A frequency-domain approach to interpolation from a nonuniform grid

Abstract We introduce a linear time-varying (LTV) system framework for the modeling and design of linear algorithms for interpolating band-limited signals from nonuniformly spaced samples. The LTV model characterizes the interpolation process in the frequency domain via the notion of bifrequency transmission function (BFTF). The BFTF provides a convenient means of assessing interpolator quality because it conveys how frequency components in the input are mapped to the frequency axis of the interpolated output. We show plots of BFTFs for several common linear interpolators, with an emphasis on a minimum-mean-squared-error method due to J.L. Yen. We prove that Yen's algorithm can be obtained as a special case of an optimal frequency-domain design of its BFTF using a weighted least-squares criterion. This design procedure is then generalized by using an unevenly weighted least-squares error measure that can incorporate knowledge of the approximate spectral shape of the original analog signal, to reduce interpolation error. Block interpolators are considered for situations where the number of signal samples is large or infinite, and methods are described for avoiding the matrix inversion to compute the interpolation. Finally, the performances of all interpolators studied in this paper are compared via simulation.

[1]  John L. Brown.  Anharmonic Approximation and Bandlimited Signals , 1967, Inf. Control..

[2]  S. Alexander,et al.  Analysis of a recursive least squares hyperbolic rotation algorithm for signal processing , 1988 .

[3]  Henry Stark,et al.  Iterative and one-step reconstruction from nonuniform samples by convex projections , 1990 .

[4]  Farrokh Marvasti,et al.  A Unified Approach to Zero-Crossings and Nonuniform Sampling of Single and Multidimensional Signals and Systems , 1987 .

[5]  S. Haykin,et al.  Adaptive Filter Theory , 1986 .

[6]  D. Wehner.  High Resolution Radar , 1987 .

[7]  Thomas W. Parks,et al.  An optimal recovery approach to interpolation , 1992, IEEE Trans. Signal Process..

[8]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[9]  K. Gröchenig.  RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING , 1992 .

[10]  Jan P. Allebach,et al.  Analysis of error in reconstruction of two-dimensional signals from irregularly spaced samples , 1987, IEEE Trans. Acoust. Speech Signal Process..

[11]  Gabor T. Herman,et al.  Image Reconstruction From Projections , 1975, Real Time Imaging.

[12]  Allan O. Steinhardt,et al.  Stabilized hyperbolic Householder transformations , 1989, IEEE Trans. Acoust. Speech Signal Process..

[13]  G. Stewart.  Introduction to matrix computations , 1973 .

[14]  Jan P. Allebach,et al.  Iterative reconstruction of bandlimited images from nonuniformly spaced samples , 1987 .

[15]  Eric John Diethorn.  A Linear, Time-Varying System Framework for Bandlimited Signal Extrapolation , 1987 .

[16]  Gene H. Golub,et al.  Matrix computations , 1983 .

[17]  David G. Messerschmitt,et al.  Adaptive Filters: Structures, Algorithms and Applications , 1984 .

[18]  Joos Vandewalle,et al.  Adaptive filters: structures, algorithms and applications , 1989 .

[19]  Giancarlo Calvagno.  A linear time-varying approach to the problem of signal reconstruction from a finite set of irregularly spaced noisy samples , 1993, 1993 IEEE International Symposium on Circuits and Systems.

[20]  Duncan J. Wingham.  The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition , 1992, IEEE Trans. Signal Process..

[21]  T. Claasen,et al.  On stationary linear time-varying systems , 1982 .

[22]  J. A. Simon.  ILLINOIS STATE GEOLOGICAL SURVEY , 1975 .

[23]  Yoram Bresler,et al.  Norm variance of minimax interpolation , 1992, IEEE Trans. Inf. Theory.

[24]  Farrokh Marvasti,et al.  Recovery of signals from nonuniform samples using iterative methods , 1991, IEEE Trans. Signal Process..

[25]  A. Macovski,et al.  Selection of a convolution function for Fourier inversion using gridding [computerised tomography application]. , 1991, IEEE transactions on medical imaging.

[26]  G. Swenson,et al.  Interferometry and Synthesis in Radio Astronomy , 1986 .

[27]  Jack Dongarra,et al.  LINPACK Users' Guide , 1987 .

[28]  Charles M. Rader,et al.  Hyperbolic householder transformations , 1986, IEEE Trans. Acoust. Speech Signal Process..

[29]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[30]  L. Levi,et al.  Fitting a bandlimited signal to given points , 1965, IEEE Trans. Inf. Theory.

[31]  H. Unz,et al.  Nonuniformly spaced arrays: The orthogonal method , 1966 .

[32]  David C. Munson,et al.  A frequency-domain characterization of interpolation from nonuniformly spaced data , 1989, IEEE International Symposium on Circuits and Systems,.

[33]  Richard P. Brent,et al.  A Note on Downdating the Cholesky Factorization , 1987 .

[34]  Frederick J. Beutler,et al.  Sampling Theorems and Bases in a Hilbert Space , 1961, Inf. Control..

[35]  G. Stewart.  The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization , 1979 .

[36]  A. Zemanian,et al.  Distribution theory and transform analysis , 1966 .