A note on the measurement of spectral flatness and the calculation of prediction error variances

The Szegö-Kolmogorov-Krein theorem is the natural basis for the spectral flatness measure commonly advocated in linear predictive speech processing and parametric spectrum analysis. From this theorem it follows that the logarithm of any normalized spectrum averages to zero. The normalization constant is the minimum prediction error for the underlying process. Jensen´s theorem is introduced as a practical method of computing prediction error and spectral flatness for rational spectrum models. The resulting computation algorithm is closely related to Fejer´s factorization of rational spectra. Thus our discussion generalizes Makhoul´s discussion of the so-called Fejer method for computing prediction error in MA models. One property of any measure of spectral flatness that would seem essential is the following: the introduction of white noise should increase spectral flatness. This property is established by appealing to a classical information-theoretic theorem.