Multi-taper methods for spectral analysis

0 Since the early 1980's there has been a revolution in the spectral analysis, due largely to a seminal paper by Thomson [39]. There is now a standard textbook on the subject, by Percival and Walden [40], to which we will frequently refer.

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:

$N$: The length of the discretely-sampled data-set, typically denoted by the integer npoints in the GRASP routines.

$NW \Delta t$: The product of total observation time $N \Delta t$ and the resolution bandwidth $W$. This dimensionless (non-integer) quantity is denoted nwdt in the GRASP routines. Note that for a conventional FFT, the frequency resolution would be $W=\Delta f = 1/N \Delta t$. This corresponds to having $NW \Delta t = 1$. The multitaper techniques are typically used with values of $W$ which are several times larger, for example $W = 3 \Delta f$, which corresponds to $NW \Delta t = 3$.

$K$: 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 $K< 2NW \Delta t$.

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.