# Adaptive Beamforming Applied to OFDM Systems
## 2 OFDM Communication System and Beamforming  
Figure 1 shows the structure of an OFDM discrete baseband system where the source of information is transmitting symbols \(a(m)\). The symbol, \(a(m)\), is part of the dataset \(A_{M}=a_{0},a_{1},\ldots\), \(a_{M-1}\) of \(M\) symbols, and they are transmitted over a period of \(t_{s}\) seconds. Each \(m\)-th symbol represents  
a word of \(B\) bits, and \(t_{s}\) can be called the sampling period or symbol interval. Table 1 lists the major notations adopted in this paper. The OFDM transmission technique consists of transforming a Single Carrier (SC) signal, of bandwidth \(W\) Hz, into a signal formed by \(C\) carriers of bandwidth \(W/C\) Hz. The OFDM scheme parallelizes one data source in several other data sources transmitted in orthogonal sub-carriers [4]. One advantage of this type of transmission is that each sub-carrier may have a bandwidth smaller than the coherence bandwidth of the channel. In other words, each OFDM symbol may have a period longer than the delay spread of the channel, improving the robustness of the system against ISI [4]. The OFDM symbol period, \(T\), is given by:  
\[T=t_{s}\cdot C, \tag{1}\]  
where the symbol interval, \(t_{s}\), also coincides with the sampling period of the OFDM symbol. Figure 2 illustrates the OFDM transmitter where the vector, \(\mathbf{a}\left(n\right)\), expressed as:  
\[\mathbf{a}\left(n\right)=\left[ \begin{array}{c}a_{0}\left(n\right)\\ \vdots\\ a_{p}\left(n\right)\\ \vdots\\ a_{C_{D}-1}\left(n\right)\end{array} \right]=\left[ \begin{array}{c}a\left(m+nC _{D}\right)\\ \vdots\\ a\left(m-p+nC_{D}\right)\\ \vdots\\ a\left(m-C_{D}+1+nC_{D}\right)\end{array} \right], \tag{2}\]  
collects \(C_{D}\) symbols, \(a(m)\), after the Serial-to-Parallel converter (S/P). In the OFDM transmitter scheme, the signal \(a_{p}(n)\) represents the \(p\)-th data carrier associated with the \(n\)-th OFDM symbol in the frequency domain. \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline GI & The guard interval. \\ \(C\) & Number of the carriers. \\ \(C_{D}\) & Number of the data carriers. \\ \(C_{P}\) & Number of the pilot carriers. \\ \(C_{ZP}\) & Zero padding size. \\ \(T\) & OFDM symbol period. \\ \(t_{s}\) & The sampling period of the OFDM symbol. \\ \(N_{GI}\) & The number of the GI samples. \\ \(L\) & The number of paths of the channel. \\ \(\alpha_{i}\) & The complex gain of the \(i\)-th path. \\ \(\tau_{i}\) & The delay of the \(i\)-th path. \\ \(K\) & The number of the antenna elements. \\ \(\Delta y\) & Space between the antenna elements. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Major notations used in this paper. Figure 1: Discrete baseband OFDM system with antenna array. The data carriers are multiplexed with \(C_{P}\) pilot carriers, \(\mathbf{o}\left(n\right)\), expressed as:  
\[\mathbf{o}\left(n\right)=\left[ \begin{array}{c}\begin{array}{c}o_{0}\left(n \right)\\ \vdots\\ o_{k}\left(n\right)\\ \vdots\\ o_{C_{P-1}}\left(n\right)\end{array}\end{array} \right]\,, \tag{3}\]  
where \(o_{k}\left(n\right)\) is the \(k\)-th pilot carrier in the \(n\)-th OFDM symbol. After pilot insertion are inserted \(C_{ZP}\) zeros (zero padding) generating a set of \(C=C_{D}+C_{P}+C_{ZP}\) carriers, characterized by:  
\[\mathbf{s}\left(n\right)=\left[ \begin{array}{c}s_{0}\left(n\right)\\ \vdots\\ s_{j}\left(n\right)\\ \vdots\\ s_{C-1}\left(n\right)\end{array} \right]=\left[ \begin{array}{c}s\left(m+nC \right)\\ \vdots\\ s\left(m-j+nC\right)\\ \vdots\\ s\left(m-C+1+nC\right)\end{array} \right] \tag{4}\]  
where,  
\[s_{j}\left(n\right)=\left\{ \begin{array}{c}o_{k}(n),\quad\textbf{if}\left(j \in\text{pilot carrier position}\right)\\ a_{p}\left(n\right),\quad\textbf{if}\left(j\in\text{data carrier position}\right)\\ 0,\quad\quad\textbf{if}\left(j\
otin\text{pilot or data carrier position}\right) \end{array} \right.\,.