Target vibration estimation in SAR based on phase-analysis method
- Weijie Xia^{1}Email authorView ORCID ID profile and
- Linlin Huang^{1}
https://doi.org/10.1186/s13634-016-0390-7
© The Author(s). 2016
Received: 3 March 2016
Accepted: 25 August 2016
Published: 7 September 2016
Abstract
An efficient vibration estimation method for synthetic aperture radar (SAR) systems based on phase-analysis method is presented. Small vibrations, which consist of multiple frequency components and are usually rather weak, introduce phase modulation in radar echoes. They contain important signatures of objects. Extracting and analyzing the phase from radar echoes is one of the significant approaches to acquire vibration characteristics. In order to separate different targets in different range cells, the proposed method acts the range compression and range migration correction on the radar echoes, followed by extracting the phase history of each target’s radar echoes. Since the phase history contains the time-varying ranges from the targets to airborne radar besides the vibration signal, we apply wavelet transform to the phase history to extract the vibration signal. Then the vibration signatures would be estimated quantitatively. The method is a more efficient and convenient approach to estimate the vibration amplitudes and frequencies of one or more targets, whose vibration signal is a single-frequency or multi-frequency signal. Moreover, it can be applied in the case where the airborne radar moves along the desirable or undesirable trajectory. Simulation results indicate that this method provides a larger range of estimable vibration frequency, higher estimation precision, and lower computation complexity.
Keywords
1 Introduction
Vibration is ubiquitous in the real world. Objects’ vibration signatures bear vital information about the type of the objects [1]. Spaceborne synthetic aperture radar (SAR) can operate over thousands of kilometers, which is an effective way to acquire vibration signatures of structures placed in remote locations. Small vibrations introduce phase modulation in the returned SAR signals, which is often referred to as the micro-Doppler effect [2]. By analyzing the micro-Doppler effect, we can identify and classify the objects according to the vibration signatures. In addition, the vibration signatures can be used to compensate the phase errors in SAR returns to eliminate the image blur [3, 4]. Therefore, target vibration signatures estimation has great significance.
Several methods have been presented in the literature which deal with the problem of how to extract the vibration signature with SAR systems. The phase-modulated signals induced by vibration with Cohen’s class time-frequency method were analyzed, and the time-frequency signatures and the SAR image of a vibrating target were studied in [5–7]. In [8] and [9], the Doppler frequency of the rotating and vibrating targets was analyzed with smoothed pseudo Wigner-Ville distribution. A novel vibration estimation method based on the discrete fractional Fourier transform (DFRFT) was presented in [10–12]. The method in [13] with sinusoidal frequency modulation Fourier transform (SFMFT) was proposed to obtain the frequency spectrum of vibration traces. In [14], the target was processing with autofocus to estimate the phase error caused by vibration, and the vibration parameters were calculated based on the phase error. Another extracting method of vibrating features based on slow time envelope (STE) signatures was presented in [15]. In [16] and [17], a micro-Doppler reconstruction method using azimuth time-frequency tracking of the phase history was proposed.
In this paper, a vibration estimation method using SAR is presented based on phase analysis. In comparison with the previous works, our approach differs in two main ways. On the one hand, this method is applicable to the case where a single scatterer vibrating in multiple frequencies and several scatterers distributed in different range cells are placed within a SAR scene. In addition, it is also suitable for the situation where the spaceborne radar moves along the undesirable trajectory. On the other hand, compared to methods which rely on the interpretation of time-frequency representations, the proposed method provides a quantitative estimation of the vibration signatures by offering the history of the instantaneous displacement and the spectrum of the vibrating object. The estimation results contain complete parameters, not only the vibration amplitudes and frequencies but also the initial phases included.
The procedure of this method is as follows. It starts with the conventional SAR processing procedure to obtain a nonstationary signal from the vibrating target. First, the returned SAR signals are demodulated. Second, range compression is acted on the demodulated SAR echoes, followed by an application of range cell migration correction to range lines. Hence, the signal from a vibrating target is focused on a range line. Since small vibrations modulate the phase in radar echoes, we extract the phase history of each range line containing vibrating objects and also analyze the restrictive condition of phase ambiguity. Next, by performing the wavelet transform, the vibration displacement is acquired and vibration amplitudes and frequencies of each target are estimated quantitatively. The influence of noises on the performance of the proposed approach is also discussed. The method is capable of extracting vibration signatures in the case where the airborne radar moves along the desirable or undesirable trajectory. Finally, the simulation is given and this method has been compared with other radar-based vibration estimation method. The results illustrate the superior of this method.
The remainder of this paper is organized as follows. In Section 2, the motion model is established and the mathematical formulas of SAR returns containing vibration signatures are derived. In Section 3, the phase-analysis algorithm including phase extraction and wavelet transform is introduced, followed by the discussion related to the signal-to-noise ratio (SNR) in Section 4. Simulations are provided and a comparison between the proposed method and some of others is presented in Section 5. Section 6 gives the conclusion.
2 Signal model
where a _{ i } is the projection of the vibration amplitude onto LOS, f _{ vi } is the vibration frequency, and φ _{ i } is the initial phase of the vibration signal.
3 Algorithm
3.1 Phase extraction
According to Nyquist sampling theorem, PRF must be larger than 2f _{ v }. And for the vibration amplitude meets a > 0, \( \arcsin \left(\frac{c}{8a{f}_0}\right) \) should be satisfied as \( 0< \arcsin \left(\frac{c}{8a{f}_0}\right)<\frac{\pi }{2} \).
3.2 Wavelet transform
Compared to Fourier transform, wavelet transform (WT) is a local transform in time and frequency domain, which can be used to extract information from signal effectively. Wavelet transform makes a multi-scale analysis of functions or signals by expansion, translation operations, so that some characteristics can be fully highlighted. It is known as mathematical microscope.
3.3 Frequency, amplitude, and initial phase estimation
By applying the Fourier transform directly to (19), the spectrum is obtained. It is convenient to get the estimated frequencies \( {\tilde{f}}_{vi} \) from the spectrogram.
where T is the total azimuth time and dη is the sample interval in azimuth direction, the estimated amplitudes are acquired.
The estimated initial phase \( {\tilde{\varphi}}_i \) is the \( {\widehat{\varphi}}_i \) in [0, 2π] which makes \( S\left({\widehat{\varphi}}_i\right) \) achieve the maximum.
4 Discussions
In real-world applications, the performance of the proposed method is affected by the presence of noise. If SAR returns are highly corrupted by noise, some errors may exist in the estimated vibration displacement. Thus, we are interested in finding the SNR threshold above which the estimation result is acceptable.
where ‖ • ‖ denotes the l _{2} norm operator.
where σ ^{2} is the power of each return echo and \( {\sigma}_w^2 \) is the variance of the additive noise.
5 Simulations
5.1 Airborne radar along desirable trajectory
5.1.1 Single point target
SAR system parameters used in the simulation
Parameter | Quantity |
---|---|
Carrier frequency | 15 GHz |
Bandwidth | 400 MHz |
Pulse duration | 0.5 μs |
Range sampling frequency | 420 MHz |
Range resolution | 0.375 m |
Plane velocity | 100 m/s |
Length of the synthetic aperture | 125 m |
Pulse repetition frequency | 880 Hz |
Comparison of real phase and estimated phase
Real phase/rad | 11π/9 (3.8397) | π (3.1416) | π/4 (0.7854) |
Estimated phase/rad | 3.7856 | 3.3929 | 0.8011 |
5.1.2 Multiple point targets in different range cells
Vibration parameters of three targets
Targets | O | P | Q |
---|---|---|---|
Amplitude/mm | 3 | 6 | 4 |
Frequency/Hz | 25 | 15 | 20 |
Initial phase/rad | 2π/3 (2.0944) | π (3.1416) | π/4 (0.7854) |
Estimated vibration parameters of three targets
Targets | O | P | Q |
---|---|---|---|
Amplitude/mm | 2.6 | 5.4 | 3.8 |
Frequency/Hz | 25.62 | 15.7 | 20.66 |
Initial phase/rad | 2.0577 | 3.2358 | 0.8482 |
5.2 Airborne radar along undesirable trajectory
In fact, the aircraft’s fluctuation frequency is very low so that it is also can be referred to as a low-frequency signal compared to the vibration signal. Hence, the proposed method still holds true in the case of the undesirable trajectory.
5.3 Comparison with TF analysis method, DFRFT-based method, and HAF-based method
As described in Section 1, several methods about extracting the vibration signatures with SAR systems have been proposed during the last few years. The time-frequency (TF) analysis method is an important tool to analyze the micro-Doppler effect. DFRFT is an effective method to estimate the time-varying accelerations, frequencies, and displacements associated with vibrating objects [10–12]. Moreover, a time-frequency tracking algorithm based on the high-order ambiguity function (HAF) is employed to reconstruct the instantaneous frequency law of the micro-Doppler [16, 17]. In this section, we compare them with phase-analysis method.
Time-frequency analysis method
DFRFT-based method
The DFRFT-based method approximates the nonstationary signal by a chirp signal in a small time window. DFRFT is applied to estimate the vibration acceleration. Then the history of the vibration acceleration can be reconstructed and the vibration frequency can be obtained by calculating DFT of the acceleration. The proposed method extracts the phase of the global data directly and applies the WT to the phase history to obtain the vibration signal.
HAF-based method
The HAF-based method tracks the instantaneous frequency laws by dividing the range line of the vibrating target in nonoverlapping windows and locally approximating the phase with a polynomial expression. In each window, HAF algorithm is applied, which is an iterative algorithm, to estimate the polynomial coefficients. After the initial reconstruction, the instantaneous frequency laws must be filtered in order to mitigate the possible gaps that may appear at the transitions between estimation windows [17]. On the other hand, the phase-analysis method needs neither iterative algorithm in each window nor filtering for estimated instantaneous frequency laws. Instead, it gives the history of the vibration displacement directly and then the vibration amplitudes, frequencies, and initial phases can be estimated. Therefore, our proposed method may have relatively low computation complexity. Besides, our method makes use of the global data so that the possible errors induced by transitions between the estimation windows may be avoided.
6 Conclusions
In this paper, a new method for radar-based vibration estimation was presented, making the radar capable of more accurate vibration measurement. We obtain the SAR phase history by applying the range compression and range migration correction to the SAR returns. Then the wavelet transform is acted on the phase history to extract the vibration displacement. This method provides quantitative estimations of the vibration signature including vibration amplitudes, frequencies, and initial phases. Some simulations provided to demonstrate that it can estimate the vibration signature of one or several targets vibrating in single or multiple frequencies successfully and it also can be used for the case where the aircraft with the radar sensor moves along desirable or undesirable trajectory. Note that in this paper, we assume that the aircraft flies at a constant speed. When the aircraft flies at the speed varied with time, the fluctuation of the velocity would generate a little interference to the phase history, but the vibration signal can be still extracted exactly. Therefore, the influence of the changed velocity can be neglected.
However, this method has limitation that targets have to be distributed in different range cells. We cannot extract the vibration displacement of each target which is in the same range cell. In our future work, the case of multiple point targets in the same range cell and the models of real-world vibrating objects will be examined carefully, which will have great significance to object recognition, classification, and SAR imaging.
Declarations
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant 61201366, in part by the Fundamental Research Funds for the Central Universities under Grant NS2016040, in part by the Fundamental Research Funds for the Central Universities under Grant NJ20150020.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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