# Adaptive Beamforming Applied to OFDM Systems
## 4 Simulations and Results  
To validate the adaptive beamforming scheme proposed here, several simulations were performed with the OFDM system. Table 2 shows the constant parameters used in all simulations where they were based on the SBTVD standard [8]. Three channel scenarios (Channel 1, Channel 2 and Channel 3) were simulated according to the parameters presented in Table 3. For each channel, the Bit Error Rate (BER) curves were created for distinct values of the number of antennas (\(K=1\), \(K=4\), \(K=6\), \(K=8\) and \(K=10\)) and GI size (\(GI=1/2\) and \(GI=1/512\)). Channel coding was not used in any simulation. Figure 6 shows the BER performance curve for Channel 1 (see Table 3). Channel 1 is a long channel, \(L\approx 104\) OFDM samples (see Equations (10) and (11)), that is \(12.7\,\mathrm{\SIUnitSymbolMicro s}/t_{\mathrm{s}}\approx 104\). Based on Equations (16)-(18), the minimum number of GI samples, \(N_{GI}\), may be greater than \(104\) (\(N_{GI}\geq 104\)) in order to avoid the OFDM symbol interference. Regarding the study of the improvements associated with the adaptive beamforming, \(GI=1/512\) (\(N_{GI}=4\)) and \(GI=1/2\) (\(N_{GI}=740\)) were used for three numbers of antennas, \(K=5\), \(K=6\) and \(K=8\). The results show that the adaptive beamforming proposal had an excellent performance where, when the \(BER=10^{-4}\), the gain was about \(10\,\mathrm{dB}\), \(15\,\mathrm{dB}\) and \(17.5\,\mathrm{dB}\) for \(K=5\), \(K=6\) and \(K=8\), respectively; even for the cases where \(N_{GI}<104\) (\(GI=1/512\)), which enables one to work with more bandwidth efficiency. It is essential to observe that there is a small gain, about \(2\,\mathrm{dB}\), between \(GI=1/2\) and \(GI=1/512\) for all beamforming results (\(K=5\), \(K=6\) and \(K=8\)), and this proves the shortest of the channel presented by Equations (29)-(31). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Space between the Antenna Elements (\(\Delta y\))** & \(\lambda\)**/2** \\ \hline Sampling period of the OFDM symbol (\(t_{s}\)) & \(0.1230\,\mathrm{\SIUnitSymbolMicro s}\) \\ Number of the data carriers (\(C_{D}\)) & \(1248\) carriers \\ Number of the pilot carriers (\(C_{P}\)) & \(156\) carriers \\ Zero padding size (\(C_{ZIP}\)) & \(644\) carriers \\ Number of the carriers (\(C\)) & \(2048\) carriers \\ OFDM symbol period (\(T\)) & \(251.9040\,\mathrm{\SIUnitSymbolMicro s}\) \\ Modulation & \(16\)-QAM \\ Pilots & Spread (SBTVDStandard [8]) \\ Channel coding & none \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constant parameters used in all simulations. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline Channel 1 & \(\mathrm{\SIUnitSymbolMicro s}\) & \(0.00\) & \(0.30\) & \(3.50\) & \(4.40\) & \(9.50\) & \(12.70\) \\  & dB & \(0.00\) & \(-12.00\) & \(-4.00\) & \(-7.00\) & \(-15.00\) & \(-22.00\) \\  & AOA & \(30^{\circ}\) & \(50^{\circ}\) & \(90^{\circ}\) & \(120^{\circ}\) & \(150^{\circ}\) & \(170^{\circ}\) \\ \hline Channel 2 & \(\mathrm{\SIUnitSymbolMicro s}\) & \(0.00\) & \(0.15\) & \(0.42\) & \(1.51\) & \(2.32\) & \(2.80\) \\  & dB & \(0.00\) & \(-3.09\) & \(-6.02\) & \(-10.45\) & \(-20.00\) & \(-26.02\) \\  & AOA & \(30^{\circ}\) & \(50^{\circ}\) & \(90^{\circ}\) & \(120^{\circ}\) & \(150^{\circ}\) & \(170^{\circ}\) \\ \hline Channel 3 & \(\mathrm{\SIUnitSymbolMicro s}\) & \(0.00\) & \(0.74\) & \(1.11\) & \(1.48\) & \(1.85\) & \(2.10\) \\  & dB & \(0.00\) & \(-20.00\) & \(-20.00\) & \(-10.00\) & \(-14.00\) & \(-18.00\) \\  & AOA & \(30^{\circ}\) & \(50^{\circ}\) & \(90^{\circ}\) & \(120^{\circ}\) & \(150^{\circ}\) & \(170^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulated channels. The BER performance curve for Channel 2 (see Table 3) is presented in Figure 7 where it is a typical wireless network channel. The length of the channel \(L\) is \(\approx\) 23 OFDM samples (see Equations (10) and (11)), that is 2.8 \(\upmu\)s/\(t_{s}\approx\) 23.