Fourier Spectra with Data Window Comparison
Fourier Spectra with Data Window Comparison
The Fourier Spectra with Data Window Comparison option in the Spectral menu or the Spectral toolbar is similar to the Fourier Spectrum with Data Window option, except that two additional windowed spectra can be simultaneously viewed. This enables direct comparison of the performance of various data tapering windows in a single spectral graph. The primary spectrum is based upon the initial window. You can select one or two reference spectra which use different windows or the same window with a different width (if it uses an adjustable parameter). Note that critical limits have not been implemented for this procedure.
Transform
AutoSignal offers the following FFT algorithms:
· Chirp-Z
The Best Exact N composite algorithm is the default. If the data size is a power of 2, the FFT Radix 2 algorithm is used. If not and the size is included in the prime-factor set, then the Prime Factor procedure is used. Otherwise, the Mixed Radix algorithm is used if the largest prime <= 509 and the Chirp-Z is used if the largest prime >= 521. This produces the fastest possible exact n FFT.
Nmin
The initial value will be the data size. To zero pad, enter any value greater than the data size. You may also select from a power of 2 sequence in the drop down box. Note that the actual size of the FFT may be greater than this value if the FFT Radix 2 or Prime Factor algorithm is used.
Increasing the zero padding will increase the number of frequency channels, which for small size data sets can aid in more accurately determining the center frequencies of spectral peaks. This will not change the basic shape of the spectrum, however. If a given peak is defined by only three frequency bins when an exact 64 point FFT is made, a 1024 point FFT will basically fill in this same shape. It is a kind of interpolation; zero padding cannot sharpen the peaks. To achieve this sharpening with an FFT, a longer set at the same sampling rate would be required. For data that are rapidly changing, or when the time series is limited in size, a non-FFT procedure is usually required for good spectral resolution.
When a 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.
· Real, abs(Re)
· Imaginary, abs(Im)
· Magnitude, sqrt(Re*Re+Im*Im)
· Phase, sine-based, Pi/2+atan(Im/Re)
· Mag/Phase, dual plot, magnitude in Y, phase in Y2
· Amplitude, 2.0*sqrt(Re*Re+Im*Im)/n
· Ampl/Phase, dual plot, amplitude in Y, phase in Y2
· 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.
Window
AutoSignal offers a broad selection of data tapering windows to minimze spectral leakage. The window selected here is the primary one and its spectrum is used for all output options. The adjust field is used to set the spectral width, and thus the dynamic range, of adjustable windows. This field will be disabled for fixed windows.
An appropriate window and adjustable width can be determined by using the Explore
Data Tapering Windows option.
Each spectrum is normalized so that its power equals the power of the input data, regardless of the data window used. In AutoSignal, a windowed FFT spectrum will contain the same power as a non-windowed spectrum, and both will match the power in the input data.
References
The Count field sets the number of references desired. Up to two reference spectra can be plotted. For each reference, select the window desired and set the adjustment if the window is adjustable.
Note that the reference spectra appear only in the graph. Except for these references appearing in an exported graph, these spectra are not a part of the output or processing options.
Peaks
The spectral peaks in the primary 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.
List
The List Data option offers an extended FFT data summary for the
primary spectrum. The listing uses the AutoSignal text
viewer facility. The FFT channel number, frequency, and magnitude are always listed. The Format
menu offers the optional selection of the following:
· Add dB
· Add Power Spectral Density, Sum Squared Amplitude
· or Add Power Spectral Density, Mean Squared Amplitude
· or Add Power Spectral Density, Time-Integral Squared Amplitude
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 a MS Word RTF file, while the extended data summaries or the current spectra can be exported to an
Excel 95 or Excel 97 file. The graphs produced in the automation will contain the reference spectra.
Numeric Summary
The Numeric
Summary offers a full FFT report for the primary spectrum. 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 will always use a landscape orientation. Beyond a certain size, the graph will
utilize a full landscape page. The exported graph will contain the reference spectra.
Explore Data Tapering Windows
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.
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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