# Blind Source Separation of Intermittent Frequency Hopping Sources over LOS and NLOS Channels +
## 3 Method
### Hidden State Filtering
#### 3.3.2 Individually Most Probable States
4:for each non-zero row of \(\hat{\mathbf{X}}_{d}\)do
5:for\(T=1:t\)do
6:\(z(t)=1_{c}\{\hat{\mathbf{x}}_{d}(t)\}\)
7:while no convergence do
8: Obtain \(\alpha_{j}(t)\) and \(\beta_{i}(t)\) using Forward-Backward procedure
9: Update \(\hat{\Omega}=(\hat{\mathbf{A}},\hat{\mathbf{B}},\hat{\pi})\) using Equations (27a)-(27c)
10:endwhile
11:\(\Omega\leftarrow\hat{\Omega}\). 12:endfor
13:endfor
```
**Algorithm 3** Learning HMM parameters

### Pairing

Given the FH and DOA estimates, the final stage is to pair them. For each time instant \(t\), the pairing stage is designed to pick a combination of DOA and FH estimates that provide the best fit to the observed data. To perform the pairing, two new dictionaries \(\tilde{\mathbf{H}}_{d}\) and \(\tilde{\mathbf{H}}_{f}\) are formed from the original respective dictionaries \(\mathbf{H}_{d}\) and \(\mathbf{H}_{f}\). The new dictionary \(\tilde{\mathbf{H}}_{d}\) is defined as a submatrix of \(\mathbf{H}_{d}\), with elements corresponding to non-zero entries in \(\tilde{\mathbf{X}}_{d}\). Similarly, the new dictionary \(\tilde{\mathbf{H}}_{f}\) is defined as a submatrix of \(\mathbf{H}_{f}\), with elements corresponding to non-zero entries in \(\hat{\mathbf{X}}_{f}\). The new dictionaries are introduced to limit the computational cost of the pairing operation and used to create a new dictionary [(43)]. The Kronecker product of \(\tilde{\mathbf{H}}_{d}\) and \(\tilde{\mathbf{H}}_{f}\) defines this new dictionary \(\tilde{\mathbf{H}}\):

\[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]3

This dictionary is a grid that contains all active frequencies for each active source over the entire observation interval. The pairing stage utilizes the newly formed dictionary \(\tilde{\mathbf{H}}\) to output matrix \(\mathbf{X}\), which contains the complex amplitudes of sources which are indexed by their hop frequencies and their DOAs. The following optimization problem is solved:

\[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]4

to obtain a sparse vector \(\mathbf{x}(t)\) whose non-zero elements are the estimated complex amplitudes. Here, \(\lambda\) is the hyperparameter that controls the sparsity of \(\mathbf{x}(t)\). The pairing of FH estimates and source activity is outlined in Algorithm 4. ```
1:Input: Matrices \(\tilde{\mathbf{X}}_{d}\), \(\hat{\mathbf{X}}_{f}\) and dictionaries \(\mathbf{H}_{d}\), \(\mathbf{H}_{f}\)
2:Output: Matrix of paired source activity and FH estimates \(\hat{\mathbf{X}}\)
3:Obtain new dictionary \(\tilde{\mathbf{H}}_{d}\) from \(\mathbf{H}_{d}\) using \(\mathbf{X}_{d}\)
4:Obtain new dictionary \(\tilde{\mathbf{H}}_{f}\) from \(\mathbf{H}_{f}\) using \(\mathbf{X}_{f}\)
5:Create dictionary \(\tilde{\mathbf{H}}\) using Equation (28)
6:Obtain \(\hat{\mathbf{X}}\) from Equation (29)
```

**Algorithm 4** Pairing of source activity and FH estimates

Source separation is obtained by labeling the sources according to DOA estimates. A source label may be associated with multiple frequency hops. Furthermore, the pairing stage is also capable of reducing false alarms in the FH and DOA estimation stages by considering a source to be active at a given time instant only if it produces a joint FH and DOA estimate. ## 4 Results

In this section, numerical results are presented to demonstrate the performance of the proposed approach to separate intermittent FH signals. We assume that the activity for each source is defined by an HMM with transition probabilities \(a_{01}=0.02\) and \(a_{10}=0.02\). A modal dictionary for DOA estimation is considered with \(G_{d}=36\) bins, each having a bin size of 5 degrees. The FH grid has \(G_{f}=40\) bins with a spacing of 50 kHz for a 2 MHz total bandwidth. The signals being transmitted are slow FH signals in which each hop contains one or more symbols. Frequency hops do not violate the narrowband assumption for DOA estimation.