## 1 Introduction
### Related Work  
In the literature, we can find different implementations of DM [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. These solutions differ in the architecture of the array feeding network and the synthesis of the weights that feed the antennas.  
Regarding the feeding networks, we can distinguish between those that use switches [10; 13], phase shifters [11; 12; 20; 31], phase shifters with switches [17; 21; 29], vector modulators (reconfigurable amplitude and phase) [18; 19; 22; 23; 24; 25; 26; 27; 28; 29; 30; 32], IQ data separately encoded transmitters [14; 27], retrodirective array [28] and a Fourier Rotman lens [15; 16].  
Some of the reported architectures [14; 15; 16; 27; 28] do not require a synthesis method for the antenna weights since the feeding network generates DM straightforwardly. Among the reported synthesis methods, we can find some that make use of brute force and explore all the possible combinations allowed by the architecture by searching for those that generate the desired constellation in the secure direction [10; 13; 20]. Other authors utilize optimization algorithms to obtain the antenna weights; some examples are genetic [11; 12], simulated annealing [17], particle swarm [18], differential evolution [29] and convex optimization [30; 31; 32]. In these cases, the optimization procedure tries to minimize a cost function subject to some constraints that are related to the architecture limitations, \(PE_{DM}\) and/or BER in certain observation angles. We can also find synthesis methods inspired by concepts [24] or methods that try to minimize the variations of BER response with respect to errors in the estimation of the secure observation angle [25]. Finally, there is the orthogonal vector approach proposed in [19] as a general framework for the synthesis of DM. The orthogonal vector approach was later adapted to include restrictions in the BER response [18], in the radiation patterns [22] and in the orthogonal artificial noise injected in some observation angles [23].  
All the above-mentioned methods need to evaluate BER during the design process and/or at the end of the synthesis in order to determine the level of security that can be expected from the DM transmission. BER evaluation techniques used in the literature are summarized in Table 1. In some cases [11; 12; 13; 18], the evaluation of BER is carried out, for every observation angle, using approximations that only apply to quadrature phase shift keying (QPSK) modulation and zero-mean artificial orthogonal Gaussian noise. However, most of the evaluation techniques [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32] use the time-consuming transmission of a stream of \(10^{6}\)-\(10^{7}\) symbols in a channel with additive white Gaussian noise (AWGN), since it can be applied to SDM and DDM with zero-mean artificial orthogonal noise.  
\begin{table}
\begin{tabular}{c c c} \hline  & **Closed-Form Equation**[11; 12; 13; 18] & **Data Stream Simulation**[10; 11; 12; 13; 18; 19; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32] \\ \hline Static DM & Only QPSK & Yes \\ \hline Dynamic DM: zero-mean Gaussian orthogonal artificial noise & Only QPSK & Yes \\ \hline Dynamic DM: zero-mean non-Gaussian orthogonal artificial noise & No & Yes \\ \hline Computation time & Low & High \\ \hline \end{tabular}
\end{table}
Table 1: Directional modulation (DM) system performance assessment: bit error rate (BER) evaluation techniques