Fourier Spectra of Segmented Data
Fourier Spectra of Segmented Data
The Fourier Spectra of Segmented Data option in the Spectral menu or the Spectral toolbar is the traditional periodogram with some additional Fourier spectral information. For non-stationary data, it is also the precursor of the Short Time Fourier Transform. This option computes an averaged frequency spectrum by taking the individual FFTs of multiple (and usually overlapping) segments of the data stream. The segmenting results in a smaller size data record, and consequently in a reduced spectral resolution, but the averaging reduces the variance that would arise in a single FFT.
Rather than assume stationarity, AutoSignal retains the spectrum of each overlapping segment. This makes it possible to check the assumption of stationarity by inspecting a graph of the individual spectra or by displaying all of the spectra in a 3D plot. It also makes it possible to generate full error bars for the averaging across segments. A true spectral component should be present to some degree in every segment, and the peak should not vanish in the range of the error bars. The spectrum is normalized to enable a comparison of the averaged data with that of the individual segments. The overall power of the averaged spectrum will exactly match that of the input data only when each segment contains the same power.
The individual spectra of the various segments and their labels can be toggled on and off with the added
reference buttons in the graph's
toolbar.
The graph’s
toolbar also has a button that offers the full selection of error
bars, as well as an error-bar toggle.
Transform
For this procedure, AutoSignal constrains the FFT algorithm to the Best Exact N option (there is no algorithm selection).
Window
AutoSignal offers a broad selection of data tapering windows to minimze spectral leakage. The adj field is used to set the parameter for adjustable windows. This field will be disabled for fixed windows.
This procedure normally employs a data tapering window since the redundancy in sampling compensates for the loss of information at the bounds of each taper.
The Explore
Data Tapering Windows option is available to assist with data window selection and adjustment. This
option plots a discrete FFT in a problem designed to illustrate the frequency widths of each window as
well as the rolloff vs. maximum sidelobe tradeoff. Up to four windows can be inspected simultaneously
and key window properties are empirically determined.
Each segment's spectrum is normalized so that its power equals the power of the input data, regardless of the data window used. This makes it straightforward to compare segments or different windows, although the averaged spectrum's power will exactly match that of the input data only when each segment contains the same power.
Segments
The length of each of segment, Seg n, and the amount of overlap, overlap %, is specified. You should set the segment size based on the resolution needed. Values for the overlap that produce the minumum variance are reported to be in the range of 50-70%.
To accommodate zero padding, the actual N of the FFT, FFT n, is also specified. When a data tapering window is used, very little spectral leakage arises from the zero-padding. Zero-padding is especially useful for interpolating peak frequencies with this algorithm, given the loss in resolution incurred by the reduced size of the segments. The Sections informational field will display the number of FFTs averaged to produce the spectrum.
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
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.
Note that the dB Norm format normalizes each of the individual spectra based upon the maximum found across all of the spectra. Thus only one of the individual spectra will likely attain the 0dB level. The averaged spectrum is normalized separately based upon the maximum in the average. The averaged spectrum will have a maximum at 0dB, and the associated peak will probably be slightly positive as a consequence of the bin interpolation.
Note also that averaging dB values is not an arithmetic average of the individual spectra, but more of a logarithmically weighted average. With a simple average of power, intermittent harmonics or noise bursts in one or more segments can overwhelm an otherwise low power trend. When dB values are averaged, the influence of intermittent elements is significantly lessened. A dB average is thus a robust measure of power, although it may miss a harmonic that appears only briefly in time. An arithmetic average in power, or even amplitude, is more likely to catch an intermittent harmonic. This is one of the reasons it is important to inspect the individual spectra and the error bars associated with the average spectrum. If non-stationarity is observed, you should consider time frequency analysis using the Short-Time Fourier Transform Spectrum, Continuous Wavelet Spectrum (3D Surface), or Continuous Wavelet Spectrum (2D Contour) options.
Peaks
The spectral peaks in the averaged spectrum are identified by a local maxima detection algorithm. Both the amplitude and the frequency locations of the detected peaks are based upon a cubic spline bin interpolation procedure.
The sig item sets the target number of peaks (signal components) to detect. Up to 50 peaks can be detected. Peaks are ranked by interpolated 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 wid item sets the bin width tolerance for defining a peak. A peak must exist across this number of FFT bins to be counted. The default is a single bin.
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
Peak-type critical limits determine the statistical significance of the largest peak present in the spectrum. Due to the complexity of segmented and overlapped FFTs, critical limits are available only for the unwindowed case. 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 extended FFT data summary. The listing
uses the AutoSignal text
viewer facility. The FFT channel number and frequency are always listed. The Format
menu offers the optional selection of the following:
· Add Avg,SD dB
· Add Avg,SD Power Spectral Density - SSA
· or Add Avg,SD Power Spectral Density - MSA
· or Add Avg,SD Power Spectral Density - TISA
The number of data averaged will appear in brackets in the standard deviation (SD) columns.
Copy
The Copy Data to Clipboard option copies all of the columns currently selected 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 all of the columns currently
selected 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.
Display as 3D Plot
The Display as 3D Plot option will generate an AutoSignal
3D surface graph using all of the individual spectra. The 3D display option is particularly useful
in discerning non-stationary trends. While the individual segment spectra readily reveal non-stationary
behavior, the 3D contour or surface plots offer a better visualization of how the spectrum changes with
time. Note that stationarity can also be readily assessed in the
Short-Time Fourier Transform Spectrum or in the
Continuous
Wavelet Spectrum (3D Surface) or Continuous
Wavelet Spectrum (2D Contour) options.
For this option to be available, there must be at least 4 segments generated. The nodes will consist of a rectangular grid formed by the individual spectra. If the spectrum is more than 128 frequencies in length, an averaging decimation is used to create the grid. Similarly, if there are more than 128 segments, only the first 128 segments are used in the grid. The rendering is by the Bicubic B-Spline algorithm.
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.
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