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Two Approaches to Solving the Problem of Smoothing Digital Signals Based on a Combined Criterion

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SERBIAN JOURNAL OF ELECTRICAL ENGINEERING
Vol. 14, No. 3, October 2017, 365-377
365
Two Approaches to Solving the Problem
of Smoothing Digital Signals
Based on a Combined Criterion
Evgenii Semenishchev
1
, Igor Shraifel
1
, Ilya Svirin
2
Abstract: The paper presents a method for smoothing signals represented by a
single realization of a finite-length random process, under conditions of a limited
amount of a priori information about the signal function and statistical
characteristics the noise component. The recommendations on the use of
parameters affecting the processing speed and the efficiency of smoothing are
given. Two solutions are presented to obtain the result of smoothing the signals.
The efficiency results are shown for the processing of digital signals. Examples
of comparison of simple methods and suggested ones are given.
Keywords: Signal processing, Denoising, Two-criteria method, Smoothing.
1 Introductions
In modern automatic control systems for data collection, processing and
transmission, intelligent sensors play a special role. These devices allow for
continuous monitoring and transmission of information to a remote terminal.
The process of signal conversion is associated with the effect on the measured
signal of a random component. To reduce noise, processing is performed
immediately after the analog interface of the sensor, and data transfer to
subsequent monitoring systems is carried out digitally [1]. Modern sensor
systems produce ADC conversion of the received signals and a reduction in the
influence of the noise component. In this connection, high demands are placed
on the sensitive element and the pre-treatment block. There are technological
limitations in the manufacture of the measuring element and the corresponding
devices of the analog interface. As a consequence, algorithms for preliminary
processing of digital signals are of special interest for increasing the reliability.
The use of digital signal processing methods has found wide application
[2]: in automation and control systems, when creating sensors with the
possibility of automatic adjustment in the event of possible aging of the sensing
element or changes in environmental parameters; in modern antenna systems, in
1
Don State Technical University, Gagarina sq. 1, Rostov-on-Don, 344000, Russia;
E-mails: sea.sea@mail.ru; shraifel17@mail.ru
2
CJSC Nordavind, Warsaw highway 125, Moscow, Russia; E-mail: i.svirin@nordavind.ru
UDC: 621.39:004.93 DOI: https://doi.org/10.2298/SJEE1703365S
E. Semenishchev, I. Shraifel, I. Svirin
366
the study of atmospheric, hydro- and lithospheric structures, as well as object
detection systems; when investigating biomechanical parameters, biometric data
collection systems located directly on the object under study; in modern systems
of automatic processing of two-dimensional signals obtained from CCD the
photo and video cameras, as well as computer vision systems, to reduce the
noise effect of video sensors associated with the operation of the
communication channel or the defect of the scanning device; in economics and
sociology in the study of trends; in information-measuring systems; in computer
technology to increase the accuracy associated with the possibility of reducing
the interference caused by noise conversion of the signal from analog to digital.
In the general case, the analysis of signals is hampered by the presence of
noise having a random character with priori unknown statistical characteristics.
Information about the useful component of the signal is also limited. The use in
the automation and control systems of the methods of considered in the works
of leading scientists, such as V.I. Tikhonov, A.I. Orlov, B.R. Levin, L. Rabiner,
B. Golden, etc., is possible only if there is a sufficient amount of a priori
information, otherwise, their effectiveness is significantly reduced [3–6].
As a rule, in these cases, the methods of processing are based on
minimizing the standard deviation criterion or maximizing the signal-to-noise
ratio. The choice of the criterion is due to the amount of a priori information
about the problem being solved. In the conditions of a limited amount of
information on the useful signal function and the statistical characteristics of
noise, the problem is sharply complicated. The presence in the useful
component of the points of discontinuity of the first kind imposes additional
requirements on the methods of processing.
In this connection, the actual task is to develop methods and algorithms for
smoothing the digital signals of measuring complexes and automatic control
systems, as well as processing devices simultaneously by several criteria in
conditions of a limited amount of a priori information about the function of the
useful and noise components.
In modern automatic systems for the collection, processing and
transmission of measurement on the signal is affected by various interference.
The process of converting a signal into a digital form also introduces a random
component into the signal. The main task in the processing of signals is the
separation of the useful and the noise component. At the same time, in practice,
the criterion of minimum mean square error or the criterion of mean-absolute
deviation is used. Each of the criteria has merits and limitations of use
depending on the task and a priori information about the components of the
input signal [7]. Actual is the task of processing a digital signal based on the
objective function of the combined criteria. Of particular interest is the use of
multicriterial methods for processing digital signals. When the signal is
Two Approaches to Solving the Problem of Smoothing Digital Signals…
367
represented by a single implementation with a limited amount of a priori
information about the useful component and statistical information about noise
characteristics.
2 Mathematical Model of Signals
Let the input signal be a discrete sequence of values of the measured
physical quantity )(
k
tY obtained at equidistant times
k
tkT
, when
1,kn
(
0T constant). This signal can be considered as an implementation of the
random process
)(
k
tY
, which is an additive mixture of useful and noise
components. A simplified mathematical model of the input signal is represented
in the form [7]:
() () ()
kk k
Yt st t
 , 1,kn . (1)
when
()
k
s
t – useful signal component; ()
k
t
– additive noise component, n
sample size.
The functional dependence of the useful component on time
()
kk
s
ts
is
unknown. The law of distribution of additive noise ( )
kk
t
 is also limited,
but it is assumed that the distribution density has a Gaussian law, with zero
mathematical expectation and constant variance [8].
To conduct test studies, we will use the signals component models:
rectangular, harmonic, triangular, parabolic and exponential.
3 Methods for Smoothing Digital Signals Based
on Combined Criteria
The derivation of the estimate
kk
s
st of the
k
s
value can be interpreted
as a decrease in the additive noise variance
k
. In the paper it is proposed to
reduce the dispersion of the measured process by decreasing the sum of the
squares of the finite differences of its values:

1
2
1
1
n
kk
k
ss
, (2)
In this case, as a measure of the divergence of the input signal and its
evaluation, the sum is used:

2
1
n
kk
k
s
Y
. (3)
To determine the estimates of
k
s
, we simultaneously, reduce the sums (2)
and (3). This goal is achieved by minimizing two-criteria target functions of the
form [10, 11]:
E. Semenishchev, I. Shraifel, I. Svirin
368
1
22
12 1
11
( , ,..., ) ( ) ( )
nn
nkkkk
kk
ss s s Y s s



, (4)
where
– constant coefficients.
It should be noted that the objective function (4) are continuous and
bounded of below on the set
n
R
, therefore, at least at one point in the interval,
the
12
( , ,..., )
n
s
ss reaches its lowest value. Let us prove the uniqueness of such a
point, of objective function of the form (4). Due to the necessary condition of
the extremum, its coordinates must satisfy the system of equations:
0,
j
s

1,jn . (5)
Of the following system of
n linear equations with n unknowns
12
, ,...,
n
s
ss:
12 1
11
1
(1 ) 0;
....
(2 ) 0, 2,3,..., 1;
...
(1 ) 0.
kk k k
nn n
ss Y
ss s Y k n
ss Y

 

 
(6)
We rewrite system (6) in the form:
211
11
1
(1 ) ;
...
(2 ) , 2,3,..., 1;
...
(1 ) 0.
kkkk
nn n
ssY
sssYkn
ss Y


 
 
(7)
Let us prove that the system of equations (7) has a unique solution. To this
end, by the method of mathematical induction, we establish the validity of the
assertion
:
k
P
«the first ( 1)k
arguments of systems (7) set variable
12
, ,...,
k
s
ss, as linear functions of the argument
1
,
s
etc.
1
j
jj
ss
, when
1
j
j
, 1,jk » at each 1,kn (use here
0
0
). When 1k
we have the
result
1
1

0 ,
1
0
, when
2k
2212
ss
, where
21
1
,
21
Y
, thus the statements
1
P
,
2
P
are correct. Assuming the validity of the
statement
k
P for some
2 kn
, we prove the assertion
1k
P
. From any
k
equation of the system (7) we obtain:
Two Approaches to Solving the Problem of Smoothing Digital Signals…
369
11111111
(2 )( ) ( ) ,
kkkkkkkk
sssYs

 
where
11
(2 )
kkkk
;
11
(2 )
kkkk
Y

.
The above calculations for
12
,,,
n
PP P are correct. Using the expression
n
P , the last equation of the system (10) reduces to the form
1
0,s where
1
(1 )
nn
 
1
() 0
nn n
  ,
1
(1 )
nn n
Y
. The
resulting equation has a unique solution
1
s
 , in which the values
1kk k
ss  , where 2,kn .
Thus, the system of equations (4) has a unique solution.
We also propose the definition of an exact analytic solution of a two-
criteria objective function of the form (4). As proved, the minimum point of the
function (4) is the unique solution of the system of linear equations (6).
We show that this solution has the form:
1
1
1
k
kk iki
i
ss y

, 1,2,...,kn
, (8)
where
1
0
1
2
k
j
k
j
kj
j





, (9)
1
1
21
k
j
k
j
kj
j





. (10)
Note. Here and in what follows we establish the notation for binomial
coefficients
0, 0,
1, 0,
( 1) ... ( 1)
,0.
!
l
m
l
l
mm ml
l
l




1
1
1
n
ni i
i
n
y
s


. (11)
Using the relations (8) and (9) for
2k
and the relation (10) for 1k , we
obtain:
22111 11 11
12 1
(1 )
02 1
s
sy sy sy

 


 
 

.
E. Semenishchev, I. Shraifel, I. Svirin
370
Substituting the result in the first equation (6), we obtain the identity:
1111
(1 ) ((1 ) )
s
sy y
.
Let us verify that the quantities (8) (under the conditions (9) – (11)) satisfy
any equation of the system (6) and with
21kn
. Thus,
12
111
00
12
(2 )
22
kk
jj
kk
jj
kj kj
ss
jj



 

 





11
0
,
2
k
j
kk
j
kj
s
y
j







where
1
1
k
kiki
i
y

. (12)
We transform the left side of the equation:

2
1
0
1
111
12
2(2)
222
(2 1) (2 ) .
k
k j
j
kk
kk k k
kj kj kj
jjj
ksvvvy


 

 




 
We simplify part of the expression on the left side, using the properties of
binomial coefficients [9]:
2
0
12
(2 )
222
k
j
j
kj kj kj
jjj

 






2
0
112
2
2222
k
j
j
kj kj kj kj
jjjj

  






22
00
12 1
2
222 2
kk
jj
jj
kj kj kj kj
jjj j



  
 


 
 


1
2
1
12 23
2
00024
122
2
22222
k
k
j
j
kk kk
k
kj kj kj kj
jj jj






  






Two Approaches to Solving the Problem of Smoothing Digital Signals…
371
1
2
1
(2 3)
11 2 2
222222
k
k
j
j
k
kj kj kj kj kj
jjj jj


   






2
1
1
2
1 1
1
212
(2 3)
21 21 22
22
(2 3) (2 3) .
22 22
k
k j
j
k
kjk
j
kj kj kj
k
jjj
kj kj
kk
jj

 








 

 





(Here and in what follows we assume that when
lm sum
m
j
jl
b
is zero).
Thus, any average equation of system (6) takes the form:
111
0(2)
kk k k
s
vv v y

 .
Taking into account expressions (10) and (12), the resulting ratio is
rewritten as follows:
121
11 1 1
12
(2 )
21 21
kki kki
jj
ii
ij i j
kji kji
yy
jj


 






1
11
.
21
kki
j
ik
ij
kji
yy
j








We simplify the left-hand side of the equation using the properties of
binomial coefficients:

2
1
11
21
2
1
11
21 2
11 1
2
1
1
1
(2 ) (2 )
21
2
(2 )
21
(2 )
21
(2 2 )
(
kki
j
ki
ij
kki
j
ikk
ij
kki k
jki
iki
ij i
k
ki ki
i
i
i
kji
yy
j
kji
yyy
j
kji
yyy
j
yki
y


























21
11
12
2)
21 21 21
kki
j
ij
kji kji kji
jjj



  








E. Semenishchev, I. Shraifel, I. Svirin
372
2
1
21
11
(2 2 2)
12
(2 ) .
21 21 21
k
ki
ki
i
kki
j
i
ij
yyki
kji kji kji
y
jjj




  








We will convert the coefficient at
i
y in the last sum:
1
1
1
1
1
112
2
21 21 21 21
112
21 21 21 21
1
21
ki
j
j
ki
j
j
k
j
k ji k ji k ji k ji
jjjj
k ji k ji k ji k ji
jjjj
kji
j



   








   











11
1
1
1
22
1
2
21
22 22
22
(2 2 2)
23 23
2
(2 2 2) .
23
iki
jj
j
ki ki
jki j
jj
ki
jki
j
kji kji
jj
kji kji
ki
jj
kji
ki
j





 







 
 

 

 







Thus, any average equation of system (6) becomes an sameness:
22
11
(2 2 2) (2 2 2) ,
2,3,..., 1.
kk
ki ki
ki i k
ii
yyki yki y
kn



 


Let us prove that the quantities (8) satisfy the last equation of the system
(6):
12
111
00
12
(1 )
22
nn
jj
nnn
jj
nj nj
s
vsvy
jj



 

 





,
or
2
1
1
0
1
12
(1 ) (1 )
22
(1 ) .
n
nj
j
nn n
nj nj
s
jj
vv y


 

 








The coefficient at
1
s
is
21
1
01
12 2
(1 )
22 22
nn
njj
jj
nj nj nj
jj j



  
 


 
 


Two Approaches to Solving the Problem of Smoothing Digital Signals…
373
21
1
11
2
11
11
22
(1 )
21 22
11
(1 ) (2 3) .
21 21
nn
njj
jj
nn
nn j j
n
jj
nj nj
jj
nj nj
n
jj




 





 

 





The equation takes the form
11
1
11 1
12
(1 ) .
21 21
nni ni
jj
ni n
ij j
nji nji
s
yy
jj



 








The expression in brackets is
1
21
1
2
1
1
212
(1 )
23 21 21
22
22 23
1
22 2
ni ni
jjni
jj
ni
ni j
j
ni
ni j
j
nji nji nji
jjj
nji nji
jj
nji nji
jj





  








 

 





 

 



1
0
.
ni
j
ni
j


Because
1
1 , to
11
1
0
n
nini
i
sy


, and this equality is satisfied, in
view of (11).
Thus, the expression (8) (when the expressions (9) (11) is substituted into
it) has a unique solution of the system of equations (6), which minimizes the
function (4). It should be noted that this function does not have other minimum
points.
To test the effectiveness of a multi-criteria method for smoothing digital
signals, the standard deviation
о
of the estimates from the values of the input
signal is used as a criterion.
Fig. 1 shows the dependencies of the root-mean-square deviation on the
values of the coefficient
obtained by processing a two-criteria objective
function, with the analytical solution and the iterative approach. As a useful
component (1), a function is used whose discrete values are described by a
parabola, with a root-mean-square noise deviation
n
= 0.15 [9].
The analysis of the curves presented in Fig. 1 allows us to conclude that the
results of the processing efficiency estimates obtained by the iterative algorithm
(4) and the analytical solution practically coincide, deviation
о
is less than
1%.
E. Semenishchev, I. Shraifel, I. Svirin
374
Investigations of the effect of coefficients on the result are presented in
[10], ranges of confidence intervals are shown coefficient
. To minimize the
value of the root-mean-square error, it is necessary to optimize the choice of the
approximation
. To determine the minimum error and to determine the effect
on the error parameter
, we fix the parameter
min
when
min
()min
o

.
Fig. 1 – Dependence of
()
о
for the analytical and iterative solution.
As a result of simulation modeling, for various models of the useful
component of functions, we the dependence of the parameter change
()
o
.
The value ( )
o
 of the root-mean-square error obtained when processing the
input realization, for fixed values of the standard deviation of the additive noise
component, is coincided. Fig. 2 shows the graphs of the change
()
o
, in the
case of using a harmonic signal with different mean square deviation of the
noise component.
The results of simulation presented in Fig. 2 make it possible to conclude
that the values of the smoothing error
()
o
at which
min
()min
o
 lies at
the point
0.001 .
In order to determine the required computationally-time costs for the
implementation of the proposed method, we estimate the number of required
m
iterations, in order to achieve the approximation parameter of the estimates of
0.001 for fixed values of
min
. During the simulation, data were obtained
showing that the number of iterations depends on the root-mean-square
deviation of the noise
n
and the shape of the useful component. Table 1
shows the values of
m obtained with the value of
min
in the case when the
Two Approaches to Solving the Problem of Smoothing Digital Signals…
375
condition for ensuring a given accuracy is achieved when solving the problem
of smoothing by multi-criteria method of smoothing digital signals in conditions
of limited volume of a priori information.
Fig. 2 – Schedule for changing the parameter
()
o
.
Table 1
The number of iterations m required to perform the smoothing operation
as a function depending the useful and noise components.
n
signal
m
The sin
function
The
exponent
function
The
parabolic
function
The
composite
function
The square
pulse
0,05
m with
min1
20 40 37 21 5
0,1
m with
min1
35 61 57 39 9
0,15
m
with
min1
45 91 73 50 14
0,2
m with
min1
57 135 103 89 24
Fig. 3 shows an example of the determination of the number of iterations
()m , at the input of the filter a mixture of sine and noise with 0.1
n
.
E. Semenishchev, I. Shraifel, I. Svirin
376
Fig. 3 – An example of calculating the number of iterations.
The results shown in Fig. 3 show that as the value of
increases, the
number of required repetitions of the calculation decreases.
4 Testing Results
We analyze on a set of test signals of the form: the sin function, the
exponent function, the parabolic function, composite signal, square pulse. We
compared the method with the standard implementations of signal processing
methods (weighted moving average, exponential smoothing, and median
smoothing) presented in the Matlab library. The results of the root-mean-square
error
о
obtained are presented in Table 2. The results are averaged over a
thousand implementations.
Table 2
Results of mean square error
о
.
Amplitude noise
method
0.05
n
0.1
n
0.15
n
0.2
n

weighted moving average 0,025 0,041 0,068 0,085
exponential smoothing 0.38 0.67 0.084 0.121
median smoothing 0,038 0,064 0,075 0,084
two-criteria method 0,024 0,041 0,055 0,069
Two Approaches to Solving the Problem of Smoothing Digital Signals…
377
The method of the weighted moving average is the closest to the developed
method in the case of small values of the root-mean-square deviation of noise.
Most methods work in the presence of a priori information about the function of
the useful signal, otherwise the root-mean-square error can dramatically
increase on 40% or more.
5 Acknowledgement
This work was supported by Don State Technical University “Grant of L.V.
Krasnichenko”, research project “The development of the theory of digital
signal processing, by creating new methods and algorithms for combining
medical images into a single content”.
6 References
[1] M. Kostić, N. Dončov, B. Stošić, B. Milovanović: Digital Filter-based 1D TLM Model of
Dispersive Anisotropic Conductivity Panel, Serbian Journal of Electrical Engineering Vol.
13, No 1, Feb. 2016, pp. 1 – 8.
[2] D.S. Antić, M.B. Milovanović, S.Lj. Perić, S.S. Nikolić, M.T. Milojković: Input Data
Preprocessing Method for Exchange Rate Forecasting via Neural Network, Serbian Journal
of Electrical Engineering Vol. 11, No. 4, Dec. 2014, pp. 597 – 608.
[3] C.E. de Souza, U. Shaked, M. Fu: Robust
Filtering for Continuous Time Varying
Uncertain Systems with Deterministic Input Signals, IEEE Transactions on Signal
Processing, Vol. 43, No .3, March 1995, pp. 709 – 719.
[4] F. Combet, L. Gelman: Optimal Filtering of Gear Signals for Early Damage Detection based
on the Spectral Kurtosis, Mechanical Systems and Signal Processing, Vol. 23, No. 3, April
2009, pp. 652 – 668.
[5] A.O. Boudraa, J.C. Cexus: EMD-based Signal Filtering, IEEE Transactions on
Instrumentation and Measurement, Vol. 56, No. 6, Dec. 2007, pp. 2196 – 2202.
[6] W.A. Gardner: Cyclic Wiener Filtering: Theory and Method, IEEE Transactions on
Communications, Vol. 41, No. 1, Jan. 1993, pp. 151 – 163.
[7] V. Marchuk, S. Makov, A. Minaev, S. Stradanchenko: Studying Accuracy of the New
Desired Signal Extraction Method by Residuals in a Priory Indeterminacy Conditions, 14
th
IEEE East-West Design and Test Symposium, Yerevan, Armenia, 14-17 Oct. 2016.
[8] V. Marchuk, S. Makov, D. Timofeev, M. Pismenskova, A. Fisunov: A Method of Signal
Estimation Error Reduction in a Priori Indeterminacy, Telecommunications Forum –
TELFOR, Belgrade, Serbia, 24-26 Nov. 2015. pp. 400 403.
[9] G.A. Korn, T.M. Korn: Mathematical Handbook for Scientists and Engineers, McGraw-Hill,
NY, USA, 1961.
[10] E. Semenishchev, V. Marchuk: Search Dependencies Parameters for Two-Criterion Method
Processing in a Sliding Window, International Conference on the Information, Signals,
System: Issues of Methodology, Analysis and Synthesis, Taganrog, Russia, 2008, Part. 4,
pp. 45 – 47. (In Russian).
[11] E. Semenishchev, I. Shraifel, V. Marchuk, I. Svirin, S. Makov: A Multi-Criteria Method for
Noise Reduction, 14
th
IEEE East-West Design and Test Symposium, Yerevan, Armenia, 14-
17 Oct. 2016, pp. 444 – 449.
 
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