# Spectrum Awareness for Cognitive Radios Supported by Radio Environment Maps: Zonal Approach +
## 3 Map Construction Techniques  
There are three main construction techniques suitable for REMs [33; 34]:* _Indirect (transmitter location based methods)_--applying transmitter location and propagation model to obtain the estimated value,
* _direct (spatial statistics based methods)_--applying interpolation techniques and sampled data,
* _hybrid_--combining the two approaches. The selection of the most appropriate map construction technique depends on its specific application. In some research works, indirect methods are preferred due to the fact that they help to construct more accurate maps if the propagation model has been selected properly and if the terrain model is accurate. On the other hand, indirect methods require a set of input data, e.g., transmitter location, the TX power, and activity pattern of the transmitter as a minimum. Indirect methods are widely applied to predict the communication range of the wireless systems or to assign frequencies for specific scenarios. Spatial statistics based methods use measurement data from many sensors. The collected measurement data is used as an input for the interpolation process, which enables us to get the data for the remaining points. Direct methods seem to be more relevant for spectrum monitoring and spectrum awareness systems. For this reason, in the following part of the paper, we deal with direct methods. When REM is considered, the most promising interpolation techniques described in the literature are as follows: Nearest Neighbor (NN), Inverse Distance Weighting (IDW), and Kriging [35; 36]. NN method uses Thiessen (or Voronoi) polygons, which are created around each point at which the measurement was taken. A characteristic feature of these polygons results from the fact that their borders are exactly in the middle of the distance between neighboring sensors. NN method is thought to be the simplest, and it offers little accuracy. IDW is based on weighting factors \(w_{i}\) that are inversely proportional to the distance between the given point \(x_{0}\) and the measurement point \(x_{i}\) and raised to the power \(p\). The power \(p\) defines how the influence of the weighting factors decreases with the distance. When the power \(p\) value is set high, the points which are closer have a stronger impact. If the power \(p\) value is set at zero, regardless of the distance, the weighting factors remain at the same level. The typical formula for IDW method is [37; 38]:  
\[\hat{V}(x_{0})=\sum_{i=1}^{N}w_{i}(x_{0})\cdot V(x_{i}), \tag{1}\]  
where:  
\(\hat{V}(x_{0})\) is the predicted signal level for point \(x_{0}\),  
\(N\) is the number of points at which the signal level was measured,  
\(w_{i}(x_{0})\) is the weighing factor for point \(x_{0}\),  
\(V(x_{i})\) is the signal level measured at location \(x_{i}\). The formula to determine the weights for \(w_{i}\) is given below [37; 38]. \[w_{i}=\frac{\left(\frac{1}{h_{i}}\right)^{p}}{\sum_{k=1}^{N}\left(\frac{1}{h_ {k}}\right)^{p}}\, \tag{2}\]  
where \(h_{i}\) is the distance between point \(x_{i}\) and point \(x_{0}\), \(h_{k}\) are distances between points \(x_{k}\) and point \(x_{0}\), and \(p\) is the power. Usually, in the literature on the topic, the power \(p\) is set to 1 or 2. In our research, we have taken into account the power \(p\) up to 6. In the rest of the paper, we use the following notation for IDW method: IDW pa where \(\alpha\) is the value of the power \(p\), e.g., for IDW p3 the value of the power is equal to 3. Kriging is one of the geostatic methods of interpolation [39; 40]. Similarly to IDW method, Kriging also uses Equation (1) and weighting factors. However, these weighting factors are determined on the basis of the semivariogram that takes into account the distance between measurement points and the variation between measurements of signal levels as a function of the distance.