Signal Reconstruction from Nonuniformly Spaced Samples Using Evolutionary Slepian Transform-Based POCS

We consider the reconstruction of signals from nonuniformly spaced samples using a projection onto convex sets (POCSs) implemented with the evolutionary time-frequency transform. Signals of practical interest have finite time support and are nearly band-limited, and as such can be better represented by Slepian functions than by sinc functions. The evolutionary spectral theory provides a time-frequency representation of nonstationary signals, and for deterministic signals the kernel of the evolutionary representation can be derived from a Slepian projection of the signal. The representation of low pass and band pass signals is thus efficiently done by means of the Slepian functions. Assuming the given nonuniformly spaced samples are from a signal satisfying the finite time support and the essential band-limitedness conditions with a known center frequency, imposing time and frequency limitations in the evolutionary transformation permit us to reconstruct the signal iteratively. Restricting the signal to a known finite time and frequency support, a closed convex set, the projection generated by the time-frequency transformation converges into a close approximation to the original signal. Simulation results illustrate the evolutionary Slepian-based transform in the representation and reconstruction of signals from irregularly-spaced and contiguous lost samples.

[1]  Paulo Oliveira,et al.  Interpolation of Signals with Missing Data Using PCA , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[2]  Anamitra Makur,et al.  A new iterative reconstruction scheme for signal reconstruction , 2008, APCCAS 2008 - 2008 IEEE Asia Pacific Conference on Circuits and Systems.

[3]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .

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

[5]  Billur Barshan,et al.  Signal recovery from partial fractional Fourier domain information and its applications , 2008 .

[6]  Werner Kozek,et al.  Reconstruction of signals from irregular samples of its short-time Fourier transform , 1995, Optics + Photonics.

[7]  M. B. Priestley,et al.  Non-linear and non-stationary time series analysis , 1990 .

[9]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[11]  Ahmet Serbes,et al.  Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains , 2010 .

[12]  Luis F. Chaparro,et al.  Reconstruction of nonuniformly sampled time-limited signals using prolate spheroidal wave functions , 2009, Signal Process..

[13]  Robert J. Marks,et al.  Block loss recovery in DCT image encoding using POCS , 2002, 2002 IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No.02CH37353).

[14]  Ryszard Stasin ' ski.  POCS-BASED IMAGE RECONSTRUCTION FROM IRREGULARLY-SPACED SAMPLES , 2000 .

[15]  Ryszard Stasinski,et al.  Improved POCS-based image reconstruction from irregularly-spaced samples , 2002, 2002 11th European Signal Processing Conference.

[16]  Boris Polyak,et al.  The method of projections for finding the common point of convex sets , 1967 .

[17]  R. Stasinski,et al.  POCS-based image reconstruction from irregularly-spaced samples , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[18]  Kareem E. Baddour,et al.  Channel estimation using DPSS based frames , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[19]  Aydin Akan,et al.  Discrete evolutionary transform for time-frequency signal analysis , 2000, J. Frankl. Inst..

[20]  Thomas Strohmer,et al.  Numerical analysis of the non-uniform sampling problem , 2000 .

[21]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[22]  Paulo Jorge S. G. Ferreira.  The stability of a procedure for the recovery of lost samples in band-limited signals , 1994, Signal Process..

[23]  M. Unser.  Sampling-50 years after Shannon , 2000, Proceedings of the IEEE.

[24]  Miki Haseyama,et al.  Adaptive Reconstruction Method of Missing Texture Based on Projection Onto Convex Sets , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[25]  Soo-Chang Pei,et al.  Reducing sampling error by prolate spheroidal wave functions and fractional Fourier transform , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..