  This paper proposes a robust reduced-rank scheme for adaptive beamforming
based on joint iterative optimization (JIO) of adaptive filters. The novel
scheme is designed according to the constant modulus (CM) criterion subject to
different constraints, and consists of a bank of full-rank adaptive filters
that forms the transformation matrix, and an adaptive reduced-rank filter that
operates at the output of the bank of filters to estimate the desired signal.
We describe the proposed scheme for both the direct-form processor (DFP) and
the generalized sidelobe canceller (GSC) structures. For each structure, we
derive stochastic gradient (SG) and recursive least squares (RLS) algorithms
for its adaptive implementation. The Gram-Schmidt (GS) technique is applied to
the adaptive algorithms for reformulating the transformation matrix and
improving performance. An automatic rank selection technique is developed and
employed to determine the most adequate rank for the derived algorithms. The
complexity and convexity analyses are carried out. Simulation results show that
the proposed algorithms outperform the existing full-rank and reduced-rank
methods in convergence and tracking performance.
