Finite Number Of Samples Research Articles

Reconstruction of signals in L p (ℝ)-space by generalized sampling series based on linear combinations of B-splines†

In her plenary lecture [Daubechies, I., 2005, Redundancy and robust A/D and D/A conversion (see ref. 28). SampTA 05, Sampling Theory and Applications. Proceedings of International Workshop, Samsun, Turkey, 2005 (Samsun: Ondokuz Mayis University, Department of Mathematics), p. 18.] at the workshop SampTa 05, Samsun, Turkey, July 2005, Daubechies, Princeton, presented three major reasons as to why the classical sampling theorem was fully impractical for real-life signal processing. One needs infinitely many samples extended over the whole real axis, the sinc-kernel decaying much too slowly, and band limitation is a too restrictive assumption. This paper presents an approach to overcome these difficulties, which actually began to develop at Aachen 1977. The sinc-function is replaced by certain simple linear combinations of B-splines, only a finite number of samples need be available. This approach can be used to process arbitrarily continuous and even discontinuous signals. Best possible error estimates in terms of the L p -average modulus of smoothness are presented. Three typical examples exhibiting the various problems involved are worked out in detail. †This paper is based in part on a plenary lecture held by the first-named author at the International Conference ‘Analytical Methods of Analysis and Differential Equations (AMADE—2006)’, 13—19 September 2006, Minsk, Belarus.

Time interval measurement device based on surface acoustic wave filter excitation, providing 1ps precision and stability

This article deals with the time interval measurement device, which is based on a surface acoustic wave (SAW) filter as a time interpolator. The operating principle is based on the fact that a transversal SAW filter excited by a short pulse can generate a finite signal with highly suppressed spectra outside a narrow frequency band. If the responses to two excitations are sampled at clock ticks, they can be precisely reconstructed from a finite number of samples and then compared so as to determine the time interval between the two excitations. We have designed and constructed a two-channel time interval measurement device which allows independent timing of two events and evaluation of the time interval between them. The device has been constructed using commercially available components. The experimental results proved the concept. We have assessed the single-shot time interval measurement precision of 1.3 ps rms that corresponds to the time of arrival precision of 0.9 ps rms in each channel. The temperature drift of the measured time interval on temperature is lower than 0.5 ps/K, and the long term stability is better than +/-0.2 ps/h. These are to our knowledge the best values reported for the time interval measurement device. The results are in good agreement with the error budget based on the theoretical analysis.

Blind Subspace-Based Signature Estimation in DS-CDMA Systems With Unknown Wide-Sense Stationary Interference

In this paper, we propose a new blind subspace-based signature waveform estimation technique for direct-sequence code division multiple-access (DS-CDMA) communication systems operating in the presence of unknown wide-sense stationary interference. Unlike the existing algorithms, the proposed technique requires just a single receive antenna and is applicable to the case of arbitrary transmitted symbol constellations. Necessary and sufficient conditions for identifiability of the proposed technique are derived. For practical scenarios where the data covariance matrix is estimated using a finite number of samples, a closed form expression for the mean-squared error (MSE) of the estimated channel is obtained. For the high signal-to-interference ratio (SIR) regime and Gaussian interference, an approximation of the MSE expression is also derived and the effects of different parameters on the performance of the proposed algorithm are analyzed. Numerical examples validate our analytical results.

Unconditional Maximum Likelihood Performance at Finite Number of Samples and High Signal-to-Noise Ratio

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This correspondence deals with the problem of estimating signal parameters using an array of sensors. In source localization, two main maximum-likelihood methods have been introduced: the conditional maximum-likelihood method which assumes the source signals nonrandom and the unconditional maximum-likelihood method which assumes the source signals random. Many theoretical investigations have been already conducted for the large samples statistical properties. This correspondence studies the behavior of unconditional maximum likelihood at high signal-to-noise ratio for finite samples. We first establish the equivalence between the unconditional and the conditional maximum-likelihood criterions at high signal-to-noise ratio. Then, thanks to this equivalence we prove the non-Gaussianity and the non-efficiency of the unconditional maximum-likelihood estimator. We also rediscover the closed-form expressions of the probability density function and of the variance of the estimates in the one source scenario and we derive a closed-form expression of this estimator variance in the two sources scenario. </para>

Estimates of the information content and dimensionality of natural scenes from proximity distributions

Natural scenes, like most all natural data sets, show considerable redundancy. Although many forms of redundancy have been investigated (e.g., pixel distributions, power spectra, contour relationships, etc.), estimates of the true entropy of natural scenes have been largely considered intractable. We describe a technique for estimating the entropy and relative dimensionality of image patches based on a function we call the proximity distribution (a nearest-neighbor technique). The advantage of this function over simple statistics such as the power spectrum is that the proximity distribution is dependent on all forms of redundancy. We demonstrate that this function can be used to estimate the entropy (redundancy) of 3x3 patches of known entropy as well as 8x8 patches of Gaussian white noise, natural scenes, and noise with the same power spectrum as natural scenes. The techniques are based on assumptions regarding the intrinsic dimensionality of the data, and although the estimates depend on an extrapolation model for images larger than 3x3, we argue that this approach provides the best current estimates of the entropy and compressibility of natural-scene patches and that it provides insights into the efficiency of any coding strategy that aims to reduce redundancy. We show that the sample of 8x8 patches of natural scenes used in this study has less than half the entropy of 8x8 white noise and less than 60% of the entropy of noise with the same power spectrum. In addition, given a finite number of samples (<2(20)) drawn randomly from the space of 8x8 patches, the subspace of 8x8 natural-scene patches shows a dimensionality that depends on the sampling density and that for low densities is significantly lower dimensional than the space of 8x8 patches of white noise and noise with the same power spectrum.

A Neural Network Solution for Identification and Classification of Cylindrical Targets above Perfectly Conducting Flat Surfaces

This paper evaluates the radar target identification and classification performance of neural networks. A set of features are derived from scattered fields calculated by using the image technique formulation and Moment Method (MoM). An RBF (Radial Basis Function) networkthat utilizes the feature set is proposed for target identification and classification. The database contains a finite number of samples of cylindrical targets at certain angles. A portion of the database is used to train the networkand the rest is used to test the performance of the neural networkfor target identification and classification. This workaims to find the heights measured from the surface and radiuses of the targets for identification of targets and determine the right target for classification of targets from the scattered field values.

Estimation of the In-Band Delta–Sigma Noise Power Based on Windowed Data

Spectral figures of merit of Delta-Sigma (DeltaSigma) converters are usually evaluated in the frequency domain by estimating the power of the narrow- and wide-band components from the power spectral density (PSD) of the modulator output. Because of the finite number of processed samples, both components are affected by a spectral leakage. Scientific literature has widely investigated the leakage of the fundamental and its higher harmonics, which can be prevented if the sampling and the input frequencies are synchronized accurately. Conversely, effect of the finite number of samples on the PSD of DeltaSigma-shaped wideband noise has not been treated in the technical literature yet. To date, windows commonly used for reducing leakage of the fundamental (i.e., the Hanning, Blackman-Harris, or flattop sequences) are also used for windowing the DeltaSigma-shaped quantization noise. By noticing that published PSDs of low-order DeltaSigma modulators are usually obtained by employing the Hanning window Norsworthy (Oversampling Delta-Sigma Data Converters: Theory, Design, and Simulation. Piscataway, NJ: IEEE Press, 1997), Markus and Temes (IEEE Trans. Circuits Syst. I, vol. 51, no. 1, p. 63, 2004), in this paper, the deviation in the estimate of the noise power induced by the use of the general class of the cosine windows-to which the Hanning one belongs-is investigated. This is employed for deriving an a priori criterion for the selection of the best cosine window that reduces the spectral leakage of the shaped wideband noise and for the choice of the minimum number of acquired samples needed to bound the relative error in the estimated in-band noise power below a given value

Blind identification/equalization using deterministic maximum likelihood and a partial prior on the input

A (semi)deterministic maximum likelihood (DML) approach is presented to solve the joint blind channel identification and blind symbol estimation problem for single-input multiple-output systems. A partial prior on the symbols is incorporated into the criterion which improves the estimation accuracy and brings robustness toward poor channel diversity conditions. At the same time, this method introduces fewer local minima than the use of a full prior (statistical) ML. In the absence of noise, the proposed batch algorithm estimates perfectly the channel and symbols with a finite number of samples. Based on these considerations, an adaptive implementation of this algorithm is proposed. It presents some desirable properties including low complexity, robustness to channel overestimation, and high convergence rate.

Fast Reconstruction Methods for Bandlimited Functions from Periodic Nonuniform Sampling

A well-known generalization of Shannon's sampling theorem states that a bandlimited function can be reconstructed from its periodic nonuniformly spaced samples if the effective sampling rate is at least the Nyquist rate. Analogous to Shannon's sampling theorem this generalization requires that an infinite number of samples be available, which, however, is never the case in practice. Most existing reconstruction methods for periodic nonuniform sampling yield very low order (often not even first order) accuracy when only a finite number of samples is given. In this paper we propose a fast, numerically robust, root-exponential accurate reconstruction method. The efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure and utilizing localized Fourier analysis. We discuss applications in analog-to-digital conversion where nonuniform periodic sampling arises in various situations. Finally, we demonstrate the performance of our algorithm by numerical examples.

Correcting the Kullback–Leibler distance for feature selection

A frequent practice in feature selection is to maximize the Kullback–Leibler (K–L) distance between target classes. In this note we show that this common custom is frequently suboptimal, since it fails to take into account the fact that classification occurs using a finite number of samples. In classification, the variance and higher order moments of the likelihood function should be taken into account to select feature subsets, and the Kullback–Leibler distance only relates to the mean separation. We derive appropriate expressions and show that these can lead to major increases in performance.

Effect of Jitter on Asynchronous Sampling With Finite Number of Samples

In modern communication systems, the conversion of analog signals into digital form [analog-digital conversion (ADC)] is one of the most critical functions. A fundamental limit of the signal-to-noise ratio (SNR) achievable in this conversion is given by the jitter of the sampling clock. The requirements on the maximum jitter tolerable are typically specified using SNR expressions which hold in the case of an infinite number of samples. However, there are good reasons to investigate the resulting SNR when only a finite number of samples is taken into account. This paper evaluates the effective impact of jitter on the SNR of the ADC process when the observation interval is limited to a finite number of samples. It will be shown that, in this case, the jitter constraints on the sampling clock can be more relaxed.

Digital time-delay beamforming with interpolated signals

In this paper, a general treatment of digital time-delay beamforming with interpolated signals is given. It covers beamforming for low-pass and band-pass signals, and popular digital beamformer structures are included in this approach. Two different topics are combined in the investigation: the first is reconstruction of complex envelope and analytic band-pass signals from a finite number of samples, the second topic applies the first to digital beamforming. Closed form expressions for the digital interpolation beamformer output and its beam-pattern are derived, and the effect of sampling and interpolation is discussed. Numerical simulations underline the power of the general approach of choosing the right sampling rate and interpolation method to realize digital beamformers with very little distortion of their beam-pattern.

Nonefficiency of Stochastic Beamforming Bearing Estimates at High SNR and Finite Number of Samples

It is well known that the stochastic maximum-likelihood (SML) method yields Gaussian and efficient estimates when the number of independent identically distributed samples tends to infinity. This letter investigates the behavior of SML bearing estimation for a single source impinging on an antenna array when the SNR tends to infinity for a fixed number of samples. We prove that, rather surprisingly, the bearing estimates are not efficient and are asymptotically distributed according to a Student law. Simulation results confirm theoretical analysis.

Exact Sampling Results for Some Classes of Parametric Nonbandlimited 2-D Signals

We present sampling results for certain classes of two-dimensional (2-D) signals that are not bandlimited but have a parametric representation with a finite number of degrees of freedom. While there are many such parametric signals, it is often difficult to propose practical sampling schemes; therefore, we will concentrate on those classes for which we are able to give exact sampling algorithms and reconstruction formulas. We analyze in detail a set of 2-D Diracs and extend the results to more complex objects such as lines and polygons. Unlike most multidimensional sampling schemes, the methods we propose perfectly reconstruct such signals from a finite number of samples in the noiseless case. Some of the techniques we use are already encountered in the context of harmonic retrieval and error correction coding. In particular, singular value decomposition (SVD)-based methods and the annihilating filter approach are both explored as inherent parts of the developed algorithms. Potentials and limitations of the algorithms in the noisy case are also pointed out. Applications of our results can be found in astronomical signal processing, image processing, and in some classes of identification problems.

Adaptive solution for blind identification/equalization using deterministic maximum likelihood

A deterministic maximum likelihood (DML) approach is presented for the blind channel estimation problem. It is first proposed in a block version, which consists of iterating two steps, each one solving a least-squares problem either in the channel or in the symbols. In the noiseless case and under certain conditions, this algorithm gives the exact channel and the exact symbol vector with a finite number of samples. It is shown that even if the DML method has a single global minimum, the proposed iterative procedure can converge to spurious local minima. This problem can be detected (under some channel diversity conditions) by using a numerical test that is proposed in the paper. Based on these considerations, we extend the maximum likelihood block algorithm (MLBA) to recursive implementations [maximum likelihood recursive algorithm (MLRA)]. The MLRA is able to track variations of the system by the introduction of an exponential forgetting factor in the DML criterion. The link between the adaptive algorithm and a soft decision feedback equalizer (SDFE) is emphasized. Low-complexity versions of the recursive and adaptive algorithm are presented.

Two deterministic blind channel estimation algorithms based on oblique projections

In this paper we consider the problem of deterministic blind channel estimation using oblique projections. Oblique projections allow to decompose a matrix into two non-orthogonal components. This property allows to develop two new blind channel estimation algorithms. The algorithms are robust to the spatial color of the additive noise and they are deterministic: if there is no additive noise, they allow perfect channel recovery with a finite number of samples if the model assumptions hold.

Misinterpretation of human electrogastrograms related to inappropriate data conditioning and acquisition using digital computers.

Despite the fact that digital techniques for data acquisition and processing were widely used in electrogastrographic (EGG) research during the last decade, inappropriate signal conditioning and digitization are still potential pitfalls threatening the reliability of the experiments. The aim of this paper was to review: (1) the importance of the antialiasing low-pass filtering for reducing recording artifacts and interferences, (2) the advantages brought by the proper choice of filter cutoff frequency and the slope for the decrement of the minimal required sampling frequency, (3) the impact of incorrectly selected sampling frequency on data interpretations, with particular attention to the percent distribution ranges, and (4) the "leakage effect" related to the finite number of samples processed simultaneously in frequency domain representation of the recordings. A model of electrogastrographic (EGG) recording was mixed with a model of electrocardiographic (ECG) artifact. The resulting finite-duration signal was low-pass filtered and then digitized with a sampling frequency of 1 Hz. The cutoff frequency of the first-order low-pass filter was altered from 0.5 to 0.1 Hz. Amplitude frequency spectra of the digitized recordings were investigated. An example with a real human electrogastrogram in which an ECG artifact was present confirmed the simulation results. When a first-order anti-aliasing filter is utilized at least a fivefold difference between the filter cutoff frequency and the sampling frequency is recommended for compliance with the Nyquist theorem of digitization. Leakage effects associated with the finite-time duration of the recordings and the use of the discrete Fourier transform should be considered when frequency domain analysis is performed. Misinterpretation of the "bradygastric" and "tachygastric" ranges in the percent distribution of EGG frequency components is possible if inappropriate signal conditioning and digitization are employed.

Noise properties of periodic interpolation methods with implications for few-view tomography

A number of methods exist specifically for the interpolation of periodic functions from a finite number of samples. When the samples are known exactly, exact interpolation is possible under certain conditions, such as when the function is band-limited to the Nyquist frequency of the samples. However, when the samples are corrupted by noise, it is just as important to consider the noise properties of the resulting interpolated curve as it is to consider its accuracy. In this work, we derive analytic expressions for the covariance and variance of curves interpolated by three periodic interpolation methods-the circular sampling theorem, zero padding and periodic spline interpolation-when the samples are corrupted by noise. We perform empirical studies for the special cases of white and Poisson noise and find the results to be in agreement with the analytic derivations. The implications of these findings for few-view tomography are also discussed.

Überabtastung und Rekonstruktion verlorener Werte bandbegrenzter Signale

A bandlimited signal sampled at a sampling rate higher than the Nyquist rate can be reconstructed uniquely from its samples even when a finite number of samples is being lost. This paper gives a solution to this problem which partially originates from Marks [1]. The solution described here is available in a parallel and a sequential (iterative) form. The parallel implementation requires the solution of a single system of linear equations. Some investigation of the numerical properties of this system is done and a numerical example is given.

Viability of Sacramento River Winter‐Run Chinook Salmon

The winter run of chinook salmon (Oncorhynchus tshawytscha) on the Sacramento River in California (U.S.A.) was the first Pacific salmon stock to be listed under the U.S. Endangered Species Act. We describe some of the characteristics of Pacific salmon populations that require special consideration in viability analysis during development of a model specific to the Sacramento River winter run of chinook salmon. Their anadromous, semelparous life history leads to a special definition of quasi‐extinction. Random variability occurs primarily in spawning or early life and is reflected in the “cohort replacement rate,” the number of future spawners produced by each spawner, a measure consistent with the common practice of characterizing salmon population dynamics in terms of stock‐recruitment relationships. We determine the distribution of cohort replacement rates from spawning abundance data and life‐history information. We then show through simulations that replacing this distribution with a lognormal distribution with the same mean and variance has a negligible effect on extinction rates, but that approximating an indeterminate semelparous life history using a determinate semelparous life history leads to inaccurate estimates of extinction rate. We derive delisting criteria that directly assess the effects of habitat improvement by explicitly including population growth rate (geometric mean cohort replacement rate ≥1.0) in addition to abundance (≥10,000 female spawners). These delisting criteria allow for the uncertainty due to limited accuracy in measuring spawner abundance and the finite number of samples used to estimate population growth rate (estimates must be based on at least 13 years of data, assuming spawner abundance is measured with less than 25% error). Because the probability of extinction will generally be very sensitive to the uncertainty involved in meeting delisting criteria, we recommend that similar uncertainty be accounted for in future recovery criteria for all endangered species.Viabilidad del Salmón Chinook Durante la Corrida Invernal en el Río SacramentoLa corrida invernal del salmón Chinook (Oncorhynchus tshawytscha) en Río Sacramento en California (E.U.A.) fue la primera población de salmón del pacífico que se incluyó en el Acta de Especies En Peligro de los Estados Unidos. Describimos algunas características de las poblaciones de salmón del Pacifico que requieren ser consideradas en el análisis de viabilidad al desarrollar un modelo específico para la corrida invernal del salmón Chinook en el Río Sacramento. Su historia de vida anádroma, semélpara conduce a una definición especial de la cuasi‐extinción. Principalmente durante el desove o en las primeras etapas de la vida ocurre variabilidad aleatoria que se refleja en la “tasa de reemplazo de cohorte,” el número de reproductores producidos por cada reproductor, que es un medida consistente con la práctica común de caracterizar a la dinámica de las poblaciones de salmón en términos de las relaciones de reclutamiento. Determinamos la distribución de tasas de reemplazo de cohortes a partir de datos de abundancia de reproductores e información de la historia de vida. Mediante simulaciones posteriormente mostramos que el cambio de esta distribución por una distribución logarítmica con la misma media y varianza tiene un efecto no significativo sobre las tasas de extinción, pero que la aproximación a una historia de vida semélpara indeterminada mediante una historia de vida semélpara determinada conduce a estimaciones inexactas de la tasa de extinción. Derivamos criterios para retirar de la lista de especies en peligro que directament evalúan los efectos del mejoramiento del hábitat al explícitamente incluir la tasa de crecimiento poblacional (media geométrica de las tasa de reemplazo de cohorte ≥ 1.0) además de la abundancia (≥10,000 hembras reproductoras). Estos criterios permiten la incertidumbre debido a la limitada precisión para medir la abundancia de reproductores y al número finito de muestras utilizadas para estimar la tasa de crecimiento (las estimaciones deben basarse en datos de por lo menos 13 años, suponiendo que la abundancia de reproductores se mide con &lt;25% de error). Debido a que la probabilidad de extinción generalmente será muy sensible a la incertidumbre presente al cumplir con los criterios para retirar de la lista de especies en peligro, recomendamos que se debe considerar la incertidumbre similar para los criterios de recuperación futura de todas las especies en peligro de extinción.