Dynamic harmonic analysis with finite impulse response filters designed with O-splines
DOI:
https://doi.org/10.29105/ingenierias24.91-18Keywords:
Cardinal splines, central Lagrange interpolation kernel, Taylor-Fourier discrete time transform, oscillatory signals, power oscillations, time-frequency analysis, multi-resolution analysis, data compressionAbstract
Splines are at the essence of signal processing. Not only in sampling and interpolation, but also in filter design, image processing, and multi-resolution analysis. A new class of splines is presented here. They are referred to as O-splines since their knots are separated by one fundamental cycle. They are used as optimal state samplers, in the sense that their coefficients provide the derivatives for the best Taylor approximation to a given signal about a time instance or the best Hermite interpolation between two of them. They are the impulse response of the filters of the Discrete-Time Taylor-Fourier Transform (DTTFT) filter bank. Lowpass O-spline coincides with the Lagrange central interpolation kernel, which converges towards the ideal Sinc function. It comes with its derivatives which in turn converge to the ideal lowpass differentiator. The bandpass O-splines are harmonic splines since they are modulations of the former kernel at harmonic frequencies. In closed-form, they reduce the computational complexity of the DTTFT and can be used to design ideal bandpass filters at a particular frequency. By increasing the order they define a ladder of spaces very useful for multi-resolution and time-frequency analysis. Examples are provided at the end of the paper. Naturally, a new family of wavelets is coming soon from these splines.
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C. de Boor, A practical Guide to Splines. Springer, 2001.
I. J. Schoenberg, Cardinal Spline Interpolation. SIAM, 1973. DOI: https://doi.org/10.1137/1.9781611970555
D. Pang, L. A. Ferrari, and P. Sankar, “B-spline FIR filters,” Circuits Systems and Signal Process, vol. 13, no. 1, pp. 31–64, 1994, available: https://doi.org/10.1007/BF01183840. DOI: https://doi.org/10.1007/BF01183840
T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imag., vol. 18, no. 11, pp. 1049–1075, Nov 1999. DOI: https://doi.org/10.1109/42.816070
S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 7, pp. 674–693, July 1989. DOI: https://doi.org/10.1109/34.192463
M. Unser, A. Aldroubi, and M. Eden, “B-spline signal processing. Part I– theory,” IEEE Trans. Signal Process., vol. 41, no. 2, pp. 821–833, Feb 1993. DOI: https://doi.org/10.1109/78.193220
——, “B-spline signal processing. Part II – efficiency design and applications,” IEEE Trans. Signal Process., vol. 41, no. 2, pp. 834–848, Feb 1993. DOI: https://doi.org/10.1109/78.193221
M. Unser and T. Blu, “Cardinal exponential splines: part I - theory and filtering algorithms,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1425–1438, April 2005. DOI: https://doi.org/10.1109/TSP.2005.843700
M. Unser, “Cardinal exponential splines: part II - think analog, act digital,” IEEE Trans. Signal Process., vol. 53, no. 4, pp. 1439–1449, April 2005. DOI: https://doi.org/10.1109/TSP.2005.843699
J. A. de la O Serna, “Analyzing power oscillating signals with the O-splines of the Discrete Taylor–Fourier Transform,” IEEE Trans. Power Syst., vol. 33, no. 6, pp. 7087–7095, Nov 2018. DOI: https://doi.org/10.1109/TPWRS.2018.2832615
J. A. de la O Serna, M. R. A. Paternina, A. Zamora-Méndez, R. K. Tripathy, and R. B. Pachori, “EEG-rhythm specific taylor-fourier filter bank implemented with O-splines for the detection of epilepsy using EEG signals,” IEEE Sensors J., pp. 1–1, 2020. DOI: https://doi.org/10.1109/JSEN.2020.2976519
E. H. W. Meijering, W. J. Niessen, and M. A. Viergever, “The Sinc-approximating kernels of classical polynomial interpolation,” in IEEE International Conference on Image Processing (ICIP’99), vol. 3, Oct 1999, pp. 652–656.
M. Unser and A. Aldroubi, “A general sampling theory for nonideal acquisition devices,” IEEE Trans. Signal Process., vol. 42, no. 11, pp. 2915–2925, Nov 1994. DOI: https://doi.org/10.1109/78.330352
E. Meijering, “A chronology of interpolation: from ancient astronomy to modern signal and image processing,” Proceedings of the IEEE, vol. 90, no. 3, pp. 319–342, March 2002. DOI: https://doi.org/10.1109/5.993400
J. Chen and Z. Cai, “Cardinal MK-spline signal processing: Spatial interpolation and frequency domain filtering,” Information Sciences, vol. 495, pp. 116 – 135, 2019. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0020025519303767 DOI: https://doi.org/10.1016/j.ins.2019.04.056
J. Chen and Z. Cai, “A new class of explicit interpolatory splines and related measurement estimation,” IEEE Trans. Signal Process., pp. 1–1, 2020. DOI: https://doi.org/10.1109/TSP.2020.2984477
G. Rogers, Power System Oscillations. Boston: Springer, 2000. DOI: https://doi.org/10.1007/978-1-4615-4561-3
A. Roman Messina, Inter-area Oscillations in Power Systems: A Nonlinear and Nonstationary Perspective. Springer, 2009. DOI: https://doi.org/10.1007/978-0-387-89530-7
M. Forouzanfar, H. R. Dajani, V. Z. Groza, M. Bolic, S. Rajan, and I. Batkin, “Oscillometric blood pressure estimation: Past, present, and future,” IEEE Reviews in Biomedical Engineering, vol. 8, pp. 44–63, 2015. DOI: https://doi.org/10.1109/RBME.2015.2434215
J. G. Proakis and M. Salehi, Digital Communications, 5th ed. Boston, MA: McGraw Hill, 2008, sec. 2.1.
M. Vetterli, K. Kovacevic, and V. K. Goyal, Foundations of Signal Processing, 3rd ed. Cambridge, UK: Cambridge University Press, 2014, available at http://www.fourierandwavelets.org/. DOI: https://doi.org/10.1017/CBO9781139839099
H. W. Meijering, “Image enhancement in digital x-ray angiography.” [Online]. Available: https://dspace.library.uu.nl/bitstream/handle/1874/367/c6.pdf
A. K. Jain, Fundamentals of Digital Image Processing. Prentice-Hall,Englewood Cliffs, New Jersey,USA, 1989.
J. A. De la O Serna, “Taylor-Fourier analysis of blood pressure oscillometric waveforms,” IEEE Trans. Instrum. Meas., vol. 62, no. 9, pp. 2511–2518, 2013. [Online]. Available: http://dx.doi.org/10.1109/TIM.2013.2258245 DOI: https://doi.org/10.1109/TIM.2013.2258245
R. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Signal Process., vol. 29, no. 6, pp. 1153–1160, December 1981. DOI: https://doi.org/10.1109/TASSP.1981.1163711
