and the different length smooth window makes different results. However, there is no definite method for length determination. Based on the proposed method, algorithms would be developed to recognize the micro-motion target through estimated amplitude and phase. Further research on how to achieve this would be carried out. From the above results and analysis, the following conclusion would be made: (i) for a micro-motion target, the aligned HRRPs would be affected by the length of smooth window; (ii) if the scattering center of target is a front-behind variable, the instantaneous range of the scattering center with the largest energy isn't a sum of sinusoidal signal and polynomial signal. Therefore, their applications to estimate micro-motion parameters interfered with translational motion are limited.

Conclusions
This work proposes a novel HRRP-based method to estimate rotational parameter under the interference of translational motion. The MFCMRP is utilized to describe the rotational dynamics by function approximation, and the relationship between the MFCMRP and translational motion is established. When the reference distance is measured by narrowband radar signal, the MFCMRP contains a periodic item associated with rotational motion and reference distance measured error item.
Taking advantage of WT transform, the ranging error of the reference could be separated to compensate the contaminated HRRPs. Then, performing autocorrelation operation on the periodic item would obtain the rotational period; compensated HRRPs could be utilized to estimate rotational amplitude and initial phase of rotational target by IRT. The results of synthetic and measured radar data shown that the proposed method is effective to estimate the period, amplitude, and phase of micro-motion with high accuracy, and it is convenient for signal processing compared to the method based on high-order difference sequence. In the method motivated by high-order difference sequence, the results of HRRPs range alignment is important for later processing, and the different length smooth window makes different results. However, there is no definite method for length determination. Based on the proposed method, algorithms would be developed to recognize the micro-motion target through estimated amplitude and phase. Further research on how to achieve this would be carried out. Figure 6. Results motivated by method proposed in [6]. (a) instantaneous range of the scattering center with largest energy; (b) high-order difference sequence with optimal spectral concentration measure; (c) the spectrum of high-order difference sequence.

Conclusions
This work proposes a novel HRRP-based method to estimate rotational parameter under the interference of translational motion. The MFCMRP is utilized to describe the rotational dynamics by function approximation, and the relationship between the MFCMRP and translational motion is established. When the reference distance is measured by narrowband radar signal, the MFCMRP contains a periodic item associated with rotational motion and reference distance measured error item.
Taking advantage of WT transform, the ranging error of the reference could be separated to compensate the contaminated HRRPs. Then, performing autocorrelation operation on the periodic item would obtain the rotational period; compensated HRRPs could be utilized to estimate rotational amplitude and initial phase of rotational target by IRT. The results of synthetic and measured radar data shown that the proposed method is effective to estimate the period, amplitude, and phase of micro-motion with high accuracy, and it is convenient for signal processing compared to the method based on high-order difference sequence. In the method motivated by high-order difference sequence, the results of HRRPs range alignment is important for later processing, and the different length smooth window makes different results. However, there is no definite method for length determination. Based on the proposed method, algorithms would be developed to recognize the micro-motion target through estimated amplitude and phase. Further research on how to achieve this would be carried out.