[Introduction]
[Signal
arithmetic] [Signals
and
noise] [Smoothing]
[Differentiation]
[Peak
Sharpening] [Harmonic
analysis] [Fourier
convolution] [Fourier
deconvolution] [Fourier
filter] [Wavelets]
[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] [Self deconvolution] [Multiple sequential deconvolution] [Segmented deconvolution] [convdeconv function] [Interactive deconvolution]

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 degradationcommonly occurs; any noise added to the signal by the systemafterthe 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 (thedenominatorin the division of the Fourier transforms) are typically very small, with some individual components often of the order of 10^{-12}or 10^{-15}, resulting a huge amplification of those particular frequencies in the resulting deconvoluted signal, which is called "ringing". (See the Matlab/Octave code example at the bottom of this page). The problem can be reduced either by low-pass filtering (smoothing) or even more simply by adding a small positive non-zero constant to the denominator, which increases the excessively small high-frequency members in the denominator without significantly increasing the much greater low-frequency members (reference 85). Smoothing or filtering reduces the amplitude of the highest-frequency components, and denominator addition reduces the amplitude of the frequencies that are the most highly amplified by deconvolution. Both methods can have a similar effect, but they work in different ways and can sometimes be more effective when used together rather than separately.

You can see the amplification of high frequency noise happening
in the example in the first example above. On the other hand, 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 many cases, the width of the physical convolution is not 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 result will
seldom be perfect, especially if the original signal is noisy, but
it is often a better approximation to the real underlying signal
than the recorded data without deconvolution.

As a method for *peak sharpening*, deconvolution can be
compared to the derivative
peak sharpening method described earlier or to the power method, in
which the raw signal is simply raised to some positive power *n.*

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

% The sum(c2) is included simply to scale the amplitude of the
result to match the original y.

% 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')

Click here for
a simple explicit example of Fourier convolution and
deconvolution, for a small 9-element vector, with the vectors
printed out at each stage.

DeconvDemo4.m (on the right) shows a Gaussian deconvoluted from a Gaussian function and an attempt to recover the original peak width. Typically this would be applied to a signal containing multiple overlapping peaks, in an attempt to sharpen the peaks to improve the resolution. Note that in this example the deconvolution width must be within 1% of the deconvolution width. In general, the wider the physical convolution width relative to the signal, the more accurately the deconvolution width must be matched the physical convolution width.

DeconvDemo5.m (shown on the left) shows an example with

In the example, shown below on the right, the underlying signal (uyy) is a Gaussian, but in the observed signal (yy) the peak is broadened exponentially resulting in a shifted, shorter, and wider peak, and then a little constant white noise is added after the broadening convolution (cc). Assuming that the exponential broadening time constant ('tc') is known, or can be guessed or measured, the Fourier deconvolution of cc from yy successfully removes the broadening (yydc), and restores the original height, position, and width of the underlying Gaussian, but at the expense of considerable noise increase. However, the noise remaining in the deconvoluted signal is "blue" (high-frequency weighted) and so is easily reduced by smoothing and has less effect on least-square fits than does white noise. (For a greater challenge, try more noise in line 6 or a bad guess of time constant ('tc') in line 7). To plot the recovered signal overlaid with underlying signal: plot(xx,uyy,xx,yydc). To plot the observed signal overlaid with with underlying signal:

plot(xx,uyy,xx,yy).

To curve fit the recovered signal to a Gaussian to determine peak parameters: [FitResults,FitError]=peakfit([xx;yydc],26,42,1,1,0,10), which yields excellent values for the original peak positions, heights, and widths. You can demonstrate to yourself that with ten times the previous noise level (Noise=.01 in line 6), the values of peak parameters determined by curve fitting are still quite good, and even with 100x more noise (Noise=.1 in line 6) the peak parameters are more accurate than you might expect for that amount of noise (because that noise is blue). Remember, there is no need to smooth the results of the Fourier deconvolution before curve fitting, as seen previously.

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):

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 the more complicated exponentially broadened Gaussian model. If the exponential factor "tc" is not known, it can be determined by iterative curve fitting, manually adjusting the exponential factor ('extra') interactively to get the best fit:

which in this particular case gives a best fit when the exponential factor "tc" is adjusted to about 69.9 (very close the correct value of 70 in this simulation).

Alternatively, you can use peakfit.m with the unconstrained 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:

The value of the exponential factor determined by this method is 69.8, again close to 70. However, if the signal is very noisy, there will be quite a bit of uncertainty in the value of the exponential factor so determined - for example, the value will vary a bit if slightly different regions of the signal are selected for measurement (e.g. by panning or zooming in ipf.m or by changing the center and window arguments in peakfit.m). See Appendix H: Fourier deconvolution vs curve fitting for another example with four overlapping Gaussians.

For example, the figure on the right (Matlab script) shows the application of self deconvolution of two overlapping Lorentzian peaks with a narrower Lorentzian function, using a

The Matlab script also displays the Fourier spectra of those four signals in their corresponding colors, in figure window 2, shown on the left. Observing these spectra can be a useful guide to adjusting the deconvolution width and the noise/ringing reduction settings. Looking at the spectrum of the original raw data (black) you can see two distinct frequency regions:

(a) below a frequency of 10

(b) above 10

In order to sharpen the peaks, we need to raise the amplitude of the high-frequency components of the signal up to 10

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" 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. As a rough but practical rule of thumb, if there is any

tstep=(endt-startt)/NumSegments;

tc=startt:tstep:endt;In cases where the signal may have been subject to two or more sequential convolutions, as described in the previous section, the reversal of that effect requires multiple sequential deconvolutions and cannot be undone

**Combination
function.** The Matlab/Octave function P=convdeconv(x,y,vmode,smode,vwidth,DAdd)
performs Gaussian, Lorentzian, or exponential convolution and
deconvolution of the signal in x,y. Set vmode=1 for convolution,
2 for deconvolution, smode=1 for Gaussian, 2 for Lorentzian, 3
for exponential; vwidth is the width of the convolution or
deconvolution function, and DAdd is the constant denominator
addition used to control ringing and noise resulting from
deconvolution. For examples of the operation of this function,
see ThreeGaussianSignalTrial.m
and ThreeLorentzianSignalTrial.m.
You can clearly see the substantial increase in peak
height and decrease in peak width resulting
from deconvolution. Although the peak heights are increased (by
deconvolution) and the peak areas are slightly reduced (by the
denominator addition), that will not be a problem in quantitative
analysis by calibration curves as long as the same
deconvolution setting are used for all samples and standards. The random noise is also increased;
nevertheless, the peak heights, positions, and widths can still
be measured precisely by least squares methods. The noise can be reduced by applying a
little subsequent smoothing.
That leaves the question of which values of vwidth and DAdd to
use. Start with a value of vwidth somewhat smaller than the
estimated width of the peaks in the signal, with DAdd set to
zero or some small number like 0.001%, then try larger values of
vwidth
to increase the peak sharpening. If noise and ringing
starts to obscure the signal, try larger values of DAdd.
Excessively large values of either can distort the peaks.

`-----------------------------------------`

`Parameters of the three Lorentzian
peaks`

`
Peak Height
FWHM Area`

`
1.0000 2.0000
2.0000 6.0929`

`
2.0000 3.0000
2.2000 10.0594`

`
3.0000 1.0000
2.2000 3.3407
`

In this example, the original signal
is shown as the dotted green line and the results of
deconvoluting it with a *Lorentzian*
deconvolution function is shown as the blue line. The
deconvolution width was adjusted as large as possible without
causing significant negative dips between the peaks, which for
many types of experimental data, would be non-physical. (Recall
that the mathematics of the deconvolution operation is
structured so that the *area
under the peaks remains unchanged*, even though the widths
are reduced and the heights are increased). Several of the peaks
shown in the zoomed-in close-up in the upper panel have
shoulders that are resolved into distinct peaks, allowing their
peak positions to be measured more accurately. Fortunately, the
amplitude of those revealed peaks is greater than the small
amount of noise remaining in the signal (thanks to the good
signal-to-noise ratio of the original signal).

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

Revised November, 2020. 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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