# Adaptive Beamforming Applied to OFDM Systems
## 2 OFDM Communication System and Beamforming
After the spatial filter, the signal, \(\mathbf{x}(n)\), is processed by the OFDM receiver, shown in details in Figure 4.  
The estimated carriers associated with the \(n\)-th OFDM symbol is given by:  
\[\widetilde{\mathbf{s}}\left(n\right)=\mathrm{DFT}\left(\mathbf{x}(n)\right)= \mathrm{DFT}\left(\mathbf{B}^{T}(n)\mathbf{A}(\boldsymbol{\theta})\mathbf{h} \right)+\mathrm{DFT}\left(\overset{K-1}{\underset{p=0}{\sum}}f_{\
u}\mathbf{r} _{\
u}(n)\right)=\mathbf{s}\left(n\right)\mathbf{c}^{T}\mathbf{I}+\mathbf{g}(n) \tag{29}\]  
where \(\mathbf{c}\) can be expressed as:  
\[\mathbf{c}\equiv\mathrm{DFT}\left(\left[ \begin{array}{c}\boldsymbol{\gamma }\\ ------\\ \boldsymbol{0}\end{array} \right]\right)=\mathrm{DFT}\left(\left[ \begin{array}[] {c}\mathbf{A}(\boldsymbol{0})\mathbf{h}\\ ------\\ \boldsymbol{0}\end{array} \right]\right). \tag{30}\]  
The vector \(\boldsymbol{\gamma}\) is a \"better version\" of the channel, \(\mathbf{h}\), and it is given by:  
\[\boldsymbol{\gamma}=\left[ \begin{array}{c}\gamma_{0}\\ \vdots\\ \gamma_{\mathcal{D}-1}\end{array} \right], \tag{31}\]  
where \(D\leq L\) and \(\boldsymbol{0}\) is a vector with \(C-D\) zeros. The vector \(\mathbf{g}(n)\) is the OFDM symbol noise in the frequency domain, and it is indicated as:  
\[\mathbf{g}(n)=\mathrm{DFT}\left(\overset{K-1}{\underset{p=0}{\sum}}f_{\
u} \mathbf{r}_{\
u}(n)\right)=\left[ \begin{array}{c}g_{0}(n)\\ \vdots\\ g_{C-1}(n)\end{array} \right]. \tag{32}\]  
The frequency equalizer (see Figure 4) compensates the channel effects, \(\mathbf{c}\), using received pilot carries, \(\bar{\mathbf{0}}\left(n\right)\), expressed as:  
\[\bar{\mathbf{0}}\left(n\right)=\left[ \begin{array}{c}\tilde{\sigma}_{0} \left(n\right)\\ \vdots\\ \tilde{\sigma}_{Cp-1}\left(n\right)\end{array} \right]=\mathbf{o}\left(n\right) \mathbf{c}^{T}\mathbf{I}+\mathbf{g}_{P}(n) \tag{33}\]  
where \(\mathbf{g}_{P}(n)\) is the pilot noise. There are several techniques and schemes for the OFDM frequency equalizer, and the conventional strategy uses the interpolation algorithm with pilot carriers to create a channel estimation [4]. With the channel information, it is possible to compensate the channel effects and estimate the transmitted data carriers, \(\bar{\mathbf{a}}(n)\). Equations (29) and (30) show that the beamforming scheme can minimize the ISI problem by reducing the attenuation (\(\mathbf{A}(\boldsymbol{\theta})\mathbf{h}\)) and length (\(D\leq L\)) of the channel, which helps the frequency domain equalizer and also enables the reduction of the GI size, \(N_{GI}<L\). Another important point is that the beamforming is the Single Input Multiple Output (SIMO) system, and the pilot carriers work with the Signal-to-Noise Ratio (SNR) over the SIMO system; this enables the reduction of the noise  
Figure 4: Illustration of the discrete baseband OFDM receiver. DFT, Discrete Fourier Transform. enhancement in the frequency equalizer [38]. Thus, the question is how to efficiently find the spatial filters' gains,  
\[\mathbf{f}=\left[ \begin{array}{c}f_{0}\\ \vdots\\ f_{K-1}\end{array} \right], \tag{34}\]  
that allow these improvements.