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A Doppler aliasing free micromotion parameter estimation method in the terahertz band
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 61 (2017)
Abstract
MicroDoppler, induced by micromotion of targets, is an important characteristic for target recognition once extracted via parameter estimation. However, microDoppler is usually too significant to result in aliasing in the terahertz band. According to this problem, a Doppler aliasing free micromotion parameter estimation method based on the modulo Generalized Hough transform is proposed in this paper. Its basic idea is to search and match the parameters of aliasing microDoppler curves in the timefrequency distribution image. The high estimation precision and excellent noise immunity of this method are verified through simulations based on CST data and experiments based on a 0.33 THz radar system.
Introduction
Terahertz (THz) waves usually refer to electromagnetic waves with frequencies between 0.1 and 10 THz. The terahertz band lies between the millimeter wave and infrared, which is a transitional band from electronics to photonics. Its position in the spectrum confers special properties and applications on terahertz waves that differ from other bands [1–3]. With breakthroughs in terahertz sources, signal detectors and other devices, the terahertz radar technology has developed rapidly, and many terahertz radar systems have been established, mainly for the study of high resolution imaging [4–7]. Some terahertz devices and radar systems are getting mature, and examples are the 0.85 THz vacuumbased power amplifier designed by Northrop Grumman Corporation [8] and the ultrahighresolution radar Miranda 300 designed by the FGAN Research Institute for High Frequency Physics and Radar Techniques [9]. However, researches on micromotion targets, as a common type of object in the real world, are still very limited, although terahertz waves, due to their short wavelengths, are more sensitive to Doppler than microwaves and hence more suited for microDoppler imaging. Also, the serious atmospheric attenuation of terahertz waves has little influence on their applications in exoatmosphere or near space. As a result, the research on parameter estimation of micromotion targets with terahertz radar is significant. Despite the advantages mentioned above, micromotion parameter estimation in the terahertz band faces a significant hurdle, i.e., the Doppler aliasing induced by the inadequate pulse repetition frequency (PRF). Terahertz radar usually utilizes frequency modulated continuous wave (FMCW) signals due to low peak power requirement. For example, both the 220 GHz COBRA radar of FGAN [6] and the 580 GHz imaging radar of Jet Propulsion Laboratory (JPL) [7] utilize FMCW. In order to ensure the linearity of the FMCW transmitted signal, the equivalent PRF can’t be too large, and therefore the limited PRF sets an upper limit to the maximal observable nonaliasing Doppler values. Specifically, the Doppler frequency can be unambiguously observed only when it lies in the interval between –PRF/2 and PRF/2, and it will be aliasing or folded up when Doppler values are outside this interval. In addition, microDoppler in the terahertz band tends towards being aliasing as the carrier frequency is much higher than that in the microwave band.
For micromotion in the terahertz band, Robert W. McMillan, et al. introduced an experimental coherent pulsed radar operating at 225 GHz early in 1991, and obtained the Doppler spectrum of a stationary truck with its engine running [10]. Li J et al. performed a theoretical analysis of typical micromotion forms to establish the microDoppler signature model, and then compared the microDoppler effect in the terahertz band and that in X band by using the joint timefrequency analysis method [11]. X. Zhengwu et al. analyzed the characteristics of micromotion and investigated methods for motion parameter estimation and microDoppler signature extraction by the Radon transform [12]. However, the parameter estimation method in microDoppler aliasing situation was not considered, just because the microDoppler values they assumed are relatively smaller than PRF/2. In practice, microDoppler aliasing in the terahertz radar system is inevitable, and thus these algorithms are inapplicable for our concerns.
Herein, we propose an algorithm for micromotion parameter estimation based on the modulo Generalized Hough transform (GHT). Its main idea is derived from the GHT in image processing field, which searches sinusoidal curves or other types of curves hidden in images. However, the GHT cannot be used in the aliasing situation directly. In order to obtain micromotion parameters in the aliasing situation, we firstly obtain the timefrequency distributions of micromotion targets, and then match the microDoppler curves of scattering centers in the timefrequency image with aliasing reference curves obtained by modulo. Finally, we map the parameters of curves to parameter spaces and estimate them by extracting positions of peaks in parameter spaces.
The paper is organized as follows. Section 2 analyzes the microDoppler characteristics of a precession warhead model in the terahertz band. In section 3, an aliasing free parameter estimation method based on modulo GHT is proposed and the detailed procedures are given. In section 4, the simulation and experimental results are shown and the performance of the method is analyzed. The conclusions are presented in section 5.
MicroDoppler characteristics of a precession warhead model
Theoretical model of echo signals
The intended target in this paper is a precession rotational symmetric warhead model, which is a simplified model of the ballistic missile target during the midcourse. The simulation data of this paper come from a 3D electromagnetic simulation software named computer simulation technology (CST). The diagram of the warhead model observed by radar is shown in Fig. 1.
A reference frame OXYZ that takes the mess center of the model as the origin is established. Considering the model precessing around OZ axis, the spin angular velocity is Ω, the precession angular velocity is ω, the precession angle is θ, the azimuth and pitch angles of the light of sight (LOS) are α and β, the initial distance between the radar and the model is R _{0}. According to the theoretical calculation and the experimental measurement, every scattering center corresponds to a discontinuity in the StrattonChu integral, that is the discontinuities of the curvature or surface from the view point of geometry. For the model shown in Fig. 1, the scattering centers include the conetop P1, the discontinuities in the intersection line of the plane Γ (plane Γ is composed of the LOS and the symmetrical axis of the model Oz) and the model P2, P3, P4, and P5.
Because the model is rotational symmetric, spin makes no difference to the echo modulation. Thus, the radial distance r(t) between the radar and any scattering center located at (x, y, z) on the target may be written as:
In this equation, A is amplitude modulated coefficient, and φ is the initial phase. The microDoppler f _{ d }(t) of the scattering center with a carrier frequency f _{0} is:
where A _{ ω } is the maximal microDoppler frequency, and φ _{ ω } is its initial phase. The microDoppler of scattering centers on the model are sinusoidally modulated with a period 2π/ω. Therefore, the extraction of precession parameters is equivalent to the estimation of periods, amplitudes and initial phases of sinusoidal curves in timefrequency images. In general, amplitudes and initial phases usually reflect the relative positions of scattering centers, which can be used for space reconstruction and imaging of micromotion targets.
Characteristics of Doppler aliasing
There is no sufficient available real data of precession targets in terahertz band to illustrate the characteristics of Doppler aliasing limiting by the current devices and conditions of our laboratory, and consequently the CST data are adopted to simulate the reality. It is obvious that microDoppler f _{ d } is in proportion to the carrier frequency f _{0} through Eq. (2). The higher the f _{0}, the more evident is the microDoppler effect. Therefore, microDoppler values in terahertz band are far larger than that in microwave band under identical motion conditions. Suppose the PRF of the transmitted signal is 1 KHz, and the maximum microDoppler value of a micromotion scattering center is about 40 Hz when the carrier frequency is 10 GHz (Fig. 2a), while at 0.33 THz it would reach 1320 Hz, which substantially exceeds the up limit of PRF/2, hence aliasing is present (Fig. 2b).
In microDoppler aliasing situations, the observed Doppler values are no longer the correct values, but are projections on the interval from –PRF/2 to PRF/2. Estimation of the real Doppler value for SAR or ISAR becomes necessary as we cannot obtain perfect SAR/ISAR images without the real Doppler value.
Aliasing effect on parameter estimation methods
Traditional parameter estimation methods such as the Fourier transform and the Inverse Radon transform (IRT) are not applicable for the aliasing situation because aliasing changes the signal properties and destroys the completeness of timefrequency curves.
The spectrum results of echo signals at 10 GHz and 0.33 THz are shown in Fig. 3, respectively. We can easily find that the abscissa values of the peaks in the spectra correspond to the maximal microDoppler values in nonaliasing situation. In aliasing situation on the contrary, abscissa values of the peaks are no longer correspond to the maximal microDoppler values but the aliasing values, and suggests that the Fourier transform is not suitable for aliasing situations.
The IRT is an effective method for the detection of sinusoidal curves. It can map sinusoidal curves in digital images to peaks in parameter spaces, and then obtain the parameters according to positions of peaks in parameter spaces. However, it is not applicable in microDoppler aliasing situations for the completeness of sinusoidal curves is destroyed. The IRT results of echo signals at 10 GHz and 0.33 THz are shown in Fig. 4, and we cannot obtain any useful information from the IRT result of aliasing microDoppler in Fig. 4b although the IRT is a powerful tool (see [13]). As a result, we have to study new parameter estimation algorithms for microDoppler aliasing situations.
Aliasing free parameter estimation based on modulo GHT
Algorithm principle
The algorithm in this paper depends on the timefrequency distributions, and we chose shorttime Fourier transform (STFT) as a basis as it has no cross term interference. There is a phase error because of the discrepancy in the start time between STFT and the signal and can be compensated during parameter estimation. We can estimate the micromotion parameters after obtaining timefrequency distribution images of echo signals. It is well known that GHT is an effective way to detect curves in digital image processing when the curve expressions are known [14–16]. The basic idea of GHT is to map the curves in measurement space to peaks in the parameter space. Curves which share the parameters correspond to the same peak in the parameter space. We can then extract the parameters by identifying the positions of peaks in parameter spaces. Returning to the problem in this paper, the primary objective is to map the aliasing sinusoidal curves in timefrequency distribution images to peaks in parameter spaces. GHT cannot be directly applied to the Doppler aliasing situation as no analytical expressions exist, necessitating an improvement in this case.
We at first establish a parameter space K = (A _{ k }, φ _{ k }, ω _{ k }) according to the expression of microDoppler f _{ d }(t). The period of sinusoidal curves is easily obtained by the autocorrelation method in time domain or by the cepstrum method in frequency domain [17], because Doppler aliasing and noise have no effect on the periodicity of signals. Therefore, the parameter space can be reduced to K = (A _{ k }, φ _{ k }). If there is a sinusoidal curve f(t) = A _{ ω } sin(ωt + φ _{ ω }) in a timefrequency image (Fig. 2a), f(t) is similar to the reference curve made up by pixels located at the coordinates (t, A _{ k } sin(ωt + φ _{ k }) ), with just different parameters that are termed as reference pixels. When the target curve and the reference curve have the same parameters, i.e., A _{ k } = A _{ ω } and φ _{ k } = φ _{ ω }, we call this reference curve the matching curve (Fig. 5a). Their shapes are match well with each other, and the values of matching pixels are larger, the average value of matching pixels is larger, too. If we map the mean of all groups of reference pixels to the parameter space, the mean corresponding to matching curves manifests as peaks in the parameter space. In the aliasing situation, the improvement on GHT is that, if the microDoppler is aliasing (Fig. 2b), then the reference curve should be equally aliasing too, i.e. the vertical ordinates of reference pixels modulo PRF (Fig. 5b), so the aliasing microDoppler still matches well with the aliasing reference curve. We then identify the appropriate search scope and step length for the searching algorithm. Finally, we extract positions of peaks and derive the parameters of timefrequency curves. Considering the computation load, we can design a variable steplength searching method. The first step would be to affirm the potential intervals of parameters by choosing a long step length and searching the space parameter. Then, we can estimate parameters accurately in these potential intervals with a short step length. The algorithm proposed in this paper is applicable in both situations, without prejudging whether aliasing is present or not. Thus, the complexity of this algorithm has been greatly reduced. We call this algorithm the modulo GHT method.
Algorithm steps
To sum up, the parameter estimation method will be carried out in six steps:

(1)
Obtain the timefrequency distribution image TF of a micromotion target by an appropriate timefrequency analysis method (such as STFT).

(2)
Estimate the period of micromotion by the autocorrelation method in order to reduce the parameter space.

(3)
Establish a parameter space K = (A _{ k }, φ _{ k }) and identify an appropriate search scope and step length based on both effectiveness and estimation precision. In this paper, we set the upper limit of the microDoppler frequency F reasonably according to the experimental conditions. The integration time is T = 1s, the microDoppler step length is Δf = 1Hz, and the phase step length Δφ = 1^{∘}.

(4)
Identify the coordinates of reference pixels in the timefrequency distribution image TF, especially in the microDoppler aliasing situations. The reference pixel coordinates can be expressed as (t, mod(A _{ k } sin(ωt + φ _{ k }) + PRF/2 , PRF)), where mod(·) is the modulo operation and that is different from GHT.

(5)
Search and match the parameters, and average the values of reference pixels in each group in order to reduce detrimental effects of noise.

(6)
Extract positions of peaks in the parameter space. Their abscissa values represent the initial phases and vertical ordinate values of the maximum microDoppler values.
Data processing results and analysis
Simulation results
We design a group of simulation experiments to validate our algorithm above. Simulation results of each experiment are shown in Figs. 6, 7, 8, 9, and 10. The relative frequency and phase errors of each experiment are shown in Fig. 11.
For each group of results, (a) is the timefrequency image of scattering centers, and (b) is the parameter space obtained through the modulo GHT method. The peaks in parameter spaces correspond to timefrequency curves of scattering centers. The results taken together show that the algorithm based on modulo GHT does well both in situations of aliasing and nonaliasing, and the noise immunity is very good from the results of No. 2 and No. 3. We can estimate the parameters even though the shapes of timefrequency distributions are no longer observed due to low SNR, i.e., we can still find the right peaks in the parameter space when the SNR is −3 dB, even though the timefrequency distribution image is polluted severely by noise. Furthermore, the algorithm has a high reliability in multiscattering centers situations from the comparison of No. 5 with others. Finally, the relative frequency and phase errors of each experiment are within the range of 1% from Fig. 11, so the estimation precision of parameters is very high.
Experimental results
Experiments based on a 0.33 THz radar system are carried out. The terahertz radar system adopted in this paper is based on linear frequency modulated (LFM) pulse principle and has a 322 GHz of central frequency with a synthetic bandwidth of 10 GHz, thereby realizing a 1.5 cm theoretical range resolution. The terahertz signal is transmitted by the coneshaped horn antenna with an azimuth beam angle 11° after 36 times frequency multiplication of an Xband sweeping generator in the transmitting chain. The pulse repetition frequency (PRF) of the transmitting signal is 1000, and the transmitting power is greater than 3 mW.
The target in this paper is a warhead model which consists of three parts: a dome cone, a cylinder, and a frustum of a cone (as shown in Fig. 12). There are two motors in the model to implement functions of spin and coning, respectively. The Range profile sequence of the precession warhead model is shown in Fig. 13.
We extract the range resolution bins that are related to the conetop and analyze them by a timefrequency method to obtain the microDoppler curves of the conetop. The microDoppler curves of conetop at two sets of experimental parameters are shown in Fig. 14, which separately represent the two situations of aliasing and nonaliasing. The parameter spaces through the modulo GHT are shown in Fig. 15, from which we can reconfirm the excellent performance of this method on Doppler aliasing free micromotion parameter estimation.
Performance analysis
Computation load analysis
The essence of our algorithm is to search the matching parameters, and the computation load is therefore closely related to the step lengths of the searching parameters. If the upper limit of the microDoppler frequency is F, and the microDoppler step length is Δf, the search requires F/Δf cycles in the frequency search. Likewise, 360/Δφ cycles are required in the phase search. There is only one averaging operation in each cycle. If the sampling time is Δt, and the integration time is T _{.} Then the number of addition operation in a cycle is (T/Δt − 1 + 1), adding a division operation which is equivalent to an addition operation. The computation load C is approximated as:
The computation load is directly proportional to the upper limit frequency and the integration time and inversely proportional to the frequency and phase precision from Eq. (3). So, we need to make a tradeoff between the computation load and parameter estimation precision or adopt the variable steplength searching strategy mentioned above.
SNR analysis
This algorithm has a good performance in noise immunity because it has a coherent integration processing. In order to analyze the relationship between the estimation precision and signaltonoise ratio (SNR), we plot the errors when SNR varies from −20 to 3 dB at 220 GHz in Fig. 16. It is clear that the algorithm possesses good robustness and high precision when the SNR is above −12.5 dB, the average errors are lower than 1%. When SNR is below −12.5 dB, the algorithm is invalid because the timefrequency distribution images are severely polluted by noise. Errors in this situation are random and there are no detectable peaks in parameter spaces. To sum up, the modulo GHT method has a good performance with SNRs above −12.5 dB.
Conclusions
Researches on parameter estimation of micromotion in the terahertz band are of great value to exploit the advantages of terahertz band and promoting the applications of terahertz radars. In this paper we have proposed a modulo GHT algorithm, and applied it to the estimation of micromotion parameters in microDoppler aliasing situation. Its essence is to search the matching parameters of microDoppler curves in timefrequency distribution images, and it is especially suited for the microDoppler aliasing situation in the terahertz band. The simulation and experimental results demonstrate the high estimation precision and excellent noise immunity, especially when the SNR is above −12.5 dB.
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Acknowledgements
The authors would like to thank the editors and reviewers for their insightful comments.
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Authors’ Contributions
The problem of Doppler aliasing resolution with terahertz radar was arisen from the discussions between BD and HW. YQ built the 0.33 THz radar system and wrote Section 1. The Doppler aliasing free micromotion parameter estimation method was derived and implemented by QY. HW wrote the Abstract and BD wrote the Conclusions. Section 2, Section 3 and Section 4 were written by QY. All authors read and approved the final manuscript.
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Yang, Q., Deng, B., Wang, H. et al. A Doppler aliasing free micromotion parameter estimation method in the terahertz band. J Wireless Com Network 2017, 61 (2017). https://doi.org/10.1186/s136380170845z
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Keywords
 Terahertz radar
 MicroDoppler
 Generalized Hough Transform
 Doppler aliasing
 Parameter estimation