# Adaptive Beamforming Applied to OFDM Systems
## 2 OFDM Communication System and Beamforming
\begin{array}{cccc}b\left(m-d _{0,C-L}\right)&\cdots&b\left(m-d_{0,C-1}\right)\\ \vdots&\ddots&\vdots\\ GI\left(m-d_{0,C-L}\right)&\cdots&GI\left(m-d_{L-1,C-1}\right)\end{array}  \right], \tag{18}\]  
and the \(\mathbf{G}\mathbf{I}\left(n\right)\) matrix describes the interference within the GI. In the \(\mathbf{B}^{\prime}\left(n\right)\) and \(\mathbf{B}^{\prime\prime}\left(n\right)\) matrices, the \(d_{i,j}\) variable is characterized as:  
\[d_{i,j}=\tau_{i}-j. \tag{19}\]  
Figure 3 shows a Linearly Equally-Spaced (LES) array where the antenna elements are arranged along the \(y\)-axis, with spacing of \(\Delta y\). It is assumed that all the multipath arrives at the array in the vertical plane, with the Angle Of Arrival (AOA) of \(\theta\) radians with respect to the to the \(x\)-axis orthogonal to the \(y\)-axis. Each \(v\)-th element of the antenna array is weighted by a complex gain, \(f_{v}\), and the spacing \(\Delta y\) should generally be greater than or equal to \(\lambda/2\). The signal, \(u(m)\), received by the \(v\)-th antenna element is expressed as:  
\[u_{v}(m)=\sum_{i=0}^{L-1}\alpha_{i}z(m-\tau_{i})e^{-j\left(v\right)\Delta y \cos(\theta_{i})}+r_{v}(m), \tag{20}\]  
Figure 3: Antenna array with \(K\) equally-spaced sensors. where \(\beta=(2\pi)/\lambda,\theta_{i}\) is the AOA associated with the \(i\)-th channel path and \(r_{v}(m)\) is the noise associated with each \(v\)-th antenna element. Rewriting the Equation (20) based on Equation (15) gives:  
\[\mathbf{u}_{v}\left(n\right)=\mathbf{B}^{T}(n)\mathbf{O}_{v}(\boldsymbol{ \theta})\mathbf{h}+\mathbf{r}_{v}(n) \tag{21}\]  
where \(\mathbf{B}(n)\) represents the signal after GI remotion, \(\mathbf{GI}(n)\) (see Figure 3 and Equation (16)), that is:  
\[\mathbf{B}\left(n\right)=\left[ \begin{array}{cc}\mathbf{B}^{\prime}\left(n \right)&\mid\mathbf{B}^{\prime\prime}\left(n\right)\end{array} \right] \tag{22}\]  
and \(\mathbf{O}_{v}(\boldsymbol{\theta})\) is a diagonal matrix expressed as:  
\[\mathbf{O}_{v}(\boldsymbol{\theta})=\left[ \begin{array}{ccc}e^{-j(v\beta) \Delta y\cos(\theta_{0})}&\ldots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&e^{-j(v\beta)\Delta y\cos(\theta_{L-1})}\end{array} \right] \tag{23}\]  
and:  
\[\boldsymbol{\theta}=\left[ \begin{array}{c}\theta_{0}\\ \vdots\\ \theta_{L-1}\end{array} \right]. \tag{24}\]  
The combined output of the signals of the \(K\) antenna elements, \(x\left(m\right)\), is represented by:  
\[x\left(m\right)=\sum_{v=0}^{K-1}f_{v}u_{v}\left(m\right)=\sum_{v=0}^{K-1}\sum _{i=0}^{L-1}f_{v}\alpha_{i}z(m-\tau_{i})e^{-j\beta v\Delta x\cos(\theta_{i})} +f_{v}r_{v}(m). \tag{25}\]  
where using Equation (21), the OFDM symbol received, after the LES, can be shown as:  
\[\mathbf{x}\left(n\right)=\sum_{v=0}^{K-1}f_{v}\mathbf{u}_{v}\left(n\right)= \sum_{v=0}^{K-1}f_{v}\mathbf{B}^{T}(n)\mathbf{O}_{v}(\boldsymbol{\theta}) \mathbf{h}+\sum_{v=0}^{K-1}f_{v}\mathbf{r}_{v}(n)=\mathbf{B}^{T}(n)\mathbf{A} (\boldsymbol{\theta})\mathbf{h}+\sum_{v=0}^{K-1}f_{v}\mathbf{r}_{v}(n) \tag{26}\]  
where:  
\[\mathbf{A}(\boldsymbol{\theta})=\sum_{v=0}^{K-1}f_{v}\mathbf{O}_{v}( \boldsymbol{\theta}) \tag{27}\]  
and \(\mathbf{x}(n)\) is expressed as:  
\[\mathbf{x}\left(n\right)=\left[ \begin{array}{c}x_{0}\left(n\right)\\ \vdots\\ x_{p}\left(n\right)\\ \vdots\\ x_{C-1}\left(n\right)\end{array} \right]=\left[ \begin{array}{c}x\left(m+nC \right)\\ \vdots\\ x\left(m-p+nC\right)\\ \vdots\\ x\left(m-C+1+nC\right)\end{array} \right], \tag{28}\]  
where \(x_{p}(n)\) is the \(p\)-th sample of the \(n\)-th OFDM symbol. The variable \(\mathbf{A}(\boldsymbol{\theta})\) is the array factor, and it determines the antenna pattern and the direction gain. Adjusting the \(K\) antenna array gains, \(f_{v}\left(v=0,\ldots,K-1\right)\), it is possible to select any direction for maximum gain [2].