**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32856

##### Piecewise Interpolation Filter for Effective Processing of Large Signal Sets

**Authors:**
Anatoli Torokhti,
Stanley Miklavcic

**Abstract:**

**Keywords:**
Wiener filter,
filtering of stochastic signals.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1079770

**References:**

[1] H. Kopka and P. W. Daly, A Guide to LATEX, 3rd ed. Harlow, England: Addison-Wesley, 1999.

[2] J. Chen, J. Benesty, Y. Huang, and S. Doclo, New Insights Into the Noise Reduction Wiener Filter, IEEE Trans. on Audio, Speech, and Language Processing, 14, No. 4, pp. 1218 - 1234, 2006.

[3] M. Spurbeck and P. Schreier, Causal Wiener filter banks for periodically correlated time series, Signal Processing, 87, 6, pp. 1179-1187, 2007.

[4] J. S. Goldstein, I. Reed, and L. L. Scharf, "A Multistage Representation of the Wiener Filter Based on Orthogonal Projections," IEEE Trans. on Information Theory, vol. 44, pp. 2943-2959, 1998.

[5] Y. Hua, M. Nikpour, and P. Stoica, "Optimal Reduced-Rank estimation and filtering," IEEE Trans. on Signal Processing, vol. 49, pp. 457-469, 2001.

[6] A. Torokhti and P. Howlett, Computational Methods for Modelling of Nonlinear Systems, Elsevier, 2007.

[7] E. D. Sontag, Polynomial Response Maps, Lecture Notes in Control and Information Sciences, 13, 1979.

[8] S. Chen and S. A. Billings, Representation of non-linear systems: NARMAX model, Int. J. Control, vol. 49, no. 3, pp. 1013-1032, 1989.

[9] V. J. Mathews and G. L. Sicuranza, Polynomial Signal Processing, J. Wiley & Sons, 2001.

[10] A. Torokhti and P. Howlett, Optimal Transform Formed by a Combination of Nonlinear Operators: The Case of Data Dimensionality Reduction, IEEE Trans. on Signal Processing, 54, No. 4, pp. 1431-1444, 2006.

[11] A. Torokhti and P. Howlett, Filtering and Compression for Infinite Sets of Stochastic Signals, Signal Processing, 89, pp. 291-304, 2009.

[12] J. Vesma and T. Saramaki, Polynomial-Based Interpolation Filters - Part I: Filter Synthesis, Circuits, Systems, and Signal Processing, Volume 26, Number 2, Pages 115-146, 2007.

[13] A. Torokhti and J. Manton, Generic Weighted Filtering of Stochastic Signals, IEEE Trans. on Signal Processing, 57, issue 12, pp. 4675-4685, 2009.

[14] A. Torokhti and S. Miklavcic, Data Compression under Constraints of Causality and Variable Finite Memory, Signal Processing, 90 , Issue 10, pp. 2822-2834, 2010.

[15] I. Babuska, U. Banerjee, J. E. Osborn, Generalized finite element methods: main ideas, results, and perspective, International Journal of Computational Methods, 1 (1), pp. 67-103, 2004.

[16] S. Kang and L. Chua, A global representation of multidimensional piecewise-linear functions with linear partitions, IEEE Trans. on Circuits and Systems, 25 Issue:11, pp. 938 - 940, 1978.

[17] L.O. Chua and A.-C. Deng, Canonical piecewise-linear representation, IEEE Trans. on Circuits and Systems, 35 Issue:1, pp. 101-111, 1988.

[18] J.-N. Lin and R. Unbehauen, Adaptive nonlinear digital filter with canonical piecewise-linear structure, IEEE Trans. on Circuits and Systems, 37 Issue:3, pp. 347 - 353, 1990.

[19] J.-N. Lin and R. Unbehauen, Canonical piecewise-linear approximations, IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, 39 Issue:8, pp. 697 - 699, 1992.

[20] S.B. Gelfand and C.S. Ravishankar, A tree-structured piecewise linear adaptive filter, IEEE Trans. on Inf. Theory, 39, issue 6, pp. 1907-1922, 1993.

[21] E.A. Heredia and G.R. Arce, Piecewise linear system modeling based on a continuous threshold decomposition, IEEE Trans. on Signal Processing, 44 Issue:6, pp. 1440 - 1453, 1996.

[22] G. Feng, Robust filtering design of piecewise discrete time linear systems, IEEE Trans. on Signal Processing, 53 Issue:2, pp. 599 - 605,2005.

[23] F. Russo, Technique for image denoising based on adaptive piecewise linear filters and automatic parameter tuning, IEEE Trans. on Instrumentation and Measurement, 55, Issue:4, pp. 1362 - 1367, 2006.

[24] J.E. Cousseau, J.L. Figueroa, S. Werner, T.I. Laakso, Efficient Nonlinear Wiener Model Identification Using a Complex-Valued Simplicial Canonical Piecewise Linear Filter, IEEE Trans. on Signal Processing, 55 Issue:5, pp. 1780 - 1792, 2007.

[25] P. Julian, A. Desages, B. D-Amico, Orthonormal high-level canonical PWL functions with applications to model reduction, IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, 47 Issue:5, pp. 702 - 712, 2000.

[26] T. Wigren, Recursive Prediction Error Identification Using the Nonlinear Wiener Model, Automatica, 29, 4, pp. 1011-1025, 1993.

[27] G. H. Golub and C. F. van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1996.

[28] T. Anderson, An Introduction to Multivariate Statistical Analysis, New York, Wiley, 1984.

[29] L. I. Perlovsky and T. L. Marzetta, Estimating a Covariance Matrix from Incomplete Realizations of a Random Vector, IEEE Trans. on Signal Processing, 40, pp. 2097-2100, 1992.

[30] O. Ledoit and M. Wolf, A well-conditioned estimator for largedimensional covariance matrices, J. Multivariate Analysis 88, pp. 365-411, 2004.