Abstract
The article presents two multicriteria methods for noise reduction. The idea of these methods based on a combination of two criteria. The first criteria is the sum of squares of the finite differences of the first and/or the second order. The second criterion is square difference of input signal and its evaluation. In the work, we prove the convergence and uniqueness of solutions obtained. We discuss influence of method’s parameters on the result of noise reduction. On the set of test signals, we show the effectiveness of smoothing and noise reduction..
1. Introduction
In modern electronic systems during transmission the signal suffer of additive noise. The process of receive and conversion of the signal to digital form is associated with addition of noise component. Particular parameters of such a noise associated to change of temperature, change of electrical components parameters, interference with other radiosources, possible impurities in sensitive elements and other issues. In most cases, the noise is additive. The main task of digital signal processing is to extract the useful signal and suppress the noise component. In the case of rapid signal changes and excesses in the useful signal the problem of signal denoising become more complicated. The preservation of these areas and noise reduction will be the subject of research. For this we use the multicriteria approach based on the simultaneous minimization of two criteria.
Over the past 50 years, represented a large number of signal processing techniques [17]. The effectiveness of methods depends on the availability of information about the signal and noise. In the case of her lack of task becomes difficult.
The relevance of developing new method is based on the necessity of flexibility in adapting the devices for a wide range of tasks. An example of this problem is the ECG data processing [14, 15]. Development of filter allows you to simultaneously allocate isoline, denoising and keep of peaks. This is very urgent task in mobile devices.
The goal of this research is to the development of the method and its adaptation for the filter with opportunity to simultaneously allocate isoline, denoising and keep of peaks. This method finds implementation in mobile devices.
2. Model of input signal
A simplified discrete mathematical model of input signal could be represented as the sum of desired signal model and additive noise :
, .
The functional dependence of the desired signal and the law of distribution of additive noise is unknown a priori. As a model of the noise component zero expectation and Gaussian distribution law, we use ( ).
As criterion of efficiency we apply the mean squared error (MSE). The estimation of the efficiency of the results of processing are carried out on a test set of signals of different forms with random additive noise. The set contains 100000 different implementations of processed signal.
3. Method of noise reduction
3.1 Objective function with employing two criteria.
In the work to reduce the noise variance, we will decrease the sum of the squares of the finitedifference of its values:
, (1)
As well as to minimize of the sum of squares of finite differences of the second order:
. (2)
At the same time as a measure of discrepancy between the input and desired signal is used:
. (3)
The result of the smoothing is obtained and show to reduce the sum of (1 and (or) 2) and (3). This is obtained by minimization of objective function employing two criteria of form:
, (4)
, (5)
and minimization of the multicriteria objective function of form:
, (6)
here and – constant adjustment factors.
We implement these methods of noise reduction and have observed that the best results based on the use of simulation is reached for , in the case of expressions (4) and (5), and , in the case of (6).
Proposed functions (46) are continuous and limited below by a set of , and at least in one point reaches its minimum value. We prove the uniqueness of this point, the example of the target function of the form (4). Due to the necessary conditions of an extremum, its coordinates must satisfy the system of equations:
, (7)
ie the following system of linear equations with unknowns :
(8)
or:
(9)
We have proved that the system of equations (9) has a unique solution. We have applied the method of mathematic induction and check the validity of the approval «The first equation of the system (9) make it possible to define variables , as linear functions of the argument ie , when , » for all (for accept ). For get , , and in the case – , when , , ie mathematical statements , are true. Assuming for approval of fidelity for all , for the prove . From the equation of (9) we obtain
here ; .
The approval is confirmed. Use the mathematical statements the last equation in (9) takes the form when , . The resulting equation has a unique solution . It is uniquely determined by the values , when .
Thus, the system of equations (4) has a unique solution. Similarly we can prove uniqueness of the solutions for the objective functions of the form (5) and (6).
To find the point of the smallest the objective function value (4), (5) and (6) use the method of steepest descent [10]. We establish accuracy with which to be found the values . As a first iteration, we will , .
3.2 Minimization method for smoothing the noise.
For all we set the value . For this we get it with the left side of the th equation of system (9).
For the objective function (5):
(10)
For the objective function (6), reduces to the solution of the system:
(11)
For the objective function (4) we set:
. (12)
For the objective function (5):
. (13)
For the objective function (6.7):
. (14)
If , in the point the function reaching the lowest value. Note that and that if and only if . If value the function it is a quadratic function with a positive second derivative. Solving the equation , find the minimum point of the objective function of the form (4):
, (15)
for the objective function (5):
,(16)
and for the objective function (6):
(17)
In the point has function derivative and be the direction vector is positive, then ; and consequently . On next step performed correction is values :
, .
After this we check the mathematical expression
. (18)
If inequality (18) holds, the required accuracy is considered to be achieved, and the calculation ends. Then , ie the distance between the last two iterations in the space does not exceed of the values . Upon failure to perform of condition (18) is repeated calculation and verification of the previous step.
The vector estimates iteratively corrected so that the objective function reached its lowest value. At some stage of the iterative process is performed expression (18) and calculating terminated. The resulting vector estimates with accuracy will be minimum point of the objective function for given initial conditions [11].
4. The impact of the method parameters on the result
For digital signal processing in real time we propose use the objective function in the window , with subsequent sliding over all input implementation to finding evaluations for multicriteria.
Choice of the value of the processing window the shown in Fig. 1,2 and is based on a minimum iteration cost, for the to get the result of processing the input signal to the , [12].
Figure. 1 Graph of standard deviation values from of the window width
Figure. 2 Graph of standard deviation values from of the step size of the sliding window
Analysis of the results presented in Fig. 1 has shown that, minimum estimation achieved with , and achieved with that weakly dependent on the function of the useful component of the Fig. 2 shows the dependences of . The values obtained by smoothing of starting implementations of multicriteria objective function (5). The starting signals is are described: the united model (curve 1), the triangular shape (curve 2), the exponential function (line 3), the parabolic function (curve 4) and the harmonic form (line 5). In the same time we use additive noise with a Gaussian distribution law and variance =0.01, ei [13].
Figure. 3 Graph parameter changes depending on the shape of the signal
Analysis of the results presented in Fig. 3 shows that the use of twocriteria of the objective function (5) allows to localize the value in one segment (tab. 1). The table shows the data processing of the input signals of different forms . Inaccuracy in selecting parameter , increases the value to on 10% .
Table. 1 shows the values of setting in which values are the minimum mean square error [9].
Table 1. The values of which MSE .





the united model

0,04

0,023502

0,21

0,025858

the triangular shape

0,02

0,023961

0,08

0,026778

the exponential function

0,01

0,022876

0,08

0,032578

the parabolic function

0,01

0,022665

0,09

0,03423

the harmonic form

0,01

0,022271

0,21

0,041156

5. Signal processing as they become available
The process of obtaining estimates in a sliding window , carried out with parallel processing of initial values in a sliding window, with use of multicriteria objective function with different processing parameters . The rule of selecting , is presented in peppers [8, 9].
The transition between estimates of the obtained with different parameters , on the condition:
when: , – the result of processing with parameters and , p – threshold which is defined experimentally and with dispersion of the additive noise is .
Fig. 4 shows an example of the result of digital signal processing (curve 1) represented in the form of an additive mixture of the useful signal (curve 2) and the noise component in the presence of impulse noise [13].
Figure. 4 Example of a digital signal smoothing in the presence of impulse noise
area of application of of the proposed method is to process ECG data. An example of the suppression of the noise component on the ECG and the allocation of isoline is shown in fig. 5.
Figure. 5 An example of the definition of isoline on the ECG (1  the result of processing, 2 – ECG with noise, 3 – the isoline).
6. Conclusion
In the paper, the multicriteria methods of noise reduction the digital signal in a limited volume of a priori information about functions signal and the statistical characteristics of the noise are developed and studied.
Use a multicriteria methods smoothing for the digital signal processing in a sliding window, gives the increased the efficiency by an average of 25% in comparison with the processing of the entire implementation. In the presence of in input implement the functions of discontinuity of the first kind or impulse noise, there is an increase in efficiency of 60% on average.
When analyzing the data ECG, the proposed method can effectively determine the the signal of heart rate and the isoline
7. Acknowlegement
This work was supported by Russian Ministry of Education and Science in frame of the Federal Program "Research and development on priority directions of scientifictechnological complex of Russia Federation in 20142020" (contract ٠14.576.21.0083 (RFMEFI57614X0083)).
8. References
[1] de Souza, Carlos E., Uri Shaked, and Minyue Fu. "Robust ℋ∞ filtering for continuous time varying uncertain systems with deterministic input signals." IEEE Transactions on Signal processing 43.3 (1995): 709719.
[2] Veeneman, D., and S. BeMent. "Automatic glottal inverse filtering from speech and electroglottographic signals." IEEE transactions on acoustics, speech, and signal processing 33.2 (1985): 369377.
[3] Boudraa, AbdelOuahab, and JeanChristophe Cexus. "EMDbased signal filtering." IEEE Transactions on Instrumentation and Measurement 56.6 (2007): 21962202.
[4] Gardner, William A. "Cyclic Wiener filtering: theory and method." IEEE Transactions on communications 41.1 (1993): 151163.
[5] Chung, S. H., and R. A. Kennedy. "Forwardbackward nonlinear filtering technique for extracting small biological signals from noise." Journal of neuroscience methods 40.1 (1991): 7186.
[6] Combet, F., and L. Gelman. "Optimal filtering of gear signals for early damage detection based on the spectral kurtosis." Mechanical Systems and Signal Processing 23.3 (2009): 652668.
[7] Vladimir Marchuk, "Increasing the Reliability of the Primary Processing of the Results of Measurements." Measurement Techniques 46.12, PP 11211126. (2003).
[8] Vladimir Marchuk, Konstantin Rumyancev, Igor Shraifel “Twocriteria method for processing measurement results” Aerospace Instrument. Vol. 12. PP.33–35. (2005).
[9] Vladimir Marchuk and others “Practical aspects of digital signal processing”. Monograph. Shakhty. P.207. (2007).
[10] Granino Korn and Theresa Korn “Mathematical Handbook for Scientists and Engineers” McGrawHill, New York, p. 695. (1961)
[11] Evgeny Semenishchev “Investigation of the effectiveness of the iterative method of isolation of the desired signal based on the twocriteria of the objective function” Proceedings of the Russian Scientific and Technical Society of Radio Engineering, Electronics and Communication named after AS Popova « Digital Signal Processing and Applications ». Moskow. Vol.X1. PP. 452455. (2008).
[12] Vladimir Marchuk and Evgeny Semenishchev “Search dependencies parameters for twocriterion method processing in a sliding window”. Proceedings of the international scientific conference "Information, signals, system: issues of methodology, analysis and synthesis". Taganrog. Vol.4 PP.4547. (2008)
[13] Vladimir Marchuk and Evgeny Semenishchev “Investigation of twocriteria of the objective function for processing digital series realtime” Articles VI Scientific and Technical Conference "Modern methods and means of spacetime processing signal". – Penza. P.568. (2008).
[14] Stadler, Robert, et al. "Method and apparatus for filtering electrocardiogram (ECG) signals to remove bad cycle information and for use of physiologic signals determined from said filtered ECG signals." U.S. Patent No. 6,115,628. 5 Sep. 2000.
[15] Vorobyov, Sergiy, and Andrzej Cichocki. "Blind noise reduction for multisensory signals using ICA and subspace filtering, with application to EEG analysis."Biological Cybernetics 86.4 (2002): 293303.
[16] Sameni, Reza, et al. "Filtering noisy ECG signals using the extended Kalman filter based on a modified dynamic ECG model." Computers in Cardiology, 2005. IEEE, 2005.