Paulo Jorge S. G. Ferreira

Interpolation and the discrete Papoulis-Gerchberg algorithm

Analyze the performance of an iterative algorithm, similar to the discrete Papoulis-Gerchberg algorithm, and which can be used to recover missing samples in finite-length records of band-limited data. No assumptions are made regarding the distribution of the missing samples, in contrast with the often studied extrapolation problem, in which the known samples are grouped together. Indeed, it is possible to regard the observed signal as a sampled version of the original one, and to interpret the reconstruction result studied as a sampling result. The authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data. They give upper and lower bounds for the error as a function of the number of iterations, together with the signals for which the bounds are attained. Also, they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix. From this analysis they infer the optimum value of the relaxation constant. They also point out, among all sampling sets with the same density, those for which the convergence rate of the recovery algorithm is maximum or minimum. For low-pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer. The worst convergence rates generally occur when the missing samples are contiguous. >

Modern sampling theory

Modern sampling theory , Modern sampling theory , کتابخانه دیجیتالی دانشگاه علوم پزشکی و خدمات درمانی شهید بهشتی

Fourier analysis of symbolic data: A brief review

We overview and discuss several methods for the Fourier analysis of symbolic data, such as DNA sequences, emphasizing their mutual connections. We consider the indicator sequence approach, the vector and the symbolic autocorrelation methods, and methods such as the spectral envelope, that for each frequency optimize the symbolic-no-numeric mapping to emphasize any periodic data features. We discuss the equivalence or connections between these methods. We show that it is possible to define the autocorrelation function of symbolic data, assuming only that we can compare any two symbols and decide if they are equal or distinct. The autocorrelation is a numeric sequence, and its Fourier transform can also be obtained by summing the squares of the Fourier transform of indicator sequences (zero/one sequences indicating the position of the symbols). Another interpretation of the spectrum is given, borrowing from the spectral envelope concept: among all symbolic-to-numeric mappings there is one that maximizes the spectral energy at each frequency, and leads to the spectrum.

The stability of a procedure for the recovery of lost samples in band-limited signals

Abstract In this paper we study the eigenvalues of a matrix S which arises in the recovery of lost samples from oversampled band-limited signals. Emphasis is placed on the variation of the eigenvalues as a function of the distribution of the missing samples and as a function of the oversampling parameter. We present a number of results which help to understand the numerical difficulties that may occur in this class of problems, and ways to circumvent them.

Genome analysis with inter-nucleotide distances

Motivation: DNA sequences can be represented by sequences of four symbols, but it is often useful to convert the symbols into real or complex numbers for further analysis. Several mapping schemes have been used in the past, but they seem unrelated to any intrinsic characteristic of DNA. The objective of this work was to find a mapping scheme directly related to DNA characteristics and that would be useful in discriminating between different species. Mathematical models to explore DNA correlation structures may contribute to a better knowledge of the DNA and to find a concise DNA description. Results: We developed a methodology to process DNA sequences based on inter-nucleotide distances. Our main contribution is a method to obtain genomic signatures for complete genomes, based on the inter-nucleotide distances, that are able to discriminate between different species. Using these signatures and hierarchical clustering, it is possible to build phylogenetic trees. Phylogenetic trees lead to genome differentiation and allow the inference of phylogenetic relations. The phylogenetic trees generated in this work display related species close to each other, suggesting that the inter-nucleotide distances are able to capture essential information about the genomes. To create the genomic signature, we construct a vector which describes the inter-nucleotide distance distribution of a complete genome and compare it with the reference distance distribution, which is the distribution of a sequence where the nucleotides are placed randomly and independently. It is the residual or relative error between the data and the reference distribution that is used to compare the DNA sequences of different organisms. Contact: vera@ua.pt

Incomplete sampling series and the recovery of missing samples from oversampled band-limited signals

It is well known that a band-limited oversampled signal is completely determined even if an arbitrary finite number of samples is lost. It is shown that an alternative simple proof of this fact carries over to generalized sampling expansions. More precisely, it is shown that any finite number of missing samples can be recovered from the remaining ones, in the case of generalized Kramer sampling expansions, if an appropriate oversampling constraint is satisfied. The recovery can be accomplished either iteratively or noniteratively. >

Unusual properties of superoscillating particles

It has been found that differentiable functions can locally oscillate on length scales that are much smaller than the smallest wavelength contained in their Fourier spectrum—a phenomenon called superoscillation. Here, we consider the case of superoscillations in quantum mechanical wavefunctions. We find that superoscillations in wavefunctions lead to unusual phenomena that are of measurement theoretic, thermodynamic and information theoretic interest. We explicitly determine the wavefunctions with the most pronounced superoscillations, together with their scaling behaviour. We also briefly address the question of how superoscillating wavefunctions might be produced experimentally.

Noniterative and fast iterative methods for interpolation and extrapolation

In this correspondence we study the band-limited interpolation and extrapolation problems for finite-dimensional signals. We show that these problems can be easily reduced to the solution of a set of linear equations with a real symmetric positive-definite matrix S with spectral radius /spl rho/(S) >

A geophysical and hydrogeological study of aquifers contamination by a landfill

Abstract Landfills are the classical solution for waste disposal. During the last years there has been a growing concern about the effect of landfills in public health, because leaching water can contaminate nearby aquifers. Hydrogeological studies are very important to investigate and remedy these contamination problems but an integrated use of both geophysical and hydrological methods can be even more effective. In this paper, the groundwater contamination by a landfill, located on very porous and permeable formations, is assessed by both geophysical and hydrogeological techniques. Bearing in mind the geological and hydrogeological characteristics of the area, electrical methods were chosen to carry out the geophysical survey. At first, electromagnetic methods were used and a ground conductivity survey was carried out to determine the contamination plume. Then several resistivity soundings were proposed and two resistivity pseudo-sections obtained so that a more detailed study of the contamination was accomplished. From the geophysical data interpretation, strategically located boreholes were drilled, water samples obtained and chemical analysis carried out. Finally, the overall results are integrated and discussed as well as the merits and disadvantages of the techniques that have been employed.

On the representability of complete genomes by multiple competing finite-context (Markov) models.

A finite-context (Markov) model of order yields the probability distribution of the next symbol in a sequence of symbols, given the recent past up to depth . Markov modeling has long been applied to DNA sequences, for example to find gene-coding regions. With the first studies came the discovery that DNA sequences are non-stationary: distinct regions require distinct model orders. Since then, Markov and hidden Markov models have been extensively used to describe the gene structure of prokaryotes and eukaryotes. However, to our knowledge, a comprehensive study about the potential of Markov models to describe complete genomes is still lacking. We address this gap in this paper. Our approach relies on (i) multiple competing Markov models of different orders (ii) careful programming techniques that allow orders as large as sixteen (iii) adequate inverted repeat handling (iv) probability estimates suited to the wide range of context depths used. To measure how well a model fits the data at a particular position in the sequence we use the negative logarithm of the probability estimate at that position. The measure yields information profiles of the sequence, which are of independent interest. The average over the entire sequence, which amounts to the average number of bits per base needed to describe the sequence, is used as a global performance measure. Our main conclusion is that, from the probabilistic or information theoretic point of view and according to this performance measure, multiple competing Markov models explain entire genomes almost as well or even better than state-of-the-art DNA compression methods, such as XM, which rely on very different statistical models. This is surprising, because Markov models are local (short-range), contrasting with the statistical models underlying other methods, where the extensive data repetitions in DNA sequences is explored, and therefore have a non-local character.