# Blind Source Separation of Intermittent Frequency Hopping Sources over LOS and NLOS Channels +
## 1 Introduction
### Main Contributions  
This work addresses the problem of BSS of FH sources, which are stationary, spatially sparse, have activity that is intermittent and follow a hidden Markov model. Three different propagation environments of the SCM channel models are considered, namely (i) LOS, (ii) single-cluster, and (iii) multiple-cluster settings.  
Blind source separation of frequency hopping sources is a problem that has been tackled in RF communications before. However, it is not sufficient to just produce a frequency versus time of observed signals; it is also necessary to associate the signals to physical sources. Current methods lack the association of multiple frequency hops to the sources they are transmitted from. We fulfill this gap in our paper by pairing the FH estimates with DOA estimates and labeling signals to their sources, irrespective of their hopped frequencies. This is performed for both LOS and NLOS channels.  
Compared to the conference version [35], this paper studies sources that transmit FH signals over both LOS and NLOS channels; and, compared to [36], this paper includes a comparative study of two filtering approaches that refine the estimations to provide improved source labeling.  
The main contributions of this paper are as follows:  
1. A sparse representation framework is introduced to determine frequency hops and DOA of propagation paths of signals emitted by physical RF sources. FH and DOA estimations are posed and solved as sparse representation problems;
2. A method is developed to associate FH and source activity of multiple sources, thus effectively achieving blind source separation;
3. It is shown that applying hidden state filtering (HSF) improves BSS performance;
4. An algorithm is developed that combines HSF with the estimation of HMM parameters, implementing the filtering without prior knowledge of the model parameters.  
The rest of the paper is organized as follows. The system model with aperiodic FH and the hidden Markov source model are presented in Section 2. In Section 3, we propose the approach to separate FH, source activity and assign labels to signals transmitted from different sources. In Section 4, simulation-based numerical results demonstrate the performance of the proposed approach, and Section 5 reports our conclusions.  
Notation: Notation \(1:T\) denotes the sequence \(1,2,\ldots,T\). Vectors are denoted by boldface lower case letters, such as \(\mathbf{x}\). All vectors are assumed to be column vectors. Matrices are denoted by boldface upper-case letters, such as \(\mathbf{X}\). The transpose of \(\mathbf{X}\) is denoted as \(\mathbf{X}^{\prime}\).  
\begin{table}
\begin{tabular}{p{85.4 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**References** & **Description of Approach** & **Limitations** \\ \hline
[20; 21; 22; 23; 30] & Time-frequency analysis to obtain representation of FH signals & Not possible to obtain good resolution in both time and frequency domains; cross-term interference; spectral leakage; high SNR requirements \\ \hline
[26] & TFA as an exploratory tool, applying particle filter on initial estimations & Limited by the assumption that only one FH signal is present \\ \hline
[27] & Blind ML-based iterative algorithm to estimate hop timing and frequency & Cannot be generalized to multiple FH signals \\ \hline
[28] & Dynamic programming ML estimator for multiple signals & Treats each hop as a distinct source; cannot be used for multiple sources that each transmit at multiple frequencies by hopping \\ \hline
[29] & Sparse linear regression for multiple FH signals & Assumes all sources are active for the entire observation period \\ \hline
[31; 32; 33] & Joint estimation of FH parameters and DOA & Assumes all hop periods are uniform \\ \hline
[34] & & Assumes all hop periods are uniform \\ \hline \hline \end{tabular}
\end{table}
Table 1: Short descriptions and limitations of approaches from current literature.