Power Line Interference Tracking in ECG Signal using State Space RLS

Maryam Butt1, Nauman Razzaq2, Ismail Sadiq3, Muhammad Salman4, Tahir Zaidi5 College of Electrical and Mechanical Engineering National University of Sciences and Technology

Rawalpindi, Pakistan 1maryambutt2011@ceme.nust.edu.pk, 2nauman.razzaq@ceme.nust.edu.pk, 3ismail_sadiq_ee@yahoo.com,

4salmanmasaud@ceme.nust.edu.pk, 5tahirzaidi@ceme.nust.edu.pk

Abstract—In this paper, we present an adaptive approach to Power Line Interference (PLI) tracking and elimination from ECG signal using State Space Recursive Least Squares (SSRLS) algorithm. The key benefit is that a separate reference power line signal is not required to track PLI signal having unknown frequency. Based on sinusoidal state-spaced model, SSRLS tracks the PLI using frequency estimate provided by a separate frequency estimator. SSRLS then locks on to the phase and amplitude of PLI for its subsequent elimination from ECG signal. Computer simulations carried out on simulated ECG signal demonstrate usefulness of the idea presented.

Keywords—ECG; PLI; SSRLS; adaptive noise cancellation;

I. INTRODUCTION Power Line Interference (PLI) is the main cause of distortion in ECG signal. Sometimes, the noise amplitude is so large that the ECG signal is totally covered by this noise. PLI is usually characterized as additive noise having properties like: A sinusoidal signal of fixed but unknown frequency (close to 50Hz) with unknown phase and amplitude [1]. PLI can disturb the clinical importance of the ECG signal.

Various techniques are given in literature for PLI removal from ECG signal. If the PLI parameters are fixed and known, then conventional notch filter is the best solution because it is easy to implement and has low computational complexity. However, it is difficult to achieve very small frequency band around the power line frequency while using notch filter. This leads to distortion of ECG frequency spectrum due to increased bandwidth of notch filter [2]. Consequently, fixed notch filter fails when the power line frequency drifts with time or with change in place. Here is when adaptive techniques help to solve the problem.

Widrow et al. introduced adaptive noise canceling concept in 1975 [3]. The adaptive filters can track the characteristics of PLI noise by updating filter coefficients. But most of the adaptive filtering techniques require a reference input, which might not always be available (as in portable equipment). The reference input also makes medical devices costly [4]. The adaptive filters like Least Mean Square (LMS) and Recursive Least Square (RLS) belong to that group of filters which requires power line signal as reference signal [5], [6].

In the literature, we also come across such adaptive noise cancelation algorithms which do not require reference power line input [1], [4] and [7]. For example, Acharaya et al. proposed the idea of tracking the amplitude; frequency and phase of noise continuously using an improved form of the short time Fourier transform [7]. Islam S. Badreldin et al. presented the adaptive removal of PLI noise and its harmonics from ECG signal without sampling an external power line reference signal in their

two separate papers [1], [4]. We have used State Space Recursive Least Squares (SSRLS) algorithm [8] to track the unknown parameters of PLI noise and finally subtracting it from noisy ECG signal to get our desired ECG signal. SSRLS also does not require the power line reference input.

II. PROPOSED METHOD FOR PLI TRACKING AND ELIMINATION

SSRLS is an adaptive algorithm which possesses good tracking performance and fast rate of convergence [9]. It uses linear model to track a signal. The SSRLS algorithm used to track PLI noise in ECG signal is explained below.

A. SSRLS Algorithm The unforced discrete time system is considered as follows

that generates y[k] which is a signal plus noise.

x[k + 1] = A[k]x[k]

y[k] = C[k]x[k] + v[k] (1)

where nx R∈ is the state vector, my R∈ is the output vector such that m ≤ n , v[k] is the observation noise and k is the discrete time index. The matrices A[k] and C[k] are supposed to be - s tepl observable and full-rank [10].

In SSRLS, the estimated state x [k ] is defined as [9] x[k] = x[k] + K[k] [k] (2)

where x[k] is termed as predicted state given by ˆx[k] = Ax[k - 1] (3) The [k] is the prediction error or innovations, given by [k] = y[k] - y[k] (4)

In (4), y[k] is the predicted output state given as follows ˆy[k] = Cx[k] = CAx[k - 1] (5) The iterative update of time average correlation matrix

[k] is given by

-T -1 T[k] = A [k - 1]A + C C (6) SSRLS filter’s memory (also termed as forgetting factor) is

represented by parameter and its value should be close to but less than 1. The estimator gain K[k] can be found as follows

1 TK[k] = [k]C− (7) PLI signal is usually modeled as sinusoidal signal so we have

adopted the following sinusoidal model for SSRLS [9]

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cos( [k]T) sin( [k]T)

A[k] =-sin( [k]T) cos( [k]T)

[ ]C = 1 0 (8) where T is the sampling time. In (8), the state transition matrix A[k] is time varying due to recursive update of frequency . The states and coefficients are related by [8]

akx[k] = Ab

(9)

Initial conditions are a = coss φ and b = sins φ . The

amplitude s , signal frequency , and initial phase φ are all assumed to be unknown. Rearranging (9), we get

ˆ

ˆˆ

a -k= A x[k]

b

(10)

In (10), -kA are computed using following properties of state transition matrix A [10]

-1A [k, j] = A[j,k]

A[k,i] = A[k, j]A[j,i]

A[k +1,k] = A[k]

(11)

From (10), we can find the initial conditions that help us to find the phase estimate by the following relation

ˆˆˆ

-1 b[k] = tana

φ (12)

In (12), we may encounter the discontinuities because of recursive update of φ , so we removed it by unwrapping the phase [11]. The frequency update equation using a stochastic gradient like algorithm is given by [8]

ˆ ˆˆ ˆ [k] - [k - 1][k] = [k - 1] + ( )φ φ (13) In (13), the parameter is termed as step-size parameter and it has significant effect on tracking results of SSRLS algorithm which is discussed in detail in parametric study section III C. The difference of φ is computed by passing it through a discrete filter having the transfer function [8]

z-1H(z) = 2z (14)

B. PLI tracking using SSRLS Algorithm The tracking scheme of PLI noise has been depicted by the block diagram in Fig.1.

Fig. 1. Scheme for tracking PLI noise using SSRLS filter

In Fig.1, u[k] represents the noise-free ECG signal, v[k] is the PLI signal and y[k] is noisy ECG signal. Mathematically, v[k] is represented as

v[k] = sin( kT + )s φ (15) In computer simulations, following parameters are used to

generate sinusoidal signal acting as PLI noise 0.1= ,s = 2 49.5× , T=1/360 and / 4φ = . The output of SSRLS filter is given to phase estimator block

which estimates the phase of PLI signal. The output of phase estimator block serves as input to frequency estimator block which gives us estimated frequency of PLI noise according to (13). The sinusoidal state-spaced model is updated with new estimate of . SSRLS uses updated sinusoidal state-spaced model. After some iteration, the SSRLS filter is able to track the PLI signal. The PLI signal y[k] estimated by SSRLS is subtracted from noisy ECG signal y[k] and we obtained estimated ECG signal represented by d[k]. This signal d[k] has high SNR as compared to y[k]. It is noted that SSRLS algorithm does not need any reference signal like the adaptive techniques mentioned in [1], [4] and [7].

III. COMPUTER SIMULATIONS In this section, the results of SSRLS algorithm in case of unknown PLI parameters are presented in comparison with fixed notch filter. The effect of variation in values of some parameters used in the SSRLS algorithm is also investigated.

We have taken noise free ECG signal sampled at 360Hz from MIT-BIH database [12] and added sinusoidal signal, keeping SNR=10dB to make noisy ECG signal. Fig. 2 shows the noise free ECG signal called pure ECG signal and Fig. 3 shows the noisy ECG signal with PLI of 49.5Hz frequency.

Fig. 2. Pure ECG signal

Fig. 3. ECG signal corrputed by PLI

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A. 50 Hz Notch Filter In this section, performance of notch filter has been presented. For a general 2nd order notch filter, the frequency response is given by

2

2z0

z

z b + b + b1 2H(z) =z + a + a1 2

(16)

We have used 50Hz notch filter with attenuation level 50dB for PLI noise removal from ECG signal. The coefficient vectors of this filter are b= [0.9472, -1.2177, 0.9472] and a= [1.0000, -1.2177,0.8944]. The frequency plot of notch filter having 50dB attenuation level is shown in Fig. 4 while output ECG signal from this notch filter is shown in Fig. 5. The residual amount of PLI noise can be seen in all the three cycles of ECG signal in Fig. 5.

Fig. 4. Frequency plot of 50 Hz Notch filter

Fig. 5. ECG signal after suppressing PLI using 50Hz Notch filter

Fig. 6 shows the error between pure ECG signal (Fig. 2) and output ECG signal from notch filter (Fig. 5). Note that the notch frequency 50Hz is intentionally kept different from PLI frequency of 49.5Hz to show our ignorance of PLI frequency knowledge in any practical situation. Fig. 5 and Fig. 6 show that if we do not know the PLI parameters exactly, then notch filter performance degrades and is unable to remove PLI noise completely from ECG signal.

Fig. 6. Error signal using 50Hz Notch filter

B. SSRLS Algorithm SSRLS algorithm is initialized using following parameters:

, the forgetting factor, is assumed to be 0.999. Two values of are used i.e. 0.01 & 0.5 and φ is initialized with 0. Initial

value of is taken as 50Hz. SSRLS is initialized with inaccurate information about PLI parameters, so as to investigate convergence behavior of SSRLS algorithm. Fig. 7 shows that SSRLS converges to correct value of PLI frequency. For higher value of parameter (0.5 in our case), SSRLS converges to true value faster as compared to smaller value of . Therefore =0.5 is used in subsequent results.

Fig. 7. Adaptation of PLI noise frequency using SSRLS algorithm

The Fig. 8 shows the esitmated ECG signal using SSRLS. It can be seen that SSRLS converges quickly to remove complete PLI noise from ECG signal.

Fig. 8. Estimated ECG signal using SSRLS algorithm

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The plot of estimation error between pure ECG signal (Fig. 2) and estimated ECG using SSRLS algorithm (Fig. 8) is shown in Fig. 9. The amplitude of error signal in case of SSRLS filter is much less as compared to error signal in case of notch filter in Fig. 6. This shows the superior performance of SSRLS algorithm.

Fig. 9. Error in estimation by SSRLS algorithm

C. Parametric Study In this section, we investigate the effect of variation in the

values of some parameters used by SSRLS algorithm in tracking PLI noise.

1) Effect of parameter : The parameter used in (13)

has an effect on convergence rate of SSRLS algorithm which is tracking unknown PLI noise in ECG signal. The higher value of

leads to faster convergence of SSRLS to accurate value of PLI frequency as compared to lower value of . Table I and Fig. 7 depict this effect. The =0.5 is found to be a suitable value for the problem at hand.

TABLE I. PARAMETRIC STUDY OF SSRLS

2) Signal to Noise Ratio(SNR): It is observed that the SSRLS based system shown in Fig. 1 is able to track PLI signal when SNR is low. This is because phase and frequency estimator blocks try to estimate phase and frequency, respectively, of PLI signal. The job of these blocks becomes more difficult in the presence of a strong ECG signal or low amplitude of noise (i.e. ECG signal having high SNR). SSRLS based system operates well for SNR values from 0 dB to 16dB approximately. To solve high SNR problem, we can use another scheme shown in Fig. 10. In this scheme, two SSRLS filters are employed. We have used a bandpass filter before first SSRLS

filter. The bandpass filter passes all frequencies between 47Hz to 52Hz and blocks all other frequencies in noisy ECG signal. First SSRLS filter estimates the PLI noise frequnecy which is given to second SSRLS filter. In this way, SSRLS algorithm is able to remove PLI from ECG signal with high SNR at the cost of increased complexity. A detailed study of this model is planned to be appeared in a future paper.

Fig. 10. Scheme for tracking PLI noise with high SNR

IV. CONCLUSION In this paper, a new method of PLI tracking and elimination from ECG signal has been introduced. SSRLS algorithm works to track PLI noise signal with all its parameters unknown. It is also shown that notch filter is unable to remove PLI noise if its frequency is unknown because notch filter requires the exact knowledge of PLI noise frequency to remove it from ECG signal. The parametric study of SSRLS algorithm is also done and optimal values of its parameters are also found for the problem at hand. SSRLS algorithm operates well for low SNR and solution for high SNR case is also provided in this paper. This scheme for high SNR will be investigated using computer simulations in some of our future work.

ACKNOWLEDGMENT We acknowledge the support of National Information &

Communication Technologies Research and Development (ICT R&D) fund and National Institute of Heart Diseases (NIHD), Pakistan for their help in this work.

REFERENCES [1] I. S. Badreldin, D. S. EI-Kholy, and A. A. EI-Wakil, “Modified adaptive noise canceler for electrocardiography with no power-line reference,” Cairo International Biomedical Engineering Conference (CIBEC'IO), Cairo, Egypt, Dec. 2010. [2] Maryam Butt, Nouman Razzaq, Ismail Sadiq, Muhammad Salman and Tahir Zaidi, “Power line interference removal from ECG signal using SSRLS algorithm”, IEEE 9th International Colloquium on Signal Processing and its Applications, in press. [3] B. Widrow et al., “Adaptive noise cancelling: Principles and applications,” IEEE proceedings, vol. 63, no. 12, pp. 1692-1716, Dec. 1975. [4] I. S. Badreldin, D. S. EI-Kholy, and A. A. EI-Wakil, “Harmonic adaptive noise canceler for electrocardiography with no power-line reference,” Electrotechnical Conference (MELECON), Mediterranean, March 2012.

Sr. No. Parameter

Convergence after samples (approx.)

1. 0.005 870

2. 0.01 550

3. 0.05 90

4. 0.1 40

5. 0.15 25

6. 0.2 18

7. 0.5 12

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[5] Md. Maniruzzaman, Kazi Md. Shimul Billah, Uzzal Biswas, and Bablu Gain, “Least-Mean-Square algorithm based adaptive filters for removing power line interference from ECG signal”, IEEE/OSA/APR International Conference on Infonnatics, Electronics & Vision, pp-737-740, 2012. [6] Chinmay Chandrakar, and M.K. Kowar, “Denoising ECG signals using adaptive filter algorithm”, International Journal of Soft Computing and Engineering (IJSCE), vol. 2, pp. 120-123, March 2012. [7] Soumyadipta Acharya, Dale H. Mugle, Bruce C. Taylor, “A fast adaptive filter for electrocardiography”, Proceedings of the IEEE 30th Annual Northeast Bioengineering Conference, April 2004. [8] Mohammad Bilal Malik and Muhammad Salman, “Adaptive Tracking of a Noisy Sinusoid/Chirp with Unknown Parameters”, IEEE ISIE 2006 , Canada, July 2006.

[9] Mohammad Bilal Malik “State-Space Recursive Least Squares: part I ,” Signal Processing, vol. 84/9, pp 1709-1728, 2004. [10] Wilson J. Rugh, Linear System Theory, 2nd Ed. Prentice Hall, 1996. [11] Richard G. Lyons, Understanding Digital Signal Processing, 2nd Ed, Pearsan Education, 2004. [12] A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. Ch.Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C. K. Peng, and H.E. Stanley. (2000) PhysioBank, PhysioToolkit, and Physionet: Components of a new research resource for complex physiologic signals. Circulation [Online] Available: Circulation Electronic Pages: http://circ.ahajournals.org/cgi/content/full/101/23/e215.

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