# Renyi Entropy-Based Adaptive Integration Method for 5G-Based Passive Radar Drone Detection
## 5 Renyi Entropy for 5G Signal Analysis  
### Theory  
Let us assume an OFDM signal model given as follows [36]:  
\[x(t)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}d_{k}\epsilon^{p2\pi f_{k}t},\quad 0<t<T, \tag{5}\]  
where \(f_{k}\) is the frequency of the \(k\)th subcarrier (from the set of \(N\) orthogonal subcarriers), \(d_{k}\) is the \(k\)th complex data symbol and \(T\) is the duration of the OFDM symbol. Equation (5) is a general formula defining signal performance depending on subcarriers. The model is valid for 5G signals and describes the inverse Fourier transform on modulated subcarriers. From (5), it is clear that the higher the number of subcarriers, the more complex the signal character is. In 5G networks, the number of subcarriers and their duration change continuously depending on the transmitted content. The distribution of subcarriers in a time-frequency grid links the amount of data to the possible resources of the system. A time-domain signal is obtained by performing the inverse Fourier transform, ultimately used for data transmission. The OFDM signal reaching the radar receiver can easily be transformed into a time-frequency grid by performing a short-time Fourier transform (STFT). This makes it possible to directly analyze the allocation of subcarriers in the network and the use of time resources without knowing the signal character and network configuration. The STFT of the complex and continuous signal \(x(t)\) and the even and real window \(h(t)\) is defined as [37]  
\[F_{x}^{h}(t,\omega)=\int_{\mathbb{R}}x(\tau)h^{*}(\tau-t)\mathrm{e}^{-j\omega( \tau-t)}\mathrm{d}\tau. \tag{6}\]  
The energy distribution, usually referred to as a spectrogram, is given as  
\[S_{x}^{h}(t,\omega)=|F_{x}^{h}(t,\omega)|^{2}. \tag{7}\]  
To express the amount of information contained in the STFT, the Renyi entropy is usually applied. The Renyi entropy was proposed in the context of measuring the content contained in the time-frequency representation [38]. It is very often used as a measure of a method's ability to concentrate distribution, e.g., for reassignment or synchrosqueezing techniques [39]. In this work, Renyi entropy is used in a different context: namely, it serves as a quantitative measure of resource utilization in a 5G network. The \(\gamma\)-order Renyi entropy of any time-frequency signal representation \(\mathrm{TF}_{x}(t,\omega)\) is expressed in bits (for simplicity, the results given in the further part of this work are presented without a unit) and defined as [38]  
\[H_{R}^{\gamma}(\mathrm{TF}_{x})=\frac{1}{1-\gamma}\log_{2}\Biggl{(}\frac{ \iint_{\mathbb{R}^{2}}(\mathrm{TF}_{x}(t,\omega))^{\gamma}\mathrm{d}t\mathrm{ d}\omega}{\iint_{\mathbb{R}^{2}}(\mathrm{TF}_{x}(t,\omega))\mathrm{d}t \mathrm{d}\omega}\Biggr{)}. \tag{8}\]  
In this work, the Renyi entropy is computed using the absolute value of the STFT, such that \(|F_{x}^{h}(t,\omega)|\), and its order was fixed to \(\gamma=3\) for all experiments. As shown in the literature [38; 40; 41; 42; 43], \(\gamma=3\) ensures stable results and is the best choice for information analysis. Thus, further considerations are carried out within this assumption. Since the signal given by (5) is an inverse Fourier transform on the modulated subcarriers, the Fourier transform performed on this signal allows the observation of variability in the number of subcarriers. Thus, it is possible to give a quantitative relationship informing the utilization of downlink resources using the Renyi entropy. An illustrative example of the Renyi entropy with different values of the downlink allocation and noise level is shown in Figure 12. In the experiment, the signal was simulated in the MATLAB environment with an assumed portion of subcarriers corresponding to the percentage of allocated resources. The results clearly show a strong relationship between the transmitted content and the entropy value.