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. (Of course, if the denominator contains any *zeros*,
the result will be a "divide-by-zero" error and the whole
operation fails). The problem of low values or zeros in the
denominator can be reduced by smoothing before convolution and by
constraining the deconvolution to a frequency region where the
denominator is sufficiently high. You can see this amplification
of high frequency noise happening in the example in Figure 12
above. However, this effect is not observed in the example in
Figure 13 because in that case the noise was present in the
original signal, *before *the convolution performed by the
spectrometer's derivative algorithm. The high frequency components
of the denominator in the division of the Fourier transforms
are typically much *larger *than in the previous example,
avoiding the noise amplification and divide-by-zero errors, and
the only post-convolution noise comes from numerical round-off
errors in the math computations performed by the derivative and
smoothing operation, which is 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')

Here is another example, shown on the left. (Download this script). In this case, the underlying signal (

xx=5:.1:65;

% Underlying signal with a single peak (Gaussian) of unknown height, position, and width.

uyy=gaussian(xx,25,10);

% Observed signal yy, with noise added AFTER the broadening convolution (ExpG)

Noise=.001;

tc=70; % Time Constant

yy=expgaussian(xx,25,10,-tc)'+Noise.*randn(size(xx));

% Guess, or use prior knowledge, or curve fit one peak to ExpG to determine

% time constant (20), then compute transfer function cc

cc=exp(-(1:length(yy))./tc);

% Attempt to recover original signal uyy by deconvoluting cc from yy

yydc=deconv([yy zeros(1,length(yy)-1)],cc).*sum(cc); % It's necessary to zero-pad the observed signal as shown here.

subplot(2,2,1);plot(xx,uyy);title('Underlying 4 Gaussian signal, uyy');

subplot(2,2,2);plot(xx,cc);title('Exponential transfer function, cc')

subplot(2,2,3);plot(xx,yy);title('observed broadened and noisy signal, yy');

subplot(2,2,4);plot(xx,yydc);title('After deconvoluting transfer function, yydc')

% Try more Noise (line 6) or a bad guess of time constant (line 11)

% plot(xx,uyy,xx,yydc) % plot recovered signal overlaid with underlying signal

% plot(xx,uyy,xx,yy) % plot observed signal overlaid with with underlying signal

% Curve fit recovered signal to a Gaussian to determine peak parameters

% [FitResults,FitError]=peakfit([xx;yydc],26,42,1,1,0,10)

% Alternatively curve fit observed signal with an exponentially broadened

% Gaussian

% [FitResults,FitError]=peakfit([xx;yy],26,50,1,5,tc,10)

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

Revised

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