A New Approach based Delay Differential Equation to Smoothing Filter Design

Document Type : Original Article

Author

Faculty of Computer Science, Fasa University, Fasa, Iran

Abstract

Among the techniques that are used for signal denoising, smoothing filters have received significant attention during the past. However, these methods are particularly suited for polynomial signal smoothing. Therefore, their performance is significantly decreased for signals that cannot be well modelled with a polynomial function. To overcome this limitation, in this paper, we propose a new approach to smoothing filter design, which is based on the delay differential equation model. In this approach, we propose to substitute the derivative of the signal with a DDE model of the signal. As an example, a delay differential equation of moving average (MA) model is used as penalty term in the optimization problem. The results indicate that a better solution can be found by appropriate balancing a trade-off between the MA model of the signal and the minimum mean square error. The proposed MA smoothing filter is analyzed in frequency domain. It is shown that the proposed MA smoothing filter displays good properties within its pass-band and stop-band bands for small values of window length. As an application, the proposed MA smoothing filter was used for electrocardiogram (ECG) signal denoising. We tested the method over data from the PhysioNet PTB database. The results show that the proposed MA smoothing filter outperforms the original smoothness priors or QV regularization.

Keywords

Main Subjects


[1] A. Shenoi, Introduction to Digital Signal Processing and Filter Design. New York, NY, USA: Wiley-Interscience, 2005.
[2] J. Blinchikoff and A. I. Zverev, Filtering in the Time and Frequency Domains. Melbourne, FL, USA: Krieger Publishing Co., Inc., 1986.
[3] Guest and N. Mijatovic, “Discrete-time complex bandpass filters for three-phase converter systems,” IEEE Trans. Ind. Electron., vol. 66, pp. 4650–4660, 2019.
[4] W. Schafer, “What is a savitzky-golay filter? [lecture notes],” IEEE Signal Processing Magazine, vol. 28, no. 4, pp. 111–117, July 2011.
[5] M. Stein, “Confidence sets for the mean of a multivariate normal distribution,” Journal of the Royal Statistical Society. Series B Methodological), vol. 24, pp. 265–296, 1962.
[6] Kopsinis and S. McLaughlin, “Development of emd-based denoising methods inspired by wavelet thresholding,” IEEE Transactions on Signal Processing, vol. 57, pp. 1351–1362, April 2009.
[7] L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc., vol. 90, pp. 1200–1224,1995.
[8] S. Crouse, R. D. Nowak, and R. G. Baraniuk, “Wavelet-based statistical signal processing using hidden markov models,” IEEE Transactions on Signal Processing, vol. 46, pp. 886–902, April 1998.
[9] L. Donoho, “De-noising by soft-thresholding,” IEEE Transactions on Information Theory, vol. 41, pp. 613–627, May 1995.
[10] Sameni, “Online filtering using piecewise smoothness priors: Application to normal and abnormal electrocardiogram denoising,” Signal Processing, vol. 133, pp. 52 – 63, 2017.
[11] Fasano and V. Villani, “Baselinewander removal for bioelectrical signals by quadratic variation reduction,” Signal Processing, vol. 99, pp. 48–57, 2014.
[12] Villani and A. Fasano, “Fast detrending of unevenly sampled series with application to hrv,” Computers in Cardiology, vol. 40, pp. 417–420, 2013.
[13] P. Tarvainen, P. O. Ranta-aho, and P. A. Karjalainen, “An advanced detrending method with application to hrv analysis,” IEEE Transactions on Biomedical Engineering, vol. 49, no. 2, pp. 172–175, 2002.
[14] Dong, D. Thanou, P. Frossard, and P. Vandergheynst, “Learning laplacian matrix in smooth graph signal representations,” IEEE Trans. Signal Process., vol. 64, no. 23, pp. 6160–6173, 2016.
[15] J. Wang, J. Li, H. Xu, H. O. Yan, J. Yuan, J. H. Li, X. H. Liu, Q. Zhou, and N. Li, “Smoothness prior approach to removing nonlinear trends from signals in identification of low frequency oscillation mode,” in Renewable Energy and Power Technology II, ser. Applied Mechanics and Materials, vol. 672. Trans Tech Publications, 11, pp. 1070–1074, 2014.
[16] Kheirati Roonizi and C. Jutten, “Improved smoothness priors using bilinear transform,” Signal Processing, vol. 169, p. 107381, 2020.
[17] Kheirati Roonizi and C. Jutten, “Forward-backward filtering and penalized least-squares optimization: A unified framework,” Signal Processing, vol. 178, p. 107796, 2021.
[18] Kheirati Roonizi and C. Jutten, “Band-stop smoothing filter design,” IEEE Transactions on Signal Processing, vol. 69, pp. 1797–1810, 2021.
[19] Kheirati Roonizi “ℓ2 and ℓ1 Trend Filtering: A Kalman Filter Approach,” IEEE Signal Processing Magazine, vol. 38, no. 6, pp. 137-145, Nov. 2021
[20] Wang, J. Liang, F. Gao, L. Zhang, and Z. Wang, “A method to improve the dynamic performance of moving average filter-based pll,” IEEE Trans. Power Electron., vol. 30, pp. 5978–5990, Oct 2015.
[21] Golestan, M. Ramezani, J. M. Guerrero, F. D. Freijedo, and M. Monfared, “Moving average filter based phase-locked loops: Performance analysis and design guidelines,” IEEE Trans. Power Electron., vol. 29, pp. 2750–2763, June 2014.
[22] J. Morales and Y. Shmaliy, “Moving average hybrid filter to the enhancing ultrasound image processing,” IEEE Latin America Transactions, vol. 8, pp. 9–16, March 2010.
[23] Salih, S. A. Aljunid, S. Aljunid, and O. Mask, “Adaptive Filtering Approach for Denoising Electrocardiogram Signal Using Moving Average Filter,” J Med Imaging Health Inform, vol. 5, pp. 1065—-1069, 2015.
[24] Rabiner and B.-H. Juang, Fundamentals of Speech Recognition. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 1993.
[25] G. Proakis and D. G. Manolakis, Digital Signal Processing (3rd Ed.): Principles, Algorithms, and Applications. Upper Saddle River, NJ,USA:Prentice-Hall, Inc., 1996.
[26] C. Gonzalez and R. E. Woods, Digital Image Processing (3rd Edition). Upper Saddle River, NJ, USA: Prentice-Hall, Inc., 2006.
[27] Kheirati Roonizi and R. Sassi, “A Signal Decomposition Model- Based Bayesian Framework for ECG Components Separation,” IEEE.Trans. Signal Process., vol. 64, pp. 665–674, 2016.
[28] Sayadi and M. B. Shamsollahi, "ECG Denoising and Compression Using a Modified Extended Kalman Filter Structure," in IEEE Transactions on Biomedical Engineering, vol. 55, no. 9, pp. 2240-2248, Sept. 2008.
[29] L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. C. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C.-K. Peng, and H. E. Stanley, “Physiobank, Physiotoolkit, and Physionet: Components of a new research resource for complex physiologic signals,” Circulation,vol. 101, pp. e215–e220, 2000.