Multiple Window Spectrum Analysis and some Applications


Power density spectrum estimation methods can be divided into two
main groups, non-parametric and parametric, where the periodogram
belongs to the former. The spectrum is usually estimated by
successively averaged periodograms or subspectra from time epochs
which is almost non-overlapping. The subspectra are then assumed
to be uncorrelated. Data is assumed to be stationary and is
weighted with some window, e.g. Hanning, Blackman or Kaiser. To
reduce the variance, Thomson has proposed the use of multiple
windows. With certain constraints on data, e.g., locally white
spectrum, the window-shapes are designed to give uncorrelated
subspectra. However, it has been shown that for a varying
spectrum, e.g., a spectrum that includes peaks, the performance of
the Thomson multiple-window method deteriorates due to
cross-correlation between the subspectra. For a predefined shape
of a peaked spectrum the multiple windows can then be given as the
Karhunen-Loéve basis functions to the corresponding covariance
matrix to give uncorrelated spectra at the peak frequency.

Two applications where the spectrum can be assumed to include
peaks are the Electroencephalogram (EEG), which is the graphic
representation of spontaneous electrical brain activity, and the
Heart Rate Variability (HRV), the variation in the heart
frequency. In both these applications transient non-stationary
behaviour could be expected which calls for methods giving
estimates with small variance as well as good time and frequency
resolution.