# Analytical Model of an Energy Detector for Ultra-Wideband Chaotic Communications
## 1 Introduction  
Designing communication devices usually requires tedious numerical simulation. Occasionally, analytical solutions can be derived, exact or approximate, that allow us to substitute numerical simulation by direct calculation of analytical formulas, which can simplify and accelerate the design procedure. However, efficient analytical solutions pertain, as a rule, to concrete schemes, signals, and processing algorithms, because they take into account the signal statistics, the circuit or algorithm peculiarities, etc. ; that is, they are not universal, and the introduction of a new type of, e.g., information carrier may require a new analytical solution. Such an introduction of ultra-wideband (UWB) signals at the beginning of the 2000s [1] and the development of communications based on such signals, in particular direct chaotic communications [2], raised interest in analytical models of the energy detector (ED) [3; 4; 5; 6; 7; 8; 9]. Much interest in using energy detectors is now paid to MIMO (multiple input - multiple output) systems, including UWB MIMO systems [10; 11; 12; 13]. The history of analytical ED models started long ago [14; 15; 16; 17; 18], and many solutions were presented since then. The structure of these models is approximately the same. As a rule, they contain a block for estimating the signal energy and a decision unit that makes a conclusion on the received information. The energy estimation function is implemented by the \"squaring device + integrator\" bundle, which corresponds to the \"diode + LPF\" (low-pass filter) radio components. With regard to digital communications, the energy of a radio pulse encoding a certain symbol is estimated on a certain time interval and, based on the energy value, the ED decides that one of several symbols is received. In the case of binary communications, the decision is usually made by comparing the energy value with a threshold (single-sample detection) [17; 18]. Often the choice of the threshold and its optimal value are also a part of the model. After the signal energy is evaluated, information about the type of the data carrier (regular, pseudo-noise, AM, FM, etc.) is lost, and its nature exhibits itself only in the statistics of the energy values corresponding to transmitted symbols. In many cases, these statistics are close to random, and they can therefore be approximated by a normaldistribution due to the central limit theorem (CLT) [18]. An analysis of the literature shows that such a transition to an approximate solution is a very popular approach [17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. To date, in addition to many existing ED models, new ones constantly appear to solve still newer tasks. For instance, optimization of the signal sampling time (or the number of sensed samples) is analyzed in [19] for the problem of spectrum sensing under the effect of noise uncertainty, and a CLT-based ED model is used to compute the signal detection and false alarm probabilities. Energy detection in the case of a signal with a random amplitude is investigated in [20] for various amplitude statistics (Rayleigh, Rice, Nakagami, etc. ); distributions of ED decision statistics are retrieved and expressions for detection probabilities are obtained. The authors of [21] notice that \"the performance analysis of ED over slow fading channels is cumbersome, because it is difficult to derive closed-form expressions for the average probability of detection... \"; in order to obtain a computationally tractable solution, they derive an approximation \"by replacing the log-normal distribution with a Wald distribution\"; as for the ED, they count on [22], which gives a direct \(Q\)-form solution for the \(\chi^{2}\)-distribution. In Ref.