Compressive Sensing Based Separation of LFM Signals Irena Orovi, Srdjan Stankovi and Ljubiša Stankovi University of Montenegro, Faculty of Electrical Engineering, 81000 Podgorica, Montenegro irenao@ac.me Abstract - A compressive sensing approach for separation of linear frequency modulated signals from non-stationary disturbance is proposed. The linear time-frequency representation is achieved using the Local Polynomial Fourier Transform (LPFT), which allows revealing data local behavior. Based on the LPFT, the frequency-chirp rate domain is used to achieve sparse signal representation. Then the LPFT is combined with the L-statistics to collect only the time-frequency points belonging to the desired signal, while the points belonging to overlapping regions and disturbance are deemed inappropriate and omitted from observations. The relationship between the measurement and sparsity domain is established in order to use the compressive sensing concept and to completely recover the desired signal. The theory is proven on examples. Keywords - time-frequency analysis, compressive sensing, L- estimation, signal separation I. INTRODUCTION Signals in real applications, such as in radars and communications, are often disturbed by different kinds of interferences that are produced by different sources and different physical processes. The components of interest (desired components) could be seriously interrupted by impulse noise, clutters, frequency hopping jammer, etc [1]-[3]. Therefore, if these components highly overlap in time and frequency, signal separation and desired signal recovery can be hardly accomplished through conventional methods and filtering techniques [4],[5]. Moreover, the time-frequency (TF) content of desired and undesired signals can reside over common TF regions and thus it is difficult to provide the separation in the TF domain using TF masking and synthesis methods. Also, the disturbance can be much stronger than the desired signal in the overlapping regions. Hence, we need to exclude all observations containing the disturbances. For this purpose, we use the L-statistics to isolate the TF points belonging only to the signal of interest [6]-[8]. Such an approach can be efficient only if the desired components have stationary nature [10]. In that sense, when dealing with time- varying frequency content as in the case of linear frequency modulated signals (LFM), we propose to use the Local Polynomial Fourier Transform (LPFT), which provides the TF representation of demodulated signal [11]. As a consequence of applying the L-statistics, we deal with missing observations in the TF domain. The theory that considers reconstruction of signals, with sparse representations in certain transform domain, using an incomplete set of samples is known as Compressive sensing (CS) [12]-[18]. Although, in the considered application, the missing samples are not due to Nyquist sampling relaxation as in the case of standard CS concept, we might benefit from the CS reconstruction algorithms. Unlike the standard CS formulations and reconstructions [16], here the observations are made in the TF domain instead of the time domain. Thus, the problem is observed as a CS application aiming at recovery of narrowband signals in interference, using  reconstruction algorithms. Unlike the other existing methods, the proposed one provides efficient results with preserved amplitudes and phases. The paper is organized as follows. The theoretical background on the L-statistics and time-frequency analysis is given in Section 2. The LFM components separation based on the LPFT, the L-estimation and the CS is proposed in Section 3. The experimental results are presented in Section 4, while the concluding remarks are given in Section 5. II. THEORETICAL BACKGROUND – L-ESTIMATION AND TIME-FREQUENCY ANALYSIS A. Problem formulation Consider the case of signal corrupted by impulse noise or certain disturbances (impurity components) that impede the analysis of useful components. For example, the narrowband signals in communications may be disturbed by a frequency hopping jammer that is of shorter duration than the considered time-interval, but may also be overlapping with narrowband signals within same intervals. The problem formulation can be stated as follows. Consider a composite signal: () () () xn fn n ε = + , (1) where f(n) represents useful signal part, while ε(n) represents the impurity components. The discrete Fourier transform (DFT) of signal x(n) can, therefore, be defined as: () () () Xk Fk k = +Ε , (2) where F(k)≠0 for k∈{k 1 ,k 2 ,…k K }, K<<N (N is the number of time samples). Furthermore, we assume that certain frequency components in E(k) could be much stronger than their counterparts in F(k):