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A Comparison of the Recursive and FFT-Based Reassignment Methods in Micro-Doppler Analysis Karol Abratkiewicz, Piotr Samczyński, Dominique Fourer

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A Comparison of the Recursive and FFT-Based Reassignment Methods in Micro-Doppler Analysis Karol Abratkiewicz, Piotr Samczyński, Dominique Fourer A Comparison of the Recursive and FFT-based Reassignment Methods in micro-Doppler Analysis Karol Abratkiewicz, Piotr Samczyński, Dominique Fourer To cite this version: Karol Abratkiewicz, Piotr Samczyński, Dominique Fourer. A Comparison of the Recursive and FFT- based Reassignment Methods in micro-Doppler Analysis. IEEE Radar Conference 2020, Sep 2020, Florence, Italy. hal-02889226 HAL Id: hal-02889226 https://hal.archives-ouvertes.fr/hal-02889226 Submitted on 3 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Comparison of the Recursive and FFT-based Reassignment Methods in micro-Doppler Analysis Karol Abratkiewicz∗, Piotr Samczynski´ Dominique Fourer◦ Institute of Electronic Systems, IBISC, EA 4526 Warsaw University of Technology, Warsaw, Poland University of Evry/Paris-Saclay, France ∗[email protected] ◦[email protected] Abstract—A brief comparison of two time-frequency (TF) a variant of the reassignment method which provides invert- reassignment methods is provided in this paper. Both techniques ible concentrated TF distribution and can be combined with use the short-time Fourier transform (STFT), however, they additional processing, e.g. filtering, components extraction, can be formulated and computed differently. The first classical method is based on the fast Fourier transform (FFT), while among others. As shown in [15], the reassignment method the second one uses a recursive filter bank which, in turn, can can be successfully used for micro-Doppler signal analysis be more efficient due to a lower time delay and a reduced and can significantly improves the readability of a spectro- computational complexity. Thanks to the proposed methodology, gram. However, the usual FFT-based approach can suffer a real-time computation of the spectrogram and the reassigned from several limitations and trade-off which exclude this spectrogram can be obtained. Hence, the reassignment method allows an almost ideal localization of the micro-Doppler signature technique from being applied in real-time systems. Recently, components in a TF distribution to be obtained. Both approaches an efficient alternative implementation of this method was are presented, investigated, and validated using real-life radar proposed in [16] where the recursive version of the reassigned signals in the form of micro-Doppler signatures originating from and synchrosqueezed STFT was introduced. Hence, this paper different targets. proposes to investigate and to compare together the classical Index Terms—Time-frequency analysis, micro-Doppler, short- time Fourier transform (STFT), reassignment, real-time. FFT-based STFT and the recursive filter-bank-based STFT in terms of results and of computational efficiency, when applied in real-time applications in modern radar systems. I. INTRODUCTION The paper is organized as follows: Section II covers the TF In recent years, micro-Doppler analysis has become one of reassignment theory, including the FFT-based and recursive the fundamental techniques in target recognition and classifica- approaches. Numerical results obtained using the real-life tion [1], [2], [3], [4], [5]. This is the result of a relatively simple radar micro-Doppler signals are presented in Section III. In analysis of the narrowband signal in the baseband, which Section IV, a short discussion is provided that explains the allows fast algorithms to be applied. Moreover, targets that re- differences in both versions of the technique. The summary flect radar signals may have individual radar signatures, which and conclusions, as well as future plans, are provided in in consequence allow for their fast and precise classification. Section V. Excellent examples are micro-Doppler signatures of drones [6], people [7] and animals [8]. Typically, micro-Doppler sig- II. TIME-FREQUENCY REASSIGNMENT natures are obtained through TF analysis. A common method for this purpose is the STFT. However, this technique suffers A. Principle from a limited resolution on the TF distribution resulting from the Heisenberg-Gabor uncertainty principle [9]. Additionally, The reassignment method is a sharpening technique which the resolution depends on the analysis parameters (e.g. the was first introduced by Kodera et al. [9] to improve the number of frequency bins, analysis window length, etc.). Fur- readability of TF representations. It can be viewed as a thermore, these parameters are related to the signal character post-processing operation which moves the values of the which can change in time, thus the micro-Doppler signature considered TF distribution to new coordinates according to may have poor resolution if the initial analysis parameters are (t; !) 7! (t^(t; !); !^(t; !)), where t^ and !^ are expected to be badly conditioned. closer to the true support of the analyzed signal. In [10], Auger In the literature, several enhancement techniques of the and Flandrin generalized this method to any TF distribution STFT have been proposed such as reassignment and syn- belonging to the Cohen class, such as the spectrogram that is chrosqueezing with their respective extensions [9], [10], [11], computed from the STFT. [12], [13], [14]. In general, thanks to these techniques a strong B. FFT-based method energy concentration of the signal in the TF plan can be obtained, which allows more accurate estimation and decom- The STFT provides a function of time t and of frequency position of multicomponent signals. The synchrosqueezing is ! = 2πf of a signal x using a differentiable analysis window h g h g Since Mx (t; !) = jyx(t; !)j and Φx(t; !) = Ψx(t; !) − !t, the reassignment operator can be reworded as [16]: g T g @Ψx yx (t; !) t^(t; !) = t − (t; !) = t − Re g ; (8) @! yx(t; !) g Dg @Ψx yx (t; !) !^(t; !) = (t; !) = Im g (9) Fig. 1: Graphical illustration of the TF reassignment technique. The @t yx(t; !) left-hand image presents a TF distribution (e.g. a spectrogram) of the @g linear chirp, the red arrows denote the reassignment operators, and with T g(t; !) = tg(t; !) and Dg(t; !) = @! (t; !). the reassigned spectrogram obtained using Eq. (4) is presented on In [16], [18], a specific analysis window is introduced and the right. allows an implementation in terms of a recursive infinite impulse response (IIR) filtering when discretized: k−1 h(t). It can be defined as: t −t=T hk(t) = e U(t); (10) Z +1 T k(k − 1)! h ∗ −j!τ jΦh(t;!) F (t; !) = x(τ)h(t−τ) e dτ = M (t; !)e x k−1 x x j!t t pt −∞ gk(t; !) = hk(t)e = e U(t) (11) (1) T k(k − 1)! where j2 = −1 and z∗ is the complex conjugate of z. with p = − 1 + j!, k ≥ 1 the filter order, T the time spread This transform allows one to compute the spectrogram of the T of the window and U(t) the Heaviside step function. analysis signal defined as jF h(t; !)j2. x Using the impulse invariance method, the z-transform of the According to [10], the reassignment operator of the spec- h filter gk(t; !) allows one to compute the filter coefficients as: trogram can be related to the phase Φx(t; !) leading to the following expressions of the reassignment operators: k−1 X b z−i @Φh F T h(t; !) i t^(t; !) = − x (t; !) = t − Re x ; (2) i=0 h Gk(z; !) = TsZ fgk(t; !)g = ; (12) @! Fx (t; !) k X @Φh F Dh(t; !) 1 + a z−i !^(t; !) = ! + x (t; !) = ! + Im x (3) i h i=1 @t Fx (t; !) dh with the z-transform Z ff(t)g = P+1 f(nT )z−n, the filter where T h(t) = th(t) and Dh(t) = dt (t) are modified n=0 s i 1 i versions of the original analysis window h. coefficients ai = Ak;i (−α) , bi = Lk(k−1)! Bk−1;k−i−1α i The last step of the reassignment consists in moving the pTs P j with α = e , L = T=Ts. Bk;i = j=0(−1) Ak+1;j(i+1− values of the spectrogram to obtain a sharpened representation k j)k denotes the Eulerian numbers and A = = k! called the reassigned spectrogram, expressed as: k;i i i!(k−i)! ZZ gk 2πm the binomial coefficients. Hence, yk[n; m] ≈ y (nTs; ) h h 2 ^ x MTs Rx(t; !) = jFx (t; !)j δ(t−t(t; !))δ(!−!^(t; !)) dtd!: 2 can be computed from the sampled analyzed signal x[n] by a R (4) standard recursive equation: where δ(t) denotes the Dirac distribution. The discretization k−1 k process based on the rectangle method leads to the following X X h h 2πm 1 yk[n; m] = bi x[n − i] − ai yk[n − i; m]: (13) approximation F [n; m] ≈ F (nTs; ), Ts = being x x MTs Fs i=0 i=1 the sampling period, n 2 Z standing for the time indices, and m 2 [−M=2; +M=2] corresponding to the discrete frequency The z-transform of the other specific impulse responses can bin. Hence, each vertical slice of the resulting discrete-time be computed as functions of Gk(z; !) at different orders k ≥ 1 h as: STFT Fx [n; m] can be computed efficiently using the FFT algorithm [17]. TsZfT gk(t; !)g = kT Gk+1(z; !) (14) C. Recursive method 1 T Z fDg (t; !)g = G (z; !) + pG (z; !) (15) The STFT as defined in Eq.
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