# Blind Source Separation of Intermittent Frequency Hopping Sources over LOS and NLOS Channels +
## 2 System Model  
### Setting and Channel Model  
We consider \(N\) sources capable of transmitting intermittent FH signals. These sources switch carrier frequency in a randomized fashion across multiple frequency hops. Signals emitted by the sources are received at a uniform linear array (ULA) with \(J\) receiving antenna sensors spaced at uniform \(d\) intervals. The total number of sources \(N\) may be larger than the number of sensors \(J\), but the number of active sources at any given time is lower than the number of sensors. The observation interval is \(T\) discrete time units, and, without loss of generality, the sampling interval corresponds to one time unit. The cumulative time a source is active is a small fraction of the observation interval. Source activity is assumed to be governed by a Markov model, as described below. The on-off patterns of each source change slowly, making transitions intermittent, but smooth. The intermittent activity of the sources is illustrated in Figure 1.  
A hop duration is the period of time between two consecutive switches of the carrier frequency of a source. The time of the \(k\)-th frequency switch is denoted \(t_{k}\), and thus the duration of the corresponding hop is \((t_{k+1}-t_{k})\). When multiple sources are active, the hop duration is determined by the source with the shortest time between frequency hops. An example is shown in Figure 2.  
Figure 1: Intermittent source activity. The filled blocks denote active sources at different time instants. Figure 2: Source activity of two FH sources depicted by filled blocks with distinct shading. Time duration \(t_{k}\leq t<t_{k+1}\) denotes a single hop with hop index \(k\) where sources with mean DOAs \(\theta_{i}\) and \(\theta_{i+1}\) transmit with frequencies \(f_{i}\) and \(f_{i+1}\), respectively. Sources are assumed to be observed over channels that follow the SCM. The SCM is a statistical channel model developed by 3 GPP for three propagation environments: (i) LOS, (ii) single-cluster, and (iii) multiple-cluster settings [15; 16; 17]. Over an LOS propagation environment, a source is observed with a single DOA. With the single-cluster environment, a source is observed over a cluster of possibly unresolvable multipath propagation paths. For the multiple-cluster environment, the multipath results in multiple clusters of propagation paths. In this work, BSS of FH sources is considered over each of the three propagation environments. For the SCM with multiple clusters, assuming synchronized sensors, the signal received at the \(j\)th sensor in the time interval \(t_{k}\leq t<t_{k+1}\) corresponding to the \(k\)th hop is given by  
\[y(j,t)=\sum_{i=1}^{N_{k}}\bigg{\{}\sum_{l=1}^{L_{i}}\sqrt{\frac{P_{i}}{M}}\sum _{m=1}^{M}c_{i,l}^{(m)}(t)a^{j}(\theta_{i,l}^{(m)})\bigg{\}}h^{t}(f_{i})+w(j,t), \tag{1}\]  
where \(N_{k}\) is the number of active sources during the \(k\)th hop (\(N_{k}\ll N\)); \(L_{i}\) is the number of clusters for the \(i\)th source; \(P_{i}\) is the power of the \(l\)th cluster of the \(i\)th source which is normalized so that the total average power for all clusters is equal to one; \(M\) is the number of unresolvable multipaths per cluster that have similar characteristics. The number of sources that could be active during the \(k\)th hop is less than the number of sensors \(J\), i.e., \(N_{k}<J\). The variable \(h(f_{i})\) is the frequency mode of the \(l\)th cluster of the \(i\)th source, and \(a(\theta_{i,l}^{(m)})\) is the spatial mode of the \(m\)th multipath of the \(l\)th cluster of the \(i\)th source. \(a^{j}(\cdot)\) and \(h^{t}(\cdot)\) denote the corresponding \(j\)th and \(t\)th powers; \(c_{i,l}^{(m)}(t)\) is the complex amplitude of the \(m\)th multipath of the \(l\)th cluster of the \(i\)th source; and \(w(m,t)\) is the additive zero-mean complex Gaussian noise with variance \(\sigma_{w}^{2}\). The frequency mode is given by \(h(f_{i})=e^{j2\pi f_{i}}\), with \(f_{i}\in[f_{min},f_{max}]\) being the carrier frequency of the \(i\)th source during the \(k\)th hop.