In this chapter, the theoretical foundation to extract the characteristic time constants from stochastic single-defect emissions, especially from RTN signals was laid out. After covering the basics of Markov processes and introducing the Markov chains of different types of defects in Sections 6.1 to 6.3 , in Section 6.4 a way to combine different defects into a combined system was developed. Section 6.5 covers the special case of coupled defects which will be used in the next chapter to estimate the electrostatic coupling between neighboring defects.
In Section 6.6, a new way to extract the characteristic time constants based on spectral maps was presented and compared to traditional methods like the histogram method and the lag plot method. The limitations of these methods, especially their inability to detect thermal states producing anomalous RTN led to the development of a HMM library able to handle systems of multiple, arbitrarily shaped defects (see also Appendix A).
The theoretical foundation for the training of the HMM training including the constraints imposed by the special case of systems of defects producing single-charge emissions is given in the preceding Section 6.7.
Despite the well-known Baum-Welch algorithm, this section discusses the treatment of multiple defects, multiple sequences and the extraction of 
of the defects and the measurement noise of the combined system. One very important problem which is often ignored for HMM, namely the long-term drift of measurement signals is treated in Section 6.7.4. All of these constraints make the presented HMM library a versatile and robust tool to extract defect properties from complex RTN signals.
The presented HMM library can be used to either simulate RTN signals for different systems or to extract their properties by training the model with a set of measurement sequences. The former case was used on many occasions within this chapter to reveal expected RTN signals produced by certain types of defects or to point out systematic deficiencies of different extraction modes. The latter case was used in the previous section to test the robustness of the library to different influences. In the next chapter the HMM library will be used to investigate the defect structure and to extract the characteristic time constants of defects producing three-level anomalous RTN in nanoscale GaN/AlGaN MIS-HEMTs.
