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Deconvolution
Deconvolution is the converse of convolution 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
occuring 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.
Figure 13. A different application of the deconvolution function is to reveal
the nature of a data transformation function that has been previously 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
derivative spectrum is deconvoluted from the original spectrum. The result
(bottom left) is the convolution function used by the differentiation algorithm
in the spectrometer. 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.
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.
SPECTRUM, the freeware signal-processing application that accompanies this tutorial, includes a deconvolution function.
Matlab has built-in function for deconvolution: deconv.
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This page is maintained by Prof. Tom O'Haver , Department of Chemistry and
Biochemistry, The University of Maryland at College Park.
Comments, suggestions and questions should be directed to
Prof. O'Haver at toh@umd.edu.