# Analytical Model of an Energy Detector for Ultra-Wideband Chaotic Communications
## 2 Energy Detector Model
### Signal Processing in the Envelope Detector  
Let us describe the signal transformation in the receiver. The schematic of the envelope detector is represented with a quadratic diode and a low-pass filter (LPF) (Figure 1), i.e., a squaring device and an integrator (a band-pass filter at the input is not shown, yet it is implied).  
An RF (radio frequency) UWB electric signal at the detector input is denoted as \(X(t)\). To detect a radio pulse on a time interval, which we denote for certainty as \([0,T]\), where \(T\) is the radio pulse duration, the detector must distinguish between the two cases:  
\[\begin{array}{ll}X_{1}(t)=s(t)+\xi(t),&\text{--symbol `1' (a radio pulse)},\\ X_{0}(t)=\xi(t)&\text{--symbol `0' (no pulse), $t\in[0,T]$},\end{array} \tag{1}\]  
where \(s(t)\) is a carrier chaotic signal generated by a deterministic dynamic system whose spectrum \(S(f)\) is confined in the frequency band \([f_{lo},f_{up}]\), and \(\xi(t)\) is the AWGN of the same frequency band \([f_{lo},f_{up}]\).  
From the output of the squaring device (a diode), the signal \(X^{2}(t)\), that is the instantaneous power of the received signal, is fed to the integrator (a low-pass filter). At its  
Figure 1: Energy detector of chaotic radio pulses.  
output, the signal is proportional to the energy of the input signal in some preceding time windows \(T_{up}\).  
The filter constant must be matched with the pulse length \(T\); therefore, if the voltage across the filter output is sampled at moment \(T\), when the filter window coincides with the interval \([0,T]\), then the envelope detector sample \(X_{D}(T)\) is equal to the radio pulse energy on the interval \([0,T]\). That is, \(X_{D}(T)=E_{1}\) when transmitting symbol \"1\" or \(X_{D}(T)=E_{0}\) when transmitting symbol \"0\". Hereinafter, the subscripts \"1\" or \"0\" of the variables and functions refer to the cases of pulse and no pulse, respectively.  
Let us expand the expression for \(E_{1}\)  
\[E_{1}\ =\ \int_{0}^{T}(s(t)+\xi(t))^{2}dt\ =\ \int_{0}^{T}s^{2}(t)dt+\int_{0}^{T} \xi^{2}(t)dt+2\int_{0}^{T}s(t)\xi(t)dt. \tag{2}\]  
Chaotic signal \(s(t)\) and noise \(\xi(t)\) are uncorrelated zero-mean variables; hence, the cross-correlation term in (2) is close to zero and can be discarded.  
Following the usual approach, we proceed from continuous to discrete signal representation:  
\[E_{1}\ =\ \sum_{i\ =\ 1}^{n}s_{i}^{2}+\sum_{i\ =\ 1}^{n}\xi_{i}^{2}. \tag{3}\]  
Here, \(s_{i}\) and \(\xi_{i}\) are independent samples on \([0,T]\). Since the continuous signals occupy a frequency band \([f_{lo},f_{up}]\), according to the sampling theorem, the samples must be taken in time intervals \(\Delta t=1/2\Delta f=1/2(f_{up}-f_{lo})\). The number of independent samples (number of freedom degrees) on the pulse interval is \(n=T/\Delta t=2T\Delta f=B\), where \(B=2\Delta f\cdot T\) is the signal base (processing gain).  
Similarly, energy \(E_{0}\) in expression (4) is the sum of the squares of random noise samples \(\xi_{i}\). Here we describe AWGN samples with a normal distribution \(N(0,\sigma_{noise}^{2})\). Accordingly, the sums of the squares of such samples have noncentral \(\chi^{2}\)-distribution with \(n\) degrees of freedom with math expectation \(E[\chi]=n\) and variance \(D[\chi]=2n\sigma_{noise}^{2}\). In agreement with the central limit theorem [18], at \(n\to\infty\) this \(\chi^{2}\)-distribution tends to a normal distribution \(N(n\sigma_{noise}^{2}\ 2n\sigma_{noise}^{4})\). In practice, DCC systems operate with the bases \(B=100\)-\(250\) and more [34; 35], so condition \(n\gg 1\) is fulfilled. With good accuracy, we can approximate \(E_{0}\) with the expression:  
\[E_{0}\ =\ \sum_{i\ =\ 1}^{n}\xi_{i}^{2}\ =\ n\sigma_{noise}^{2}+\xi_{noise}, \tag{4}\]  
where \(\xi_{noise}\) is described by a zero-mean distribution \(N(0,\ 2n\sigma_{noise}^{4}\ )\).