Fourier Multitaper Spectra

The Fourier Multitaper Spectra option in the Spectral menu or the Spectral toolbar uses a series of orthogonal data tapers to generate a Fourier spectrum, utilizing the information at the edges of the data and reducing the variance of the spectral estimate.

Multitaper Spectral Analysis

A multitaper spectrum is produced by averaging multiple windowed FFTs generated with a set of orthogonal data tapering windows known as discrete prolate spheroidal sequences (DPSS), Slepian functions, or eigentapers. Since each of the windows in a specific sequence is uncorrelated, an unbiased average spectrum can be produced and an F-ratio test is offered for determining the significance of any given peak in the spectrum. The seven tapers available for an nPi of 4 are graphed below. Note that only the first of the tapers has the traditional data tapering window profile.

A multitaper spectrum offers no greater frequency resolution than a single tapered spectrum. In fact, the spectral peaks resulting from the algorithm have a flat-topped envelope shape which makes the central frequency determination more difficult. What is gained is a reduced-variance spectral estimator that retains a high dynamic range and which utilizes all of the data in the record.

The Slepian data tapers are sometimes called eigentapers since they are generated using an eigenvector routine. There are two primary parameters, one that controls the frequency width of the window (nPi) and the other controls the number of windows in the sequence (Nwin). There are limits to the number of windows that can be used as spectral leakage increases as the sequence progresses. For a width of 2, up to 3 sequences are permitted. For a width of 3, up to 5 sequences are allowed, and for a width of 4, up to 7 sequences.

The first window in the Slepian sequence is included in the Windowed FFT option as the DPSS window. It is an excellent data tapering window. The subsequent windows in a Slepian series will emphasize the data better near the edges, but they do not offer the same spectral leakage resistance and they do not compute the true frequency. Instead, they contribute to the overall envelope of a multitaper spectral peak.

AutoSignal displays the spectra of all of the constituent sequences as separate references. It is thus easy to see the diminished spectral leakage performance as the window sequence increases.

Because the peaks in an averaged spectrum have this flat-top appearance, it is difficult to isolate the central peak locations from the spectrum. Instead, AutoSignal uses the peaks in the F-ratio plots to isolate spectral frequencies. Unlike the other FFT options, the Multitaper procedure offers an optional dual plot for every format, the upper graph containing an F-ratio plot.

The F-ratio plot not only enables a better isolation of the central frequency, but also offers a means for gauging the significance of the spectrum at each frequency computed (when more than one sequence is used). These F-ratios are analogous to the F-to-add thresholds used in multivariate analysis. These values are used to threshold the spectrum for peak detection. Just as with the conventional FFT procedures, a refined central peak location can be achieved via zero padding. Here though, AutoSignal does not interpolate the F-ratio spectrum.

For more information, see Jonathan Lees and Jeffrey Park, "Multiple Taper Spectral Analysis", Computers and Geosciences, v21, p199, 1995.

Viewing Individual Spectra

The individual spectra of the various tapers and their labels can be toggled on and off with the added reference buttons in the graph's toolbar.

Algorithm

There are two algorithms, and their distinction is how the individual spectra are averaged. The Eigen Wts is a weighting by eigenvalue. The Adaptive algorithm is generally a better choice since it is less adversely influenced by the higher sequence spectra which evidence more spectral leakage.

nPi specifies the main lobe width of the data tapering windows and Nwin specifies the number of windows to use within the sequence. For high dynamic range applications, do not automatically use the highest permitted number of windows. Rather inspect the individual spectra. For example, with an nPi value of 4, the 6th and 7th tapers within the sequence might be harming the results rather than aiding them.

To accommodate zero padding, Nmin specifies the minimum FFT. As is true when any data tapering window is used, very little spectral leakage arises from the zero-padding.

Plot

The frequency domain information can be plotted in a variety of formats. In the following table, Re is the real component of the FFT at a given frequency, Im is the imaginary component, n is the data set size, dx is the sampling interval, and var is the variance of the data series.

· Magnitude, sqrt(Re*Re+Im*Im)

· Amplitude, 2.0*sqrt(Re*Re+Im*Im)/n

· dB, decibels, 10.0*log10(Re*Re+Im*Im)

· dB Norm, decibels, normalized to 0 for frequency channel with maximum power

· PSD SumSq, Power as Sum Squared Amplitude, 2.0*(Re*Re+Im*Im)/n

· PSD MeanSq, Power as Mean Squared Amplitude, 2.0*(Re*Re+Im*Im)/n/n

· PSD TimeInt, Power as Time-Integral Squared Amplitude, 2.0*dx*(Re*Re+Im*Im)/n

· Variance, Power normalized by variance, (Re*Re+Im*Im)/n/var

Each of these formats is also available in a version which also displays the F-ratio plot. In an amplitude plot, you see the actual amplitude of sine components. In a normalized decibel plot, the highest peak is at 0dB, a peak at -3dB would have half the power, and a peak at -6dB would have half the amplitude. The PSD TISA (time-integral squared amplitude power) is the actual integral under the curve defined by the square of the raw data.

Peaks

The spectral peaks in the averaged spectrum are identified by a local maxima detection algorithm in the F-ratio and averaged frequency spectrum. The frequency locations of the detected peaks are drawn from the F-ratio maxima, whilte the amplitudes are taken from the values of the amplitude spectrum at these frequencies.

The sig item sets the target number of peaks (signal components) to detect. Up to 50 peaks can be detected. Peaks are ranked by amplitude. Note that this target signal component count may not be realized as fewer peaks than this target may be detected. Note also that the frequency analysis and linear sinusoidal fits reported in the Numeric Summary use the component count and frequencies from this peak identification.

The F item sets the F-ratio tolerance for accepting a peak. An F-ratio maxima must exceed this value in order to be processed as a spectral peak. The default is F=2.0.

The Display Maxima option is used to step through the options for displaying spectral peak labels: frequencies, spectral magnitudes, both frequencies and spectral magnitudes, or none.

AR(1) Background

AutoSignal offers peak-type critical limits to determine the statistical significance of the largest peak present in the spectrum. These limits are computed for all windows, including those with adjustable parameters. The default background used for this null hypothesis is white (Gaussian or normally distributed) noise, AR(1)=0.0. A lag-1 autoregressive spectrum can also be specified. An AR(1) coefficient greater than 0.0 can often model red noise (where the noise power decreases with increasing frequency).

The AI Expert option will set the AR(1) value to the lag-1 normalized autocorrelation. This should only be used as a preliminary estimate. For a better estimate of the background, use the one of the Fourier Filtering and Reconstruction, Eigendecomposition Filtering and Reconstruction, or Wavelet Filtering and Reconstruction options to remove the spectral components from the signal, leaving only the background. Then use the AR (AutoRegressive) Spectrum option with an order of 1 to fit the AR model. The desired AR(1) coefficient is listed in AR procedure's numeric summary.

The Show Significance Levels option is in the graph's toolbar. This button is used to toggle the significance levels on and off.

List

The List Data option offers an FFT data summary with the frequency and spectral quantity being plotted. The listing also include the F-values, and the DOF (degrees of freedom) at each frequency. The list uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the columns in the List Data option to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors). You can generally find a Paste As option in most applications if you want specific control over the format imported.

Save

The Save Data to Disk option writes the columns in the List Data option to a supported file format. These formats include ASCII, Excel 97, Excel 95, Lotus WK3, Lotus WK1, SPSS, or Systat.

Production Facility

The Autosignal Automation facility allows unattended processing of large numbers of data sets. The data sets can be consolidated in an Excel file or acquired using a DLL. The numeric summaries and graphs can be exported to an MS Word RTF file, while the extended data summaries or the current spectra can be exported to an Excel 95 or Excel 97 file.

Numeric Summary

The Numeric Summary offers a full FFT report. The report optionally includes a listing of the interpolated spectral peaks, a frequency analysis, and a linear sinusoidal least-squares fit summary.

Non-Linear Optimization

The Non-Linear Optimization offers the means to refine the parameter estimates given in the linear sinusoidal fit that is reported in the Numeric Summary. Constrained least-squares and robust (maximum likelihood) non-linear fitting is available with either sinusoid or damped sinusoid models.

Rich-Text Format Export

The Export Numeric Summary and Graph to RTF File option writes the numeric summary and spectral plot to an RTF file. The numeric portion of the file is based upon the current settings in the Numeric Summary option. The text data will be written to portrait orientation pages. The graph uses the current settings and size of the spectral plot, and is inserted as a Windows Metafile. The graph always uses a landscape orientation. Beyond a certain size, the graph utilizes a full landscape page.

Local Options

A local option changes the data set for the duration of the current procedure only. The main data table is not altered. AutoSignal offers four local options in most of the spectral procedures.

Section the data to isolate specific regions for processing.

Detrend for removing mean or subtracting a least-squares trend model.

Fourier Filtration for isolating spectral components by frequency.

Eigendecomposition Filtration for isolating spectral components by signal strength.

The Reset button restores the data to its state when first entering the procedure. Note that if you implement sequential local procedures, all of the revisions are discarded upon reset. If an Automation Session is in progress, the Reset button can be used to terminate the automated processing.