## 5 Characterization of the _On/Off_ Periods  
In this section, we first characterize the active and inactive periods of the selected birds' sounds. We first obtain the absolute value of the amplitude of the spectrogram as shown in Figure 5. Observing this bird song, we can see that even in the silent period, the sound is present, different from 0, although clearly, the bird is not singing during this time. As such, we have to establish a threshold to digitalize the signal to obtain the lapses of time when the bird is singing or in silence as we show in Figure 5(a) where we show the analog signal magnified and in Figure 5(b) we show the digital representation where the _On/Off_ periods are now clearly identified. Note that the digital representation is only used to characterize active and inactive periods, not to digitalize the bird's song which will be done in a subsequent section. We measured each of these active (silent) times, which to us, they represent the times when sensor nodes will be able to transmit data packets (remain silent), effectively following  
Figure 5: Spectrum analysis and absolute value of the amplitude of the spectrum of the _Myadestes occidentalis_ birdsong. (**a**) Audio spectrum of _Myadestes occidentalis_ birdsong. (**b**) Absolute value of spectrum of _Myadestes occidentalis_ birdsong. a close behavior of the natural bird song, concealing the system to potential intruders. From these measurements, we obtain the histograms, and from the histograms we obtain the probability distribution function (pdf). From this, we can characterize the random variables of both the active/inactive times by measuring all the \(On\) and \(Off\)times of the bird's sounds and then we calculate the mean ((6) and (7)), variance ((8) and (9)), standard deviation (10), and Coefficient of Variation (11) obtained respectively as:  
\[E(x)_{On}=\sum_{-\infty}^{\infty}xP(X=siging)=\lambda_{On} \tag{6}\]  
\[E(x)_{Off}=\sum_{-\infty}^{\infty}xP(X=silence)=\lambda_{Off} \tag{7}\]  
\[\sigma_{On}^{2}=\sum_{-\infty}^{\infty}(x-\lambda_{On})^{2}P(X=singing) \tag{8}\]  
\[\sigma_{Off}^{2}=\sum_{-\infty}^{\infty}(x-\lambda_{Off})^{2}P(X=silence) \tag{9}\]  
\[\sigma_{On(Off)}=\sqrt{\sigma_{On(Off)}^{2}(x)}=\sqrt{\int_{-\infty}^{\infty} (x-\lambda_{On(Off)})^{2}P(X=singing(silence)} \tag{10}\]  
\[CoV=\frac{\sigma_{x}}{E(x)} \tag{11}\]  
By identifying the statistical characteristics of these periods, we now propose different phase-type distributions to approximate to these empirical distributions. Namely, we use the Exponential distribution, given by (12), to approximate distributions that have a \(CoV=1\); the Hyper-Exponential distribution, depicted in (15) to approximate probability distributions with a \(CoV>1\); and an Erlang distribution, given by (20), to approximate distributions that have a \(CoV<1\). To characterize these distributions we match the first moment and the second central moment of these distributions to the observed measurements. \[f_{x}(x)=\lambda e^{-\lambda x}\to E[X]=\sum_{0}^{\infty}x\lambda e^{- \lambda x}=\frac{1}{\lambda}\rightarrow\lambda=\frac{1}{E[x]} \tag{12}\]  
Figure 6: Sampled spectrum of birdsong and spectrum digitalized signal of the _Myadestes occidentalis_ birdsong. (**a**) Sampled spectrum of the _Myadestes occidentalis_ birdsong. (**b**) Spectre digitalized of the _Myadestes occidentalis_ birdsong. And the second central moment (variance) is calculated as \(\sigma_{x}^{2}(x)=E[x^{2}]-(E[x])^{2}\), then we have Equation (13):  
Figure 5:0  
Hence, the \(CoV\) is given in Equation (14):  
Figure 5:1  
The Hyper-Exponential is composed by two exponential distribution, with parameters \(\lambda_{1}\) and \(\lambda_{2}\) which are selected with probability \(p_{1}\) and \(p_{2}\) respectively where \(p_{1}=p\), and \(p_{2}=1-p\). As such, the pdf of the Hyper-Exponential distribution can be expressed as Equation (15):  
Figure 5:2  
and the mean is obtained as in Equation (16):  
condition \(\lambda_{2}>\lambda_{1}\) with \(0<p<1\) generally leads to a CoV greater than 1.