Continuous Wavelet Spectrum Time Range

The Continuous Wavelet Spectrum Time Range option in the Spectral menu or the Spectral toolbar offers the Continuous Wavelet Transform (CWT) in a specialized wavelet procedure that computes the power across frequencies for a specified time band. This option combines most of the functionality of the Continuous Wavelet Spectrum (2D Contour) procedure and the Power Analysis - Time Range feature into a single step.

This option presents the wavelet spectrum in an AutoSignal 2D graph with a default 32 color blue to red gradient contour. The Y2 axis contains the powers resulting from the integrations.

The global wavelet spectrum, which is similar to a smoothed FFT, is given by using the full time range.

Wavelet

The Morlet, Paul, and GaussDeriv wavelets are available for CWT spectral analysis. The adjustable parameter (Adj) for the Morlet is its wavenumber (from 6 to 20). For the Paul wavelet it is an order that can vary from 4 to 40. For the Derivative of Gaussian wavelet, it is the order of the derivative (from 2 to 80). The wavelets are normally complex, but a real form can be used if Complex is unchecked.

The View Mother Wavelet option can be used to select the wavelet and set its properties graphically.

The Nmin option sets the actual size of the FFT that is used. The difference between this value and the data size specifies the amount of zero padding. The CWT uses an FFT-based fast convolution procedure that requires zero padding in order to be free of wraparound effects. Since it is often possible to zero pad to the next power of 2 and find negligible wraparound effects and also achieve the fastest FFT performance, this is the Nmin initially presented. The number of time values in the spectrum is always the data count, irrespective of zero padding.

Frequency

The CWT offers the means to generate a wavelet spectrum using any set of frequencies desired. The Full Range item locks the frequency range from the lowest unit frequency to the Nyquist frequency. When this option is not checked, the start and end frequencies must be specified.

The Ln Steps item specifies that the frequencies should use a logairithmic spacing. This is useful when most of a signal's energy is at lower frequencies. When this option is not checked, the frequency spacing will be linear.

The n field specifies the count of frequencies in the wavelet spectrum. The default of 35 usually gives a respectable coverage, although it may be insufficient to catch closely spaced low frequency components when a log spacing is used, or closely spaced high frequency components when a linear spacing is used. AutoSignal supports up to 100 frequencies. Bear in mind that each frequency requires a separate FFT of Nmin length, so computation times and memory requirements for large data sets will go up appreciably when high frequency counts are specified.

Surface Decimation

The CWT spectrum is graphically rendered by evaluating a bivariate B-spline interpolant. Powers are also computed by integrating this interpolant.For perfomance reasons and to conserve memory, this B-spline interpolant is limited to a total of 16384 nodes. If the CWT generates a grid (data size x frequency count) with more than this number of values, an averaging decimation is used to reduce the nodal count before the interpolant's coefficients are computed. The decimation is not normally a problem for CWT spectra since it is not possible to directly view power or amplitude in the contours, and the averaging has minimal impact on power computations.

Memory Issues

Separate FFTs are made for each scale or frequency in the CWT. For memory reasons, the number of evaluated CWT frequencies is limited to a maximum of 100. In the CWT, zero padding is only used to prevent wraparound effects in the convolution. No additional memory is used as a consequence of zero padding. The spectral data is fitted to a bicubic B-spline for contour rendering and surface integration. The CWT surface is stored as a grid of B-spline coefficients that consumes considerable memory.

The amount of physical memory (RAM) free for AutoSignal's use is shown in the main status window in the Mem field. When dealing with large data sets, particularly WAV files, it is not difficult to exhaust this memory. When this happens, Windows uses the hard disk for memory operations. Excessive disk activity and extremely slow processing and procedure closure times will result if the physical memory is insufficient.

In the CWT, the memory relationship is linear. Doubling the frequency count doubles the amount of physical memory needed. Typically, there is little to gain beyond 50-60 CWT frequencies. If you are unable to prevent the hard disk thrashing and drastically diminished performance that results from exhausting physical memory, you can try breaking up the large data stream into smaller separate data sets. Given the relatively low cost of RAM, upgrading to 64, 96, or 128 Mb may be a good investment if you will be doing a good deal of non-stationary analysis of large data streams using the CWT.

Plot

The time-frequency spectrum can be plotted in a variety of power formats. In the following table, Re is the real component of the CWT at a given time and 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.

· MagnitudeSq, Re*Re+Im*Im

· Variance, variance normalized CWT, (Re*Re+Im*Im)/var

· Int=PSD SSA, Surface Integral is Sum Squared Amplitude Power, 2.0*(Re*Re+Im*Im)

· Int=PSD MSA, Surface Integral is Mean Squared Amplitude Power, 2.0*(Re*Re+Im*Im)/n

· Int=PSD TISA, Surface Integral is Time-Integral Squared Amplitude Power, 2.0*dx*(Re*Re+Im*Im)

Contour Options

The contour type is set using the last item in the AutoSignal graph's toolbar. The 32 color blue to red gradient contour was chosen as the default because it tends to complement the power curve.

Analysis Curve

The integration is carried out in frequency steps that span the full frequency range of the data. The ndf field sets the number of frequency steps. The powers will each reflect a frequency increment df=(frequency range)/ndf. The t init and t final fields specify the starting and ending times for each of the double integrations. Note the power curve represents frequency snapshots of the power in the signal between the times specified. It is not a cumulative curve.

The peaks in the power curve are identified by a local maxima detection algorithm. The pks item sets the target number of power peaks to detect. Up to 50 peaks can be detected. Peaks are ranked by power. Note that this target count may not be realized as fewer peaks than this target may be detected.

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

Power

The Power information field reports the total power from the integration that uses the t init and t final starting and ending times. This value will be the sum of the individual power points in the curve.

List

The List Data option lists the index, frequency, and power in a three column table. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the frequency and power values to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors).

Save

The Save Data to Disk option writes the frequency and power values 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.

Numeric Summary

The Numeric Summary offers a CWT Spectrum report. The report optionally includes a peak map where the three highest times at each frequency are listed.

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.

Evaluate 3D Surface

The Evaluation offers a full-featured numeric evalation of the interpolated CWT bicubic B-spline surface, partial derivatives, roots, and volumes as well as offering a means for generating a table or file of any size using a generated XY grid or by importing XY data from supported file formats. You can use this option to integrate any portion of the time-frequency surface in order to determine the power present. Evaluations outside the bounds of the data will map to the bounds. The integration limits should thus be at or within the data boundaries.

Fast 3D Evaluation

The Quick Evaluation offers the means to evaluate the Z of the surface at any X,Y. It also reports the X,Y,Z representing the surface minimum and maximum.

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.