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Reconstruction from Periodic Nonstationary Sampling (PNS) without Resampling
Lacaze B.
Periodic Nonuniform Samplings (PNS) are concatenations of periodic samplings of same periods but which are shifted. They allow to suppress aliasings, dead times of samplers by devices which provide a resampling of data. The obtained periodic sequence is treated through usual methods. In this paper, we show that the resampling stage is unnecessary, that PNS allow good estimations without resampling, at the same time for the sampled process, for linear filterings and for power spectra.
Design Tools for Random Sampling in Multistandard Radio Signal Processing
Grati K., Ben-Romdhane M., Rebai C., Ghazel A., Desgreys P., Loumeau P.
This paper proposes new extensions to random sampling analytical formulations suitable for radio signal spectral analysis. A Fourier transform spectrum estimator is established and verifies the alias-free condition independently of probability density functions for the additive random sampling scheme. Moreover, an energy spectrum density function formulation is proposed for an additive randomly sampled deterministic signal. The obtained theoretical results show random sampling alias suppression. Also, a time-quantized additive random sampling (TQ-ARS) based zero-padding spectral analysis tool is developed. Alias attenuations are computed to design the baseband stage of a TQ-ARS-based software defined radio multistandard receiver. Good performances are shown in terms of relaxing the anti-aliasing filter order, avoiding the automatic gain control, reducing the mean sampling frequency and reconstructing the quantized output spectrum.
Converse Sampling and Interpolation
Higgins J.R.
The present note focuses on ideas of converse sampling and related ideas of interpolation. The presentation is in the setting of reproducing kernel Hilbert space. Sources are found in the classical literature, going back to 1920.
Cadzow Denoising Upgraded: A New Projection Method for the Recovery of Dirac Pulses from Noisy Linear Measurements
Condat L., Hirabayashi A.
We consider the recovery of a finite stream of Dirac pulses at nonuniform locations, from noisy lowpass-filtered samples. We show that maximum-likelihood estimation of the unknown parameters amounts to a difficult, even believed NP-hard, matrix problem of structured low rank approximation. To solve it, we propose a new heuristic iterative algorithm, based on a recently proposed splitting method for convex nonsmooth optimization. Although the algorithm comes, in absence of convexity, with no convergence proof, it converges in practice to a local solution, and even to the global solution of the problem, when the noise level is not too high. Thus, our method improves upon the classical Cadzow denoising method, for same ease of implementation and speed.
Data Driven Sampling of Oscillating Signals
Bidegaray-Fesquet B., Clausel M.
The reduction of the number of samples is a key issue in signal processing for mobile applications. We investigate the link between the smoothness properties of a signal and the number of samples that can be obtained through a level crossing sampling procedure. The algorithm is analyzed and an upper bound of the number of samples is obtained in the worst case. The theoretical results are illustrated with applications to fractional Brownian motions and the Weierstrass function.
Sampling and Reconstruction of Solutions to the Helmholtz Equation
Chardon G., Cohen A., Daudet L.
We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain Ω from their values at scattered points x1, …, xn ⊂ Ω. This problem typically arises when sampling acoustic fields with n microphones for the purpose of reconstructing this field over a region of interest Ω contained in a larger domain D in which the acoustic field propagates. In many applied settings, the shape of D and the boundary conditions on its border are unknown. Our reconstruction method is based on the approximation of a general solution u by linear combinations of Fourier-Bessel functions or plane waves. We analyze the convergence of the least-squares estimates to u using these families of functions based on the samples (u(xi))i=1,.…,n. Our analysis describes the amount of regular-ization needed to guarantee the convergence of the least squares estimate towards u, in terms of a condition that depends on the dimension of the approximation subspace, the sample size n and the distribution of the samples. It reveals the advantage of using non-uniform distributions that have more points on the boundary of Ω. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.
Equivalent Circuits for the PNS3 Sampling Scheme
Lacaze B.
Periodic nonuniform sampling of order three (PNS3) is a sampling scheme composed of three periodic sequences with the same period. It is well-known that this sampling scheme can be useful to remove aliasing. Previous studies have provided conditions on the spectrum support for exact reconstruction in the case of functions. This paper deals more generally with the best mean-square interpolation for stationary processes with any known power spectrum, from PNS3 and possibly aliasing. We show that the best estimation is based upon particular linear filters, which depend on the gap between the sampling sequences. The mean-time error also depends on this gap. The errorless interpolation is a particular case. It requires the knowledge of the spectral support rather than the spectral true values.
Recovery of Missing Samples from Multi-Channel Oversampling in Shift-Invariant Spaces
Kang S., Kwon K.H., Lee D.G.
It is well known that in the classical Shannon sampling theory on band-limited signals, any finitely many missing samples can be recovered when the signal is oversampled at a rate higher than the minimum Nyquist rate. In this work, we consider the problem of recovering missing samples from multi-channel oversampling in a general shift-invariant space. We find conditions under which any finite or infinite number of missing samples can be recovered when they are from a single or two channels.
Filtering from PNS2 Sampling
Lacaze B.
Periodic Nonuniform Sampling of order 2 (PNS2) is defined by two sequences with same period and some delay between them. PNS2 is known to suppress aliasing of multiband signals. Even if PNS2 reconstructs the signal by a linear combination of samples, the problem of retrieving a filtered version of the signal is more complicated. The simplest solution begins by a signal reconstruction through a sampling formula, followed by an analog filter. However new sampling formulas can solve this problem in a single stage. We give equivalent digital circuits which provide these formulas. Examples are provided, particularly when looking to retrieve subbands in communications.
Representation of Operators by Sampling in the Time-Frequency Domain
Dörfler M., Torrésani B.
Gabor multipliers are well-suited for the approximation of certain time-variant systems. However, this class of systems is rather restricted. To overcome this restriction, multiple Gabor multipliers allowing for more than one synthesis windows are introduced. The influence of the choice of the various parameters involved on approximation quality is studied for both classical and multiple Gabor multipliers. Fairly simple error estimates are provided, and the study is supplemented by numerical simulations. This paper is an extended and improved version of [6].
Non-Uniform Filter Interpolation in the Frequency Domain
Bidégaray-Fesquet B., Fesquet L.
We propose a filtering technique which takes advantage of a specific non-uniform sampling scheme which allows the capture of a very low number of samples for both the signal and the filter transfer function. This approach leads to a summation formula which plays the same role as the discrete convolution for usual FIR filters. Here the formula is much more complicated but it can easily be implemented and the evaluation of these more elaborate expressions is compensated by the very low number of samples to process.
About Bi-periodic Samplings
Lacaze B.
To superpose delayed samplings with a unique period is a worthwhile technique. The first contribution in this domain comes from J.L. Yen, half a century ago, and improvements were due to A. N. Papoulis theorem and J. R. Higgins who proved that functions with energy spectrum in subbands can be recovered by samplings with same periods and with lags. In this paper, we study the case of two periodic samplings with equal or different periods, rational or irrational between them. This kind of sampling is natural in systems where the Doppler effect acts. The best linear estimation in the meansquare sense is given under an integral form. Solutions can be analytically obtained, for example, when the power spectrum is distributed in few bands or when the sampling periods are multiples.
Abdul Jerri: An appreciation on his seventy-seventh birthday
Butzer P.L., Higgins J.R., Nashed M.Z.
The Ghost Sampling Sequence Method
Lacaze B.
An interpolation formula involves samples of the observed phenomenon and functions depending on the sampling times. For instance, the classic Shannon formula is based on the observations at periodic times and the values of the cardinal series at these times. But, in real computations, only a finite number of these observations are taken into account, in a limited time window. Because what is measured outside the window does not intervene, it is often possible to use a sampling time sequence made of the real times inside the window, and completed by an arbitrary time sequence taken outside the window. The latter is a “ghost sequence” because the values of the object to reconstruct at these times are not known and are not used in real computations.
Linear Interpolation and a Clay Tablet of the Old Babylonian Period
Higgins J.R.
In a brief historical note we look for the earliest occurrence of interpolation that the literature can provide. The search leads us to a problem of compound interest on a clay tablet from the Old Babylonian period (c. 2000–1700 BC), and to commentaries on it by Neugebauer in [5] and [6], and by Thureau-Dangin in [7] and [8]. The scribe who wrote this tablet does not give an explicit account of his method; nevertheless, after discovering and correcting an error by the scribe, the analyses of Neugebauer and Thureau-Dangin establish beyond reasonable doubt that linear interpolation must have been used to solve the compound interest problem. These commentaries are not easily accessible, and hitherto they have been known only to specialists in the field. Our present purpose is to draw them together in a review which may help to make an ancient mathematical episode more generally known, and place it at the historical beginnings of interpolation.
Comparison of Numerical Methods for the Computation of Energy Spectra in 2D Turbulence. Part II: Adaptative Algorithms
Bruneau C.H., Fischer P., Peter Z., Yger A.
The first part of this work was devoted to the comparison of direct numerical methods for the computation of energy spectra in 2D turbulence. Here such direct methods are mixed together and combined with adaptative algorithms such as matching pursuit or orthogonal matching pursuit. It appears curiously that the proper orthogonal decomposition basis is sometimes less adapted to the reconstruction process than cosine or wavelet packets dictionaries.
A Theoretical Exposition of Stationary Processes Sampling
Lacaze B.
This paper introduces the sampling theory of stationary processes. In this context, the knowledge of the power spectrum is generally assumed, given for example by description of physical devices. It is the main difference with the deterministic case where the energy spectrum is unknown. We will study the best mean-square estimation of processes with any power spectrum, in the case of either uniform (periodic) or irregular sampling. In this last domain, the difference is emphasized between “observed” and “unobserved” sampling, that is to say, between cases where the sampling times are known and usable in formulae or not. When sampling times are unknown, formulae need to be based on statistical properties and errors of reconstruction will increase.
Comparison of Numerical Methods for the Computation of Energy Spectra in 2D Turbulence. Part I: Direct Methods
Bruneau C.H., Fischer P., Peter Z., Yger A.
The widely accepted theory of two-dimensional turbulence predicts a direct enstrophy cascade with an energy spectrum that behaves in terms of the frequency range k as k−3 and an inverse energy cascade with a k−5/3 decay. However, the graphic representation of the energy spectrum (even its shape) is closely related to the tool which is used to perform the numerical computation. With the same initial flow, eventually treated thanks to different tools such as wavelet decompositions or POD representations, the energy spectra are computed using direct various methods: FFT, auto-covariance function, auto regressive model, and wavelet transform. Numerical results are compared to each other and confronted with theoretical predictions. In a forthcoming part II some adaptative methods combined with the above direct ones will be developed.
Reconstruction formula for irregular sampling
Lacaze B.
Samples of functions or of stationary processes with spectrum in (−π, π) give enough information for exact reconstruction when the mean interval (if exists) between them is smaller than one. In this paper, we give formulas usable for irregular sampling. The proof uses auxiliary uniform sampling, the influence of which disappears in the limit as the number of samples tends to infinity.
Some groupings of equivalent results in analysis that include sampling principles
Higgins J.R.
It is shown that Tschakalov's sampling theorem, the generalised Van-dermonde -Chu convolution formula, Gauss's summation of the hypergeo-metric series, Dougall's bilateral sum, and the partial fractions expansion for the cotangent function are all equivalent. It is also shown that the classical sampling theorem is equivalent to the cotangent expansion, but our proofs do not allow these two groupings of equivalent results to be amalgamated.
Random process reconstruction from multiple noisy source observations
Lacaze B., Mailhes C.
The problem addressed in this paper is the reconstruction of a continuous-time stationary random process from noisy sampled observations coming from different sources. An optimal solution in terms of linear filtering of observed samples is derived and the ex- pression of the corresponding minimum reconstruction error power is given. Moreover, two equivalent reconstruction schemes are given. The first one is recursive, involving two filter banks. Its main interest is that adding or suppressing an input does not af- fect the whole scheme. The second scheme is symmetric and uses only one filter bank. However, to add a new input requires a com- plete modification of all the filter transfer functions. Simulation examples are given to prove the application of the reconstruction scheme.
A sampling technique based on transforms associated with a group representation
Sainte-Marie J.
In this paper, the properties of transforms associated with a group representation are used to derive sampling techniques based on atomic decompositions [6, 8]. The proposed algorithms allow a recovery of a signal from a sequence of its samples and extend an earlier result proposed by Xiao and Zou [21] in the context of the Fourier analysis.
Reconstruction of sampled complex processes with timing jitter
Lacaze B., Mailhes C.
This paper studies the effects of timing jitter on complex processes, whether the sampling condition is verified or not. The problem of recovering a continuous-time process from observations subjected to jitter is considered. A solution in terms of linear filtering of sampled real and imaginary parts is derived. Simulation examples are given to demonstrate the application of the proposed scheme.
