Cepstral transforms and harmonic identification

See also machine listening, system identification.

Just as you can generalise linear models for i.i.d observations you can generalise linear models for time time series. If you do this in particular with the power-spectral representation of the time series, then you are about to invent cepstral representation of the series, which represents the power-spectrogram using a log link function.

I haven’t actually read the foundational literature here, (Bogert, Healy, and Tukey 1963) just used some algorithms; but it seems to be mostly a hack for rapid identification of correlation lags where said lags are long.

For a generalized modern version, see Proietti and Luati (2019).

In this chapter we consider a class of parametric spectrum estimators based on a generalized linear model for exponential random variables with power link. The power transformation of the spectrum of a stationary process can be expanded in a Fourier series, with the coefficients representing generalised autocovariances. Direct Whittle estimation of the coefficients is generally unfeasible, as they are subject to constraints (the autocovariances need to be a positive semidefinite sequence). The problem can be overcome by using an ARMA representation for the power transformation of the spectrum. Estimation is carried out by maximising the Whittle likelihood, whereas the selection of a spectral model, as a function of the power transformation parameter and the ARMA orders, can be carried out by information criteria. The proposed methods are applied to the estimation of the inverse autocorrelation function and the related problem of selecting the optimal interpolator, and for the identification of spectral peaks.

This is no longer a popular technique. probably because we don’t need to shoe-horn everything into FFTs these days, but modern researchers can still encounter it in the mysteriously-still popular MFCC analysis used in machine listening.

Bogert, B P, M J R Healy, and J W Tukey. 1963. “The Quefrency Alanysis of Time Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum and Saphe Cracking.” In, 209–43.

Childers, D. G., D. P. Skinner, and R. C. Kemerait. 1977. “The Cepstrum: A Guide to Processing.” Proceedings of the IEEE 65 (10): 1428–43. https://doi.org/10.1109/PROC.1977.10747.

Kemerait, R., and D. Childers. 1972. “Signal Detection and Extraction by Cepstrum Techniques.” IEEE Transactions on Information Theory 18 (6): 745–59. https://doi.org/10.1109/TIT.1972.1054926.

Noll, A. Michael. 1967. “Cepstrum Pitch Determination.” The Journal of the Acoustical Society of America 41 (2): 293–309. https://doi.org/10.1121/1.1910339.

Oppenheim, A. V., and R. W. Schafer. 2004. “From Frequency to Quefrency: A History of the Cepstrum.” IEEE Signal Processing Magazine 21 (5): 95–106. https://doi.org/10.1109/MSP.2004.1328092.

Proietti, Tommaso, and Alessandra Luati. 2019. “Generalised Linear Cepstral Models for the Spectrum of a Time Series.” Statistica Sinica. https://doi.org/10.5705/ss.202017.0322.