# On Coding and Decoding Reconfigurable Radiation Pattern Modulation Symbols
## 3 Results
### BER for AWGN Wireless Channel  
Let us consider a communication system that is transmitting radiation pattern symbols over an AWGN channel. Let \(x_{i}(\theta,\ \phi)\) be an orthogonal radiation pattern symbol from the \(RP\) number of radiation patterns. During each period \(\log_{2}RP\) bits are transmitted. For each period, \(\eta\) additive noise will be added to the transmitted signal, i.e.,  
\[y(\theta,\ \phi)=\ x_{i}(\theta,\ \phi)+\ \eta, \tag{4}\]  
where \(y(\theta,\ \phi)\) is the radiation pattern with the noise added. Afterward, Algorithm 2 is used to decode the received \(y(\theta,\ \phi)\) into the \(x^{\prime}(\theta,\ \phi)\) demodulated radiation pattern. Whenever \(x(\theta,\ \phi)\
eq x^{\prime}(\theta,\ \phi)\), an error will be counted. ```
Result: ML demodulator for RRPM symbols. For a radiation pattern symbols set of length \(RP\), \(Y(\theta,\ \phi)=\ [\ ]\); for\(j=1:RP\)do \(Y_{j}(\theta,\phi)=\sqrt{y(\theta,\phi)^{2}-x_{j}(\theta,\phi)^{2}}\). end
```  
**Algorithm 2**Maximum likelihood detector. In this contribution, RRPM is an analog of the space-shift-keying spatial-modulation [31], where the only symbols transmitted are the radiation patterns. However, this methodology can also be applied for composite systems where an M-ary digitally modulated signal is used. A Monte-Carlo simulation with \(N=1\ \times\ 10^{6}\) random binary bits using the four antennas array from Section 2 is made under different system conditions. A maximum number of \(n=200\) errors is chosen, as stated in Ref. [30]. The effect of the resolution \(\rho\) and the fraction of the space considered for the measurement is investigated. For a realistic characterization, a low resolution is required to simulate scenarios outside of an an anechoic chamber. The resolutions considered for the simulation are \(\rho=[\ 2,\ 4,\ 5,\ 8,\ 10]\). A resolution of 10 restricts the number of measured points to 100, which is still a relatively high number of antennas. Realistic approaches will use resolutions equal to 2 or 4. As a result of the low resolution in comparison to the area of measurement, the radiation patterns will not be precisely plotted when the considered measured spherical space is ample. Three different fractions of the space are evaluated: the typical total radiation space \((0\leq\phi\leq 2\pi\ \&\ 0\leq\theta\leq\pi)\), the upper part \((0\leq\phi\leq 2\pi\ \&\ 0\leq\theta\leq\pi/2)\), and a minimal fraction \((0\leq\phi\leq\pi/8\ \&\ 0\leq\theta\leq\pi/8)\). The BER is simulated for a signal to noise ratio (SNR) sweep between \(-20\) dB to 20 dB. For each resolution and fraction of the space measured, Algorithm 1 is used to select the symbols; a fixed \(\gamma=1.3\) and \(\gamma^{\prime}=4\) are used. As is depicted in Figure 4, the resolution, space, and threshold restrictions in multiple orthogonal radiated symbols are shown. In Figure 4, the symbols obtained for \(\rho=4\) and the three considered fractions of the space are shown. The spatial restrictions used for the case of Figure 4a resulted in four different radiation symbols (2 bits), while for the cases considered in Figure 4b,c, only a pair of symbols is obtained for each (1 bit). A proper selection of the resolution and the space considered can lead to a higher number of successfully transmitted bits, as is the case in Figure 4b, where measuring half the space is preferred in terms of the orthogonality between symbols. The BER results are shown and contrasted in Figure 5. In the three fractions of the space considered, \(\rho~{}>~{}4\) is preferred to operate perfectly under typical SNR values for wireless channels (10 dB \(<SNR<\) 20 dB). Higher resolutions are unnecessary, as can be seen in Figure 5 for the three cases (their effect is almost negligible for normal noise ranges). A resolution of four can be appropriately used if the total radiation space measurement is avoided.