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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

, Igor Shraifel

, Ilya Svirin

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

Don State Technical University, Gagarina sq. 1, Rostov-on-Don, 344000, Russia;

E-mails: [email protected]; [email protected]

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 )(

tY obtained at equidistant times

tkT



, when

1,kn

(

0T   constant). This signal can be considered as an implementation of the

random process

)(

, 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

()

t – useful signal component; ()



– additive noise component, n –

sample size.

The functional dependence of the useful component on time

()

ts

unknown. The law of distribution of additive noise ( )



 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





st of the

value can be interpreted

as a decrease in the additive noise variance



. 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:













, (2)

In this case, as a measure of the divergence of the input signal and its

evaluation, the sum is used:









. (3)

To determine the estimates of

, 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

12 1

( , ,..., ) ( ) ( )

nkkkk

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

, therefore, at least at one point in the interval,

the

( , ,..., )

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:







1,jn . (5)

Of the following system of

n linear equations with n unknowns

, ,...,

ss:

12 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

(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

«the first ( 1)k



arguments of systems (7) set variable

, ,...,

ss, as linear functions of the argument

etc.



, when

j

, 1,jk » at each 1,kn (use here



 ). When 1k



we have the

result

1



0 ,



 , when



–

2212



, where



,

Y

, thus the statements

are correct. Assuming the validity of the

statement

P for some

2 kn





, we prove the assertion



. From any

equation of the system (7) we obtain:

Two Approaches to Solving the Problem of Smoothing Digital Signals…

369

11111111

(2 )( ) ( ) ,

kkkkkkkk

sssYs





        

where

(2 )

kkkk

;

(2 )

kkkk





.

The above calculations for

,,,

PP P are correct. Using the expression

P , the last equation of the system (10) reduces to the form

0,s where

(1 )

nn

   

() 0

nn n



   ,

(1 )

nn n





. The

resulting equation has a unique solution



  , 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:

kk iki

ss y





 



, 1,2,...,kn



, (8)

where









 







, (9)



















. (10)

Note. Here and in what follows we establish the notation for binomial

coefficients

0, 0,

1, 0,

( 1) ... ( 1)

,0.

mm ml





























ni i













. (11)

Using the relations (8) and (9) for



and the relation (10) for 1k  , we

obtain:

22111 11 11

12 1

(1 )

02 1

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 ) )

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,

111

(2 )

kj kj







 

 

  

   

 

  

  

 















 







where

kiki











. (12)

We transform the left side of the equation:



111

2(2)

222

(2 1) (2 ) .

k j

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 )

222

kj kj kj

jjj







  

 







 

 





112

2222

kj kj kj kj

jjjj







   

 





 

 





12 1

222 2

kj kj kj kj

jjj j







   

  





  

  





12 23

00024

122

22222

kk kk

kj kj kj kj

jj jj





 

  



  



  



   

  





  



  





Two Approaches to Solving the Problem of Smoothing Digital Signals…

371

(2 3)

11 2 2

222222

kj kj kj kj kj

jjj jj









    

  





  



  





1 1

212

(2 3)

21 21 22

(2 3) (2 3) .

22 22

k j

kjk

kj kj kj

jjj

kj kj











  



   













 



  













(Here and in what follows we assume that when

lm sum





is zero).

Thus, any average equation of system (6) takes the form:





111

0(2)

kk k k

vv v y





    .

Taking into account expressions (10) and (12), the resulting ratio is

rewritten as follows:

121

11 1 1

(2 )

21 21

kki kki

ij i j

kji kji

 

  

 

  

  

  



  

 

kki

kji





















We simplify the left-hand side of the equation using the properties of

binomial coefficients:



21 2

11 1

(2 ) (2 )

(2 )

(2 2 )

(

kki

ikk

kki k

jki

iki

ij i

ki ki

kji

yyy

kji

yyy

yki













 



 



 







   











   

























 



21 21 21

kki

kji kji kji

jjj







  

 







 



 





E. Semenishchev, I. Shraifel, I. Svirin

372

(2 2 2)

(2 ) .

21 21 21

kki

yyki

kji kji kji

jjj









     



  

 

  



 



 







We will convert the coefficient at

y in the last sum:

112

21 21 21 21

112

21 21 21 21

k ji k ji k ji k ji

jjjj

k ji k ji k ji k ji

jjjj

kji















   

 



 



 



 





   

 





 



 

















22 22

(2 2 2)

23 23

(2 2 2) .

iki

ki ki

jki j

jki

kji kji

kji





















 



  











 

 

 

 



 

















Thus, any average equation of system (6) becomes an sameness:

(2 2 2) (2 2 2) ,

2,3,..., 1.

ki ki

ki i k

yyki yki y





  





Let us prove that the quantities (8) satisfy the last equation of the system

(6):

111

(1 )

nnn

nj nj

vsvy







 

 

  

      

 

  

  

 



(1 ) (1 )

(1 ) .

nn n

nj nj

vv y









 

 

      



 



 



  



The coefficient at

12 2

(1 )

22 22

njj

nj nj nj

jj j









  

 

       



 



 





Two Approaches to Solving the Problem of Smoothing Digital Signals…

373

(1 )

21 22

(1 ) (2 3) .

21 21

njj

nn j j

nj nj













 











 



       









The equation takes the form

11 1

(1 ) .

21 21

nni ni

ni n

ij j

nji nji

 

 



 

 

   



 



 



 

The expression in brackets is

212

(1 )

23 21 21

22 23

22 2

ni ni

jjni

ni j

nji nji nji

jjj

nji nji





















  



















 



    











 



   





















Because

1 , to

nini







  



, 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



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



= 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



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





Fig. 1 – Dependence of

()



 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

()

.

The value ( )

 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

()





, 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

()



 at which

min

()min



 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

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



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

()



 .

Table 1

The number of iterations m required to perform the smoothing operation

as a function depending the useful and noise components.



signal

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

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



 .

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

 obtained are presented in Table 2. The results are averaged over a

thousand implementations.

Table 2

Results of mean square error



Amplitude noise

method

0.05



 0.1



 0.15



 0.2



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”.

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