# 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
The number of unresolveable multipaths per cluster is \(M=20\) and the deviation from mean DOA of each cluster has a variance of \(\sigma_{\theta}^{2}=2\) degrees. In the following figures, we consider two clusters per source while referring to sources in the multiple-cluster model. The weights of the various optimization problems included in this paper are hyperparameters, and they are chosen in accordance with methods expanded in [37]. Experiments are run on Matlab R2020a on a computer with Intel(r) CoreTM 5 CPU and Windows OS. The performance criterion chosen is the receiver operating characteristic (ROC), in which the probability of correct detection \(P_{d}\) is plotted against the probability of false alarm \(P_{fa}\). The probability of correct detection for source activity is computed as the ratio of the number of correctly detected sources to the number of true active sources. A source activity is deemed a correct detection if it has a \"similar\" source activity over the total observation interval to a true active source. Multiple activity patterns associated with different DOAs are considered \"similar\" if they match over a prescribed fraction of the time samples at which their values are one. Through experimentation, this fraction is chosen to be 0.95. The probability of false alarm for source activity is the ratio of the number of spuriously detected sources over the entire observation interval to the number of true active sources. Source activity is spurious if a source is detected to be active when no true source is active. Analogous definitions apply to \(P_{d}\) and \(P_{fa}\) of frequency hops and to paired activity. Figure 5 shows the ROC of FH estimation for 5 sources with an SNR per source of 10 dB. For \(P_{fa}=0.3\), FH estimation \(P_{d}=0.98\). The performance of the FH estimation affects the source separation, as the pairing uses the estimations from both the previous stages to pick a pair of source activity and FH pattern that best fits the received signals at each time instant. This is demonstrated later, in the ROC of paired activity. Figure 6 demonstrates the effect of HSF on the correct detection of activity for 5 LOS sources with an SNR per source of 10 dB. The figure shows the ROC performance with two HSF techniques. The ROC is obtained by solving Equation (6), followed by the thresholding

\[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]5 \(P_{d}\) versus \(P_{fa}\) of FH estimates (5 sources, \(J=20\) sensors, \(T=1000\) samples, SNR = 10 dB). of \(\hat{\mathbf{X}}_{d}\). The threshold ranges from the smallest to the largest values in matrix \(\hat{\mathbf{X}}_{d}\) obtained by solving Equation (6). Each point of the ROC, consisting of a pair of probability of correct detection and probability of false alarm, corresponds to one of these threshold values. Without filtering, for \(P_{fa}\) of activity \(=0.3\), \(P_{d}\) of activity \(=0.67\). For the same \(P_{fa}\), with MPE applied, \(P_{d}=0.86\) when \(\Omega\) is learned, and \(P_{d}=0.87\) when \(\Omega\) is known. With MAP applied, \(P_{d}=0.96\) when \(\Omega\) is learned, and \(P_{d}=0.97\) when \(\Omega\) is known. Figures 7 and 8 demonstrate the effect of HSF on the correct detection of activity for single-cluster and multiple-cluster propagation environments, respectively. DOA estimates in these figures are considered \"similar\" when the fraction measuring similarity is 0.95. Improvement in performance is observed in activity when HSF is applied. In the next figures, we apply MAP filtering with unknown HMM parameter \(\Omega\) only. \[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]6 \(P_{d}\) versus \(P_{fa}\) of activity with and without HSF for LOS sources (5 sources, \(J=20\) sensors, \(T=1000\) samples, SNR = 10 dB).