[Introduction]
[Signal
arithmetic] [Signals
and
noise] [Smoothing]
[Differentiation]
[Peak
Sharpening] [Harmonic
analysis] [Fourier
convolution] [Fourier
deconvolution] [Fourier
filter] [Peak
area
measurement] [Linear
Least
Squares] [Multicomponent
Spectroscopy] [Iterative
Curve
Fitting] [Hyperlinear
quantitative
absorption spectrophotometry] [Appendix
and
Case Studies] [Peak
Finding
and Measurement] [iPeak]
[iSignal]
[Peak
Fitters] [iFilter]
[iPower]
[List
of
downloadable software] [Interactive
tools]

[Applications] [Matlab/Octave]

Fourier deconvolution is
the converse of Fourier convolution
in the sense that division is the converse of multiplication. If
you know that * m* times

In practice, the deconvolution of one signal from another is usually performed by point-by-point

^{}

by the rules for the division
of
complex numbers. Many computer languages will perform this
operation automatically when the two quantities divided are
complex.

**Note:** The word "deconvolution"
can have two meanings, which can lead to confusion. The Oxford
dictionary defines it as "A process of resolving something into
its constituent elements or removing complication in order to
clarify it", which in one sense applies to Fourier deconvolution.
But 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 proposed peak model to the
data set. However, that process is actually conceptually distinct
from *Fourier* deconvolution, because in Fourier
deconvolution, the underlying peak shape is unknown but the
broadening function is assumed to be known; whereas in iterative
least-squares curve fitting it's just the reverse: the peak shape
must be known but the width of the broadening process, which
determines the width and shape of the peaks in the recorded data,
is unknown. Thus the term "spectral deconvolution" is ambiguous:
it might mean the Fourier deconvolution of a response function
from a spectrum, or it might mean the decomposing of a spectrum
into its separate additive peak components. These are different
processes; don't get them confused.

The practical
significance of Fourier deconvolution in signal processing is that
it can be used as a 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. In some cases the
physical convolution can be measured experimentally by applying a
single spike impulse ("delta") function to the input of the
system, then that data used as a deconvolution vector.
Deconvolution can also be used to determine the form of a
convolution operation that has been previously applied to a
signal, by deconvoluting the original and the convoluted signals.
These two types of application of Fourier deconvolution are shown
in the two figures below.

**Fourier 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)
must be known and 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 compared to the recorded signal, because
the Fourier deconvolution operation is simply recovering the
original signal before the low-pass filtering, noise and all. (Matlab/Octave script)*

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

**A different application of Fourier deconvolution***
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, the figure in the top left
is a uv-visible absorption spectrum recorded from a commercial
photodiode array spectrometer (X-axis: nanometers; Y-axis:
milliabsorbance). The figure in the top right is the first derivative of this
spectrum produced by an (unknown) algorithm in the software
supplied with the spectrometer. The objective here is to
understand the nature of the differentiation/smoothing
algorithm that the instrument's software uses. 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 Fourier
deconvolution to experimental data, for example to remove the
effect of a known broadening or low-pass filter operator caused by
the experimental system, there are four serious problems that
limit the utility of the method:

(1) the convolution occurring in the physical domain might not be accurately modeled by a mathematical convolution;

(2) the width of the convolution - for example the time constant of a low-pass filter operator or the shape and width of a spectrometer slit function - must be known, or at least adjusted by the user to get the best results;

(3) a serious signal-to-noise degradation commonly occurs; any noise added to the signal by the system

afterthe 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);

(4) if the denominator vector contains

even a single zero value, 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 Fourier deconvolution to a frequency region where the denominator is sufficiently high.

You can see the amplification of high frequency noise happening
in the example in the first example above. However, this effect is
*not *observed in the second example, 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.

In practice, the width of the physical convolution is seldom known exactly, so the deconvolution must be adjusted empirically to yield the best results. Similarly, the width of the final smooth operation must also be adjusted for best results. The results will seldom be perfect, but are often an improvement over the original.

As a method for *peak sharpening*, deconvolution can be
compared to the methods
described earlier, in particular the much simpler power method, in
which the raw signal is simply raised to some positive power *n.*
The Matlab/Octave script PowerMethod.m
compares these two methods for a pair of overlapping peaks,
showing that power method (with *n*=5) is capable of
reducing the peak width and increasing the separation-to-width
ratio better than deconvolution, but it does not restore the
original peak positions, heights, and widths as does the
deconvolution method (graphic).

SPECTRUM, the freeware signal-processing application for Mac OS8 and earlier, includes a Fourier deconvolution function.

**Matlab**
and **Octave** have a built-in function for Fourier
deconvolution: *deconv*. An example
of its application is shown below: the vector *yc* (line 6)
represents a noisy rectangular pulse (*y*) convoluted with a
transfer function *c* before being measured. In line 7, *c*
is deconvoluted from *yc*, in an attempt to recover the
original *y*. This requires that the transfer function *c*
be known. The rectangular signal pulse is recovered in the lower
right (*ydc*), complete with the noise that was present in
the original signal. The Fourier deconvolution reverses not
only the signal-distorting effect of the convolution by the
exponential function, but also its low-pass noise-filtering
effect. As explained above, there is significant amplification of
any noise that is added *after* the convolution by the
transfer function (line 5). This script demonstrates that there is
a big difference between noise added *before* the
convolution (line 3), which is recovered unmodified by the Fourier
deconvolution along with the signal, and noise added *after*
the convolution (line 6), which is amplified compared to that in
the original signal. Execution time: 0.03 seconds in Matlab; 0.3
seconds in Octave. Download this
script.

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 **before **the convolution

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
**after **the convolution

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

% deconvoluting c from yc

`% Plot all the steps`

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')

Alternatively, you could perform the Fourier deconvolution
yourself *without *using
the built-in Matlab/Octave "deconv" function by dividing the
Fourier transforms of *yc *and *c* using the built-in
Matlab/Octave "fft.m" function and inverse transform the result
with the built-in Matlab/Octave "ifft.m" function. Note that *c*
must be zero-filled
to match the size of *yc*. The results are essentially the
same (except for the numerical floating point precision of the
computer, which is usually negligible), and it's actually ten
times faster than using the deconv function:

`ydc=ifft(fft(yc)./fft([c
zeros(1,2000)])).*sum(c);
`

The script DeconvDemo3.m is similar to the above, except that it demonstrates

DeconvDemo4.m (on the right) shows a Gaussian deconvoluted from a Gaussian function and an attempt to recover the original peak width. This is an example of "self deconvolution", so-called because the shape of the deconvolution function is the same as the shape of the peaks in the signal, in this case both

In all of the above simulations, the deconvolution method works as well as it does because the signal-to-noise ratio of the "observed signal" (upper right quadrant) is quite good; the noise is not even visible on the scale presented here. In the absence of any knowledge of the width of the deconvolution function, finding the right deconvolution width depends upon experimentally minimizing the wiggles that are seen when the deconvolution width is incorrect, and a poor signal-to-noise ratio will make this much more difficult. Of course smoothing can reduce noise, especially high-frequency (blue) noise, but smoothing also slightly increases the width of peaks, which works counter to the point of deconvolution, so it must not be over used. The image on the left shows the widths of the peaks (as full width at half maximum); the width of the deconvoluted peaks (lower right quadrant) are only slightly larger than in the (unobserved) underlying peaks (upper left quadrant) either because of imperfect deconvolution or the broadening effects of the smoothing needed to reduce the high frequency noise. As a rough but practical rule of thumb, if there is any

Here is another example, shown below on the left. (Download this script). 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);

% Compute observed signal yy, using the expgaussian function with time

% constant tc, adding noise added AFTER the broadening convolution (ExpG)

Noise=.001; % <<<< Change the noise here

tc=70;

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

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

% determine time constant (tc), then compute transfer function cc

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

% Use "deconv" to recover original signal uyy by deconvoluting cc

yydc=deconv([yy zeros(1,length(yy)-1)],cc).*sum(cc);

% Plot the signals and results in 4 quadrants

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(xx,uyy,xx,yy)

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

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

% Or curve fit observed signal with an exponentially broadened Gaussian

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

An alternative to the above deconvolution approach is to use iterative curve fitting (which is covered in a further section) to fit the observed signal directly with an exponentially broadened Gaussian (shape number 5)

>> [FitResults,FitError]=peakfit([xx;yy], 26, 50, 1, 5, 70, 10)

Both methods give good values of the peak parameters, but the Fourier deconvolution method is faster, because fitting the deconvoluted signal with a simple Gaussian model is faster than iteratively curve fitting the observed signal with a exponentially broadened Gaussian model. Also, if the exponential factor "tc" is not known, it can be determined by iterative curve fitting, adjusting the exponential factor ('extra') interactively to get the best fit:

Alternatively, you can use peakfit.m with the constrained variable exponentially broadened Gaussian (shape 31), which will automatically find the best value of "tc", but in that case the best results will be obtained if you give it a rough starting guess ("start") at least within a factor of two or so of the correct values:

FitResults =

1 25.006 0.99828 10.013 10.599 69.83

GoodnessOfFit =

0.15575 0.99998

tstep=(endt-startt)/NumSegments;

tc=startt:tstep:endt;

**iSignal **version 5.7 has a **Shift-V**
keypress that displays the menu of Fourier convolution and
deconvolution operations that allow you to convolute a Gaussian or
exponential function with the signal, or to deconvolute a Gaussian
or exponential function from the signal, and asks you for the
width or the time constant (in X units).

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

Revised June, 2017. This page is part
of "**A
Pragmatic Introduction to Signal Processing**", created
and 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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