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A constrained iterative deconvolution technique with an optimal filtering:

Application to a hydrocarbon concentration sensor

Article in IEEE Transactions on Instrumentation and Measurement · September 2000

DOI: 10.1109/19.863937 · Source: IEEE Xplore

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852 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 4, AUGUST 2000

A Constrained Iterative Deconvolution Technique

with an Optimal Filtering: Application to a
Hydrocarbon Concentration Sensor
Philippe Neveux, E. Sekko, and Gérard Thomas

Abstract—A deconvolution method for estimating unburned hy-

drocarbon concentration in boiler smokes is presented. In order to
qualify a boiler regarding to the ecological European Standards
an iterative constrained estimation algorithm including a filtering
step has been set up. The application of such technique to both syn-
thetic signals and experimental data has shown its robustness in Fig. 1. Distortion process.
regard to measurement noise and its reliability to restore a signal.
Index Terms—Concentration sensor, deconvolution, inverse
problem, optimal control, signal processing. In this paper, a filtering step is introduced in an iterative con-
strained deconvolution algorithm in the time domain. This kind
I. INTRODUCTION of approach has been already used in Sekko et al. [8] in a state
space framework. In contrast, the impulse response of the dis-

T HESE last years, for the sake of ecology, new constraints

have been imposed to industries, such as international
norms limiting the rejection rate of polluting smokes in the
tortion process is used in our deconvolution procedure. In this
point of view, the transfer function identification step (especially
a delay estimation procedure) is avoided. In Section II, the nota-
atmosphere [2]. In order to satisfy these constraints, one has tions and the representation used to solve the problem are given.
various possibilities. For example, in a first step, one can decide In Section III, the problem will be precisely set up and solved.
to build up a new process or modify the old one. This approach Finally, in Section IV, this new deconvolution technique is ap-
being relatively expensive, in a second step, one has to verify plied to a hydrocarbon concentration sensor.
whether the measures recorded are reliable. Then, one can see
the reliable measures as the input of a distortion process (the
data acquisition device) whose output is the recorded measures II. NOTATIONS
corrupted by noise. In such point of view, with the knowledge
of the data acquisition device, one can try to restore the reliable Let be the causal unknown signal to be restored and
measures and compare them to the norm specifications in order the noisy measured output of a distortion process sup-
to make a decision. This technique of signal restoration for posed to be linear and described by its impulse response
linear time invariant systems is referred to as deconvolution. (see Fig. 1). This distortion process is supposed to be stable and
Different processing methods for solving the deconvolution causal. Therefore, after discretization (with a sampling period
problem are available in the literature, that is to say, iterative of 1 s), the input–output relation can be written as
and noniterative techniques both exist. Introducing constraints
on the restored signal in the deconvolution algorithm implies
iterative resolution. In the recent years, numerous works have (1)
been achieved on iterative deconvolution [3]–[8], [10]–[12]. The
main idea of iterative constrained deconvolution is to give the
best estimate of an original signal satisfying constraints. Usu- where
ally, the measured signal is previously filtered in order to lessen • , , , and are the discretized counter
the level of noise; then, this filtered version is introduced in the part of , , , and , respectively;
deconvolution algorithm. Sekko et al. [9] proposed, in the fre- • and are supposed to be independent Gaussian
quency domain, to introduce the filtering step in the deconvo- white noises with variance and , respectively. In ad-
lution algorithm. The main drawback of this approach is that it dition, the signals , and are supposed to be mutually
forbids the introduction of level constraints. uncorrelated.
Therefore, after recorded samples, with , we
have the matrix relation
Manuscript received April 1, 1999; revised March 14, 2000.
The authors are with the Université Claude Bernard Lyon I, LAGEP, U.M.R.
CNRS 5007, 69622 Villeurbanne cedex, France (e-mail: neveux@lagep.univ- (2)
lyon1.fr).
Publisher Item Identifier S 0018-9456(00)06349-X. (3)
0018–9456/00$10.00 © 2000 IEEE
NEVEUX et al.: CONSTRAINED ITERATIVE DECONVOLUTION TECHNIQUE 853

Fig. 2. Wiener block diagram for deconvolution.

where

.. .. ..
. . .
.. .. .. .. Fig. 3. Optimal linear filtering block diagram.
. . . .
.. .. ..
. . . The minimization of criterion (6) leads to the following expres-
.. .. .. .. sions for the gain matrices and :
. . . .
.. .. .. ..
. . . . with (8)

(4) and gives rise to the block diagram in Fig. 3.

Proof: The criterion (6) can be rewritten just as follows:
A deconvolution technique combining optimal filtering and
classical regularization techniques has now to be built up. (9)
The innovation which corresponds to the error of estimation is
III. THE DECONVOLUTION PROCEDURE defined as
Let be the transfer function of the discretized distor- (10)
tion process.
Then, consider the Wiener configuration in the stationary case Using (2), (7), and (10), one obtains the expression
for deconvolution [5] described formally by Fig. 2.
(11)
The filter is given by
(see (12) at the bottom of the next page). The estimate should
(5) be unbiased, then it turns out that

(13)
, and are the power spectrum density of the signals
, , and , respectively. Then, from the initial assumptions, the expression of is easily
In this technique, a strong assumption on the signal to be re- obtained
stored is made. As a matter of fact, this signal has to be a second-
order random process. However, this hypothesis is not realistic. (14)
Indeed, this property is not verified in reality because of the Introducing (16) in (12), one obtains the new expression of the
deterministic nature of measurement signals. Consequently, as criterion
measurement signals are deterministic, the incorporated optimal
filtering of this deconvolution method has to be redesigned.
(15)
A. Optimal Filter Design
The optimal value of is such that
In this section, the signal is supposed to be known. Then,
from the knowledge of , and the variance of the noises, (16)
it is possible to define an optimal filter. A filter is said to be
optimal if its gains minimize a specified criterion representing From (15) and (16), the following expression is obtained:
the fidelity of the filtered signal to the ideal one. As a matter of
fact, filtering is an operation used to reduce the noise component (17)
of a measured signal. Hence, the following criterion is defined: Therefore, one can extract the expression of
(6)
(18)
where the structure of the searched optimal linear filter is However, by definition
(7) (19)
854 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 4, AUGUST 2000

Fig. 4. Modified Wiener method.

Then, the final expression of is obtained

with (20)

This completes the proof.

B. Optimal Estimate of
Fig. 5. Data acquisition device.
After designing an optimal filter in order to lessen the level of
measurement noise, an algorithm giving the best estimate of the
signal has to be designed. In order to use the optimal linear
filtering, presented in the previous section, in an input restora-
tion algorithm, the following technique is proposed. As shown
in Fig. 4, in the filtering step, the unknown input signal is
replaced by its estimate.
As it has been shown in [9], using this scheme, the optimal
estimate of is such that
(21)

where is used to penalize the high-frequency spectrum of

which results in a smoothing constraint on the restored signal;
and is the regularization parameter. The optimal estimate of
is given by [9]
(22)
In the deconvolution algorithms, one can choose to take into ac-
count or not a priori information he holds on the signal to be
restored. In many applications, the magnitude of the original Fig. 6. Unburned C H measured concentration in ppm.
signal is known to be bounded by an inferior and/or superior
value. In order to introduce this knowledge on the signal, the
equation (22) will be solved with an iterative method (the con- emitted by a boiler during its cracking. The measured signal is
jugate gradient [5]) and the solution will then be projected onto distorted by the measurement system (see Fig. 4). The aim of
the constraint set. the deconvolution is to reduce the distortion effect and to have
Remark: It has to be outlined that the solution to be found access to the polluting agents instantaneous concentration.
through this algorithm will be a suboptimal solution. An optimal A boiler is qualified to the European Standards of Safety if
one could be found by introducing the projection operator in the instantaneous concentration of unburned is inferior
criterion (21). to 10 ppm, 20 s after its cracking. From Fig. 5, it is very difficult
to locate the cracking time; thus, it is hardly possible to say
IV. APPLICATION TO ESTIMATE POLLUTING AGENTS whether the boiler will conform or not. Consequently, signal
processing on these data should be performed.
A. Description of the Measurement Process As it is shown in Fig. 6, the nominal value of the unburned
The proposed method is applied to the reconstruction of the concentration is about 5.5 ppm. In order to apply a pos-
instantaneous polluting agents (unburned ) concentration itivity constraint, this nominal concentration has been

(12)
NEVEUX et al.: CONSTRAINED ITERATIVE DECONVOLUTION TECHNIQUE 855

Fig. 7. Synthetic input (dotted line) and output for SNR = 5 dB (full line).
subtracted to the present signal. In the algorithm implemen-
tation, the following positivity constraint has been considered
[11]:
Let a strictly positive real, arbitrarily small then:
• if then
• if then
• if then
It should be outlined that any other continuously differentiable
projection operator (onto a strictly positive set) can be used in
our algorithm instead of the one proposed in this work.
Remark: The experimental data displayed in Fig. 6 are cor-
rected ones. As a matter of fact a natural delay exists in the
process that has been canceled for the processing.
In order to show the improvement obtained by this approach,
it will be first applied to a synthetic signal. Then, in a second Fig. 8. (a) Deconvolution with filtering and without filtering; (b) original
signal (dotted line), restored signal (full line).
step, the case of real data will be treated.

B. Simulations with Synthetic Signals

From Fig. 8, it is clear that introducing a filtering step in the
restoration algorithm enhances the restoration quality. First, it
reduces artifacts amplitude and secondly, as a consequence, it
provides a much more reliable input estimate. As a matter of
fact, the relative mean square error (rmse) with the proposed
approach is 0.0103 whereas it is 0.0444 without the filtering
step.The synthetic signal considered is shown in Fig. 7 together
with the system response. It can be seen that this signal is pos-
itive. In consequence, the developed method will be applied to
this case for a Signal-to-Noise Ratio (SNR) of 5 dB. The results
obtained for deconvolution with and without filtering are shown
in Fig. 8.
Remark: The rmse is defined as the ratio

Fig. 9. Estimation of the unburned C H concentration in ppm without

filtering and with positivity constraint.
C. Experimental Data
Having shown the ability of the present algorithm to work estimation is treated. From Fig. 6, one can easily see that de-
with noisy data, the problem of unburned concentration termining cracking time and norm adequacy from the recorded
856 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 4, AUGUST 2000

REFERENCES
[1] M. Athans and P. L. Falb, Optimal Control. New York: McGraw-Hill,
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[3] D. Commenges, “The deconvolution problem fast-algorithms including
the preconditionned conjugate-gradient to compute a MAP estimator,”
IEEE Trans. Automat. Contr., vol. AC-29, no. 3, pp. 229–243, 1984.
[4] Y. Ichioka, Y. Takubo, K. Matsuoka, and T. Suzuki, “Iterative image
restoration by a method of steepest descent,” J. Opt. Paris), vol. 12, no.
1, pp. 35–41, 1981.
[5] A. K. Jain, Fundamentals of Digital Image Processing, ser.
NJ. Englewood Cliffs: Prentice-Hall, 1989.
[6] P. A. Janson, Deconvolution With Application to Spectroscopy. New
York: Academic, 1984.
[7] R. Prost and R. Goutte, “Discrete constrained iterative deconvolution
algorithms with optimized rate if convergence,” Signal Process., vol. 7,
pp. 209–230, 1984.
[8] E. Sekko, P. Sarri, and G. Thomas, “Robust constraint deconvolution,”
in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, vol.
3, Atlanta, GA, 1996, pp. 1796–1798.
Fig. 10. Estimation of the unburned C H concentration in ppm with [9] E. Sekko and G. Thomas, “A deconvolution technique using optimal
filtering and positivity constraint Wiener filtering and regularization,” Signal Process., vol. 72.
[10] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Prob-
lems. New York: Halsted, 1977.
signal is a very difficult task. Consequently, the recorded signal [11] G. Thomas and R. Prost, “Iterative constrained deconvolution,” Signal
has to be treated in order to obtain a workable information. Process., vol. 23, pp. 89–98, 1991.
[12] J. K. Tugnait, “Constrained signal restoration via iterative extended
The solution given by the constrained deconvolution without fil- Kalman filtering,” IEEE Trans. Acoust., Speech, Signal Processing,
tering is given in Fig. 9. It permits to determine the cracking time vol. ASSP–33, pp. 472–475, 1985.
in the boiler ( s) but unfortunately because of artifacts, the
restored signal cannot allow us to judge the boiler adequacy to
the norm.
Philippe Neveux was born in 1970. He received the
Applying the proposed technique to the same data provides Thèse de Doctorat degree in automatics and signal
the estimate given in Fig. 10. It can be seen that both cracking processing from the Université de Lyon in 2000.
time and boiler adequacy can be stated. As a matter of fact, Since 1996, he has been with the Laboratoire
d’Automatique et de Génie des Procédé (LAGEP),
boiler cracking can be determined due to slope shift at s, Lyon, France. His main research activities are in the
and as the artifact magnitude is very low, one can easily see field of deconvolution, minimax optimization and
(by zooming on the region of interest) that the polluting agents robust filtering.
concentration is under 10 ppm after 23 s. This means that 19 s
after the boiler cracking the polluting agents concentration is
under 10 ppm. Consequently, the boiler is declared to conform
to the European Standards of Safety.
E. Sekko was born in Cotonou, Benin, in 1971. He re-
V. CONCLUSION ceived the M.Sc. and the doctoral degree in electrical
engineering from Université Claude Bernard, Lyon,
A new constrained deconvolution technique incorporating France, in 1993 and 1998, respectively.
optimal filtering in a level constrained restoration algorithm Since 1999, he has been with the Technology Insti-
tute of Engineering Science at Université d’Orleans,
has been proposed. The distortion process has been modeled by France, as an Assistant Professor. His research inter-
its impulse response, avoiding a transfer function identification ests include linear optimal H =H filtering and de-
step. In a first step, an optimal filter has been designed. Then, convolution, fractal structure, prediction of rolling el-
ement. bearing’s failure.
the combination of filtering and constrained deconvolution
has been performed. This inverse filter has been applied to a
problem of polluting agents concentration estimation. In a first
step, the noise robustness of the proposed technique has been
Gérard Thomas was born in 1947. He graduated
shown using synthetic signals. And, in a second step, it has from Ecole Centrale de Lyon, France, in 1971
shown its ability to deal with real data. and received the Doctorat d’Etat degree in applied
mathematics from the Université de Lyon in 1981.
Since 1988, he has been with the Laboratoire
ACKNOWLEDGMENT d’Automatique et de Génie des Procédé (LAGEP),
Lyon, France. His main research activities are in the
The authors would like to thank the CETIAT (Centre Tech- field of Signal Processing (deconvolution).
nique des Industries Aérauliques et Thermiques) for the pro-
vided experimental data.

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