# Analytical Model of an Energy Detector for Ultra-Wideband Chaotic Communications
## 4 Discussion  
In order to understand the reason for the discrepancy between the analytical and the simulation curves in Figures 4 and 5, the function of a \(\chi^{2}\)-distribution and its approximation by the normal distribution should be considered.  
In Figure 6 overlapping \(\chi^{2}\)-distributions for \(E_{0}\) and \(E_{1}\) obtained in numerical simulation (solid-color areas) are presented along with the two curves that depict the normal distributions that approximate them. For clarity, the distributions are depicted in logY scale.  
Figure 5: BER as a function of \(E_{b}/N_{0}\) and processing gain \(B\).  
Figure 6: \(\chi^{2}\)-distributions and their approximations for \(E_{0}\) and \(E_{1}\).  
The optimal threshold that ensures the minimum of \(P_{\mathit{err}}\) is the solution to (9), i.e., the point of intersection of the two distribution functions. As can be seen, the normal distribution functions are symmetric, whereas the functions of noncentral \(\chi^{2}\)-distributions are tilted to the left. For this reason, the point of intersection of the two approximating normal distributions never coincides with the point of intersection of the \(\chi^{2}\)-distributions; it is always shifted to the left, thus giving a suboptimal threshold.  
As can be seen in Figure 6, normal distributions fit the vertex area of \(\chi^{2}\)-distributions sufficiently well, but the farther from the vertex, the greater the discrepancy between the \(\chi^{2}\) and normal distributions. This is what we see clearly in Figure 4 for small values of \(B\). Therefore, at large values of SNR, when the point of the distributions' intersection shifts farther and farther towards the \"tails\", the \"approximate\" threshold shifts farther from the optimal one. This is a key moment that is connected with the accuracy of the approximation, and it cannot be corrected other than using either \(\chi^{2}\)-distribution itself or using distributions the form of which approximates \(\chi^{2}\) better than the normal distribution.  
In this paper, we considered an energy detector with quadratic characteristics (its squaring device can be implemented, e.g., with a semiconductor diode). However, the described model can also be applied to the log detector. In practice, a log detector is technically more advantageous, since it has a much greater dynamic range. The voltage at the output of the log detector is proportional to the logarithm of the input power. However, the statistic of the log detector output values remains random-like, and due to the CLT, it also tends to a normal distribution. Simulation shows that even the outlook of the distribution functions in Figure 2 is preserved (only the Y axis is scaled). Obviously, all further mathematics hold and the results of the above model also describe the performance of the log detector.  
When talking about the applicability of this model to other types of UWB signals, such as ultrashort pulses or chirps, we must keep in mind that all of them are signals with fixed waveforms; therefore, the variance of the energy of such pulses is close to zero and the statistic of \(E_{1}\) energy values is determined only by the AWGN variance \(\sigma_{noise}^{2}\).  
In the 802.15.4a standard, in order to increase the transmission energy, the symbols are encoded by trains of ultrashort pulses, not the single pulses; in order for the spectrum of the emitted signal to be continuous, the pulses are put in random positions within the train (i.e., on the symbol time interval), and their polarity is also set randomly. As can be seen, both \"randomization\" methods do not change the energy of the emitted signal, so the statistic of the pulse energy values in the receiver of ultrashort pulses is determined mostly by the internal noise of the receiver. Thus, the models of energy detectors for these types of UWB signals can be taken from the bank of narrowband models.