## 2 Renyi Entropy Based Spectrum Sensing  
In our region, GSM-850 is used since 2G and 5G technologies coexist in this frequency. A central frequency is used according to the absolute radio frequency channel number (ARFCN) assigned to the operator [3]. In this case, the ARFCN frequency lies in the GSM-850 MHz band [27]. The GSM-850 MHz frequency range was measured and it was found that ARFCN 166, which is 876.8 MHz in Downlink (DL) and 831.8 MHz in Uplink (UL), was free. They were used for experiments, since they are not used by the operator in the laboratory zone. However, if a PU uses this frequency, the cognitive system releases it and changes the frequency of the SU. A PU signal is considered and described below:  
\[y(t)=\left\{ \begin{array}{cc}n(t)&\quad\quad\text{noise}\\ h(t)*s(t)+n(t)&\quad\quad\text{PU}\end{array} \right. \tag{1}\]  
where \(n(t)\) is noise, \(h(t)\) is the impulse response of the system, \(s(t)\) is the signal received from a PU and \(y(t)\) is the incoming signal comprised of noise and the impulse response [28]. It is considered a system without a priori knowledge of the signal. A binary hypothesis test is used to find out the presence of the PU signal, \(H_{0}\) is the noise in the absence of a PU signal and \(H_{1}\) indicates the presence of a PU signal [29], as described below:  
\[y(n)=\left\{ \begin{array}{cc}u(n)&H_{0}\\ s(n)+u(n)&H_{1}\end{array} \right. \tag{2}\]  
where \(s(n)\) is the PU signal, \(u(n)\) is the noise and \(n=0,1,2,\ldots,\) N\(-1\), N is the sample size. It is assumed that the channel has AWGN and the noise has mean of 0 and a variance of 1. The entropy of \(y(n)\) in the time domain is sensitive to noise uncertainty while the entropy detector in the frequency domain is non-sensitive to noise. Our system is considered to improve the detection of PU presence in the network and operates with low SNR [11]. A Discrete Fourier Transform (DFT) is applied to (2) and (3) is obtained [11]:  
\[\overline{Y}(k)=\left\{ \begin{array}{cc}\overline{U}(k)&H_{0}\\ S(k)+\overline{U}(k)&H_{1}\end{array} \right. \tag{3}\]where \(\overline{Y}(k)\), \(\overline{S}(k)\) and \(\overline{U}(k)\) are the complex spectrum of the received signal, PU signal and noise, the DFT size \(K=N\,.H_{0}\) indicates the absence of a PU signal and \(H_{1}\) indicates the presence of a PU Signal [15]. The input of the system is a random variable since the symbols, amplitude and modulation scheme change over time and users of the cellphone system have a dynamic location and variable speed [3]. If we analyze the signal over time, classic energy detectors are obtained, yet the proposed solutions based on [11] translate the signal into frequency domain and calculate Renyi entropy instead of the Shannon entropy analyzed on [11]. A statistic test also contributes to detect the presence of the PU. Additionally, the designed algorithms were implemented on an SDR as explained in Section 3 to compare results with simulations and theory. Renyi entropy is described below [30]:  
\[H_{\alpha}(Y)=\frac{1}{1-\alpha}\times log\left(\sum_{i=1}^{n}p_{i}{}^{\alpha}\right) \tag{4}\]  
Given Y discrete events, \(P_{i}\) is the probability of occurrences in the ith bin, \(\alpha\) is the Renyi entropy order where \(\alpha\geq 0\) and \(\alpha\
eq 1\). Using the histogram method to estimate the received signal probabilities, for a given number of bins L, Renyi entropy is estimated by (5) for a statistic test [11]:  
\[T(Y)=\frac{1}{1-\alpha}\times log\left(\sum_{i=1}^{L}\frac{k_{i}{}^{\alpha}}{ N}\right)\left\{ \begin{array}{l}\leq\lambda:H_{1}\\ >\lambda:H_{0}\end{array}