# Adaptive Beamforming Applied to OFDM Systems
## 3 Proposed Method for OFDM Adaptive Antenna Array  
Figure 5 shows the adaptive beamforming proposed in this work, where the \(K\) spatial filter gains, \(\mathbf{f}(m)\), are adjusted in each \(m\)-th sample of the OFDM symbol using the LMS algorithm. The error signal, \(e(m)\), is calculated in time domain and is given as:  
\[e\left(m\right)=\hat{x}\left(m\right)-x\left(m\right) \tag{35}\]  
where \(\hat{x}\left(m\right)\) is part of the vector \(\mathbf{\hat{x}}\left(n\right)\) expressed as:  
\[\mathbf{\hat{x}}\left(n\right)=\left[ \begin{array}{c}\hat{x}_{0}\left(n \right)\\ \vdots\\ \hat{x}_{C-1}\left(n\right)\end{array} \right]=\text{IDFT}\left(\mathbf{\hat{s }}(n)\right). \tag{36}\]  
The reference vector, \(\mathbf{\hat{s}}(n)\), is created in the frequency domain by the module called Reference Signal Control (RSC) (see in Figure 5) and is expressed as:  
\[\mathbf{\hat{s}}\left(n\right)=\left[ \begin{array}{c}\hat{s}_{0}\left(n \right)\\ \vdots\\ \hat{s}_{j}\left(n\right)\\ \vdots\\ \hat{s}_{C-1}\left(n\right)\end{array} \right]=\left[ \begin{array}{c}\hat{s} \left(m+nC\right)\\ \vdots\\ \hat{s}\left(m-j+nC\right)\\ \vdots\\ \hat{s}\left(m-C+1+nC\right)\end{array} \right] \tag{37}\]  
Figure 5: Adaptive antenna array proposed for the OFDM system. where,  
\[\hat{s}_{j}\left(n\right)=\left\{ \begin{array}{l}o_{k}(n),\,\textbf{if}\left(j\in \text{pilot carrier position}\right)\\ \\ 0,\qquad\textbf{if}\left(j\in\text{data carrier position}\right)\textbf{and} \left(e_{p}^{DD}(n)\geq E^{DD}\textbf{or}\,n\leq N^{DD}\right)\\ \\ \hat{a}_{p}\left(n\right),\,\textbf{if}\left(j\in\text{data carrier position}\right)\textbf{and} \left(e_{p}^{DD}(n)<E^{DD}\textbf{or}\,n>N^{DD}\right)\\ \\ 0,\qquad\textbf{if}\left(j\
otin\text{pilot or data carrier position, zero padding}\right)\end{array} \right. \tag{38}\]  
where \(\hat{a}_{k}\left(n\right)\) is the \(p\)-th pilot carrier, \(\hat{a}_{p}\left(n\right)\) is the \(p\)-th decided data carrier (or decided symbol), the signal \(e_{p}^{DD}(n)\) is the \(p\)-th direct decision error expressed as:  
\[e_{p}^{DD}(n)=\hat{a}_{p}\left(n\right)-\hat{a}_{p}\left(n\right)=\text{slice} \left\{\hat{a}_{p}\left(n\right)\right\}-\hat{a}_{p}\left(n\right) \tag{39}\]  
where \(\hat{a}_{p}\left(n\right)\) is the \(p\)-th estimated data carrier (or estimated symbol) and the \(E^{DD}\) and \(N^{DD}\) are parameters managed by RSC. The RSC has three inputs: the pilot carriers, \(\textbf{o}\left(n\right)\), the decided data carriers, \(\textbf{\hat{a}}\left(n\right)\), given by:  
\[\textbf{\hat{a}}\left(n\right)=\left[ \begin{array}{c}\hat{a}_{0}\left(n \right)\\ \vdots\\ \hat{a}_{C_{D}-1}\left(n\right)\end{array} \right]=\left[ \begin{array}{c} \text{slice}\left\{\hat{a}_{0}\left(n\right)\right\}\\ \vdots\\ \text{slice}\left\{\hat{a}_{C_{D}-1}\left(n\right)\right\}\end{array} \right]= \left[ \begin{array}{c}\text{slice}\left\{\hat{a}\left(m+nC_{D}\right)\right\} \\ \vdots\\ \text{slice}\left\{\hat{a}\left(m-C+1+nC_{D}\right)\right\}\end{array} \right] \tag{40}\]  
and the direct decisions errors, \(\textbf{e}_{p}^{DD}(n)\), expressed as:  
\[\textbf{e}^{DD}(n)=\left[ \begin{array}{c}\textbf{e}_{0}^{DD}\left(n\right) \\ \vdots\\ \textbf{e}_{C_{D}-1}^{DD}\left(n\right)\end{array} \right]. \tag{41}\]  
The RSC uses the rules shown in Equation (38), and it makes a decision if the \(p\)-th decided data carrier (or data symbol), \(\hat{a}_{p}\left(n\right)\), will contribute with the reference signal, \(\textbf{\hat{s}}\left(n\right)\). The decision is based on two parameters, \(E^{DD}\) and \(N^{DD}\). The parameter \(E^{DD}\) is the upper bound of the direct decision error, and it decides if the \(p\)-th data carrier (or data symbol), \(\hat{a}_{p}\left(n\right)\), has an appropriate value to use in the reference signal, \(\textbf{\hat{s}}\left(n\right)\), and the parameter \(N^{DD}\) controls the time (from that OFDM symbol) the data carriers can start to contribute to the reference signal \(\textbf{\hat{s}}\left(n\right)\).