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
[41]. 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 [40]. The sets of tapered windows are defined by three parameters. These are, in the notation of Percival and Walden:

- : The length of the discretely-sampled data-set, typically
denoted by the integer
`npoints`in the GRASP routines. - : The product of total observation time and the resolution bandwidth . This dimensionless (non-integer)
quantity is denoted
`nwdt`in the GRASP routines. Note that for a conventional FFT, the frequency resolution would be . This corresponds to having . The multitaper techniques are typically used with values of which are several times larger, for example , which corresponds to . - : The number of Slepian tapers (or window functions) used,
typically denoted
`nwin`in the GRASP routines. Note that it is highly recommended (see page 334 of [40] and the final two figures on page 339) that the number of tapers .

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 [40]) 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 [40], 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.