Note: The word "deconvolution" appears in the Oxford dictionary, where its meaning is "A process of resolving something into its constituent elements or removing complication in order to clarify it". The same word is also sometimes used for the process of resolving or decomposing a set of overlapping peaks into their separate additive components by the technique of iterative least-squares curve fitting of a putative peak model to the data set. However, that process is actually conceptually distinct from

The practical significance of Fourier deconvolution in signal processing 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 Fourier deconvolution are shown in Figures 12 and 13.

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.

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 convolution by 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. (See the Matlab/Octave code example at the
bottom of this page). 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. You can see this happening in the example
in Figure 12. However, this is not observed in the
example in Figure 13 because in that case the noise is only
in the original signal. The high frequency components of the
denominator in the division of the Fourier transforms are
typically much larger than in the previous example, and the only
additional noise comes from numerical round-off errors in the math
computations performed by the derivative and smoothing operation,
which are almost always much smaller than the noise in the
original experimental signal.

SPECTRUM, the freeware signal-processing application that accompanies this tutorial, includes a deconvolution function.

Matlab and Octave have a built-in function for deconvolution: deconv. An example of its application is shown below: the vector yc (line 6) represents a noisy rectangular pulse (

x=0:.01:20;y=zeros(size(x));

y(900:1100)=1; % Create a rectangular function y, 200 points wide

y=y+.01.*randn(size(y)); % Noise added

c=exp(-(1:length(y))./30); % exponential trailing convolution function, c

yc=conv(y,c,'full')./sum(c); % Create exponential trailing rectangular function, yc

% yc=yc+.01.*randn(size(yc)); % Noise added

ydc=deconv(yc,c).*sum(c); % Attempt to recover y by deconvoluting c from yc

subplot(2,2,1);plot(x,y);title('original y');subplot(2,2,2);plot(x,c);title('c')

subplot(2,2,3);plot(x,yc(1:2001));title('yc');subplot(2,2,4);plot(x,ydc);title('recovered y')

This page is also available in French, at http://www.besteonderdelen.nl/blog/?p=41, courtesy of Natalie Harmann.

Revised January, 2014. 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.

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