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A new block-type adaptive-filtering algorithm is presented. This new block adaptive filter differs from the frequency-domain block adaptive filters of Ferrara (1980), and of Clark, Mitra, and Parker (1980), in that the new method applies a deterministic time-domain least-squares criteria within each of the data blocks. Information is carried from block to block via a weighted initial condition. This new block fast transversal filters (BFTF) algorithm is a numerically stable algorithm and can also be used to perform efficient least-squares system identification on any one data block, in which case it shows a moderate computational advantage over the previous most-efficient single-data-block algorithms of Morf et al. (1977), of Marple (1981), and of Kalouptsidis, Manolakis, and Carayannis (1984-1985). Mechanisms for tracking and varying block length from block to block are also presented and evaluated. Finally, we indicate how the new algorithm could be pipelined for maximum throughput with delay proportional to the number of parameters, after computation of the sample correlation lags.