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Low complexity realizations of Least Mean Squared (LMS) error, Generalized Sidelobe Cancellers (GSCs) applied to adaptive beamforming are considered. The GSC method provides a simple way for implementing adaptive Linear Constraint Minimum Variance (LCMV) beamformers. Low complexity realizations of adaptive GSCs are of great importance for the design of high sampling rate, and/or small size and low power adaptive beamforming systems. ; The LMS algorithm and its Transform Domain (TD-LMS) counterpart are considered for the adaptive processing task involved in the design of optimum GSC systems. Since all input signals are represented by complex variables, complex valued arithmetic is utilized for the realization of GSC algorithms, either on general purpose computers, or on dedicated VLSI ASICs. ; Using algorithmic strength reduction (SR) techniques, two novel algorithms are developed for efficient realizations of both LMS GSCs and TD-LMS GSC schemes. Both of the proposed algorithms are implemented using real valued arithmetic only, whilst reducing the number of multipliers by 25% and 20%, respectively. ; When VLSI implementation aspects are considered, both the proposed algorithms result in reduced power dissipation and silicon area realizations. The performance of the proposed realizations of the LMS based GSC methods is illustrated in the context of typical beamforming applications.