absorption spectrophotometry] [Appendix
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[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 x equals
n, where m and n are known
but x is unknown, then x equals n
divided by m. Conversely if you know that m
convoluted with x equals n,
where m and n are known but x
is unknown, then x equals m deconvoluted
In practice, the deconvolution of one signal from another is usually performed by point-by-point division of the two signals in the Fourier domain, that is, dividing the Fourier transforms of the two signals point-by-point and then inverse-transforming the result. Fourier transforms are usually expressed in terms of complex numbers, with real and imaginary parts representing the sine and cosine parts. If the Fourier transform of the first signal is a + ib, and the Fourier transform of the second signal is c + id, then the ratio of the two Fourier transforms is
by the rules for the division
complex numbers. Many computer languages will perform this
operation automatically when the two quantities divided are
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 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, 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.
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
% Create a rectangular function y, 200 points wide
Noise added before
% exponential trailing convolution function, c
% Create exponential trailing rectangular
% function, yc
Optional noise added after the convolution
% 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 in separate subplots
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);
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.
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,
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
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
Parameters of the three deconvoluted peaks
Peak Height FWHM Area
1.0000 5.6963 0.9624 5.9684
2.0000 7.6741 1.0711 9.3350
3.0000 2.7056 0.8623 3.0578
In , for Matlab only, the downloadable interactive multipurpose signal processing Matlab function, you can press Shift-V to display the that allow you to convolute or to deconvolute a Gaussian, Lorentzian or exponential function. It will ask you for the initial width or time constant of the deconvolution function (in X units), then you can use the 3 and 4 keys to decrease or increase the width by 10% (or Shift-3 and Shift-4 to adjust by 1%). This version of iSignal includes an additional way to reduce ringing and noise in the deconvoluted signal, by adding a constant to the denominator (reference 86) and adjusting it with the 5 and 6 keys to decrease or increase the constant by 10% (or Shift-5 and Shift-6 to adjust by 1%). Here's an application to a real experimental signal:
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 August, 2021. 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 firstname.lastname@example.org.
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