# 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
\[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]7 \(P_{d}\) versus \(P_{fa}\) of activity with and without HSF for single-cluster propagation environment (5 sources, \(J=20\) sensors, \(T=1000\) samples, SNR = 10 dB). The performance of pairing is demonstrated next. Figure 9 shows the ROCs of paired activity for all propagation environments, LOS, single cluster and multiple cluster, when the fraction measuring similarity is 0.95. For \(P_{fa}=0.3\), \(P_{d}=0.94\) for the LOS propagation environment, \(P_{d}=0.84\) for the single-cluster propagation environment, and \(P_{d}=0.83\) for the multiple-cluster propagation environment. Figure 10 plots the probability of detection as a function of the number of LOS intermittent sources that are detected as active during the observation interval. The detection threshold is set at a probability of false alarm \(P_{fa}=0.15\). It is observed that the performance of both the FH and DOA estimation stages affect the performance of the pairing stage. \[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]8 \(P_{d}\) versus \(P_{fa}\) of activity with and without HSF for multiple-cluster propagation environment (5 sources, \(J=20\) sensors, \(T=1000\) samples, SNR = 10 dB). \[\hat{s}_{\text{MPE}}(t)=\operatorname*{arg\,max}_{i=0,1}\gamma_{i}(t). \tag{16}\]9 \(P_{d}\) versus \(P_{fa}\) of paired activity for LOS, single-cluster and multiple-cluster propagation environments (5 sources, \(J=20\) sensors, \(T=1000\) samples, SNR = 10 dB). ## 5 Discussion

In this paper, an approach is proposed to solve the BSS problem for FH sources that are stationary, spatially sparse and have intermittent activity observed over channels that follow the spatial channel model for three propagation environments. The source memory is utilized in the filtering of source activity to enhance performance. Current methods in the literature do not perform the association of multiple frequency hops to the sources they are transmitted from. We bridge this gap by pairing the FH estimates with DOA estimates and labeling signals to their sources, irrespective of their hopped frequencies. The pairing stage uses the estimations from the previous stages to pick a pair of the DOA and FH patterns that best fits the received signals at each time instant. The pairing assigns source labels to the signals and is capable of reducing false alarms that arise in the individual stages, thus improving the accuracy of the proposed approach as confirmed by numerical results. Numerical results demonstrate that HSF improves BSS performance, and that MAP HSF outperforms MPE HSF. Conceptualization, A.G. and A.H.; data curation, investigation, methodology and software, A.G. and A.D.; formal analysis and validation, A.H. and O.S. ; funding acquisition, J.D. ; writing--original draft and visualization, A.G.; writing--review & editing, A.G., A.H. and O.S. All authors have read and agreed to the published version of the manuscript. The work of [PERSON] and [PERSON] was supported in part by the U.S. Department of Defense (Navy--Naval Information Warfare Center) under Agreement no. N66001-20-P-6010. Not applicable. The data presented in this study are available in this article and the article titled \"Interference Mitigation in Blind Source Separation by Hidden State Filtering\", in proceedings of the 57 th Annual Conference on Information Sciences and Systems (CISS), Baltimore, MD, USA, 22-24 March 2023. The authors declare no conflict of interest. ## References

* (1) [PERSON]; [PERSON] _Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar_; Cambridge University Press: Cambridge, MA, USA, 2005. * (2) [PERSON]; [PERSON] Chaotic Standard Map Based Frequency Hopping OFDMA for Low Probability of Intercept. _IEEE Commun. Lett._**2011**, _15_, 1019-1021.