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There are many significant applications of nonlinear adaptive digital filters such as the cancellation of echoes and intersymbol interference, the equalization of transmission channels, adaptive noise cancellation and design of optimal predictors in communication systems. In this paper, we introduce a class of efficient architectures for adaptive quadratic digital filters based on the LMS algorithm and on rank compressed lower-upper (LU) triangular decomposition method. The architectures exhibit high parallelism as well as modularity and regularity. They are mapped into parallel pipeline and systolic array implementations and are evaluated on hardware cost (in bits), and data throughput delay.