Sparse Detection in the Chirplet Transform: Application to FMCW Radar Signals Fabien Millioz, Michael Davies

Sparse Detection in the Chirplet Transform: Application to FMCW Radar Signals Fabien Millioz, Michael Davies To cite this version: Fabien Millioz, Michael Davies. Sparse Detection in the Chirplet Transform: Application to FMCW Radar Signals. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engi- neers, 2012, 60 (6), pp.2800 - 2813. 10.1109/TSP.2012.2190730. hal-00714397 HAL Id: hal-00714397 https://hal.archives-ouvertes.fr/hal-00714397 Submitted on 4 Jul 2012 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. 1 Sparse detection in the chirplet transform: application to FMCW radar signals Fabien Millioz and Michael Davies Abstract—This paper aims to detect and characterise a signal by the matching-pursuit framework making it impractical for coming from Frequency Modulation Continuous Wave radars. real time deployment. Here we are interested in a minimal The radar signals are made of piecewise linear frequency computation approach. Chirplet chains [5], [6] are based on modulations. The Maximum Chirplet Transform, a simplification of the chirplet transform is proposed. A detection of the relevant the search of a single best path in the time-frequency domain, Maximum Chirplets is proposed based on iterative masking, and are applied in the low Signal-to-Noise Ratio (SNR) an iterative detection followed by window subtraction that does context of gravitational wave detection. Only one non-linear not require the recomputation of the spectrum. This detection chirp is detected in the signal, consequently this approach is is designed to provide a sparse subset of Maximum Chirplet inappropriate for the multiple chirps of model (1). A more coefficients. The chirplets are then gathered into linear chirps whose starting time, length, and chirprate are estimated. These general chirplet chain based on a parametric model is proposed chirps are then gathered again back into the different Frequency by Dugnal et al. [7], which uses local maxima to start chirplet Modulation Continuous Wave signals, ready to be classified. An chains, and a single criterion based on the smoothness of the illustration is provided on synthetic data. frequency modulation. Other approaches to detec and estimate Index Terms—LPI radar, FMCW radar, chirplet transform, the parameters of chirps exist, based for example on higher parameter estimation order moments [8] or evolutionary algorithm [9]. However, these methods have the same computational problems than the MP methods. I. INTRODUCTION For our problem, we have strong a priori information that Low Probability of Intercept (LPI) Radars are a type of radar the chirps are piece-wise linear, and in a multi-signal case, designed to hide their emissions from hostile receivers. It aims chirps may be crossing. This paper proposes a new algorithm to see without being seen, which is critical on battlefields. This for detecting and estimating the parameters of the chirps goal may be achieved by several techniques, such as power constituting multiple FMCW radars. This method is based on management, antenna side lobe reduction or frequency agility the chirplet transform, presented in the section II. An approx- [1]. imation of the chirplet transform is proposed, based on the This paper aims to detect and identify Frequency Modula- difference between the signal’s chirprate and the chirprate of tion Continuous Wave (FMCW) radars. This class of LPI radar the analysing chirplet. However, the chirplet transform has one signals may be modelled by more dimension than the usual time-frequency representation. N Compared to the Short Time Fourier Transform, the additional X r(t) = An cos(2πfn(t)t + φn); (1) dimension leads to more coefficients and spreads the signal’s n=1 energy on more coefficients. More memory is required to deal with the chirplet transform, and more coefficients are detected with fn(t) a piecewise linear function. In practice, some chirps may be very short with a very high or very low chirprate, as containing a part of the signal’s energy. Section III proposes and thus very difficult to handle. To extend the possibility of the Maximum Chirplet Transform, time-frequency representa- the model, we consider both continuous and non-continuous tion based on the maximum over the chirprates of the chirplet coefficients. The spectrum of a chirp in this representation is functions fn(t), to avoid the problem of very sharp chirps. In the context of a radar interceptor, a sensor should detect studied, as well as the distribution of coefficients for noise only a signal s(t) coming from several LPI radars, embedded in a data. A detection method, based on a single chirplet transform white Gaussian noise n(t) of variance σ2 followed by an iterative subtraction of chirp’s spectrum is proposed, with the aim to detect as few relevant coefficients I X as possible without recomputation of the chirplet transform s(t) = n(t) + ri(t) (2) between the iterations. i=1 The goal is then to amalgamate the chirplets coming from where I radars emit the signals ri(t). a chirplet transform into different chirps, corresponding to Given such a model, the chirplet transform [2] seems to be a the instantaneous frequency lines fn(t) from the model (1). natural tool to analyse the signal and several methods already To do this, section IV uses the output of the detection in exist based on the chirplet transform. The matching-pursuit the Maximum Chirplet Transform. These detected coefficients (MP) based method [3] employed by Leveau et al. searches are gathered into chirps, using a time-frequency-chirprate for chirps in the signal and iteratively subtract them from the criterion. The case of chirps with a chirprate outside the range signal. This technique was applied to the detection of radar analysed by the chirplet transform is also studied. signals in [4]. The main drawback is the computations required Finally, these chirps are amalgamated back into the signals, 2 using a time-frequency proximity criterion. A result of the We consider here cases where c0 − c 6= 0, the case c0 = c complete algorithm is given on a synthetic signal imitating an leading to the classical Fourier transform. antenna receiving four FMCW radar signals, with Signal-to- The integrals of the form (9) can be approximated by the Noise ratio from -3dB to -18dB. stationary phase approximation [10] if AφT (τ) > 0 and (τ) 1 are both C , and if AφT (τ) varies slowly compared to the II. CHIRPLET TRANSFORM oscillations controlled by the phase (τ). In this case, negative and positive values of ej (τ) tend to cancel each other, except The chirplet transform [2] Cx(t; f; c) of a signal x(t) is determined by near points where the phase is stationary. To determine these points, we search for the time τ0 such that the derivative of +1 the phase is zero: Z c 2 −j2π 2 τ −j2πfτ Cx(t; f; c) = x(t + τ)φT (τ)e e dτ _ τ=−∞ (τ0) = 0 (10) (3) 2π (τ0(c0 − c) + f0 − f + c0t) = 0 (11) with t, f are the time and frequency indices respectively, f − f0 − c0t τ0 = (12) φT (τ) a smoothing window of length T centred on time τ = 0 c0 − c of maximum value 1. Given that ¨(τ ) = 2π(c − c), this stationary point is non- The chirplet transform may be interpreted twofold. By 0 0 j2π c τ 2 degenerate. In the general case, the phase can be approximated considering a chirped window φ (τ) = φ (τ)e 2 , the T;c T by a Taylor expansion. In the case of linear chirps (1), the chirplet transform has the same definition as the usual Short Taylor expansion leads to the exact result Time Fourier Transform. A fast computation using a Fast Fourier Transform algorithm is thus possible. On the other ¨(τ ) (τ) = (τ ) + (τ − τ )2 0 (13) hand, it may be interpreted as the projection of the signal x(t) 0 0 2 −j2πfτ+ c τ 2 onto a set of atoms e 2 , corresponding to chirplets More generally, the stationary phase approximates equation of different chirprates c, centred on time t and frequency f. (9) by +1 ¨ Z 2 (τ0) There is also a direct link to the Fractional Fourier Trans- jτ0 j(τ−τ0) Cx(t; f; c) ≈ AφT (τ0)e e 2 dτ: (14) form Xα(u), defined as τ=−∞ Z p jπ(τ 2 cot '−2uτ csc '+u2 cot ') With the change of variables Xα(u) = 1 − j cot ' x(τ)e dτ ¨ 2 2 (τ0) (4) u = (τ − τ0) ; (15) Z 2 p jπu2 cot ' j2π cot ' τ 2 −j2πuτ csc ' s = 1 − j cot ' e x(τ)e 2 e dτ 2 dτ = du; (16) ¨ (5) (τ0) πα with ' = 2 . The chirplet transform is equal to a short approximation (14) becomes time fractional Fourier transform, with a chirp parameter s +1 cot ' 2 Z 2 c = − 2 and the frequency f = u csc ', up to a factor of j (τ0) ju p 2 Cx(t; f; c) ≈ AφT (τ0)e e du: (17) jπu cot ' ¨ proportionality 1 − j cot 'e , depending on the frac- (τ0) u=−∞ tional Fourier transform parameter. Consequently, the methods Using the Fresnel integral proposed in this paper could also be applied to fractional Z +1 r r Fourier Transform as well as to the chirplet transform.

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