# Analytical Model of an Energy Detector for Ultra-Wideband Chaotic Communications
## 1 Introduction
[23], the author invents novel signal-processing techniques to improve the detection performance by choosing the value of the power operation of the signal sample according to the system settings; under certain conditions \"new energy detectors outperform the conventional energy detector.\" ED for baseband sampling is derived in [24], where detection and false alarm probabilities are calculated using a CLT approximation for the \(\chi^{2}\)-distribution; the motivation is that for \(N>200\) \"the Marcum \(Q\)-function is difficult to calculate or to take its inverse.\"  
ED models are derived in [25] for various signal distributions that actually correspond to different wireless channels (multipath, fading, etc. ); the author uses CLT and claims to have obtained more accurate approximations of the test signal statistic. In [26], instead of using CLT to proceed to normal distribution, the authors use more complex approximations of \(\chi^{2}\)-distribution, which improves the accuracy of ED performance by several percent. Energy detectors are also widely used, and their models are designed in fiber optics and visible light communications (VLC), e.g., [27; 28; 29]. In this paper, we propose an analytical model for the energy detector of ultra-wideband chaotic radio pulses. Such a detector is a part of the scheme of Direct Chaotic Communications (DCC) [2], which is the only practical implementation of chaotic communications at the moment. Early days of chaotic communications were associated mostly with chaotic synchronization [30; 31]; however, this phenomenon proved to be delicate and sensitive to signal distortion in the channel. Since classical correlation receivers are very difficult (if possible) to apply to chaos, differential correlation methods have been proposed, such as DCSK (differential chaos shift keying) [32; 33], in which the reference signal is transmitted along with the modulated signal through the same channel. The practical realization of DCSK systems has met technical difficulties (e.g., lack of long UWB time-delay components); therefore, at present such systems are mostly a subject of theoretical studies. Direct chaotic communications are based on the following ideas: the communication circuit must be as simple as possible, so the chaotic oscillations must be produced, modulated, and demodulated directly within the same frequency band that is given for the communication channel; no chaotic synchronization, because of its low immunity to the signal distortion; no up-and-down frequency conversion; and an incoherent receiver [34]. The transmitted information is encoded with a stream of chaotic radio pulses. The main element of this scheme, a chaotic radio pulse, has a number of unique features: it is wide-band or ultra-wideband by origin, because it is a property of the oscillations of the dynamic system which can be reliably controlled; variation of the pulse duration \(T\) has little or no effect on its spectrum; autocorrelation time of the UWB pulse is extremely small because it is inversely proportional to frequency bandwidth \(\Delta f\); the base of the pulse (or the number of its freedom degrees) \(B=2\Delta f\cdot T\gg 1\); and it can be varied arbitrarily by simply adjustingthe pulse duration \(T\) (in contrast, another popular UWB signal, an ultrashort pulse, is an elementary signal with the base \(B=1\)). Here we derive an analytical model of the energy detector of UWB chaotic radio pulses (Figure 1) in the static AWGN (additive white Gaussian noise) channel. The system behavior in a multipath environment needs special consideration; some of its aspects were investigated numerically in [34]. The structure of this paper is as follows. In Section 2, we describe the energy detector and the received signal transformation; we discuss chaotic signal models and derive approximate expressions for the BER (bit error ratio) and the optimal threshold. In Section 3, we compare the analytical BER values with the results of numerical simulation.