# Adaptive Beamforming Applied to OFDM Systems
## 2 OFDM Communication System and Beamforming
\tag{5}\]  
The vector, \(\mathbf{s}\left(n\right)\), is then processed by the IDFT generating a new \(n\)-th OFDM symbol, \(\mathbf{b}\left(n\right)\), characterized as:  
\[\mathbf{b}\left(n\right)=\text{IDFT}\left(\mathbf{s}\left(n\right)\right), \tag{6}\]  
where:  
\[\mathbf{b}\left(n\right)=\left[ \begin{array}{c}b_{0}\left(n\right)\\ \vdots\\ b_{j}\left(n\right)\\ \vdots\\ b_{C-1}\left(n\right)\end{array} \right]=\left[ \begin{array}{c}b\left(m+nC \right)\\ \vdots\\ b\left(m-j+nC\right)\\ \vdots\\ b\left(m-C+1+nC\right)\end{array} \right], \tag{7}\]  
Figure 2: Illustration of the discrete baseband OFDM transmitter. S/P, Serial-to-Parallel converter; IDFT, Inverse Discrete Fourier Transform; P/S, Parallel-to-Serial converter; GI, Guard Interval. the signal, \(b_{j}(n)\), is the \(j\)-th sample of the \(n\)-th OFDM symbol. The GI (see the GI mux block in Figure 2) insertion in a vector representation can be characterized by:  
\[\mathbf{GI}\left(n\right)=\left[ \begin{array}{c}G\,I\left(n-C\right)\\ \vdots\\ G\,I\left(n-C-N_{GI}+1\right)\end{array} \right], \tag{8}\]  
where \(N_{GI}\) is the number of the GI samples. The combination of Matrices 7 and 8 gives:  
\[\mathbf{z}\left(n\right)=\left[ \begin{array}{c}\mathbf{b}\left(n\right)\\ ------\\ \mathbf{GI}\left(n\right)\end{array} \right], \tag{9}\]  
where \(\mathbf{z}(n)\) is the OFDM symbol composed of \(C+N_{GI}\) samples. Usually, the size of GI is expressed as a fraction of the data carriers, \(C_{D}\), where \(G\,I=1/(C_{D}/N_{GI})\). The symbols are transmitted (by the OFDM transmitter) through a channel, \(h(m)\), subject to the ISI and additive white Gaussian noise (AWGN), \(r(m)\). The channel impulse response, \(h(m)\), can be expressed as:  
\[h(m)=\sum_{i=0}^{L-1}\alpha_{i}\delta\left(m-\tau_{i}\right), \tag{10}\]  
where \(L\) is the number of paths of the channel, \(\alpha_{i}\) is the complex gain of the \(i\)-th path and \(\tau_{i}\) is an integer value representing the delay of the \(i\)-th path at time \(m\). The receiver, shown in Figure 1, processes the signal \(u(m)\), resulting from the channel and expressed by:  
\[u\left(m\right)=\sum_{i=0}^{L-1}\alpha_{i}z(m-\tau_{i}), \tag{11}\]  
where \(z(m)\) is the symbol at the output of the OFDM transmitter or the sample of the OFDM symbol. Equation (11) can also be written in vector form, as follows:  
\[u\left(m\right)=\mathbf{h}^{T}\mathbf{z}\left(m\right), \tag{12}\]  
where \(\mathbf{h}\left(m\right)\) is the vector of complex channel gains, with ISI, length \(L\), given by:  
\[\mathbf{h}=\left[ \begin{array}{c}\alpha_{0}\\ \vdots\\ \alpha_{L-1}\end{array} \right], \tag{13}\]  
and \(\mathbf{z}\left(m\right)\) is the vector of channel delays applied to the transmitted signal \(z(m)\), given by:  
\[\mathbf{z}\left(m\right)=\left[ \begin{array}{c}z\left(m-\tau_{0}\right)\\ \vdots\\ z\left(m-\tau_{L-1}\right)\end{array} \right]. \tag{14}\]  
Rewriting Equation (12) for the \(n\)-th OFDM symbol and assuming that \(N_{GI}\geq L\), the \(C+N_{GI}\) samples of the received OFDM symbol are given by:  
\[\mathbf{u}(n)=\mathbf{Z}^{T}(n)\mathbf{h}, \tag{15}\]where \(\mathbf{Z}(n)\) is the delay matrix formed by the vectors shown in Equation (24), which can be represented by:  
\[\mathbf{Z}\left(n\right)=\left[ \begin{array}{cccc}\mathbf{B}^{\prime}\left(n \right)&\mid&\mathbf{B}^{\prime\prime}\left(n\right)&\mid&\mathbf{G}\mathbf{I} \left(n\right)\end{array} \right], \tag{16}\]  
where \(\mathbf{B}^{\prime}\left(n\right)\) is the ISI portion within the OFDM symbol characterized as:  
\[\mathbf{B}^{\prime}\left(n\right)=\left[ \begin{array}{cccc}b\left(m-d_{0,0} \right)&\cdots&b\left(m-d_{0,C-L-1}\right)\\ \vdots&\ddots&\vdots\\ b\left(m-d_{L-1,0}\right)&\cdots&b\left(m-d_{L-1,C-L-1}\right)\end{array}  \right], \tag{17}\]  
whereas \(\mathbf{B}^{\prime\prime}\left(n\right)\) is the ISI portion between the OFDM and GI samples, being described as:  
\[\mathbf{B}^{\prime\prime}\left(n\right)=\left[