## 4 Evaluation of BER for Dynamic Directional Modulation
### Zero Mean Artificial Gaussian Noise: Results and Discussion  
A straightforward choice for the zero mean artificial noise is to follow a Gaussian distribution. In that case, we can compute \(SNR_{av}(\phi)\) through (12) and evaluate BER using well-known expressions that can be found in the literature [34; 35] or the berawgn MATLAB function [36].  
Configurations with different modulation schemes, \(SNR(\phi_{0})\) and \(PE_{DDM}\) were tested, and the average radiation patterns of (9)-(11) are plotted in Figure 2. The artificial noise average patterns (\(rp_{av\_W}(\phi)\)) of Figure 2 reveal that the synthesis strategy proposed in [19] spreads the artificial noise uniformly in all the observation angles outside of the main beam. As a consequence, \(SNR_{av}(\phi)\) in the side lobes and the information beam width cannot be adjusted independently. This strategy is also inefficient in terms of \(PE_{DDM}\), since artificial noise is transmitted along some directions, such as the radiation nulls of the conventional radiation pattern, where no information is sent.  
Figure 3 shows the BER computed with our approach (for \(p\gg 1\)) and expressions in Table 2 compared to the BER evaluated through the transmission of \(10^{6}\) symbols following the procedures described in [9; 19]; all the results show very good agreement. Considering that in [9; 19], eavesdroppers follow a training stage before demodulating, the scenario under evaluation is very unfavorable from the point of view of security. Although the information beam width also depends on the modulation scheme, in Figure 3, we can also check that the narrower information beam width corresponds to the scenario with the lowest \(PE_{DDM}\) in spite of having the highest \(SNR(\phi_{0})\) (Figure 3c).  
Figure 4 compares BER evaluated using a stream of \(10^{6}\) symbols to our approach as the number of available radiation patterns per symbol \(P\) increases. Obviously, for SDM (\(P=1\)), it cannot be assumed that the added artificial noise follows a Gaussian distribution, therefore, we can find differences between both methods in the side lobe region and in the edges of the information beam. However, both BER evaluations methods converge rapidly as \(P\) increases (DDM), and four different radiation patterns per transmitted symbol are enough to produce similar results.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Modulation Scheme** & **QPSK** & **16-QAM** & **8-PSK** \\ \hline \(BER(\phi)\) & \(Q(\sqrt{SNR_{av}(\phi)})\) & \(\frac{3}{4}Q(\sqrt{\sharp SNR_{av}(\phi)})\) & \(\frac{2}{3}Q(\sin(\frac{\pi}{8})\sqrt{2 SNR_{av}(\phi)})\) \\ \hline \end{tabular}
\end{table}
Table 2: BER for uncoded data over AWGN channels [34; 35].  
Figure 2: Array of four omnidirectional antennas along x-axis, spacing \(\lambda/2\). Average radiation patterns in XY plane for dynamic directional modulation (DDM) (\(P=64\)). Zero mean artificial Gaussian noise. Only half-plane is plotted due to symmetry. **(a)** QPSK modulation, \(\phi_{0}=90^{\circ}\), SNR(\(\phi_{0}\)) = 12 dB, \(PE_{DDM}=75\%\). **(b)** 16-QAM modulation, \(\phi_{0}=120^{\circ}\), SNR(\(\phi_{0}\)) = 18 dB, \(PE_{DDM}=90\%\). **(c)** 8-PSK modulation, \(\phi_{0}=45^{\circ}\), SNR(\(\phi_{0}\)) = 21 dB, \(PE_{DDM}=50\%\).  
Figure 3: Array of four omnidirectional antennas along x-axis, spacing \(\lambda\)/2. BER in XY plane for DDM (\(P=64\)). Zero mean artificial Gaussian noise. Only half-plane is plotted due to symmetry. (**a**) QPSK modulation, \(\phi_{0}=90^{\circ}\), signal-to-noise ratio (SNR)(\(\phi_{0}\)) = 12 dB, \(PE_{DDM}\) = 75%. (**b**) 16-QAM modulation, \(\phi_{0}=120^{\circ}\), SNR(\(\phi_{0}\)) = 18 dB, \(PE_{DDM}\) = 90%. (**c**) 8-PSK modulation, \(\phi_{0}=45^{\circ}\), SNR(\(\phi_{0}\)) = 21 dB, \(PE_{DDM}\) = 50%.