Int. J. Communications, Network and System Sciences, 2011, 4, 297-303
doi:10.4236/ijcns.2011.45034 Published Online May 2011 (
Copyright © 2011 SciRes. IJCNS
Fuzzy Digital Filtering: Signal Interpretation
Juan Carlos García Infante1, José de Jesús Medel Juárez2, Juan Carlos Sánchez García1
1Professional School of Mechanical and Electrical Engineering, National Polytechnic Institute, Coyoacan, México
2Centre of Computing Research, National Polytechnic Institu te, Vallejo, México
Received February 15, 2011; r ev is e d March 2, 2011; accepted March 27, 2011
The paper makes a description of the fuzzy filter properties considering its operational principles. A digital
filter interacts with a reference model signal into real process in order to get the best corresponding answer,
having the minimum error at the filter output using the mean square criterion. Adding into this filter structure
a fuzzy mechanism, to obtain an intelligent filtering because adaptively select and emit a decision answer
according with the external reference signal changes, in order to actualize the best correct new conditions
updating a process dynamically. The interpretation of the input signal level describes the operation of the
reference model, to update the filter weights giving the answers approximation in accordance with the refer-
ence signal in natural form. Finally the paper shows the simulations results of the fuzzy filter into the Kal-
man structure using the Matlab© tool.
Keywords: Digital Filters, Fuzzy Systems, Signal Processing, Estimation, Probability
1. Introduction
To develop an intelligent system requires a deduction of
its own natural external environment, using a fuzzy me-
chanism that has the classification of its own set of an-
swers in order to discover dynamically the changes of an
external process. Interpreting the operation levels to de-
duce and select the best response updating its answers
weights, in order to follow the best approximation cond i-
tion in accordance to the external process changes by
taking a decision about the actual condition to correct or
update a system [1].
Actually, one of the best tools used to solve this, is a
recursive digital filter to adjust dynamically its parame-
ters weights in order to give limited answers with respect
to the mean square error criterion (1). A digital filter is a
logic algorithm programmed into computer software or
electronic chip in order to eliminate the noise of a system,
takes specific data, identification system or predict a
system [2].
The main problem in conventional filtering operation
is that it can’t characterizes and infer its operation levels
with respect with a reference system changes; inter-
preting its external environment changes in order to se-
lect its answers with the smallest error and considering
different operation levels in dynamical sense, following
the natural evolution of the reference system that inter-
acts with the filter. In addition, a conventional filter has
not a linguistic descriptio n as levels to display its answer
conditions as human interface, expressing it as levels
rank, making difficult to know the corresponding answer
level as non-intelligent process, which have some prob-
lems to develop capacities with high changing processes
The systems related to artificial intelligent mecha-
nisms should use into its architecture fuzzy tools (as
fuzzy neural net and evolutive systems) in order to get its
own perception giving the best decision answers, using
this to solve complex problems, actualizing an d adapting
its perceptions and answers in accordance with a refer-
ence dynamical system in order to get knowledge [3,4].
The fuzzy intelligent tools into a digital filtering is an
option for to obtain different decision answer levels, to
interacts with a reference model dynamics, adapting its
answers to the possible changes by selecting the best
values in order to get the necessary convergence condi-
tions, which should has the best operation each time [5]
[6]. The goal of this kind of filter is the characterization
of a system that has uncertainties in its operation, de-
scribing the natural process, with a rule set. This requires
a feedback law in order to follow the basic properties of
a desired input signal, adjusting its parameters to give a
correct solution dynamically to minimize the error crite-
rion response [7]. According to this, the filter chooses the
best decision answer to update the system.
In accordance with this perspective, the paper de-
scribes the FDF (Fuzzy Digital Filter) operation and the
characterization of the Kalman filter internal structure to
give answers with respect to the operation levels in order
to follow the natural reference model [8].
2. Fuzzy Digital Filtering
The fuzzy filter is an intelligent filter that has two stages
basically; the conventional adaptive filter structure con-
sidering all its conditions, and the fuzzy stage integrated
to the filter structure in order to characterize its op eration
to get a best signal approximation by levels. This stages
integration works together minimizing the error criterion
difference having a natural signal description in accor-
dance with the reference model changes [9].
In another hand, a digital filter structure without this
fuzzy stage cannot have a characterization of an external
process to have a normal operation without get an answer
into levels, making difficult to select the best signal ap-
proximation with the minimum error. Having this is dif-
ficult to interpret the input signal of the external process
and to get the best parameter value in order to update the
process with the nearest value to have a better natural
signal descri pt i on [10].
According with this a fuzzy digital filter is an adaptive
filter adding a fuzzy stage that classifies the input signal
k, into fuzzy grades to select the best filter parame-
ter value
ak from the knowledge base in order to
update the filter weights dynamically trough time to get
the best signal approximation respect to the operation
level. A fuzzy filter classifies, search and associate in-
formation giving the corresponding answer value ac-
cording to the desired signal from the reference process
at its input. This classifies a reference process in order to
get a dynamical filter answers. The goal is to give a de-
sired operation condition each time with stable op eration
because this filter uses as limit answer value the mean
square error criterion (1) as adaptive filtering, but the
fuzzy stage minimize the error difference and gets the
best signal approximation. The membership functions
(parameters value set) are limited into the knowledge
base [11].
 
JkE ek (1)
The criterion
k to reduce the filter error de-
scribes the mean square of the error between the desired
k and the filter output
k, allows finding
the corresponding membership function that is the best
signal approximation in order to minimize the filter cri-
terion [12].
The Fuzzy filter with adaptive properties has an itera-
tive searching methodology using the back-propagation
(BP) learning algorithm, which updates its parameters
per iteration dynamically by degrees classifying its
membership functions into the knowledge base in accor-
dance to the difference between the desired response
k and the actual filter output
k described as
the error
ek. The Figure 1 shows the fuzzy filter
The FDF interacts with an external reference model
(real process) changes and the filter will selects the best
answer to approximate the signal with minimum error.
Then, the fuzzy filter has next stages [13]:
The fuzzy filter previou sly has the classification of
the reference process model conditions as opera-
tion levels; having the knowledge of all the possi-
ble levels.
It interprets the desired signal
k operation lev-
els with the error value
ek in prob abilistic form
using the inference mechanism (with the logic
connector if).
It selects a corresponding membership function
ak (parameter value) from the knowledge base
(using the logic connector then) according to the
error level
The parameter value
ak selected, updates the
filter operation, adapting automatically its weigh
values to give a correct answer
k, min imizing
the least mean square criterion, with the best ap-
proximation value of the desired signal
Figure 1. Fuzzy filter operation description.
Copyright © 2011 SciRes. IJCNS
The filter emits the best answer
k, describing
its responses as linguistic operational levels and
continually repeats this cycle.
This computational model described as intelligent
process may interacts with natural processes as biologic
signals, using processing elements making connections
to constitute an automatic updating selection to follow
the best condition of a process.
The control area
T is a set of answers that the filter
has with respect to all set of desired signals that the ref-
erence system emit; the control area (2) is into member-
ship intervals inside the knowledge base; in accordance
to a filtering criterion (mean square error). The set of
membership function (parameters value) represents all
the correct responses into the Knowledge Base (KB),
according to an objective law, predefined by this natural
reference process, the filter inference chooses the best
correct response from the knowledge base to each refer-
ence model change.
 
The desired signal
k at the filter input is chang-
ing its operation conditions continuously, described in
levels. The filter interprets the corresponding level inter-
val and selects the best parameter value
ak (mem-
bership function) from the knowledge base. Updating the
filter operation to give a correct response
k, the goa l
of the filtering process as predictor is to follow the de-
sired signal
k to describe the reference process into
operation grades, and then the output filter
k will
be approximately the same signal [14]. The description
of the signal into a fuzzy structure is using ranks limited
by the reference system in order to characterize the sig-
nal levels as:
First using the logic connector if to get the member-
ship grade of the input
k, second using the logic
connector then to get the corresponding parameter value
, third the fuzzy mechanism updates the process to
a new condition, and gets the output value described as
k with the best approxi mation.
This filtering process determines the best value to up-
date de filter conditions and give a response with the
smallest error.
3. Fuzzy Filtering Properties
Considering the operational characteristics of a digital
filtering, the fuzzy filter has the same principles as ex-
ample: the mean square error criterion. Below there is a
description of the main operation principles of the fuzzy
filtering adding some other properties in order to give
more intelligence to the answer selection with a classifi-
cation into levels to minimize the error and get the best
signal approximation.
3.1. Rule Mechanism
The filtering process adding the fuzzy inference has a
rules base in order to learn, recall, associate and compare
the new information at its input in accordance with the
reference model changes and the filter signal levels use
its variance as operational limit in probabilistic form
The fuzzy rules base has a set of logical connectors
(if-then), to interpret the reference process conditions
into grades and select the corresponding membership
function fro m the knowledge ba se with its correspond ing
level. All this rule mechanism has as operation limit the
filter criterion, which has previously all th e possible data
to process in accordance with the reference model (real
process), that is interacting with the filter [16].
Using the logic connectors (if-then) in the fuzzy stage
with respect to desired signals set
k to deduce the
signal level as indicator to select the corresponding
membership function (parameter value) in order
to update the filter weights dynamically.
Each rule of the fuzzy filtering determines a specific
membership function value with the connector then, in
accordance with the input level deduced with the con-
nector if. All this structure works inside the error crite-
rion as limit, to maintain a correct operation, using the
error distribution to define the intervals in order to de-
scribe the filter operation into levels. The Figure 2, show
the elements of the rule mechanism.
First, the filtering process using the fuzzy connector if
infer the input signal level (desire signal)
k, from the
reference model. The filter mechanism finds into its in-
ternal database the corresponding new membership value
ak with the fuzzy connector then to update the pa-
rameter value having the operation level. Then the fuzzy
rule selects the corresponding parameters from the know-
ledge base in order to adjust the filter to give the correct
k at th e filter output.
3.2. Error Criterion
The fuzzy filter, works with the same basic concepts as a
conventional filter using the error mean square (1) as its
own criterion to delimit its answer levels. Additionally
the fuzzy filter integrates the fuzzy stage that character-
izes and interprets the input signal (desired signal), hav-
ing all the information required to process the data into
the knowledge base deducing the required membership
function to update the filter to a new required condition.
The fuzzy mechanism reduce the value of the filter
error criterion to its minimum difference described as
Copyright © 2011 SciRes. IJCNS
Figure 2. The fuzzy rule base selection.
getting the n earest value answer to update the filter
process, having the best signal approximati on at the filter
output of the desired signal
k, in this conditions the
error must be closest to
, that is a limit interval [0, 1].
Considering the response levels of the filter, the mem-
bership functions have a rank representation by levels
with minimum and maximum intervals limited with spe-
cific metrics previously defined in order to describe the
filter operation inside the error distribution func tion [17].
The previous classification represents the characteri-
zation of the reference model (real process) that interacts
with the filter. Using a fuzzy filter method avoids the
initial instability because the use of supervised learning
as artificial intelligence technique to reduce the process
uncertainty [18].
The minimization of the filter error criterion reduce
the distance between the desired signal
k and the
filter output
k, this allows having the best signal
approximation in accordance with the reference model,
updating the filter operation getting the correct answer.
Figure 3, shows the representation of the filter criterion
described as the convergence.
The filter convergence describes the error minimiza-
tion dynamically using th e mean square criterion. This is
a deduction of the error filter from its initial value (that is
considerably high) to a desired error value (minimiza-
tion), that accomplishes the filter requirements with re-
spect to the reference process. Having this, the fuzzy
filter gets the best answer value to update a process to a
new best condition [19].
3.3. Parameter Selection
The membership function to select into the fuzzy stage is
the corresponding parameter value respect to the
input signal level, in order to update the filter weights.
The goal of this parameter selection is to have a minimi-
zation of the error difference between the input signal
and the filter output, in order to get the best approxima-
tion of the signal using the better weight value to update
dynamically the filter process.
The membership function set into the knowledge base
bounds into the error criterion too, in order to get the
filter response inside of the reference model require-
ments [20].
The Figure 4, has th e me mb er sh ip f un ct ion
ak es-
timation in accordance with the input value
k and
the filter output
k. This is the parameter dynamical
selection using operation lev els by the fuzzy filtering.
The figure above shows the fuzzy stage that makes a
classification process describing the parameter values
into levels and dynamically selects the best correspon-
dence with the reference process changes having the best
signal approximation considering the lowest error value.
The parameter selection of the fuzzy filter allow to
have the best signal approximation dynamically, chang-
ing the parameter values to update the filter weights,
having the desired signal values deduction at the filter
input. The advantage of this metho d is that minimizes th e
error criterion value; having a selection process in ac-
cordance with the external changes, in other filter cases
Figure 3. Filter convergence.
Figure 4. Fuzzy parameter selection process.
Copyright © 2011 SciRes. IJCNS
without the fuzzy stage co nsiders this parameter value as
a constant [21].
The objectives of the knowledge base parameter selec-
tion are:
Has an automatic classification of the filter opera-
tion: having previously all the information into le-
Generates knowledge about the parameter value
about the reference model (in this case, th e desired
k), and the fuzzy filter selects the logi-
cal action, reflected in the filter out put
k value.
The adaptation rules modify dynamically the
membership functions, renewing and updating its
values in accordance with the reference model and
the error criterion.
Each membership value establishes the maximal
correspondence between the desired signal and the
filter output, minimizing the error of the filter.
The fuzzy filtering process using the criterion of
selects from a knowledge base (KB)
the parameter
to each value of
k, this per-
mits that
 
. The parameter selection pro-
cess is realized in heuristic form, which establishes pa-
rameters selection ranks in accordance with the error func-
tional ranks; where
, describing
2level , 3level . For each level of
k the filter
finds a specific value of , describing the opera-
tional filter levels and having as a goal to obtain
 
To make the selection of the parameter from the KB in
accordance with
k levels, it needs the next consid-
erations: 1) the experimentation process needs to be
fixed in accordance of its inputs and outputs, 2) To select
a value parameter proposed into the KB, 3) Observe the
error functional response, selecting parameters values
into the stability region described for the system, probing
it into the identification system described as ob-
serving the error difference with respect to ˆ()yk
k, se-
lecting the value parameter that minimizes the error
functional. For other new
klevel, based on the first
parameter value, we look for a new parameter near of the
first parameter that minimizes the error functional in
order to fix the KB of the filter [22].
4. Simulation Results
For the simulation of the fuzzy filter, we integrate the
Kalman filter structure with the fuzzy stage in order to
get a characterization of the filter operation, showing
graphically the fuzzy filter operation [23]. The Kalman
filter in this case uses the identification configuration,
having a dynamical parameter selection in order to get
the best-desired signal (from the reference model) ap-
proximation. The filter has a set of membership function s
describing all the filter changes limited by the error crite-
rion (1).
The reference model into the simulation is an ARMA
model (Autoregressive Mobile Average), which interacts
with the fuzzy filter in order to get the best answers. This
has a description in discrete states space, expressed by
the first order difference as:
 
kakxkw k (3)
Its output has t he nex t description :
: ,,
 (4)
k is the reference model interna l state,
is the matrix parameters sequence, is the refer-
ence model noise,
k is the desired signal from the
reference model to the filter input,
ck is a system pa-
rameters, and
vk = is the output vector noises.
With the fuzzy filter, the different operation levels
proposed must match inside the filter erro r criterion , with
respect with the desired signal
k, the membership
parameter selection
ak into the filter structure, and
the filter output
k. In accordance with a mathe-
matical and dynamical selection into th e fuzzy filter, this
process has the next description:
kayk k
 (5)
The Figure 5, shows the desired signal
k approxi -
mation with the fuzzy filter output described as .
In accordance with the desired signal levels, the fuzzy
stage makes a selection process by the fuzzy rules to get
dynamically the parameter value (membership value)
changing its values through time, as you can see in the
Figure 5. Desired signal approximation.
Copyright © 2011 SciRes. IJCNS
previous Figure 4.
With the fuzzy inference using the error levels into the
set of fuzzy rules The Figure 6, shows the membership
error classification in order to select the best answer.
In accordance with the fuzzy filter selection process,
The Figure 7, shows the different operational levels of
the filter output described by
Finally we verify the reference model probabilistically
with its fuzzy filter description using the density function
into The Figure 8.
The FDF describes precisely a real signal into opera-
tional levels minimizing the error difference into the fil-
ter mechanism, as we can see the graphics above, the
FDF can classify a signal, having an inference that inter-
pret the input (desired signal) and get the best parameter
value to approximate to the changes.
5. Conclusions
The fuzzy filtering is a type of signal characterization in
Figure 6. The error levels classification.
Figure 7. The filter service level.
Figure 8. Validation of the probabilistic reference model.
order to obtain the best answers in accordance with the
changes on its environment. Which as artificial intelli-
gent systems, with signal reasoning describing its opera-
tion into levels or grades by the fuzzy stage that use the
inference to interpret the input from an external process,
and get the best membership value selected dynamically
into the knowledge base, this updates the filter weights in
order to get the best signal approximation [23].
The paper describes the fuzzy filter properties that al-
low having an inference that classifies and deduces the
filter answers by the error criterion as limit, in order to
search and select the filter parameters and update its
weights to give a correct response dynamically as a nat-
ural linguistic answer. This work establishes how to con-
struct and characterize the membership values of the
knowledge base in a probabilistic manner by its distribu-
tion function.
Finally, this work shows a simulation of the FDF op-
eration using the Matlab tool and the Kalman filter
structure to integrate the fuzzy mechanism, considering
three answer levels, having an accurate filtering time
response with respect to the reference model with the
minimum error difference as natural process.
The next trend in applications for this kind of intelli-
gent filters would be the integration with technologies
that needs reasoning into its architecture. For example
the autonomous system that considers its external envi-
ronment changes in order to have an accurate response
according to it. The next systems should help us to do
our lives more dynamic and comfortable taking the deci-
sions with the best action to do in accordance with dif-
ferent kind of situations in our live.
6. References
[1] B. Rajen and M. Gopal, “Neuro-Fuzzy Decision Trees,”
Copyright © 2011 SciRes. IJCNS
Copyright © 2011 SciRes. IJCNS
International Journal of Neural Filters, Vol. 16, No. 1,
2006, pp. 63-68. doi:10.1142/S0129065706000470
[2] K. M. Passino, “Fuzzy Control,” Addison Wesley, Bos-
ton, 1998.
[3] M. Margaliot and G. Langholz, “New Approaches to
Fuzzy Modeling and Control Design and Analysis,”
World Scientific, Singapore, 2000.
[4] S. Haykin, “Adaptive Filtering,” Prentice Hall, Upper
Saddle River, 2001.
[5] T. Yamakawa, “A Survey on Fuzzy Information Proc-
essing Hardware Systems,” 1995 IEEE International
Symposium on Circuits and Systems, Seattle, 28 April - 5
March 1995, pp. 1310-1314.
[6] J. J. Medel, J. C. García and J. C. Sánchez, “Real-Time
Fuzzy Digital Filters Properties for SISO Systems,” Au-
tomatic Control and Computer Sciences, Vol. 41, No. 1,
2008, pp. 26-34.
[7] S. Mollov, R. Babuska, J. Abonyi, and H. Verbruggen,
“Effective Optimization for Fuzzy Model Predictive
Control,” IEEE Transactions on Fuzzy Systems, Vol. 12,
No. 5, 2004, pp. 661-675.
[8] R. Ash, “Real Analysis and Probability,” Academic Press,
Cambridge, 1970.
[9] L. Zadeh, “Maximizing Sets and Fuzzy Markoff Algo-
rithms,” IEEE Transactions on Systems, Man, and Cy-
bernetics-Part C: Applications and Reviews, Vol. 28, No.
1, 1998, pp. 9-15. doi:10.1109/5326.661086
[10] G. Feng, “A Survey on Analysis and Desing of Model-
Based Fuzzy Control Systems,” IEEE Transactions on
Fuzzy Systems, Vol. 14, No. 5, 2006, pp. 676-697.
[11] J. García, J. Medel and L. Guevara, “Filtrado Difuso en
Tiempo Real,” Computación y Sistemas, Vol. 11, No. 4,
2008, pp. 390-401.
[12] J. García, J. Mede l and J. Sá nchez, “Evolutive Neural Net
Fuzzy Filtering: Basic Description,” Journal of Intelligent
Learning Systems and Applications, Vol. 2, No. 1, 2010,
pp. 12-18. doi:10.4236/jilsa.2010.21002
[13] B. Kosko, “Fuzzy Engineering,” Prentice Hall, Upper
Saddle River, 1997.
[14] E. Mamdani, “Applications of Fuzzy Algorithms for
Control of Simple Dynamic Plant,” Proceedings of IEEE,
Vol. 121, No. 12, 1974, pp. 1585-1588.
[15] T. Amble, “Logic Programming and Knowledge Engi-
neering,” Addison Wesley, Boston, 1987.
[16] T. Takagi and M. Sugeno, “Fuzzy Identification of Sys-
tems and Its Applications to Modelling and Control,”
IEEE Transactions and Systems, Man, and Cybernetics,
Vol. 15, No. 1, 1985, pp. 116-132.
[17] M. Shannon, “A Mathematical Theory of Communica-
tion,” Bell Systems Technical Journal, Vol. 27, 1948, pp.
379-423 and pp. 623-656.
[18] J. Smith and A. Eiben, “Introduction to Evolutionary
Computing,” Springer, Cambridge, 2003.
[19] L. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8,
No. 3, 1965, pp. 338-353.
[20] H. S. Ali, “Fundamentals of Adaptive Filters,” John Wi-
ley & Sons, Hoboken, 2003.
[21] E. Onieva, V. Milanés, J. Pérez and T. Pedro, “Estima-
ción de un Control Lateral Difuso de Vehículos,” Inicio,
Vol. 7, No. 2, 2010, pp. 91-98.
[22] D. Marcek, “Stock Price Forecasting: Statistical, Classi-
cal and Fuzzy Neural Networks,” The 8th International
Conference on Modeling Decisions for Artificial Intelli-
gence, Barcelona, 2-4 August 2004, pp. 41-48.
[23] F. Gustafsson, “Adaptive Filtering and Change Detec-
tion,”John Wiley and Sons, Ltd., Hoboken, 2000.