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Itakura and Saito  used the maximum likelihood (ML) method to derive a spectral matching criterion for autoregressive (i.e., all-pole) random processes. In this paper, their results are generalized to periodic processes having arbitrary model spectra. For the all-pole model, Kay's  covariance domain solution to the recursive ML (RML) problem is cast into the spectral domain and used to obtain the RML solution for periodic processes. When applied to speech, this leads to a method for solving the joint pitch and spectrum envelope estimation problems. It is shown that if the number of frequency power measurements greatly exceeds the model order, then the RML algorithm reduces to a pitch-directed, frequency domain version of linear predictive (LP) spectral analysis. Experiments on a real-time vocoder reveals that the RML synthetic speech has the quality of being heavily smoothed.