Among the most useful of these techniques are the so-called ``multitaper" methods. These make use of a special set of windowing functions, called Slepian tapers. For discretely-sampled data sets, these are discrete prolate spheroidal sequences, and are related to prolate spheroidal functions. The GRASP package contains (a modified version of) a public domain package by Lees and Park, which is described in . Further details of this package may be found at http://love.geology.yale.edu/mtm/ http://love.geology.yale.edu/mtm/. Note however that we have already included this package in GRASP; there is no need to hunt it down yourself.
For those who are unfamilar with these techniques, we suggest reading Chapter 7 of . The sets of tapered windows are defined by three parameters. These are, in the notation of Percival and Walden:
In addition to providing better spectral estimation tools, the multi-taper methods also provide nice techniques for spectral line parameter estimation and removal. When the sets of harmonic coefficients are generated for different choices of windows, one can perform a regression test to determine if the signal contains a sinusoid of fixed amplitude and phase, consistent across the complete set of tapers. The GRASP package uses this technique (the F-test described on page 499, and the worked-out example starting on page 504 of ) to estimate and remove spectral lines from a data-set. This can be used both for diagnostic purposes (i.e., track contamination of the data set by the 5th line harmonic at 300Hz) or to ``clean up" the data (i.e., remove the pendulum resonance at 590 Hz).
As an aid in understanding these techniques, we have included with GRASP a section of the data-set from the Willamette River appearing on pg 505 of Percival and Walden , and an example program which repeats and reproduces the results in Section 10.13 of that textbook. This demonstrates the use of multi-taper methods in removing ``spectral lines" from a data set.