9. DECONVULATION

Deconfulation is the converse of confulation in the sense that division is the converse of multiplication^*. In fact, the deconvolution of one signal from another is usually performed by dividing the two signals in the Fourier domain^**. The practical significance of deconvolution is that it can be used as an artificial (i.e. computational) way to reverse the result of a convolution occurring in the physical domain, for example, to reverse the signal distortion effect of an electrical filter or of the finite resolution of a spectrometer. Two examples of the application of deconvolution are shown in Figures 12 and 13. 

Figure 12. Deconvolution is used here to remove the distorting influence of an exponential tailing response function from a recorded signal (Window 1, top left) that is the result of an unavoidable RC low-pass filter action in the electronics. The response function (Window 2, top right) is usually either calculated on the basis of some theoretical model or is measured experimentally as the output signal produced by applying an impulse (delta) function to the input of the system. The response function, with its maximum at x=0, is deconvoluted from the original signal . The result (bottom, center) shows a closer approximation to the real shape of the peaks; however, the signal-to-noise ratio is unavoidably degraded.

Note that this process in figure 12 has an effect that is visually similar to resolution enhancement, although the later is done without knowledge of the broadening function that caused the peaks to overlap.

Figure 13. A different application of the deconvolution function is to reveal the nature of an unknown data transformation function that has been applied to a data set by the measurement instrument itself. In this example, Window 1 (top left) is a uv-visible absorption spectrum recorded from a commercial photodiode array spectrometer (X-axis: nanometers; Y-axis: milliabsorbance). Window 2 (top right) is the first derivative of this spectrum produced by an (unknown) algorithm in the software supplied with the spectrometer. The signal in the bottom left is the result of deconvoluting the derivative spectrum (top right) from the original spectrum (top left). This therefore must be the convolution function used by the differentiation algorithm in the spectrometer's software. Rotating and expanding it on the x-axis makes the function easier to see (bottom right). Expressed in terms of the smallest whole numbers, the convolution series is seen to be +2, +1, 0, -1, -2.  This simple example of "reverse engineering" would make it easier to compare results from other instruments or to duplicate these result on other equipment.

When applying deconvolution to experimental data, to remove the effect of a known broadening or low-pass filter operator caused by the experimental system, a very serious signal-to-noise degradation commonly occurs. Any noise added to the signal by the system after the broadening or low-pass filter operator will be greatly amplified when the Fourier transform of the signal is divided by the Fourier transform of the broadening operator, because the high frequency components of the broadening operator (the denominator in the division of the Fourier transforms) are typically very small, resulting in a great amplification of high frequency noise in the resulting deconvoluted signal. This can be controlled but not completely eliminated by smoothing and by constraining the deconvolution to a frequency region where the signal has a sufficiently high signal-to-noise ratio.

Note: The term "deconvolution" is sometimes also used for the process of resolving or decomposing a set of overlapping peaks into their separate components by the technique of iterative least-squares curve fitting of a putative peak model to the data set.  The process is actually conceptually distinct from deconvolution, because in deconvolution the underlying peak shape is unknown but the broadening function is assumed to be known; whereas in iterative least-squares curve fitting the underlying peak shape is assumed to be known but the broadening function is unknown.  

* If you know that mx = n, where m and n are known but x is unknown, then x = n/m.  Conversely if you know that m convoluted with x = n, where m and n are known but x is unknown, then x = m deconvoluted from n.  

** Fourier transforms are usually expressed in terms of complex numbers, with real and imaginary parts. If the Fourier transform of the first signal is a + ib, and the Fourier transform of the second signal is c + id, then the ratio of the two Fourier transforms is

a + ib       ac + bd       bc - ad
 ^_________  =   ^_________  + i  ^_________
c + id       c² + d²       c² + d²