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The periodogram is not a consistent estimator of the true power spectral
density (PSD) of a wide-sense stationary process. To reduce the variability in the periodogram —
and thus produce a consistent estimate of the PSD — the multitaper method averages modified
periodograms obtained using a family of mutually orthogonal windows or
tapers. In addition to mutual orthogonality, the tapers also have
optimal time-frequency concentration properties. Both the orthogonality and time-frequency
concentration of the tapers are critical to the success of the multitaper technique. See Discrete Prolate Spheroidal (Slepian) Sequences for a brief description of the Slepian
sequences used in Thomson’s multitaper method.
The multitaper method uses K modified periodograms, each one obtained
using a different Slepian sequence as the window. Let
Sk(f)=Δt|∑n=0N−1gk(n)x(n)e−j2πfnΔt|2
denote the modified periodogram obtained with the kth
Slepian sequence, gk(n). In its
simplest form, the multitaper method simply averages the K modified
periodograms to produce the multitaper PSD estimate:
S(MT)(f)=1K∑k=0K−1Sk(f).
Thomson's multitaper approach, introduced in [4], resembles
Welch’s overlapped segment averaging method, in that both average over approximately
uncorrelated estimates of the PSD. However, the two approaches differ in how they produce these
uncorrelated PSD estimates. The multitaper method uses the entire signal in each modified
periodogram. The orthogonality of the Slepian tapers decorrelates the different modified
periodograms. Welch’s approach uses segments of the signal in each modified periodogram, and the
segmenting decorrelates the different modified periodograms.
The equation for S(MT)(f) corresponds to the 'unity' option in
pmtm. However, as explained in Discrete Prolate Spheroidal (Slepian) Sequences, the Slepian sequences do not possess
equal energy concentration in the frequency band of interest. The higher the order of the
Slepian sequence, the less concentrated the sequence energy is in the band [–W,W] with the concentration given by the eigenvalue. Consequently, it can be
beneficial to use the eigenvalues to weight the K modified periodograms prior
to averaging. This corresponds to the 'eigen' option in
pmtm.
Using the sequence eigenvalues to produce a weighted average of modified periodograms
accounts for the frequency concentration properties of the Slepian sequences. However, it does
not account for the interaction between the power spectral density of the random process and the
frequency concentration of the Slepian sequences. Specifically, frequency regions where the
random process has little power are less reliably estimated in the modified periodograms using
higher-order Slepian sequences. This argues for a frequency-dependent adaptive process, which
accounts not only for the frequency concentration of the Slepian sequence but also for the power
distribution in the time series. This adaptive weighting corresponds to the
'adapt' option in pmtm and is the default for
computing the multitaper estimate.
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