# On Coding and Decoding Reconfigurable Radiation Pattern Modulation Symbols
## 2 Materials and Methods
### Symbol Selection  
As shown in Figure 2, the full spherical space is considered (\(0\ \leq\ \phi\ \leq\ 2\pi\ \&\ 0\ \leq\ \theta\ \leq\ \pi\)) with a high resolution (\(\rho\ =\ 40\)) to reconstruct the radiation symbols. By inspection, only six out of the sixteen states from Figure 2 are unique. The level of uniqueness is quantified by the degree of linear independence or orthogonality between each pair of symbols. To quantify this, we consider a procedure to determine the linear independence of radiation pattern symbols according to the number of elements of the array, their position in the plane, and the phase shift each element introduces. The linear independence will be determined using the Euclidean distance between a pair of radiation patterns as a figure of merit. A higher distance will represent a higher level of independence, reducing the risk of error in a noisy channel. Each radiation pattern is a point in the Euclidean space, i.e., \(L^{\rho\times\rho}\); the Euclidean distance is then defined as  
\[d_{E}=\sqrt{\sum_{i=1}^{\rho\times\rho}\left(AF_{1}(\theta_{i},\phi_{i})-AF_{2 }(\theta_{i},\phi_{i})\right)^{2}} \tag{2}\]  
where \(AF_{1}\) and \(AF_{2}\) denote the array factors of the two considered radiation patterns. Communication systems are, in fact, noisy, and in this work, the simplest model accounting for noise is assumed, i.e., additive white Gaussian noise (AWGN) [28]. Rayleigh models are not considered in this case, under the assumption that the system is static. Selecting highly independent symbols reduces thus the probability of the symbol being corrupted by additive noise. For a set of total \(RP\) radiation patterns, we propose Algorithm 1 as a method for selecting the symbols with a higher level of independence. Figure 3: Measuring and coding information bits into radiation patterns. (**a**) Degraded radiation pattern gain (dBi) reconstruction assuming a limited resolution for symbol 00,000 shown in Figure 2. (**b**) Orthogonal symbols for an eight-antenna system with \(\rho=4\) versus the parameters \(\gamma\) and \(\gamma^{\prime}\). Two thresholds are used in Algorithm 1. The first threshold \(\gamma\) refers to the minimum distance between a pair of radiation patterns that can be considered large enough to avoid symbol corruption when noise is added. The second threshold \(\gamma^{\prime}\) refers to the number of times a symbol satisfies \(\gamma\) when the whole set is considered. The threshold selection represents the main parameter to determine the vulnerability of the communication scheme. Using \(\gamma=1.1\) and \(\gamma^{\prime}=4\), the algorithm identifies four symbols like the ones that satisfy these two conditions. Increasing \(\gamma\) to 1.3 leads to a reduction of the information bits by half. More stringent thresholds result in fewer numbers of radiation patterns highly independent and less likely to be corrupted by noise. The most straightforward way to increase the number of symbols while having high thresholds is to increase the number of point source antennas. As an example, the system can be expanded to an eight-antenna configuration positioned in the wavelength-normalized \(XY\)-plane at \([\lambda/4,\lambda/4],[-\lambda/4,\lambda/4],[-\lambda/4,\ -\lambda/4],[\lambda/4,\ - \lambda/4],[\lambda/4,\ -\lambda/4],[\ \beta\lambda/4,\ 3\lambda/4],\ [-3\lambda/4,\ -3\lambda/4],\ [-3\lambda/4,\ -3\lambda/4],\)\(and\ [3\lambda/4,\ -3\lambda/4]\). Assuming the same phase conditions as in the previous case, \(\phi_{ON}\ =180^{\circ}\) and \(\phi_{OFF}\ =\ 0^{\circ}\), the number of possible phase combinations for the eight antennas assuming that each one can be tuned independently is \(RP=2^{8}=256\). As was the case before, not all the 256 sets sets of symbols will be unique. Considering a realistic resolution \(\rho=4\) for measuring the radiation patterns in the whole spherical domain (\(0\ \leq\ \phi\ \leq\ 2\pi\ \&\ 0\ \leq\ \theta\ \leq\ \pi\)), the number of orthogonal symbols is extracted for a variable \(\gamma\).