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In this paper, we give a square-root normalized ladder algorithm for the residual energy ratio (RER) method of PARCOR coefficient estimation. The derivation uses a Hilbert space projection approach similar to that of Lee and Morf, but requires an additional related Hilbert space and its corresponding projection operators. The resulting partial correlation estimates guarantee a stable synthesis filter. In addition, we show how the algorithm may be cast in a form suitable for implementation by a CORDIC processor. Finally, we show how the normalized covariance form corresponds to an RER formulation where the prediction error energies used are chosen properly.