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OPTIMAL RANDOM SAMPLING FOR SPECTRUM ESTIMATION IN DASP APPLICATIONS
"... In this paper we analyse a class of DASP (Digital Aliasfree Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The e ..."
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In this paper we analyse a class of DASP (Digital Aliasfree Signal Processing) methods for spectrum estimation of sampled signals. These methods consist in sampling the processed signals at randomly selected time instants. We construct estimators of Fourier transforms of the analysed signals. The estimators are unbiased inside arbitrarily wide frequency ranges, regardless of how sparsely the signal samples are collected. In order to facilitate quality assessment of the estimators, we calculate their standard deviations. The optimal sampling scheme that minimises the variance of the resulting estimator is derived. The further analysis presented in this paper shows how sampling instant jitter deteriorates the quality of spectrum estimation. A couple of numerical examples illustrate the main thesis of the paper.
On The Stability Of Certain Interpolation Problems
, 1995
"... In this work we present a number of results concerning the finite dimensional bandlimited interpolation problem. Our aim is to understand how the stability of the interpolation problem is affected by changes in the positions of the interpolation knots. We give upper and lower bounds for the eigenva ..."
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In this work we present a number of results concerning the finite dimensional bandlimited interpolation problem. Our aim is to understand how the stability of the interpolation problem is affected by changes in the positions of the interpolation knots. We give upper and lower bounds for the eigenvalues of the interpolation matrix, as functions of the gaps between missing samples. The upper bound is especially useful, since it determines the convergence rate of an iterative interpolation procedure, and the conditioning of an equivalent noniterative method. The interplay between convergence rate, stability, normalized bandwidth of the data, and minimum gap between missing samples is clarified.
Analysis and Design of
"... Abstract — We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix that is specified for each ..."
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Abstract — We consider a class of interpolation algorithms, including the leastsquares optimal Yen interpolator, and we derive a closedform expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix that is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical ill conditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution consisting of a sinckernel interpolator with specially chosen weighting coefficients. The newly designed sinckernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinckernel interpolator is shown to perform better than the sinc interpolator with Jacobian weighting. I.
REFERENCES
"... examples), although it is difficult to assign a simple interpretation to the magnitude of this statistic. The second Donoho–Johnstone synthetic signal is shown in Fig. 2(a). This signal is qualitatively similar to a nuclear magnetic resonance spectrum. The corresponding noisy signal is shown in Fig. ..."
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examples), although it is difficult to assign a simple interpretation to the magnitude of this statistic. The second Donoho–Johnstone synthetic signal is shown in Fig. 2(a). This signal is qualitatively similar to a nuclear magnetic resonance spectrum. The corresponding noisy signal is shown in Fig. 2(b). The LSM reconstruction is shown in Fig. 2(c), whereas the wavelet shrinkage reconstruction is shown in Fig. 2(d). Comparing the individual peaks in the two reconstructed signals with the corresponding peaks in the noisefree signal, it is clear that the LSM method yields quantitatively superior results in addition to producing a visually pleasing reconstruction. Fig. 3(a) shows the third Donoho–Johnstone synthetic signal consisting of a sine wave that is divided into two parts with the second part shifted upwards by a constant amount. The corresponding noisy signal is shown in Fig. 3(b). The LSM reconstruction is shown
Acknowledgements
, 2007
"... I would like to thank Robert Boyer, my parents, Adam and Muffasir for their help and support over all these years. ii Contents List of Figures v ..."
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I would like to thank Robert Boyer, my parents, Adam and Muffasir for their help and support over all these years. ii Contents List of Figures v
A Class of Eigenvalue Problems in Interpolation, Extrapolation, and Sampling
, 1995
"... The reconstruction of signals or images from incomplete or noisy data is a recurring problem in signal theory, with important practical applications. The bandlimited interpolation and extrapolation problems, in which the signals or images are assumed to be bandlimited, are among the most often stu ..."
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The reconstruction of signals or images from incomplete or noisy data is a recurring problem in signal theory, with important practical applications. The bandlimited interpolation and extrapolation problems, in which the signals or images are assumed to be bandlimited, are among the most often studied reconstruction problems. We discuss certain aspects of these problems, concentrating on the stability of the reconstructions as a function of the sampling set. To do this, the reconstruction problems are formulated in a way which easily leads to several iterative and noniterative algorithms, which seem to have a number of advantages over some previously known methods. The spectrum of the reconstruction operators is studied, yielding a number of bounds which enable the stability of the reconstruction problem to be predicted in advance, as well as the convergence rate of some of the iterations. 1. INTRODUCTION The reconstruction of signals and images from sets of nonuniform samples is ...