Spectral density estimation

In statistical signal processing, the goal of spectral density estimation is to estimate the spectral density (also known as the power spectrum) of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. The purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

SDE should be distinguished from the field of frequency estimation, which assumes a limited (usually small) number of generating frequencies plus noise and seeks to find their frequencies. SDE makes no assumption on the number of components and seeks to estimate the whole generating spectrum.

Techniques

Techniques for spectrum estimation can generally be divided into parametric and non-parametric methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an auto-regressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure.

Following is a partial list of spectral density estimation techniques:

• Periodogram, a classic non-parametric technique
• Welch's method
• Bartlett's method
• Autoregressive moving average estimation, based on fitting to an ARMA model
• Multitaper
• Maximum entropy spectral estimation
• Least-squares spectral analysis, based on least-squares fitting to known frequencies

References

• Porat, B. (1994). Digital Processing of Random Signals: Theory & Methods. Prentice Hall. ISBN 0-13-063751-3.
• Priestley, M.B. (1991). Spectral Analysis and Time Series. Academic Press. ISBN 0-12-564922-3.

• P Stoica and R Moses, Spectral Analysis of Signals. Prentice Hall, NJ, 2005 (Chinese Edition, 2007). AVAILABLE FOR DOWNLOAD.