## 1 Introduction  
Direction of arrival (DOA) estimation is an important problem in many real-life applications such as MIMO communication and radar signal processing. Multiple Signal Classification (MUSIC) is a well-known algorithm for DOA estimation that first computes the covariance matrix of the received signals and then performs Eigen decomposition of the covariance matrix to estimate the DOA [1, 2, 3]. Time-frequency (TF) MUSIC algorithm further enhances the MUSIC algorithm by employing spatial time-frequency distributions (STFDs) that allow the selection of high energy points in the TF domain thus improving the estimate of the covariance matrix [4, 5, 6, 7, 8, 9]. Conventional time-domain MUSIC algorithm or its TF extension are only applicable in the over-determined scenario, i.e., when the number of sensors is higher than the number of sources. STFDs can be employed to estimate DOA in under-determined scenarios, i.e., when the number of sensors is not higher than the number of sources. STFDs are used to extract the TF signatures of sources and then estimate their DOA separately thus enabling the Spatial TF approach to become applicable in the under-determined scenario [9, 10]. TF filtering is another computational and memory-efficient approach that first separates sources and then employs MUSIC algorithm to estimate their DOAs [11, 12, 13]. In order to separate sources either through TF filtering or employing STFD based approach, we need to find out the TF signature of sources [13, 14, 15, 16, 17]. The TF signature of sources can be extracted using instantaneous frequency (IF) estimation methods like connected component linking [18] or blind source separation based methods [19, 20]. However, these methods are not applicable to signals that intersect each other as the IF estimation method may switch to a wrong path after the intersection point e.g., Figure 1 illustrates a scenario when signal components intersect each other and when signal components do not intersect. For such signals, one approach can be to use morphological image processing techniques to extract TF signatures of sources [14, 15]. However, morphological operations are sensitive to noise. DOA can be estimated in an under-determined scenario by using an orthogonal projection operator in combination with short-time Fourier transform [21]. This method assumes that at any given TF point number of sources is less than or equal to 2 [21]. Parametric methods like Hough transform are another option to estimate DOA for more than 2 signals intersecting each other but this approach is limited only for linear frequency modulated signals [22]. Matching pursuit has also been used to estimate the TF signature of sources but this method requires signals to follow some parametric model [23]. In our earlier studies [24; 25], two DOA estimation methods have been developed for sources that intersect each other in the TF domain. One of these methods first enhances STFD with directional smoothing and then employs the Viterbi algorithm for IF estimation followed by source localization [24; 26]. The major limitation of this method is the extensive computational cost due to both the computation of the STFD and the Viterbi algorithm [27]. The other method uses ridge detection and tracking-based IF estimation method for source separation followed by DOA estimation [25; 28]. The method is thus computationally efficient as compared to the Viterbi algorithm, but it still requires the computation of adaptive directional time-frequency distribution (ADTFD), which is a computationally expensive step. In short, the DOA estimation in an under-determined scenario at a low signal-to-noise ratio becomes a challenging problem when signal components cross each other because the existing methods are either too computationally expensive or they require signal IF laws to follow any particular signal model. In this study, we propose a computationally efficient method to estimate DOA in an under-determined scenario.