# Analytical Model of an Energy Detector for Ultra-Wideband Chaotic Communications
## 2 Energy Detector Model
### Models of Chaotic Signals  
As was mentioned earlier, after calculating (or measuring) the energy of the radio pulse, information about the nature of the information carrier (chaotic, regular, modulation type, etc.) is actually lost. However, it exhibits itself in the statistic of \(E_{1}\). Radio pulses are usually described with three variables: amplitude, frequency, and phase. Chaotic radio pulses (i.e., fragments of chaotic oscillations generated by a nonlinear dynamic system) can be chaotic in every pulse variable. For example, some chaotic systems give chaotic-phase oscillations with nearly constant amplitude, so they are usually referred to as \"phase chaos\" [36; 37]. However, the most common are the chaotic signals, in which amplitude, phase, and frequency are chaotic all together; they are usually called \"amplitude chaos\" (e.g., [38; 39]). These types of chaotic signals and the corresponding models of such signals were analyzed in [40; 41], where the performance of communication systems was estimated by means of numerical modeling.  
Three statistic models of a band-limited chaotic signal were investigated: (a) a \"telegraph\", (b) a Gaussian, and (c) a uniform [40]. The \"telegraph\" signal, whose samples are a random sequence of \"-1\" and \"1\", is a model of the phase chaos, i.e., oscillations with a constant amplitude but a chaotically oscillating phase. It is easy to see that in this case, the energy of radio pulses is practically constant, and its statistic is indistinguishable from the statistic of classical radio pulses with a sinusoidal filling. The second signal model, a band-limited Gaussian noise, is used to describe the general chaotic-amplitude type of signals, in which all three oscillation variables are chaotic. Despite the general nature of this model, it does not fully correspond to practically feasible chaotic signals, since it allows arbitrarily large (albeit improbable) amplitudes, whereas the signals of real physical systems cannot take on infinite values. To achieve a better fit, the authors of [40] proposed a third model that represented a random signal whose samples have a limited uniform distribution.  
In chaotic communications, the \"phase chaos\" was shown to provide better BER than the \"amplitude chaos\", because pulse-to-pulse energy variations of the phase-chaotic signal are insignificant due to nearly constant amplitude, whereas for chaotic-amplitude pulses such variations are essential, which adds interference to the system noise [40; 41].  
In this study, we consider the \"amplitude chaos\" model as a more general case.  
Chaotic signal \(s_{i}\) with a flat spectral characteristic with good accuracy can be modeled with a white noise of the same frequency band [\(f_{lu},f_{up}\)]; therefore, its samples can be described with a normal distribution \(N\left(0,\ \sigma_{chaos}^{2}\right)\). Consequently, a noncentral \(\chi^{2}\)-distribution for \(E_{1}\) (3) can also be approximated by a normal distribution, i.e.:  
\[E_{1}\ =\ \sum_{i\ =\ 1}^{n}s_{i}^{2}+\sum_{i\ =\ 1}^{n}\xi_{i}^{2}\ =\ n\left(\sigma_{chaos}^{2}+\sigma_{noise}^{2}\right)+\xi_{ chaos}+\xi_{noise}\ =\ \xi_{\Sigma}, \tag{5}\]  
where \(\xi_{\Sigma}\) are samples with normal distribution \(N(n(\sigma_{chaos}^{2}+\sigma_{noise}^{2}),\ 2n(\sigma_{chaos}^{2}+\sigma_{noise}^{2})^{2})\) or, in a simpler form, \(N(n\sigma_{noise}^{2}(1+\gamma),\ 2n\sigma_{noise}^{4}(1+\gamma)^{2})\), where \(\gamma\ =\ \frac{c_{\max}^{2}}{c_{noise}^{2}}\).