# Performance Evaluation of Directional Antennas in ZigBee Networks under NLOS Propagation Conditions
## 2 Materials and Methods
### Extraction of the Received Signal  
In NLOS environments, the power received by an antenna can be mainly due to reflections and other propagation phenomena. Knowing the distribution of the input signal at the receiver as a function of the azimuth angle allows us to evaluate the main directions in space that contribute to the received power. [PERSON] et al. [36] developed a method applied to vegetation to determine the density of the transmitted power just before reaching the receiver antenna. We used this procedure in the environments described in Section 2.1 to obtain the signal distribution at the receiving antenna input. The technique assumed that the scattered wave trains are uncorrelated in phase. Thus, the power can be summed at the receiver in real quantities. Measuring the signal distribution requires a very-high-gain antenna. The problem with this procedure is related to the antenna radiation diagram because it is not an ideal pulse.  
Considering \(i(\theta)\) as the directional intensity of the incoming signal as a function of the azimuth angle \(\theta\) and \(g(\theta)\) as the radiation pattern of the receiver antenna, the measured power is given by a convolution:  
\[p(\theta)=g(\theta)*\ i(\theta) \tag{1}\]  
where \(*\) represents the convolution operation. The Fourier Transform is  
\[P(\omega)=G(\omega)I(\omega) \tag{2}\]  
Knowing \(G(\omega)\), \(I(\omega)\) can be extracted by deconvolution. However, very high errors may occur when \(I(\omega)\) is determined from \(P(\omega)/G(\omega)\) because of the division by values close to zero. Following the technique proposed in [36], a better estimate of \(I(\omega)\) can be given by  
\[I_{e}(\omega)=P(\omega)\frac{G^{*}(\omega)}{\left|G(\omega)\right|^{2}+\lambda} \tag{3}\]  
where * is the complex conjugate. The parameter \(\lambda\) was determined by evaluation of the following error function:  
\[e(\theta)=p(\theta)\ -\ g(\theta)*\ i_{e}(\theta) \tag{4}\]  
We evaluated the error in the frequency domain for simplicity. From Equation (3), there is no effect in the deconvolution error when \(\lambda\) is too low. On the other hand, the signal distortion is high if \(\lambda\) is too high [37]. A decision criterion was to choose a value of \(\lambda\) that produced a root mean square of the normalized error of 0.1.  
The received power was measured as a function of the azimuth angle with an antenna of 17.5 dBi. This antenna consists of an array of 4 \(\times\) 2 suspended patches. The radiation pattern of this antenna, \(g(\theta)\), was measured inside an anechoic chamber, giving the result shown in Figure 4a. Figure 4b shows the radiation pattern of the directional antenna presented in the previous section.  
We attached the receiving antenna of 17.5 dBi to a stepper motor mounted on a pole to measure the signal distribution. The transmitter consisted of a radio-frequency generator connected to the antenna. We used the Lab Brick Signal Generator from Vaurix Technology Corporation [38], model LSG-602. This generator allows the transmission of a signal between 1500 MHz and 6000 MHz, with an output power between \(-45\) dBm and 10 dBm. We used the spectrum analyzer FSH8 from Rohde and Schwarz [39] connected to the 17.5 dBi gain antenna as a receiver. The received power was recorded on a personal computer connected to the spectrum analyzer. The bandwidth considered in the measurements was 30 kHz. The system obtained about 300 samples in each complete rotation of the receiver antenna.