Wavelet Filtering and Reconstruction

The Wavelet Filtering and Reconstruction option in the Process menu or the Process toolbar is a continuous wavelet transform processing procedure that offers the means to reconstruct signals from spectral components that have been isolated in the time-frequency domain. When a signal's spectral content varies across time, this option can readily isolate components that appear and disappear. Components that undergo changes in amplitude and frequency with time can also be characterized. This procedure also offers a greater flexibility than the Wavelet Smoothing and Denoising option for noise removal.

The option presents a dual AutoSignal graph with the time-frequency wavelet contour plot in the upper plot and the output time domain data in the lower graph. The default contour is a 24-color Spectrum gradient that is rendered on a 60x60 display grid. The properties of the CWT are fixed in order to achieve close to full precision data reconstruction.

The graph's toolbar has a button that can toggle on and off and change the proportions of the two graphs in the dialog. The default has the wavelet graph using the upper half of the region, and the output graph using the lower half.

Algorithm

This procedure first generates the wavelet spectrum and renders it in a contour plot in the upper graph. The Y-scale will initially be logarithmic since the frequencies required for reconstruction use a log spacing. The wavelet is adjustable to support the different time-frequency resolution tradeoffs needed for optimal filtering. The dB gradient used in the contour is also adjustable to aid visualization of signal and noise regions.

After a wavlet has been selected, the main task is the definition of the range, region, or volume of the time-frequency spectrum that is to be used for CWT data reconstruction. Although a 2D contour plot is used, the wavelet spectrum is a 3D surface. Filtering thresholds can be set in time, frequency, and spectral magnitude.

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.

Time-Frequency Thresholds

There are two primary ways to set the thresholds for wavelet filtering. The first is to define a region of time-frequency space. This is primarily used to isolate and reconstruct signal components. The Time1 and Time2 fields set limits in time. Similarly, the Freq1 and Freq2 fields define limits in frequency. The time-frequency boundaries of this region can either be included (Incl) or excluded (Excl). If a time-frequency region is included, this means that only the CWT nodes that fall within it are used for the data reconstruction. If the region is excluded, only those nodes that fall outside the region are used for the reconstruction.

The times and frequencies can be entered numerically. If both fields for either time or frequency are left blank, the full range of the variable is assumed for the Incl option, and no range is specified for the Excl option. If only a lower limit is entered, the variable range is assumed to be from this limit and greater for the Incl option, and all values below this limit if the Excl option is set. If only an upper limit is entered, the variable range is assumed to be from this limit and lower for Incl, and all values above this limit for Excl.

It is generally easier to set a time-frequency region graphically. When the XY Sectioning mode is selected, the mouse is used in the upper graph to enclose the time-frequency region of interest. To include the region that is to be graphically defined, click and hold down the left mouse button at one corner of the time-frequency rectangle in the upper graph and drag the rectangle so that it encloses the region desired. This will automatically enter the times and frequencies and set the process state to include the nodes within the region. To exclude the region that is to be graphically defined, the same procedure is followed except that the right rather than the left mouse button is used. In this case, the process state will be set to exclude the nodes within the region.

It is possible to reconstruct only one time-frequency region at a time.

The Clear button is used to clear all of the threshold fields.

The Apply button can be used to immediately update the processing rather than wait for the automatic update.

Spectral Thresholds

The other primary way to set the thresholds for wavelet filtering involves defining spectral thresholds. This is primarily used to separate signal components by power. It can also be used for noise removal, replicating the functionality in the Wavelet Smoothing and Denoising option, or for signal removal, generating a signal-free background for subsequent analysis.

Unlike the time-frequency thresholds where any spectral format can be used for displaying power, the normalized dB format should be used when setting spectral thresholds. This is the only way to visually judge signal and noise regions from a contour.

The Spec1 and Spec2 fields set the spectral limits. These can only be entered numerically. If only Spec1 is set, all values at this dB and above will be used in the reconstruction if Incl is set, and all below this dB will be used if Excl is checked. If only Spec2 is set, all values at this dB and below will be used in the reconstruction if Incl is set, and all above this dB will be used if Excl is set. For this option, the normalized dB values must use the minus sign.

Plot

The time-frequency spectrum can be plotted in five different 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, and dx is the sampling interval.

· 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)

When setting spectral thresholds, only the dB Norm format should be used.

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 gradient graph 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.

Even with 24 colors in the gradient mapped to a separate color for each 1dB delta, it is not a simple matter to discern the exact dB threshold for best partitioning signal and noise. The dBlim adjustment can be used to limit the rendered wavelet spectrum to a specified dB range. The contours rendered would thus represent only signal (noise would not be rendered). This dB value can then be used as the the threshold.

Contour Options

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

List

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

Copy

The Copy Data to Clipboard option copies the time and output signal 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 time and output 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 graphs can be exported to a MS Word RTF file, while the processed data can be exported to an Excel 95 or Excel 97 file.

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. Use this option to find the minimum and maximum power if you want to set spectral thresholds using a format other than dB Norm.

Popup Information

The Toggle Popup Information Window option is used to report estimated noise and power reductions as well as a correlation coefficient between the filtered and unfiltered data.

AutoSignal offers a robust noise estimation procedure that may be of some value for low-frequency signals. A cubic polynomial interpolation is made for each point using the two points to the left and the two to the right (excluding the current point). The difference between the interpolated and signal values is used to generate a measure of the white noise present in the signal. This assumes that the signal can be locally characterized by a smooth cubic interpolant. Also, the signal component(s) should exist only in the lower quarter of the Nyquist range. If a high frequency signal component is present, these estimates of noise will be invalid. The Noise In value reports the estimated white noise in the incoming data, the Noise Out value the estimated white noise for the filtered signal. The Noise % is given as the amount of estimated noise remaining after filtration.

The TISA In value reports the TISA power in the incoming data. The TISA Out value is the TISA power for the filtered signal. The TISA % is given as the amount of power remaining after filtration.

The r-squared correlation coefficient is also reported. An rē of 1 is a perfect correlation while a value of 0 means the filtered and unfiltered signals are completely uncorrelated.

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.

When exiting this procedure with the OK button, an option will be presented to update AutoSignal's main data table with the filtered data.