Deconvolve Exponential Response Function

The Deconvolve Exponential Response Function option in the Process menu or the Process toolbar is used to manage the instance where a signal is smeared by a one-sided or unidirectional exponential response function. Many measurement systems have a first-order kinetic response where there is a lag or delay between the signal's appearance and its record in the measurement system. The deconvolution seeks to recover the true signal that would have been measured using an ideal sensing system, one with an instantaneous response. In the deconvolution, signal amplitudes will increase and phase shifts will occur.

Since deconvolution adds significant high frequency noise, a low pass filtration is required to extract a clean reconstruction of the signal. Both the width of the response function and the extent of the low pass filtration are critical. This procedure should not be used when high frequency components are present since they will be removed by this low pass filtration..

In this option, the incoming and processed data are shown in an AutoSignal Graph. The deconvolved or convolved data are shown in the Y-axis plot, and the incoming data are plotted on the Y2 axis. The initial defaults show the input data as points, the deconvolved or convolved data as connected lines, all on a common scaled graph.

Deconvolution Example

The example below was generated using the sample4.sig file in the Generate Signal option. The 256 point data set contains 3 random amplitude and phase sinusoids spread randomly in three frequency ranges across the lower quarter of Nyquist range. 10% white noise has been added. If we assume that these data have been imperfectly detected due to slow kinetics of detection that can be described by an exponential time constant of 0.0008, the following exponential deconvolution, using a filter setting of 70%, represents a reasonable recovery of the true signal:

Note that this "sharpening" of the signal results in an increased amplitude of the overall waveform as well as the peaks shifting to earlier points in time. The TISA power of the signal increased from 468 to 2279 as a consequence of the deconvolution. Further, the distribution of power amongst the components changes as a consequence of deconvolution.

The non-linear optimization fit of three sinusoids to the original (non-deconvolved data) yields the following parameters:

 Comp Frequency Amplitude Phase Power %

    1 101.221813 53.4511351 0.19293544 72.5396103 15.4365400

    2 269.517055 96.1143316 2.10539318 233.216665 49.6288628

    3 497.614539 80.3945939 2.17047184 164.165161 34.9345972

The results for the deconvolved data in this example are as follows:

 Comp Frequency Amplitude Phase Power %

    1 100.815532 67.7891376 0.68920083 118.991261 5.13771846

    2 269.767123 177.747927 2.82366925 800.517077 34.5641464

    3 497.778287 233.742386 3.02259097 1396.52478 60.2981351

Note the higher power for each component in the deconvolved data. Also, note the very different distribution of power. The frequencies are generally preserved, but the phases and amplitudes are appreciably altered. The phase shift arising from the components appearing sooner in time is not constant across the components, although it is primarily determined by the time constant used for the deconvolution. The lowest frequency sinusoid would be least impacted by exponential smearing and is thus least changed by the undoing of this smearing that deconvolution offers. Conversely, the highest frequency sinusoid would be most impacted or attenuated by exponential smearing, and it is thus the most altered by the deconvolution.

Convolution and Deconvolution

In terms of signal non-idealities, convolution is the smearing of a data set by a given instrument response function. Deconvolution is the procedure of undoing that smearing in an effort to see what the data would look like had the instrument or measuring system perfectly rendered it.

Whereas convolving data with a response function produces smoother data, the deconvolution process of undoing the smearing produces exactly the opposite effect. Noise is introduced, usually in drastic amounts, and especially at higher frequencies. AutoSignal's exponential deconvolution procedure seeks to deal with this major growth in noise by adding a Fourier domain filter which removes most of the noise added in the deconvolution procedure. For data not yet smoothed or denoised, this deconvolution procedure can actually result in reduced noise.

Deconvolution Pitfalls

Deconvolution is fraught with many pitfalls, and the Fourier domain procedure is not always successful. If you attempt to deconvolve a response function with an unreasonably large time constant, the result is generally nonsense. Even with effective frequency domain noise filtration, noise present at lower frequencies can produce something other than a smooth deconvolution. When the exponential time constant is too high, the deconvolution can introduce extraneous sinusoidal components.

If the system delay element is described by anything other than a first-order exponential response, the deconvolution is likely to produce incorrect results.

Time Constant

When this option is used for the first time, AutoSignal will supply an initial response function time constant based upon the range and midpoint of the data. Depending on the data, this value may be quite low or quite high, perhaps significantly greater than the wavelength of one or more components. While you are offered a wide range of entry, consider restricting the time constant to no more than half the wavelength of the highest frequency component that is present in the data.

This response function width will be saved across sessions in order to preserve the entry of a carefully measured or inferred instrument response width. Note that if you import data from a different instrument, or if the data consist of higher frequencies, this previous width may not represent even a viable starting point.

Filter

You may enter any value between 1 and 99% for the Filter setting, or you may set it by using the up and down spin buttons, or by clicking and holding the right mouse button down while on the edit field or spin buttons, and then selecting the value from the popup menu.

The filter is a simple Fourier domain truncation filter. For this procedure, the scale is a non-linear one. Typically, values between 65 and 85% will be required. The impact of setting too high a filter will be the loss of the highest frequency signal components.

A little exploration quickly reveals the significance of the increased noise from the Fourier deconvolution procedure and the vast importance of effective filtration. At low filter settings, with the IRF width a significant percentage of the shortest wavelength component, the deconvolution is likely to produce nonsense. Also note that small differences in the filter setting can significantly affect the resulting deconvolution.

AI Expert

The AI Expert option will seek to determine a starting filter setting for the deconvolution. Using scanning and smoothing techniques with the frequency data, an estimate is made for the optimum frequency threshold between signal and noise, that is, the point where the two are equal. The procedure is hardly foolproof, although it works reasonably well until response function widths become too great.

Equivalent Noise %

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 deconvolved and filtered signal. The percent is given as the amount of estimated noise remaining after filtration.

List

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

Copy

The Copy Data to Clipboard option copies the time and processed data 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 processed data 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

This procedure does not offer local options. If a response function deconvolution is to be made, it should be done as the first step after acquiring the data.

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 deconvolved and filtered data.