Elsevier

Signal Processing

Volume 198, September 2022, 108598
Signal Processing

Two dimensional local maximum synchroextracting chirplet transfrom and application of characterizing micro-Doppler signals

https://doi.org/10.1016/j.sigpro.2022.108598Get rights and content

Highlights

  • Highly concentrated energy distribution in characterizing micro-Doppler signals.

  • Explore the local maxima principle on frequency-chirp-rate plane.

  • Reassignment on the frequency-chirp-rate plane.

Abstract

Methods related to reassignment have attracted great attention in recent years due to their efficiency in improving the readability of classical time-frequency (T-F) representations (e.g. the spectrogram and scalogram). Since most of the reassigning methods require the signal components separated by sufficient frequency distances, they are rarely applied to analyze the micro-Doppler (m-D) signals whose components commonly overlap in the T-F domain. To address this issue, we propose a chirplet transform based reassignment technique that performs the local maximum synchroextracting operation in both frequency and chirp-rate dimensions. The underlying principle is that m-D modes with instantaneous frequencies cross one another are actually separable as long as their chirp-rates are different. Also, the accuracy of the local maximum based reassignment estimator is analyzed with the guarantee of some error bounds. Compared with several cutting-edge techniques, the numerical experiments show that the proposed method is better at characterizing the m-D signals, particularly at the crossover regions in the T-F plane.

Introduction

The m-D effect arises in radar return of additional frequency modulation induced by micro motions [1]. It is often used to determine the dynamic properties of the radar targets such as human activities [2] and unmanned aerial vehicles [3] and analyze their characteristics. Since m-D effects are represented by highly non-stationary signals, a common technique for m-D analysis is the time-frequency analysis (TFA), which is able to decompose different m-D modes and reveal their time-varying features.

Classical TFA methods including short-time Fourier transform (STFT), Wigner-Ville distribution and wavelet transform are restricted by Heisenberg uncertainty principle. Although different trade-offs can be made between the temporal and frequency resolution, none is ideal. As a result, the time-frequency representations (TFR) generated by conventional methods are blurry. The time-frequency reassignment (RM) is often introduced as a post-processing technique of classical TFA methods in order to improve the readability of their TFRs [4]. The RM first calculates a meaningful T-F location based on phase information and then it reallocates the original T-F coefficients to the calculated positions. While RM is capable of representing a multi-component signal on the T-F plane, the method is non-invertible. To address this issue, Maes and Daubechies [5] developed the synchrosqueezing transform (SST) which is a special case of RM and possesses the additional advantage of allowing for mode reconstruction. Meanwhile, Huang N.E. et al. [6] introduced a radically different proposal, coined as Empirical Mode Decomposition (EMD) method, designed to extract the nonstationary components in a data-driven manner. Due to the simplicity and effectiveness, EMD algorithm has been mostly exploratory since then [7,8]. Unfortunately, the algorithm lacks solid mathematical foundations, making it hard to analyze its accuracy or the limitations of its applicability. In this context, the SST has recently resurfaced as a more formalized alternative which captures the flavor and philosophy of the EMD approach [9].

Ever since Daubechies et al. [10] provided the systematic investigation of the performance of SST, many achievements and new developments have been made, significantly extending the applicability of the reassignment both conceptually and computationally [11], [12], [13], [14]. Enhancements of SST are primarily found in the reassignment operator [15], [16], [17], [18], frequency estimator [19], [20], [21], [22] and both [23], [24], [25]. However, these methods work well only when different modes from the analyzed signal are separated by sufficient frequency distances. This significantly hinders the capability of the reassignment to characterize the m-D signals whose components commonly overlap in the T-F plane. At the same time, the adaptive chirplet transform (ACT) based reassignment methods have gained popularity among researchers seeking further improvement of the T-F energy distribution [26], [27], [28], [29]. This is mainly because the ACT is able to provide a sparser TFR than the classical TFA methods if the selected chirp-rate matches the characteristics of the signal [30]. Although the reassignments of ACT assume that the instantaneous frequency (IF) of different components are separated by distinct distances, the introduction of the chirp-rate dimension certainly demonstrates great potential to characterize crossover IFs that could hardly be distinguished in the T-F dimensions [30,31,32]. More recently, Zhu et al. [33,34] mapped signals into the time-frequency-chirp-rate (T-F-C) space via conventional chirplet transform (CT). By reassigning the CT coefficients on the frequency-chirp-rate (F-C) plane, a sharpened three-dimensional representation can be obtained. Such attempts indeed break the aforementioned separability limitation of the T-F post processing methods. Nonetheless, the hypotheses used in the two studies are impractical and failed to consider the overlapping interferences on the F-C plane. This makes the theoretical foundation very sketchy. Furthermore, the instantaneous feature of individual modes has to be extracted by a ridge detection algorithm proposed in [35]; this involves the implementation of additional procedures and could result in significant estimation errors if inappropriate parameters are chosen. Their reassignment method also suffers from large computational burden because the frequency and chirp-rate equations (Eq. (15) in [33] and Eq. (17) in [34]) have to be solved at each time slice. On the other hand, Chui et al. [36,37] proposed the CT based separation scheme that effectively retrieves modes with crossover frequency curves. The group of authors provided deep insights into the T-F-C space and developed rigorous theorems concerning the recovery of the adaptive harmonic model. Their pioneering work paves the way for extending the reassignments into the T-F-C space because the local behavior of CT coefficients around the modes’ IF and instantaneous chirp-rate (IC) are well defined.

In the present paper, we propose a novel reassignment technique to achieve a highly concentrated TFR of m-D signals. In order to achieve this, we first define a linear frequency modulation (LFM) model based well-separation condition. After mapping the signal into the T-F-C space via CT, a local maximum F-C estimator is proposed to approximate the ground truth IFs and ICs of signal modes. In the next step, the synchroextracting operator is employed to extract the CT coefficients on the F-C plane. Finally, we project the sharpened T-F-C representation into T-F domain to obtain a high frequency-resolution TFR. Our theoretical foundation is derived from the theorems established in Refs. [36,37] which claim that the separation power of CT depends on the instantaneous amplitude (IA), IF and IC between modes. Compared with the F-C reassignments in [33,34], the proposed method is more practical in its mathematical model and simpler to carry out.

The paper is structured as follows: in Section 2, we revisit the SST and analyze its deficiency in characterizing overlapped IFs. The Section 3 defines a class of LFM signal model and discusses the separation conditions as well as the local extrema of F-C representation. The proposed method is then introduced along with its implementation. Numerical examples demonstrating the performance of our approach are illustrated in Section 5. The conclusions are drawn in Section 6.

Section snippets

Theoretical backgrounds

The section provides a brief review of the SST. We also analyze its deficiency when signal modes do not meet the separation requirement.

The proposed method

In this section, the separation conditions on F-C plane are discussed. We also prove that the local maximum within the F-C domain can be the effective estimator to approach the IF and IC of signal modes. The proposed method is then introduced along with its implementation. Finally, signal reconstruction is described.

Algorithm implementation

The practical implementation and algorithm procedures of our method are described in this section. The implementation of the three-dimensional ridge detection algorithm mentioned in Section 3.5 is also provided.

In discrete processing, the signal s is of finite length and is assumed to be uniformally discretized. We then rewrite Csh(ω,β,t) as Csh[m,l,n] with m,n{0,,N1} and l{0,,L1}. The discrete form of CT can be found in [29].

Regarding practical applications, the thresholding |Csh(ω,β,t)|>

Performance and discussion

In this section, we focus on the comparisons between the proposed method and several cutting-edge reassignment methods in addressing m-D signals.

In order to quantify the representation quality, the generated TFRs will be evaluated by means of Rényi entropy. The Rényi entropy is an effective indicator that measures signal information and complexity on the T-F plane [15,16,19,42,43]. The smaller this entropy, the more concentrated the TFR.

We also introduce the mean square error (MSE) to evaluate

Conclusion

In this paper, we develop a novel reassignment method for analyzing m-D signals made of crossover IFs. It simply consists of a local maximum based F-C estimator (30) and a two-dimensional synchroextracting operator (33), taking advantage of mode separation in the T-F-C space. The simulations carried out on test-signals show that the new reassignment is much better at characterizing m-D signals in terms of TFR and invertibility than several state-of-the-art methods.

Data availability statement

N/A.

Funding statement

This research is supported by the National Key R&D Program of China (Grant No. 2016YFD0700302).

CRediT authorship contribution statement

Ran Zhang: Conceptualization, Methodology, Software, Validation. Xingxing Liu: Conceptualization, Methodology, Software, Validation. Yu Tan: Conceptualization, Methodology, Software, Validation. Xincheng Yang: Software, Validation. Lina Zhang: Software, Validation.

Declaration of Competing Interest

This manuscript has not been published and is not under consideration for publication elsewhere. We have no conflict of interest to disclose.

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      Citation Excerpt :

      On the other hand, as presented, the penalty function above does not offer any algorithmic means to compute the ridges. In applications listed in [15,27,29,31,37,42], the iterative procedure of forward/backward searching are suggested to execute the penalty function (53). Our detector begins with the random segmentation and selects the extrema within each segment as the starting point; the ridges are then located by the forward/backward procedure.

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