Generalizations of the sampling theorem: Seven decades after Nyquist

The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples. Applications in multiresolution computation and in digital spline interpolation are also reviewed

Streaming Reconstruction from Non-uniform Samples

We present an online algorithm for reconstructing a signal from a set of non-uniform samples. By representing the signal using compactly supported basis functions, we show how estimating the expansion coefficients using least-squares can be implemented in a streaming manner: as batches of samples over subsequent time intervals are presented, the algorithm forms an initial estimate of the signal over the sampling interval then updates its estimates over previous intervals. We give conditions under which this reconstruction procedure is stable and show that the least-squares estimates in each interval converge exponentially, meaning that the updates can be performed with finite memory with almost no loss in accuracy. We also discuss how our framework extends to more general types of measurements including time-varying convolution with a compactly supported kernel

The role of zero crossings in speech recognition and processing

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Reconstruction of multidimensional signals from zero crossings

Originally presented as author's thesis (Ph. D.--Massachusetts Institute of Technology), 1985.Bibliography: p. 90-93.Supported in part by the Advanced Research Projects Agency monitored by ONR under contract no. N00014-81-K-0742 Supported in part by the National Science Foundation under grant ECS-8407285Susan Roberta Curtis

Mass spectrometry data processing using zero-crossing lines in multi-scale of Gaussian derivative wavelet

Motivation: Peaks are the key information in mass spectrometry (MS) which has been increasingly used to discover diseases-related proteomic patterns. Peak detection is an essential step for MS-based proteomic data analysis. Recently, several peak detection algorithms have been proposed. However, in these algorithms, there are three major deficiencies: (i) because the noise is often removed, the true signal could also be removed; (ii) baseline removal step may get rid of true peaks and create new false peaks; (iii) in peak quantification step, a threshold of signal-to-noise ratio (SNR) is usually used to remove false peaks; however, noise estimations in SNR calculation are often inaccurate in either time or wavelet domain. In this article, we propose new algorithms to solve these problems. First, we use bivariate shrinkage estimator in stationary wavelet domain to avoid removing true peaks in denoising step. Second, without baseline removal, zero-crossing lines in multi-scale of derivative Gaussian wavelets are investigated with mixture of Gaussian to estimate discriminative parameters of peaks. Third, in quantification step, the frequency, SD, height and rank of peaks are used to detect both high and small energy peaks with robustness to noise

Estimation of Time-Limited Channel Spectra From Nonuniform Samples

This paper deals with the estimation of a time-invariant channel spectrum from its own nonuniform samples, assuming there is a bound on the channel’s delay spread. Except for this last assumption, this is the basic estimation problem in systems providing channel spectral samples. However, as shown in the paper, the delay spread bound leads us to view the spectrum as a band-limited signal, rather than the Fourier transform of a tapped delay line (TDL). Using this alternative model, a linear estimator is presented that approximately minimizes the expected root-mean-square (RMS) error for a deterministic channel. Its main advantage over the TDL is that it takes into account the spectrum’s smoothness (time width), thus providing a performance improvement. The proposed estimator is compared numerically with the maximum likelihood (ML) estimator based on a TDL model in pilot-assisted channel estimation (PACE) for OFDM.This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) under Project TEC2011-28201-C02-02

A Multiple Parameter Linear Scale-Space for one dimensional Signal Classification

In this article we construct a maximal set of kernels for a multi-parameter linear scale-space that allow us to construct trees for classification and recognition of one-dimensional continuous signals similar the Gaussian linear scale-space approach. Fourier transform formulas are provided and used for quick and efficient computations. A number of useful properties of the maximal set of kernels are derived. We also strengthen and generalize some previous results on the classification of Gaussian kernels. Finally, a new topologically invariant method of constructing trees is introduced.Comment: arXiv admin note: text overlap with arXiv:2305.1325