# On Coding and Decoding Reconfigurable Radiation Pattern Modulation Symbols
## 2 Materials and Methods
### Radiation Pattern Symbols  
An antenna formed by multi-elements is known as an array. Antenna arrays are mostly used to direct radiated power towards the desired direction with a high gain. The number of elements, the geometrical configuration, current, amplitude, and phase will influence the behavior of the whole system. The discrete sources radiate individually, but the total radiation pattern is built up by the coherent addition of each element. The relative amplitude and phase of the excitation currents on each element will determine the behavior of the resulting field [23]. The array factor (\(AF\)) is the far-field radiation intensity, and is obtained for an array of \(N\) elements located at the position \(\overset{\rightarrow}{r_{n}}\) as  
\[AF(\hat{r})=\sum_{n=1}^{N}a_{n}\cdot e^{\hat{\jmath}\hat{\jmath}\cdot\overset{ \rightarrow}{r_{n}}+\hat{\phi}\hat{\jmath}_{n}} \tag{1}\]  
where \(\hat{\jmath}\) is the direction (normal) unit vector to \(\overset{\rightarrow}{r_{n}}\), \(a_{n}\) is the complex excitation coefficient, \(\phi_{n}\) is the added phase component, and \(k\) is the wavenumber. The \(AF\) accounts for the variation of the power radiated as a function of \(\phi\) and \(\theta\) (components of \(\hat{r}\) in a spherical coordinate system). In the far-field, the radial component of the electric field is zero; receiving antennas have thus only the requirement of being further than \(R=2D^{2}/\lambda\) where \(D\) is the largest dimension of the antenna [19]. A reconfigurable antenna array is an antenna with a tunable \(AF\)[24; 25; 26; 27]. Without loss of generality, a straightforward model of reconfigurable antennas is used in this work: a point source isotropic antenna with a tunable phase. For an isotropic antenna, the radiation intensity is constant in all directions. The only difference between each antenna element will be their relative position in the space and the added phase component. As a first case, four antennas are collocated at a wavelength-normalized (\(XY-\) plane) at the positions \([\lambda/4,\ 0],[0,\lambda/4],[-\lambda/4,\ 0]\), and \([0,\ -\lambda/4]\) (See Figure 1). Each antenna introduces a variable phase shift of \(\phi_{ON}~{}=~{}180^{\circ}\) or \(\phi_{OFF}=0^{\circ}\). Sixteen different combinations are obtained for this \(2\times 2\) array. Considering the whole space, and measuring the radiated power gain (in spherical coordinates) at a considerable number of points, the sixteen combinations are shown in Figure 2. As shown in Figure 2, the radiation patterns have space symmetry for the upper and lower halves of the space; however, this is not the case for all radiation systems, which can be the case for complex reconfigurable arrays or metasurfaces. The purpose of an RRPM is to reconstruct the radiated symbols to decode information bits. Depending on the configuration of the receiver setup, the reconstruction of the radiation pattern may be affected. Two variables profoundly alter the measurement of the radiation symbols, i.e., the resolution used and the fraction of the space measured. Resolution: The resolution \(\rho\) takes into account the positions in \(\phi\) and \(\theta\) that will be measured. When \(\rho=N\), \(N\) positions are measured in \(\phi\), and \(N\) positions are measured in \(\theta\), leading to a total of \(\rho^{2}\) measurements. Under realistic circumstances, one static antenna will measure one point; a total number of \(\rho^{2}\) antennas are thus required. A high  
Figure 1: Two by two reconfigurable antenna array in a wavelength-normalized \(XY\)-plane. resolution requires an excessive number of antennas. The degradation of the radiation pattern symbols due to a low resolution is depicted in Figure 3a. Covered region: There are not many useful applications for this modulation scheme if the entire radiation pattern has to be measured. Under realistic conditions of operation, only certain parts of the total radiation pattern may be used to encode information.