A Block Double Proportionate Sign Adaptive Filter: Derivation and Convergence Analysis

This paper proposes the Block Double Proportionate Sign Adaptive Filter (BDP-SAF) for sparse system identification in impulsive noise environments. The algorithm brings three mechanisms that are individually established in the adaptive filtering literature, namely block-level proportionate adaptation, zero-attraction and the sign-error update, and combines them with a single block-level method. The algorithm is derived from a regularized cost function combining $\ell _{2}$ and block-level $\ell _{1}$ penalty terms, which yields stationary-point expressions for a block proportionate gain matrix and a block zero-attractor; these are then reduced using approximations to a pair of low-cost implementable gains. The resulting update rule simultaneously accelerates adaptation in active blocks and suppresses inactive ones through a unified block energy estimate, while the sign-error update improves robustness to heavy-tailed disturbances under only a symmetry assumption on the noise distribution, without requiring knowledge of its higher-order moments or parametric form. To handle practical scenarios where the block partition is unavailable or misaligned with the true impulse response support, an adaptive block boundary detection scheme is developed that progressively realigns the partition online with negligible computational overhead. A convergence analysis establishes sufficient conditions for mean stability, and a semi-analytical steady-state mean-square deviation (MSD) expression is derived, together with an explicit statement of the assumptions on which both results rely. Simulation results are consistent with the theoretical findings and demonstrate that BDP-SAF achieves lower steady-state MSD with or without abrupt channel changes compared to the considered competing algorithms, including the case of several simultaneously active blocks.