On the Eigenvalues of Matrices for the Reconstruction of Missing Uniform Samples
TL;DR: The relationship between the eigenvalues associated with the matrices of the minimum dimension time-domain and frequency-domain approaches used for reconstructing missing uniform samples and the weighted Toeplitz matrix is derived.
Abstract: In this correspondence, we derive the relationship between the eigenvalues associated with the matrices of the minimum dimension time-domain and frequency-domain approaches used for reconstructing missing uniform samples. The dependency of the eigenvalues of the weighted Toeplitz matrix on positive weights are explored. Simple bounds for the maximum and minimum eigenvalues of the weighted Toeplitz matrix are also presented. Alternative matrices possessing the same nonzero eigenvalues as that of the weighted Toeplitz matrix are provided. We verify the theory by the examples presented.
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13 Mar 2013
TL;DR: This paper presents an educational tool to be used in signal processing interpolation-related subjects that allows its users to apply three error patterns geometry to the signals and test minimum dimension and maximum dimension signal reconstruction algorithms.
Abstract: This paper presents an educational tool to be used in signal processing interpolation-related subjects. Besides the consolidation of acquired theoretical knowledge, the tool allows its users to apply three error patterns geometry to the signals and test minimum dimension and maximum dimension signal reconstruction algorithms. In the specific case of minimum dimension problems it can be solved using different solvers, iterative and direct linear equations methods. The developed tool allows the problem conditioning analysis through the spectral radius of the system matrix, the condition number and others parameters available in some specific methods. This feature gives the possibility to alter the problem definitions to the desired goal before the reconstruction begins and to choose the optimal method, depending on each problem constraints The time unit that measures the algorithms performance is expressed in terms of one Fourier Transform (FFT) calculation time. In this way the data is presented not in an absolute way but in a relative measure independent from the machine's architecture.
Cites methods from "On the Eigenvalues of Matrices for ..."
...We are planning to implement new methods and features namely semi-iterative methods, for example the Steepest Descent and the Conjugate gradient methods, as well other signal reconstruction frequency-domain formulations [14][15]....
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TL;DR: This work presents a newer and more complete version of an educational tool to be used in signal processing interpolation-related subjects and allows now its users to apply three error geometry patterns, test minimum or maximum dimension signal reconstruction algorithms and problem conditioning through the analysis of the matrix spectral radius or the condition number.
Abstract: This work presents a newer and more complete version of an educational tool to be used in signal processing interpolation-related subjects. Besides the consolidation of acquired theoretical knowledge, the tool allows now its users to apply three error geometry patterns, test minimum or maximum dimension signal reconstruction algorithms and problem conditioning through the analysis of the matrix spectral radius or the condition number: these new features gives the possibility to alter the problem definitions to the desired goal before the reconstruction begins. The time unit that measures the algorithms performance is ¶(nlogn) thus independent from the machine’s architecture. A video of the developed tool can be seen in: https://www.dropbox.com/s/t6yiiuy31ramxse/FILME_1.avi.
Cites methods from "On the Eigenvalues of Matrices for ..."
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...In this paper a new version of the “Signal Processing Interpolation Educational Tool” (SPIEW) is presented (Costa et al., 2012)....
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References
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TL;DR: Two novel block-based algorithms are presented for the reconstruction of uniform samples given the nonuniform samples and it is shown that both of the block- based algorithms provide nearly perfect reconstruction for a class of practically time and bandlimited signals.
Abstract: Two novel block-based algorithms are presented for the reconstruction of uniform samples given the nonuniform samples. The first algorithm uses a sinc interpolator whereas the second one uses a DFT-based interpolator. It is shown that the proposed algorithms are stable and the error due to noise and sampling jitter is bounded by the corresponding error norms of noise and jitter, respectively. We show that both of the block-based algorithms provide nearly perfect reconstruction for a class of practically time and bandlimited signals. Boundary effects are considered and single and multiblock processing is discussed. A modified block-based algorithm is developed by using the windowing technique in order to improve the mean-squared error (MSE) performance for nonbandlimited signals. It is shown that this algorithm performs better than a group of alternative algorithms, including Yen's third algorithm, for a variety of signal, noise, and sampling grids
25 citations
05 Nov 2007
TL;DR: A fast and robust method for approximation of contour from a set of non-uniformly distributed sample points is described here and it is revealed that iterative algorithm of selecting irregular samples is satisfactory.
Abstract: This paper is concerned with the problem of non-uniform sampling and reconstruction of data A fast and robust method for approximation of contour from a set of non-uniformly distributed sample points is described here Data is represented by irregular samples distributed in space The samples are chosen along the contour with spacing determined by the error between the original contour and the contour reconstructed by uniform set of sample points Lagrange's interpolation is used to reconstruct the contour back from non-uniform set of samples The algorithm iteratively finds the minimum number of samples along the contour The error analysis between original and reconstructed contour reveals that iterative algorithm of selecting irregular samples is satisfactory
6 citations