Algebraic Signal Processing Theory: 1-D Nearest Neighbor Models

We present a signal processing framework for the analysis of discrete signals represented as linear combinations of orthogonal polynomials. We demonstrate that this representation implicitly changes the associated shift operation from the standard time shift to the nearest neighbor shift introduced in this paper. Using the algebraic signal processing theory, we construct signal models based on this shift and derive their corresponding signal processing concepts, including the proper notions of signal and filter spaces, z-transform, convolution, spectrum, and Fourier transform. The presented results extend the algebraic signal processing theory and provide a general theoretical framework for signal analysis using orthogonal polynomials.

[1]  Jean-Bernard Martens,et al.  Local orientation analysis in images by means of the Hermite transform , 1997, IEEE Trans. Image Process..

[2]  A. J. Jerri,et al.  Integral and Discrete Transforms with Applications and Error Analysis , 1992 .

[3]  R. Merletti,et al.  Hermite expansions of compact support waveforms: applications to myoelectric signals , 1994, IEEE Transactions on Biomedical Engineering.

[4]  Anna R. Karlin,et al.  Random Walks and Undirected Graph Connectivity: A Survey , 1995 .

[5]  Jelena Kovacevic,et al.  Compression of QRS complexes using Hermite expansion , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[6]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[7]  Gabriele Steidl,et al.  Fast algorithms for discrete polynomial transforms , 1998, Math. Comput..

[8]  Harry C. Andrews,et al.  Multidimensional Rotations in Feature Selection , 1971, IEEE Transactions on Computers.

[9]  P. Laguna,et al.  Adaptive estimation of QRS complex wave features of ECG signal by the hermite model , 2007, Medical and Biological Engineering and Computing.

[10]  M. Puschel,et al.  Fourier transform for the spatial quincunx lattice , 2005, IEEE International Conference on Image Processing 2005.

[11]  Olle Pahlm,et al.  A Method for Evaluation of QRS Shape Features Using a Mathematical Model for the ECG , 1981, IEEE Transactions on Biomedical Engineering.

[12]  Dennis M. Healy,et al.  Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs , 1997, SIAM J. Comput..

[13]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[14]  A. Sandryhaila,et al.  Nearest-neighbor image model , 2012, 2012 19th IEEE International Conference on Image Processing.

[15]  Markus Püschel.  Algebraic Signal Processing Theory: An Overview , 2006 .

[16]  Jelena Kovacevic,et al.  Algebraic Signal Processing Theory: Sampling for Infinite and Finite 1-D Space , 2010, IEEE Transactions on Signal Processing.

[17]  Aliaksei Sandryhaila,et al.  Algebraic Signal Processing: Modeling and Subband Analysis , 2010 .

[18]  Jerry D. Gibson,et al.  Handbook of Image and Video Processing , 2000 .

[19]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[20]  Jean-Bernard Martens.  The Hermite transform-applications , 1990, IEEE Trans. Acoust. Speech Signal Process..

[21]  Jean-Bernard Martens,et al.  The Hermite transform-theory , 1990, IEEE Trans. Acoust. Speech Signal Process..

[22]  Walter A. Robinson.  Modeling Dynamic Climate Systems , 2001, Modeling Dynamic Systems.

[23]  Giridhar D. Mandyam,et al.  The discrete Laguerre transform: derivation and applications , 1996, IEEE Trans. Signal Process..

[24]  Erik A. van Doorn,et al.  Birth-death processes and associated polynomials , 2003 .

[25]  T. Hughes,et al.  Signals and systems , 2006, Genome Biology.

[26]  Jelena Kovacevic,et al.  Efficient Compression of QRS Complexes Using Hermite Expansion , 2012, IEEE Transactions on Signal Processing.

[27]  P. Fuhrmann.  A Polynomial Approach to Linear Algebra , 1996 .

[28]  Jelena Kovacevic,et al.  Algebraic Signal Processing Theory: Cooley-Tukey-Type Algorithms for Polynomial Transforms Based on Induction , 2010, SIAM J. Matrix Anal. Appl..

[29]  Sariel Har-Peled,et al.  Random Walks , 2021, Encyclopedia of Social Network Analysis and Mining.

[30]  Nikhil Balram,et al.  Recursive structure of noncausal Gauss-Markov random fields , 1992, IEEE Trans. Inf. Theory.

[31]  Christine Connolly,et al.  Handbook of Image and Video Processing 2nd Edition (Hardback) , 2006 .

[32]  W. Washington,et al.  An Introduction to Three-Dimensional Climate Modeling , 1986 .

[33]  José M. F. Moura,et al.  DCT/DST and Gauss-Markov fields: conditions for equivalence , 1998, IEEE Trans. Signal Process..

[34]  M. Puschel,et al.  Algebraic Signal Processing Theory: 2-D Spatial Hexagonal Lattice , 2007, IEEE Transactions on Image Processing.

[35]  Markus Püschel,et al.  Algebraic Signal Processing Theory: Cooley–Tukey Type Algorithms for Real DFTs , 2008, IEEE Transactions on Signal Processing.

[36]  José M. F. Moura,et al.  Algebraic Signal Processing Theory: Cooley–Tukey Type Algorithms for DCTs and DSTs , 2007, IEEE Transactions on Signal Processing.

[37]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[38]  Bradley K. Alpert,et al.  A Fast Algorithm for the Evaluation of Legendre Expansions , 1991, SIAM J. Sci. Comput..

[39]  José M. F. Moura,et al.  Algebraic Signal Processing Theory , 2006, ArXiv.

[40]  G. Mandyam,et al.  Application of the discrete Laguerre transform to speech coding , 1995, Conference Record of The Twenty-Ninth Asilomar Conference on Signals, Systems and Computers.

[41]  A.K. Krishnamurthy,et al.  Multidimensional digital signal processing , 1985, Proceedings of the IEEE.

[42]  P. Yip,et al.  Discrete Cosine Transform: Algorithms, Advantages, Applications , 1990 .

[43]  Markus Püschel,et al.  Algebraic Signal Processing Theory: Foundation and 1-D Time , 2008, IEEE Transactions on Signal Processing.

[44]  José M. F. Moura,et al.  Algebraic Signal Processing Theory: 1-D Space , 2008, IEEE Transactions on Signal Processing.