# Linear Algorithms for Radioelectric Spectrum Forecast
## 3 Case Study and Experiment Procedure  
The decision to carry out this study was made during the spectrum measurements campaign held in Bogota-Colombia where we obtained the measurements employed here from spectrum occupancy study previously carried out [1; 32]. The band analyzed was the GSM 850 MHz, as it is a band constantly used and viable for analysis in time function with conventional equipment, like a spectrum analyzer. The measurements used in this study correspond to a week, from 23 December to 29 December 2012. In some studies [33], it has been indicated that a reasonable option to obtain representative data without any a priori information about a band is to consider measurement periods of at least 24 h in order to avoid under or overestimating frequency bands occupancy with some temporary patterns. While a 24-h measurement period could be thought of as adequate in order to properly characterize the activity of determined spectrum bands [34], in this research 7 days were analyzed, including patterns for workdays and weekends. Additionally, this time period is sufficient to measure occupancy in mobile networks with low use, as indicated in [2; 34]. The channels to be modeled were selected after measuring the duty cycles of 60 channels at GSM band. From these, three channels with different occupancy levels (high, medium and low), were chosen. Figure 1 presents results of power measure for three downlink channels during a week, with different power level. Spectrum analyzer configuration for this band was the following: a resolution bandwidth of 100 kHz with a sweep time of 290 ms, which guarantees GSM signal detection with a bandwidth of 200 kHz. Daily duty cycles from PUs at selected channels are shown in Figure 2. Threshold (\(\lambda\)) used, which for this event is of \(-89\) dBm, was obtained from Equation (6) with a probability of false alarm (Pfa) of 1% [35]. \(\lambda\) which is above of the detected noise floor of \(-102\) dBm [1],  
\[P_{fa}=\frac{\Gamma\left(m,\frac{\lambda}{2}\right)}{\Gamma\left(m\right)} \tag{6}\]where \(\Gamma(. )\) y \(\Gamma(.\),.) are complete and incomplete gamma functions, respectively, and \(m\) is the product of time times bandwidth. In this work the HNM is calculated from the differences between measurements of received power performed outside at street level and indoors, in a building. These measurements were performed with a discone-type antenna and spectrum analyzer. The calculation allows analysis of the margin between a non-licensed device indoors, and a PU outdoors without interference due to shadowing. Table 1 shows the HNM found for each channel. Figures 3-5 present histograms corresponding to opportunities distribution during time periods of GSM band channels; it is observed that such opportunities have an exponential behavior, whose approximate equations and the coefficient of determination (R\({}^{2}\)) are exhibited in each figure. Thus, the occurrence increases for the channel occupancy, especially for shorter time periods of use. For low, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Low Channel HNM (dB)** & **Medium Channel HNM (dB)** & **High Channel HNM (dB)** \\ \hline
9.2 & 11.9 & 6.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hidden node margin (HNM) for three different frequency channels for an urban environment. Figure 1: Power measurements for three global systems for mobile communications (GSM) band downlink channels. Figure 2: Duty cycles for three GSM band downlink channels. medium and high occupancy channels, total times of opportunities were approximately 84 h, 81 h and 78 h, respectively, which indicates relatively low occupancy [2]. Following this, we proceeded to analyze the time series of measured channels over a week, which was equivalent to 1,062,514 samples. To do this, ACF is initially presented, as observed in Figure 6.  
Figure 4: Time period opportunities distribution for medium channel. Figure 5: Time period opportunities distribution for high channel.