# Adaptive Beamforming Applied to OFDM Systems
## 1 Introduction  
In digital communication systems, signals are corrupted by various factors, the most common including white noise and the Intersymbol Interference (ISI) [1]. White noise, which can be characterized as Gaussian random variables, can be minimized with the use of channel encoders [1]. However, the ISI, which is caused by different reflections of transmitted signals, is not efficiently handled by these encoders. The ISI represents the overlapping symbols from the same information source, that is they interfere with each other in the time domain. The ISI limits channel capacity and is one of the greatest problems of the current digital communication systems [1]. However, the receivers can use the spatial filters with the beamforming strategy to reduce the ISI [2,3]. The spatial filters are sensors of antenna arrays, and they can improve a signal from a specific direction with the beamforming strategy. The OFDM systems have reduced transmission capacity due to the use of the Guard Interval (GI), which in some cases is about 25% of the signal band [4]. On the other hand, the use of a cyclic prefix as GI decreases the ISI, reducing the problem to changes in phase and amplitude that can be minimized with the channel equalizer in the frequency domain [4]. The channel equalizer, which is used on typical OFDM devices, requires adaptive interpolation algorithms associated with frequency domain multiplexed pilot signals. The referenced pilot signals can be distributed, such as the Ultra High Definition Television (UHDTV) with the Integrated Services Digital Broadcasting-Terrestrial (ISDB-T) system [5], Digital Video Broadcasting-Terrestrial (DVB-T), DVB-Second Generation Terrestrial (DVB-T2-Lite) [6], the Brazilian digital television system (Sistema Brasileiro de TV Digital-SBTVD), ISDB-T International (ISDB-T\({}_{\text{B}}\)) [7, 8, 9], and Long-Term Evolution (LTE-A) [10] or in sequence, as in IEEE 802.11 [11, 12] and IEEE 802.15.3c [10, 13]. It is essential to observe that a new solution for optimizing the OFDM system also can be applied to the next generation networks based on Single-Carrier Orthogonal Frequency Division MultipleAccess (SC-OFDMA). The SC-OFDMA is another approach technique that is based on the OFDM [14; 15; 16; 17; 18]. Adaptive beamforming strategies on OFDM receivers have been the focus of many previous papers that addressed different methods. The methods can differ as to the positioning of the gains of the adaptive beamforming that is placed at pre-FFT (Fast Fourier Transformation) or post-FFT [19; 20; 21; 22]. After the FFT, there is the open eye condition in the received symbols; however, for an OFDM beamforming with \(K\) antennas and \(C\) subcarriers, the post-FFT scheme needs to execute \(K\) FFT operations, and the adaptive algorithm needs to update \(K\times C\) gains. In the pre-FFT project, only two FFT operations and \(K\) gains' updating are necessary [19; 22]. There are also methods called analog beamforming in which the gains of the array are adjusted by Variable Gain Amplifiers (VGAs) in the Radio Frequency (RF) part [23; 24]. The analog beamforming is an attractive technique when the number of antenna elements increases. In the works presented in [25; 26], the MMSEbeamforming algorithm for the pre-FFT OFDM system was applied to a channel assumed to be frequency selective fading. However, the scheme needs the channel estimation to calculate the parameters of the spatial filter. The pre-FFT beamforming based on eigenanalysis and post-FFT subcarrier diversity was proposed in [27; 28]. In this case, the eigenvalues and the corresponding eigenvectors were used to determine the parameters of the spatial filter. Similar to the proposal in [25], the work shown in [29] needed the knowledge of the Direction Of Arrival (DOA) and channel fading information. In [30], a method is proposed that uses the Channel State Information (CSI) matrix, and [22] developed a performance evaluation of pre- and post-FFT beamforming.