# Blind Source Separation of Intermittent Frequency Hopping Sources over LOS and NLOS Channels +
## 3 Method
### FH Estimation  
The estimation of the frequency hops at each time instant \(0\leq t\leq T\) is posed as a sparse representation problem. In the following, we define a dictionary matrix, a matrix of unknowns, a measurement matrix, and a noise matrix. For setting up the dictionary, we let set \(F=\{f_{1},f_{2},\ldots,f_{G_{f}}\}\), with cardinality \(G_{f}\gg N_{k}\), comprise all possible hop frequencies \(f_{i}\). The FH estimation stage samples the frequency by using this grid of frequencies \(F\). The frequency grid is used to define the \(G_{f}\)-length modal vector \(\mathbf{\hat{h}}_{f}(t)\) at sampling instant \(t\):  
\[\mathbf{\hat{h}}_{f}(t)=[h^{t}(f_{1}),h^{t}(f_{2}),\ldots,h^{t}(f_{G_{f}})]. \tag{2}\]  
Next, this vector is expanded to include all \(T\) sampling instants in a \(TG_{f}\)-length vector \(\mathbf{h}_{f}(t)=[\underbrace{\mathbf{0}_{G_{f}}^{\prime},\ldots,\mathbf{0}_ {G_{f}}^{\prime}}_{t},\mathbf{\hat{h}}_{f}^{\prime}(t),\underbrace{\mathbf{0 }_{G_{f}}^{\prime},\ldots,\mathbf{0}_{G_{f}}}_{T-t-1}]^{\prime}\), where \(\mathbf{0}_{G_{f}}\) is a vector containing \(G_{f}\) zeros. The FH modal dictionary \(\mathbf{H}_{f}\) is then defined at the \(T\times TG_{f}\) matrix  
\[\mathbf{H}_{f}=[\mathbf{h}_{f}(1),\mathbf{h}_{f}(2),\ldots,\mathbf{h}_{f}(T) ]^{\prime}. \tag{3}\]  
The selection of matrix \(\mathbf{H}_{f}\) is determined by the choice of frequency grid \(F\) via the relation between \(\mathbf{\hat{h}}_{f}(t)\), \(\mathbf{h}_{f}(t)\) and \(\mathbf{H}_{f}\). We let \(\mathbf{x}_{f}(j,t)\) be the \(G_{f}\)-length vector of unknown complex amplitudes associated with the grid frequencies at time \(t\) and sensor \(j\). This vector is then expanded to form the \(TG_{f}\)-length vector that includes all \(T\) time instants \(\mathbf{x}_{f}^{\star}(j)=[\mathbf{x}_{f}^{\prime}(j,1),\mathbf{x}_{f}^{\prime }(j,2),\ldots,\mathbf{x}_{f}^{\prime}(j,T)]^{\prime}\). Across all \(J\) sensors and \(T\) time instants, the \(TG_{f}\times J\) matrix of complex amplitudes is defined as \(\mathbf{X}_{f}=[\mathbf{x}_{f}^{\star}(1),\mathbf{x}_{f}^{\star}(2),\ldots, \mathbf{x}_{f}^{\star}(J)]\). The solution to the problem being formulated relies on the assumption that vector \(\mathbf{x}(j,t)\) has sparsity. Sparsity entails vector \(\mathbf{x}(j,t)\) having non-zero entries corresponding only to the active frequencies at each time instant \(t\).  
A sparse estimate of the frequency hopping pattern per source \(\mathbf{X}_{f}\) (or equivalently an estimate of \(\mathbf{x}_{f}(j,t)\) for all \(j\) and all \(t\)) is denoted as \(\mathbf{\hat{X}}_{f}\) (or \(\mathbf{\hat{x}}_{f}(j,t)\)), and it is found by solving the following optimization problem:  
\[\mathbf{\hat{X}}_{f}=\operatorname*{arg\,min}_{\mathbf{x}_{f}}\lVert\mathbf{Y }^{\prime}-\mathbf{H}_{f}\mathbf{X}_{f}\rVert_{2}^{2}+\lambda_{f}\sum_{m=1}^{ M}\sum_{t=1}^{T}\left\lVert\mathbf{x}_{f}(j,t)\right\rVert_{1}. \tag{4}\]  
The \(\ell_{1}\)-norm in (4) enforces the sparsity constraint. Hyperparameter \(\lambda_{f}\) controls the sparsity of the solution. A large \(\lambda_{f}\) increases the penalty of non-zero elements of \(\mathbf{x}_{f}(j,t)\).  
Figure 4: Block diagram of the proposed algorithm to separate multiple FH sources.