Transforms and Statistical Estimation

Deterministic Regression Smoothness Priors TVAR Modelling

Authors:

Jari P Kaipio,
Marko T Juntunen,

Page (NA) Paper number 1119

Abstract:

In this paper we propose a method for the estimation of time-varying autoregressive processes. The approach is essentially to regularize the heavily underdetermined unconstrained prediction equations with a smoothness priors type side constraint. The implementation of nonhomogenous smoothness properties is straightforward. The method is compared to the usual determistic regression approach (TVAR) in which the coefficient evolutions are constrained to a subspace. It is shown that the typical transient oscillations of TVAR can be avoided with the proposed method.

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An Extension of an Interior-Point Method for Entropy Minimization

Authors:

Irina F Gorodnitsky,

Page (NA) Paper number 2211

Abstract:

The field of linear optimization (LP) has undergone explosive development initiated by the introduction of Affine Scaling Transformation based methods by Karmarkar 15 years ago. This paper's contribution is two fold. I propose an algorithm that generalizes the original Affine Scaling Transformation algorithm, termed the Generalized Affine Scaling Transformation (GAST), and show that such GAST based optimization methods form a natural extension to solving problems of entropy optimization. I present a family of entropy functions for which the proposed algorithm exhibits super-quadratic convergence, that is, its convergence rate is superior to that of the existing comparable interior-point methods. The relationship of the proposed algorithm to the recently developed FOCUSS algorithm is also elucidated. The problem of entropy optimization addressed in the paper is relevant in many areas of engineering, including but not limited to signal compression, coding, estimation, and resource scheduling.

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A Krylov Subspace Method for Large Estimation Problems

Authors:

Michael K Schneider,
Alan S Willsky,

Page (NA) Paper number 1049

Abstract:

Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is difficult because the error variances are the diagonal elements of a complicated matrix. This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a converging approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a novel estimation-theoretic interpretation of the CG algorithm. The paper discusses this interpretation and convergence issues and presents numerical examples.

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Estimating the Entropy of a Signal with Applications

Authors:

Jean-François Bercher,
Christophe Vignat,

Page (NA) Paper number 2241

Abstract:

We present an estimator of the entropy of a signal. The basic idea is to adopt a model of the probability law, in the form of an AR spectrum. Then, the law parameters can be estimated from the data. We examine the statistical behavior of our estimates of laws and entropy. Finally, we give several examples of applications: an adaptive version of our entropy estimator is applied to detection of law changes, blind deconvolution and sources separation.

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The Filter Bank Approach for the Fractional Fourier Transform

Authors:

Der-Feng Huang,
Bor-Sen Chen,

Page (NA) Paper number 1692

Abstract:

In this work, we develop an equivalent filter bank structure for the computation of the fractional Fourier transform (FrFT). The purpose of this work is to provide an unified approach to the computation of the FrFT via the filter bank approach.

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The Discrete Fractional Fourier Transform

Authors:

Çagatay Candan,
Alper M Kutay,
Haldun M Ozaktas,

Page (NA) Paper number 2308

Abstract:

We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.

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A Data-Driven Scheme for the Approximated Computing of Alias-Free Generalized Discrete Time-Frequency Distributions

Authors:

Thuyen Le, Darmstadt University of Technology, Germany (Germany)
Manfred Glesner, Darmstadt University of Technology, Germany (Germany)

Page (NA) Paper number 1200

Abstract:

Time-Frequency Distribution (TFD) based on Cohen's class has significant potential for the analysis of a number of non-stationary signals. One of the discrete formulations is the recently introduced Alias-Free Generalized Discrete-Time TFD (AF-GDTFD). The spectral decomposition of the kernel allows the computation of AF-GDTFD as a weighted sum of spectrograms. The partial sum has been shown to offer a vehicle to trade-off between exactness and computational load. This paper proposes a scheme which exploits local approximations by adapting dynamically the accuracy of spectrograms to the eigenvalue magnitudes. The approach employs the wavelet packet transform followed by a block-recursive Fourier transform and a compensation network. Adaptive selection of subbands for further processing reduces substantially the computational cost while still preserving an acceptable quality. The approach is attractive in terms of VLSI aspects due to the modular structure, local connections and stream processing.

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Periodically Nonuniform Bandpass Sampling as a Tapped-Delay-Line Filtering Problem

Authors:

Dan Scholnik,
Jeffrey O Coleman,

Page (NA) Paper number 2101

Abstract:

In this paper we consider systems for demodulation/modulation which use periodically nonuniform sampling (of arbitrary order) of the bandpass signal to circumvent the carrier-frequency restrictions of uniform sampling. The design of a particular tapped-delay-line (demodulation) or piecewise-constant-impulse-response (modulation) equivalent filter determines both the actual implementation filters and system performance. The tap spacing of the former and the transition times of the latter are periodically nonuniform. Following a characterization of the equivalent filter response, the special case of second-order sampling is examined for insight into the choice of sampling offset. A set of example designs demonstrates that, while nonuniform sampling permits carrier frequencies not allowed with uniform sampling, the resulting system performance is limited by the choice of carrier frequency.

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New Realization Method for Linear Periodic Time-Varying Filters

Authors:

Alban Duverdier,
Bernard Lacaze,

Page (NA) Paper number 1606

Abstract:

For channel modelisation, modulation and analogue scrambling, the modern telecommunications use often linear periodic time-varying filters. The authors recall the definition of these filters. In particular, it is shown that a stationary process subjected to a linear periodic filter becomes cyclostationary. In this paper, we show that any linear periodic filter can be realized by means of periodic clock changes. An original implementation method is then introduced. An example illustrates the periodic clock change implementation and presents the advantages of the new implementation technique in comparison to the classical one.

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Wavelet Based Estimator For The Self-Similarity Parameter Of Alpha-Stable Processes

Authors:

Patrice Abry, CNRS URA 1325 - Laboratoire de Physique - ENS Lyon - 46 allee d Italie - 69364 Lyon cedex - France (France)
Lieve Delbeke, KU Leuven - Dept. of mathematics - Celestijnenlaan 200 B, 3001 Heverlee, Belgium (Belgium)
Patrick Flandrin, CNRS URA 1325 - Laboratoire de Physique - ENS Lyon - 46 allee d Italie - 69364 Lyon cedex - France (France)

Page (NA) Paper number 1153

Abstract:

We, here, study self-similar processes with possibly in finite second-order statistics and long-range dependence. To do so, we detail the statistical properties of the wavelet coefficients of alpha-stable self similar processes, used as a paradigm for those situations. We, then, propose a wavelet-based estimator for the self-similarity parameter and analyse its statistical performance both theoretically and numerically. We show that it is unbiased, that its variance decreases as the inverse of the length of the data and that it can be easily implemented.

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