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This paper shows how to apply square root Kalman filtering to high speed data transmission on fading dispersive high-frequency (HF) channels. High data-rate HF transmission can be achieved by use of a single carrier bandwidth efficiency modulation that is demodulated with decision-feedback equalization (DFE). The DFE coefficients are updated adaptively by square root Kalman algorithms. Theoretical formulation and mechanization procedures for the square root Kalman algorithms are given. Computer simulation indicates that good error-rate performance can be achieved with these algorithms for rapid fading HF radio channels.