Fourier Spectrum
Fourier Spectrum
The Fourier Spectrum option in the Spectral menu or the Spectral toolbar offers a basic Fast-Fourier Transform (FFT) spectrum of the current data table. This option is the simplest of the Fourier spectral procedures. No data window tapering is available, there is no averaging of segments, and the data stream should be uniformly sampled (constant sample increment).
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 Nmin 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 since 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.
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.
Peaks
The spectral peaks 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
AutoSignal offers peak-type critical limits to determine the statistical significance of the largest peak present in the spectrum. 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 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, 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
The amplitude and phase of each component in the FFT is derived from sine-based conversion. Each of the components in the FFT can be reconstructed using:
Y=Amplitude*sin(2*PI*Frequency*X+Phase)
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 will always use a landscape orientation. Beyond a certain size, the graph will
utilize 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.
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