Sleep is an essential process maintaining the life cycle of the human. In parallel with physiological, cognitive, subjective, and behavioral changes that take place during the sleep, there are remarkable changes in the electroencephalogram (EEG) that reflect the underlying electro-physiological activity of the brain. However, analyzing EEG and relating the results to clinical observations is often very hard due to the complexity and a huge data amount. In this article, I introduce several linear and non-linear tools, developed to analyze a huge time series data in many scientific researches, and apply them to EEG to characterize various sleep states. In particular, the spectral analysis, detrended fluctuation analysis (DFA), and synchrony analysis are administered to EEG recorded during nocturnal polysomnography (NPSG) processes and daytime multiple sleep latency tests (MSLT). I report that 1) sleep stages could be differentiated by the spectral analysis and the DFA ; 2) the gradual transition from Wake to Sleep during the sleep onset could be illustrated by the spectral analysis and the DFA ; 3) electrophysiological properties of narcolepsy could be characterized by the DFA ; 4) hypnic jerks (sleep starts) could be quantified by the synchrony analysis.