2018 International Conference on Modeling, Simulation and Optimization (MSO 2018) ISBN: 978-1-60595-542-1
The Application of Improved Tracking-differentiator Filter in ECG Data
Jin-ping FENG, Guang-yu LI, Wei WANG
*and Xun LIANG
School of Information, Renmin University of China, 100872, China *Corresponding author
Keywords: Tracking-Differentiator filter, ECG, Frequency interference, Filtering.
Abstract. Denoising is very important in signal processing. Electrocardiogram (ECG) data always contains 50 or 60 Hz power line interferences (PLI), PLI may affect the detection of P wave, QRS complex and T wave. So the pre-processing of ECG data is necessary. In order to remove PLI, in this paper, we use two improved tracking-differentiator filters (ITDFs), which are signed ITDF1 and ITDF2 respectively. The simulation results for ECG signals obtained from MIT-BIH database indicate that ITDFs are good methods for filtering noises, and ITDF2 does better than ITDF1.
Introduction
ECG is a basic human physiological signal, and it has important clinical diagnostic value. In the detection of ECG signals, which are susceptible to the impact of equipment, human activities, etc., the detected ECG is often accompanied with some interference, which is mainly in the following three kinds: frequency interference, baseline drift and Electromyography interference. The interference has a great impact on the analysis and processing of ECG data. Therefore, it is an important task of ECG signal processing by using digital filtering technology to filter out the various interferences and to obtain accurate detection of ECG signals. It is known that there are many denoising methods have been proposed, the most useful technique is wavelet denoising method [1-3]. But there exist some limitations in the wavelet method, such as lack of shift invariance, poor direction selectivity [3-4]. In this paper, we intend to deal with the denoising problem by using some of the improved tracking-differentiator filters (ITDFs), which are introduced by Feng et al. [5], simply referred to as ITDF1, and another ITDF proposed by Feng et al. [6], simply signed as ITDF2.
As a matter of fact, tracking-differentiator (TD) is a useful tool in tracking signal and differential extraction. It was first introduced by Han-Wang [7]. Later, Han-Yuan [8] provided its discrete form. B.Z. Guo et al. [9-10] gave the strict mathematical proof of the TD from Han-Wang [7]. There are many improved methods to increase the accuracy of TD [6, 11-12]. In addition to above properties, TD also could be applied in filtering, and simulation results showed that its filtering performance is good [12]. From the simulation results from the early researchers, there are phase lags in the outputs of TD. Many advanced algorithms are proposed, such as TD which using an ingenious technique based on Taylor’s expansion [5, 6, 13], the time lagging phenomenon is reduced effectively. In this paper, we mainly consider frequency interference filtering technique introduced by references [5, 6], and we also compare their filtering performance.
The paper is organized as follows. The second section provides the details of two improved tracking-differentiator filters. Simulation analysis is presented in the third section, and conclusions are provided at the end.
The Improved Tracking-Differentiator Filters [5, 6]
Both of the two improved tracking-differentiator filters are based on Taylor’s formula, they can reduce the time lagging phenomenon effectively. In the following, we first give the ITDF1 and ITDF2 respectively.
1 1 2
2 2
( 1) ( ) ( )
( 1) ( ) ( ( ), )
x k x k hx k
x k x k hrsat g k (1)
where h is the sample step, r is one parameter that could control the speed of the tracking
differentiator. ( )g k and are given as follows
1 1 2 2 2 1 2 1 ( ) ( ) ( ) ( ) ( ) ( )
8 | ( ) |
( )
( ) ( ( )) ,| |
( ) 2
( ) ( ) , | | hr h
e k x k v k
z k e k hx k
z k
r h h
r
x k sign z k z
g k
z k
x k z
h
(2)
And sat( , ) is a saturate function
( ),| | ( , ) , | |
sign x z
sat x x
z (3)
Here rand h are design parameters, through amounts of experiments, we can set the optimal
design parameters to obtain the desired results.
The basic filter method can be carried out by the following steps [5]: Firstly, we should deal with the input signal v t( ) with n-order cascade of DNTD. Secondly, we get the output signalsx t1i( ) (the
tracking signals) and i-order differential signals x2i( )t from DNTD att1, wherex t1i( ) is the filter and
approximation of ( 1)i , 1, 2, ,
y i n . Lastly, we get the compensated signal which also is the
approximate valuey tˆ( )1 of the real valuey t( )1 .
1( 1) 1 1
1 1 12 1
1 11 1 1 2 1 2 1 2
( ) ( )
( )
ˆ( ) ( ) ( ) ( ) ( )
1! ! ( 1)!
xn t n x n t n
x t
y t x t t t t t t t
n n (4) ITDF2. The second ITDF is the one with a high-gain tracking-differentiator, and it is the improved tracking-differentiator by Feng et al. [6].
Consider the following classical high-gain tracking differentiator [14]:
3
1 2 1
2
2 3 1
1 3 1 ( ) ( ) [ ( ) ( )] ( ) ( ) [ ( ) ( )] ( ) [ ( ) ( )] a
x t x t x t v t
a
x t x t x t v t
a
x t x t v t
(5)
where a1, a2, a3are constants such that s3a s3 2a s2 a1 0 is Hurwitz. It is shown that for any
0
The outputs from TD in (5) exist time lagging phenomenon, Feng-Li [6] proposed an improved method based on Taylor’s formula. The third-order high-gain tracking differentiator introduced by Feng-Li [6] as follows,
2 (3) 3
1 1
( ) ( ) ( ) ( ) ( )
2 2
x t x t x t x t v t (7)
where ( )v t is the input, and x( )i ( )t is the output which will be used to track v( )i ( ),t i0,1, 2.
Theorem. Suppose that x t( ) is a solution of system (5) with any given initial condition. Assume
that ( )( ) , 0,1, 2.
vi t i Then for any
0 0
t , we have
( ) ( )
0
lim[ ( ) ( )] 0, [ , ].
i i
x t x t uniformly in t t (8)
More precisely, the transfer function and the phase-frequency characteristics from the input ( )i ( )
v t
to the output ( )i ( )
x t , which are denoted by Gi( )s and Gi(j) respectively, satisfy
4 4
( ) 1 ( ), ( ) ( ), 0.
i i
G s o G j o as (9)
The proof of this theorem can be seen in reference [6].
Next, we will try to use the ITDFs for the filtering of ECG data.
Simulation Analysis
In this section, to indicate the efficiency of denoising with the ITDFs, we will provide some of the simulation results.
First, the electrocardiogram (ECG) signal is taken from ECG record 105 of MIT-BIH arrhythmia database [15], we sign it as y. It contains 48 half-hours long two-channel ECG recordings sampled at
360 Hz with 11-bit resolution over a 10mV range.
When power frequency interference by the appropriate amplitude of f 50Hz sine wave
simulation, i.e., the noisy signal y0.1sin(2 ft). In this situation, the parameters of ITDF1 are
1500
r , h1 / 360. The related results are given in Figure 1.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time/s
-1 0 1 2
V
oltage/mv
Original signal
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time/s
-1 0 1 2
V
oltage/mv
[image:3.612.164.440.477.702.2]Filtering signal based on ITDF1
The parameter of ITDF2 is 0.02. Related results are given in Figure 2. The simulation results indicate the improved tracking-differentiator filters performance well when deal with the noisy ECG data. The filtering results, which can be seen in Fig. 1 and 2, indicate that the ITDFs work well. It is obviously that the filtering performance of ITDF2 is better than that of ITDF1. Note that, if it is given proper design parameters, ITDF1 could obtain better results. Based on these outputs, further analysis could be done.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time/s -1
0 1 2
V
oltage/m
v
Original signal with noise
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time/s -1
0 1 2
V
oltage/m
v
[image:4.612.145.459.157.411.2]Filtering signal based on ITDF2
Figure 2. The Filtering result of ECG record 105 with noise of MIT-BIH arrhythmia database based on ITDF2.
Conclusions
The improved tracking-differentiator filters based on Taylor’s formula are very easy and convenient tools for the filtering of ECG data. Simulation results show that these filters have good filtering performance, even though the noisy signal is with high frequency. And the performance of ITDF2 is better than that of ITDF1.
Acknowledgement
This work was supported by National Natural Science Foundation of China (Grant No.71531012).
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