Wavelet Smoothing and Denoising

The Wavelet Smoothing and Denoising option in the Process menu or the Process toolbar is a specialized continuous wavelet transform filtration procedure that seeks to threshold the CWT so that only the signal nodes in the CWT are reconstructed. This option is used exclusively to remove noise via the CWT and its main use is for denoising non-stationary data.

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. For this procedure, the contour is fixed with a 24-color Spectrum gradient, and the contour 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 normalized decibel 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.

As in the Fourier Smoothing and Denoising option, the primary task is to set a threshold, either in frequency or spectral magnitude, whereby the noise that is present in the signal can be removed. When a Frequency threshold is selected, the reconstruction ignores all nodes in the CWT whose frequencies exceed this value. It is far more common to select a dB threshold. In this case, the CWT reconstruction ignores all nodes whose power falls below this specified signal threshold.

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.

Plot

The dBlim field 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. 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.

Thresholding

A dB threshold is normally used. In this instance, the CWT reconstruction uses only those nodes whose power matches or exceed the specified dB signal threshold. A Frequency threshold causes the CWT reconstruction to use only those nodes in the CWT whose frequencies are less than the specified value. The Frequency and dB threshold values must be entered numerically.

The dB value can be either positive or negative. A dB value of 12 and -12 would be interpreted in the same way, setting the signal threshold as 12dB below the spectral maximum.

The Reset Previous button restores the threshold value last used in this procedure.

Estimated Noise Reduction

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 In value reports the estimated white noise in the incoming data, the Out value the estimated white noise for the filtered signal. The percent is given as the amount of estimated noise remaining after filtration.

Power Reduction

In this section the In value reports the TISA power in the incoming data. The Out value is the TISA power for the filtered signal. The percent is given as the amount of power remaining after filtration. For most S/N ratios, the reduction in power should be minimal. These value can alert you when signal components are being discarded.

Correlation Coefficient

The r-squared correlation coefficient should also remain high. An rē of 1 is a perfect correlation while a value of 0 means the filtered and unfiltered signals are uncorrelated. Low rē values are also indicative of signal components being lost in the thresholding.

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.

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 denoised data.