## 1 Introduction  
Adaptive beamforming algorithms often encounter problems when they operate in dynamic environments with large sensor arrays. These problems include steering vector mismatches, high computational complexity and snapshot deficiency. Steering vector mismatches are often caused by calibration/pointing errors, and a high complexity is usually introduced by an expensive inverse operation of the covariance matrix of the received data. High computational complexity and snapshot deficiency may prevent the use of adaptive beamforming in important applications, like sonar and radar [1; 2]. The adaptive beamforming techniques are usually required to have a trade-off between performance and complexity, which depends on the designer's choice of the adaptation algorithm. In order to overcome this computational complexity issue, adaptive versions of the linearly-constrained beamforming algorithms, such as minimum variance distortionless response (MVDR) with stochastic gradient and recursive least squares [1; 2; 3], have been extensively reported. These adaptive algorithms estimate the data covariance matrix iteratively, and the complexity is reduced by recursively computing the weights. However, in a dynamic environment with large sensor arrays, such as those found in radar and sonar applications, adaptive beamformers with a large number of array elements may fail in tracking signals embedded in strong interference and noise. The convergence speed and tracking properties of adaptive beamformers depend on the size of the sensor array and the eigen-spread of the received covariance matrix [2]. Regarding the steering vector mismatches often found in practical applications of beamforming, they are responsible for a significant performance degradation of algorithms. Prior work on robust beamforming design [4; 5; 6; 7] has considered different strategies to mitigate the effects of these mismatches. However, a key limitation of these robust techniques [4; 5; 6; 7] is their high computational cost for large sensor arrays and their suitability to dynamic environments. These algorithms need to estimate the covariance matrix of the sensor data, which is a challenging task for a system with a large array and operates in highly dynamic situations. Given this dependency on the number of sensor elements \(M\), it is thus intuitive to reduce \(M\) while simultaneously extracting the key features of the original signal via an appropriate transformation. Reduced-rank signal processing techniques [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17] provide a way to address some of the problems mentioned above. Reduced dimension methods are often needed to speed up the convergence of beamforming algorithms and reduce their computational complexity. They are particularly useful in scenarios in which the interference lies in a low-rank subspace, and the number of degrees of freedom required to mitigate the interference through beamforming is significantly lower than that available in the sensor array. In reduced-rank schemes, a rank reduction matrix is introduced to project the original full-dimension received signal onto a lower dimension. The advantage of reduced-rank methods lies in their superior convergence and tracking performance achieved by exploiting the low-rank nature of the signals. It offers a large reduction in the required number of training samples over full-rank methods [2], which may also addresses the problem of snapshot deficiency at low complexity. Several reduced-rankstrategies for processing data collected from a large number of sensors have been reported in the last few years, which include beamspace methods [7], Krylov subspace techniques [13,14] and methods of joint and iterative optimization of parameters in [15, 16, 17]. Despite the improved convergence and tracking performance achieved with Krylov methods [13,14], they are relatively complex and may suffer from numerical problems.