Summary of LP Spectral Envelopes

In summary, the spectral envelope of the th spectral frame, computed by linear prediction, is given by

where is computed from the solution of the Toeplitz normal equations, and is the estimated rms level of the prediction error in the th frame.

The stable, all-pole filter

can be driven by unit-variance white noise to produce a filtered-white-noise signal having spectral envelope . We may regard (no absolute value) as the frequency response of the filter in a source-filter decomposition of the signal .

It bears repeating that is zero mean when is monic and minimum phase (all zeros inside the unit circle). This means, for example, that can be simply estimated as the mean of the log spectral magnitude .

For best results, the frequency axis ``seen'' by linear prediction should be warped to an auditory frequency scale, as discussed in Appendix F. This has the effect of increasing the accuracy of low-frequency peaks in the extracted spectral envelope, in accordance with the nonuniform frequency resolution of the inner ear.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.