Measuring Smoothness of Trigonometric Interpolation Through Incomplete Sample Points

In this paper we present a metric to assess the smoothness of a trigonometric interpolation through an in-complete set of sample points. We measure smoothness as the power of a particular derivative of a 2π-periodic Dirichlet interpolant through some sample points. We show that we do not need to explicitly complete the sample set or perform the interpolation, but can simply work with the available sample points, under the assumption that any missing points are chosen to minimise the metric, and present a simple and robust approach to the computation of this metric. We assess the accuracy and computational complexity of this approach, and compare it to benchmarks.

[1]  Ian K. Proudler,et al.  Measuring Smoothness of Real-Valued Functions Defined by Sample Points on the Unit Circle , 2019, 2019 Sensor Signal Processing for Defence Conference (SSPD).

[2]  Chi Hieu Ta,et al.  Shortening the order of paraunitary matrices in SBR2 algorithm , 2007, 2007 6th International Conference on Information, Communications & Signal Processing.

[3]  Ian K. Proudler,et al.  Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvalues , 2021, IEEE Transactions on Signal Processing.

[4]  Ian K. Proudler,et al.  Efficient Implementation of Iterative Polynomial Matrix EVD Algorithms Exploiting Structural Redundancy and Parallelisation , 2019, IEEE Transactions on Circuits and Systems I: Regular Papers.

[5]  A. Bunse-Gerstner,et al.  Numerical computation of an analytic singular value decomposition of a matrix valued function , 1991 .

[6]  Stephan Weiss,et al.  Maximally Smooth Dirichlet Interpolation from Complete and Incomplete Sample Points on the Unit Circle , 2019, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[7]  Kunio Tanabe,et al.  An Algorithm for Computing the Analytic Singular Value Decomposition , 2008 .

[8]  John G. McWhirter,et al.  Row-shift corrected truncation of paraunitary matrices for PEVD algorithms , 2015, 2015 23rd European Signal Processing Conference (EUSIPCO).

[9]  J. Selva.  FFT Interpolation From Nonuniform Samples Lying in a Regular Grid , 2015, IEEE Transactions on Signal Processing.

[10]  John G. McWhirter,et al.  Design of FIR Paraunitary Filter Banks for Subband Coding Using a Polynomial Eigenvalue Decomposition , 2011, IEEE Transactions on Signal Processing.

[11]  W. Wirtinger.  Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen , 1927 .

[12]  John G. McWhirter,et al.  Relevance of polynomial matrix decompositions to broadband blind signal separation , 2017, Signal Process..

[13]  Stephan D. Weiss,et al.  An extension of the MUSIC algorithm to broadband scenarios using a polynomial eigenvalue decomposition , 2011, 2011 19th European Signal Processing Conference.

[14]  John G. McWhirter,et al.  An EVD Algorithm for Para-Hermitian Polynomial Matrices , 2007, IEEE Transactions on Signal Processing.

[15]  Ian K. Proudler,et al.  Iterative Approximation of Analytic Eigenvalues of a Parahermitian Matrix EVD , 2019, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[16]  Ian K. Proudler,et al.  On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix , 2018, IEEE Transactions on Signal Processing.

[17]  Stephan D. Weiss,et al.  MVDR broadband beamforming using polynomial matrix techniques , 2015, 2015 23rd European Signal Processing Conference (EUSIPCO).

[18]  John G. McWhirter,et al.  Sequential Matrix Diagonalisation Algorithms for Polynomial EVD of Parahermitian Matrices , 2014 .

[19]  Stephan D. Weiss,et al.  Corrections to "On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix" , 2018, IEEE Trans. Signal Process..

[20]  K. Wright.  Differential equations for the analytic singular value decomposition of a matrix , 1992 .

[21]  Timo Eirola,et al.  On Smooth Decompositions of Matrices , 1999, SIAM J. Matrix Anal. Appl..

[22]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .