Continuous Wavelet Spectrum (2D Contour)

The Continuous Wavelet Spectrum (2D Contour) option in the Spectral menu or the Spectral toolbar presents the Continuous Wavelet Transform (CWT) in a limited 2D spectral format. The CWT is a multiresolution time-frequency technique that is useful for exploring data known to be non-stationary. It is also of value for determining whether data can be safely judged wide-sense stationary as required by many of the spectral procedures.

If you prefer to view wavelet spectra exclusively as contour plots, as opposed to 3D surfaces, this option will be somewhat faster, use less memory, and will offer more contour plot graphing flexibility. For viewing wavelet spectra as 3D surfaces, use the Continuous Wavelet Spectrum (3D Surface) option. It also offers contour plots, but as a subset of the 3D rendering engine.

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 logarithmic 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 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

· Magnitude, sqrt(Re*Re+Im*Im)

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

· Phase, sine-based, Pi/2+atan(Im/Re)

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

· dB Norm, decibels, normalized to 0 for time-frequency node with maximum power

· 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)

In wavelet spectra, because of multiresolution analysis, the contour magnitudes (spectral peak heights) are not linearly proportional to power. If you need this property, and your data series is of sufficient length, you should use the Short-Time Fourier Transform Spectrum option. The most useful options for CWT spectra are the dB formats for visualization and the PSD formats where the surface can be integrated to provide power measurements.

The dBlim field is active only when the dB or dB Norm formats are used. A dB limit is used to specify the exact z-gradient that will be rendered in the contours. The default of 24, in conjunction with the default Spectrum 24 contour type means that a different color will be used for each 1dB delta in the spectrum. Since the contour types consist of 24, 32, and 48 color gradients, it it often convenient to select a dBlim value that produces meaningful differentiations. For example, if the Spectrum 48 contour is used, a dBlim of 12 would result in a separate color for each 1/4 dB delta in the spectrum. Below the lower threshold of the dB range, the limit color is used.

The Add Data item is used to overlay the nodes defining the interpolated surface. The data grid will consist of the CWT grid if its size (data size x frequency count) is less than 16384. If the CWT grid is larger, the data grid will be decimated by averaging that number of adjacent time nodes necessary to achieve a size smaller than 16384. When data are plotted in any CWT graph, points within the cone of influence are displayed in the inactive color. The cone of influence defines the spectral region where edge effects can be present.

Contour Options

The contour type is set using the last item in the AutoSignal graph's toolbar.

AR(1) Bkgrnd

AutoSignal implements peak-type critical limits rather than the traditional confidence limits. A 95% critical limit means that in only 1 of 20 similar size random data sets would the largest CWT spectral peak attain this height strictly by chance.

If the AR(1) Bkgrnd option is checked, the contours will consist of critical limits for the time-frequency spectra rather than a z-gradient coloring. The wavelet critical limit gradients are the following colors by default: 8-level grayscale from 10 to 50%, 8-level cyanscale from 50% to 90%, 8-level greenscale from 90% to 95%, 8-level yellowscale from 95% to 99%, and 8-level redscale from 99% to 99.9%.

The AR(1) Bkgrnd numeric value is used to adjust the critical limits for a first order autoregressive background. When this value is zero, a data value is assumed to have no correlation with its predecessor and the critical limit gradients will test a white noise background null hypothesis. When this AR coefficient value is greater than zero, a red noise background model assumption is tested. Persistence in natural systems often results in red noise backgrounds that can be modeled with an AR(1) coefficient in the vicinity of 0.5 to 0.8.

The CWT critical limits should be considered approximate since the spectral peak heights (contour magnitudes) can be impacted by the variable smearing in the time-frequency domain arising from multiresolution analysis.

Displaying Period

If the YR box is checked in the Labels section of the 2D View Options dialog, an inverse scale is drawn on the right Y axis. This scale will correspond with the frequency axis and will not be uniformly spaced. If the YR box is checked in the Titles section of the 2D View Options, the title "Period" will be used for the right Y axis if the left title is "Frequency". Otherwise the title will reflect the inverse of the current Y title used in the plot.

List

The List Data option lists the index, time, frequency, and the spectral quantity currently plotted in a four column table. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the time, frequency and the spectral quantity currently plotted 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 time, frequency and the spectral quantity currently plotted 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 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 always uses a landscape orientation. Beyond a certain size, the graph utilizes 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.

Power Analysis - Frequency Range

The Power Analysis - Frequency Range option computes the power across time for a specified frequency band by integrating the interpolated wavelet spectrum surface. The Continuous Wavelet Spectrum Frequency Range option in the Spectral menu is a specialized wavelet procedure that combines the CWT and this power analysis into a single step.

Power Analysis - Time Range

The Power Analysis - Time Range option computes the power across all frequencies for a specified range in time by integrating the interpolated wavelet spectrum surface. The global wavelet spectrum, which is similar to a smoothed FFT, is given by using the full time range. The Continuous Wavelet Spectrum Time Range option in the Spectral menu is a specialized wavelet procedure that combines the CWT and this power analysis into a single step.

Overall Power

The Power informational field reports the power of the wavelet spectrum arising from integrating the interpolated surface. This power will be reported only when one of the power formats is selected.

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.