## 2 Composite Signal's Distribution and the Dataset
### Distribution Fitting  
To determine a suitable statistical distribution for representing the voltage levels of composite signals, we first fit continuous statistical distributions provided in SciPy 1.11.3 library [27] to the data using RSS [3]. The RSS score quantifies how well the fitted distribution replicates the individual sample data points by measuring the sum of the squared differences between the observed values (\(y_{k}\)) and the values predicted by the fitted distribution (\(y_{k}\)). The RSS score, \(R_{SS}\), is defined as  
\[R_{SS}=\sum_{k=1}^{K}(y_{k}-y_{k})^{2} \tag{3}\]  
where \(K\) is the number of data samples, \(y_{k}\) is the \(kth\) data sample, and \(y_{k}\) is an estimate of \(y_{k}\) from the fitted distribution. The parameters for each candidate distribution are optimized by minimizing \(R_{SS}\), which provides the closest fit of the statistical distribution to the data. Once the parameters of the fitted distributions were optimized using RSS, KL divergence was calculated to evaluate how well the theoretical distribution, generated with the optimal estimated parameters, aligned with the composite signal's voltage distribution [3]. KL divergence measures the relative entropy between two probability distributions that indicates how much information is lost when the fitted distribution (\(Q\)) is used to approxi \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Range** \\ \hline Modulation order & \(\{2,4,8,16,32,64,128,256\}\) \\ Data rate & \([1,10]\) kbps \\ Power level & \([-30,0]\) dBm \\ Nomalized frequency & \([-60,60]\) kHz \\ Number of component signals & \([2,10]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Range of parameters used to generate composite signals [3]. Figure 4: Frequency domain representation of a composite signal [3]. mate the true distribution of the signal (\(P\)). For discrete probability distributions \(P\) and \(Q\), the KL divergence score from \(Q\) to \(P\) is defined as:  
\[D_{KL}(P\parallel Q)=\sum_{x\in\Phi}P(x)log\bigg{(}\frac{P(x)}{Q(x)}\bigg{)} \tag{4}\]  
where \(P\) represents the composite signal's samples, and \(Q\) represents theoretical distribution with optimal estimated parameters, and \(\Phi\) is the probability space. A lower \(D_{KL}\) value indicates a better approximation, with \(D_{KL}=0\) meaning that the fitted distribution \(Q\) is identical to the true distribution \(P\). To compare the quality of each fitted distribution, we computed the mean, \(\mu\), and standard deviation, \(\sigma\), for both \(R_{SS}\) and \(D_{KL}\) scores across 1000 instances of composite signals. The standard deviation, \(\sigma\), describes consistency of fit; a lower \(\sigma\) suggests that a distribution performed consistently well across different composite signal instances. Table 2 highlights the four distributions with the lowest \(R_{SS}\) and \(D_{KL}\) scores, signifying their close approximation to the composite signals' behavior. Figure 5 illustrates the probability density function (PDF) of these distributions when fitted to a composite signal comprised of ten component signals, illustrating the efficacy of each in modeling the signal's statistical properties. Given that the GGD consistently yielded the lowest combined \(R_{SS}\) and \(D_{KL}\) scores, we use it to represent the voltage distribution of composite signals. For a voltage level, \(v\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution** & **RSS (\(\mu\pm\sigma\))** & **KL Divergence Score (\(\mu\pm\sigma\))** \\ \hline Beta & 0.2637 \(\pm\) 0.3544 & 0.0020 \(\pm\) 0.0018 \\ Chi & 0.5200 \(\pm\) 0.6494 & 0.0065 \(\pm\) 0.0047 \\ Generalized Gamma & 0.0922 \(\pm\) 0.2189 & 0.0007 \(\pm\) 0.0009 \\ Rice & 0.2342 \(\pm\) 0.4192 & 0.0025 \(\pm\) 0.0031 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SSE and KL divergence scores for the four distributions fitted to composite signals [3]. Figure 5: Four common distributions that closely fit composite signals [3].