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15
Time–frequency formulation, design, and implementation of timevarying optimal filters for signal estimation
 IEEE TRANS. SIGNAL PROCESS
, 2000
"... This paper presents a time–frequency framework for optimal linear filters (signal estimators) in nonstationary environments. We develop time–frequency formulations for the optimal linear filter (timevarying Wiener filter) and the optimal linear timevarying filter under a projection side constraint ..."
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Cited by 14 (5 self)
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This paper presents a time–frequency framework for optimal linear filters (signal estimators) in nonstationary environments. We develop time–frequency formulations for the optimal linear filter (timevarying Wiener filter) and the optimal linear timevarying filter under a projection side constraint. These time–frequency formulations extend the simple and intuitive spectral representations that are valid in the stationary case to the practically important case of underspread nonstationary processes. Furthermore, we propose an approximate time–frequency design of both optimal filters, and we present bounds that show that for underspread processes, the timefrequency designed filters are nearly optimal. We also introduce extended filter design schemes using a weighted error criterion, and we discuss an efficient time–frequency implementation of optimal filters using multiwindow shorttime Fourier transforms. Our theoretical results are illustrated by numerical simulations.
Linear TimeFrequency Filters: Online Algorithms and Applications
, 2002
"... This chapter discusses practical discretetime methods for the timefrequency (TF) design of linear timevariant (LTV) filters. The filters are specified via a prescribed TF weight function (timevarying transfer function). We consider both explicit TF filter designs where a TF representation of the ..."
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Cited by 11 (3 self)
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This chapter discusses practical discretetime methods for the timefrequency (TF) design of linear timevariant (LTV) filters. The filters are specified via a prescribed TF weight function (timevarying transfer function). We consider both explicit TF filter designs where a TF representation of the LTV filter is matched to the specified TF weight function, and implicit TF filter designs that use an analysisweightingsynthesis procedure involving a linear TF signal representation. All filter designs allow for efficient online implementations and are thus suited to realtime applications. Our theoretical development is complemented by detailed descriptions of online algorithms, discussions of the choice of design parameters, and estimates of computational complexity and memory requirements. The performance and selected applications of the various TF filters are illustrated via numerical simulations.
Nonstationary spectral analysis based on timefrequency operator symbols and underspread approximations
 IEEE TRANS. INF. THEORY
, 2006
"... We present a unified framework for timevarying or time–frequency (TF) spectra of nonstationary random processes in terms of TF operator symbols. We provide axiomatic definitions and TF operator symbol formulations for two broad classes of TF spectra, one of which is new. These classes contain all m ..."
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Cited by 9 (4 self)
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We present a unified framework for timevarying or time–frequency (TF) spectra of nonstationary random processes in terms of TF operator symbols. We provide axiomatic definitions and TF operator symbol formulations for two broad classes of TF spectra, one of which is new. These classes contain all major existing TF spectra such as the Wigner–Ville, evolutionary, instantaneous power, and physical spectrum. Our subsequent analysis focuses on the practically important case of nonstationary processes with negligible highlag TF correlations (socalled underspread processes). We demonstrate that for underspread processes all TF spectra yield effectively identical results and satisfy several desirable properties at least approximately. We also show that Gabor frames provide approximate Karhunen–Loève (KL) functions of underspread processes and TF spectra provide a corresponding approximate KL spectrum. Finally, we formulate simple approximate input–output relations for the TF spectra of underspread processes that are passed through underspread linear timevarying systems. All approximations are substantiated mathematically by upper bounds on the associated approximation errors. Our results establish a TF calculus for the secondorder analysis and timevarying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes.
Kernels and multiple windows for estimation of the WignerVille spectrum of gaussian locally stationary processes
 IEEE Trans. on Signal Processing
, 2007
"... Abstract—This paper treats estimation of the WignerVille spectrum (WVS) of Gaussian continuoustime stochastic processes using Cohen’s class of timefrequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman’s ..."
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Cited by 3 (0 self)
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Abstract—This paper treats estimation of the WignerVille spectrum (WVS) of Gaussian continuoustime stochastic processes using Cohen’s class of timefrequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman’s sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalueeigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter. Index Terms—Locally stationary circular symmetric processes, multitaper spectral estimation, optimal estimation, timefrequency analysis, WignerVille spectrum (WVS). I.
Wideband Weyl Symbols for Dispersive TimeVarying Processing of Systems and Random Signals
, 2002
"... We extend the narrowband Weyl symbol (WS) and the wideband PHWeyl symbol (PHWS) for dispersive time–frequency (TF) analysis of nonstationary random processes and timevarying systems. We obtain the new TF symbols using unitary transformations on the WS and the PHWS. For example, whereas the WS is m ..."
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Cited by 3 (0 self)
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We extend the narrowband Weyl symbol (WS) and the wideband PHWeyl symbol (PHWS) for dispersive time–frequency (TF) analysis of nonstationary random processes and timevarying systems. We obtain the new TF symbols using unitary transformations on the WS and the PHWS. For example, whereas the WS is matched to systems with constant or linear TF characteristics, the new symbols are better matched to systems with dispersive (nonlinear) TF structures. This results from matching the geometry of the unitary transformation to the specific TF characteristics of a system. We also develop new classes of smoothed Weyl symbols that are covariant to TF shifts or time shift and scaling system transformations. These classes of symbols are also extended via unitary warpings to obtain classes of TF symbols covariant to dispersive shifts. We provide examples of the new symbols and symbol classes, and we list some of their desirable properties. Using simulation examples, we demonstrate the advantage of using TF symbols that are matched to the changes in the TF characteristics of a system or random process. We also provide new TF formulations for matched detection applications.
Time–frequency signal processing: a statistical perspective
 in: Proceedings of the Workshop on Circuits, Systems and Signal Processing, Mierlo, The
, 1998
"... Abstract—Timefrequency methods are capable of analyzing and/or processing nonstationary signals and systems in an intuitively appealing and physically meaningful manner. This tutorial paper presents an overview of some timefrequency methods for the analysis and processing of nonstationary random s ..."
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Cited by 1 (1 self)
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Abstract—Timefrequency methods are capable of analyzing and/or processing nonstationary signals and systems in an intuitively appealing and physically meaningful manner. This tutorial paper presents an overview of some timefrequency methods for the analysis and processing of nonstationary random signals, with emphasis placed on timevarying power spectra and techniques for signal estimation and detection. We discuss two major definitions of timedependent power spectra— the generalized WignerVille spectrum and the generalized evolutionary spectrum—and show their approximate equivalence for underspread random processes. Timedependent power spectra are then applied to nonstationary signal estimation and detection. Specifically, simple expressions and designs of signal estimators (Wiener filters) and signal detectors in the stationary case are extended to underspread nonstationary processes. This results in timefrequency techniques for nonstationary signal estimation and detection which are intuitively meaningful as well as efficient and stable. I.
Nonparametric Wavelet Methods for Nonstationary Time Series
, 1998
"... This article gives an overview on nonparametric modelling of nonstationary time series and estimation of their timechanging spectral content by modern denoising (smoothing) methods. For the modelling aspect localized decompositions such as various local Fourier (spectral) representations are discus ..."
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This article gives an overview on nonparametric modelling of nonstationary time series and estimation of their timechanging spectral content by modern denoising (smoothing) methods. For the modelling aspect localized decompositions such as various local Fourier (spectral) representations are discussed, among which wavelet and local cosine bases are most prominent ones. For the estimation of the possibly timevarying coefficients of these local representations wavelet denoising algorithms are applied and their particular properties in the context of time frequency and timescale analysis is discussed. In particular nonlinear wavelet thresholding, including recent developments of generalizing its usefulness for nonidentically, correlated and even nonstationary noise, is briefly reviewed as this is the unifying component of the estimation algorithms. The introduced procedures are illustrated by application to various simulated and real data examples from timefrequency and time...
ON SOME CLASSES OF LINEAR TIMEVARYING PARAMETRIC FILTERS
"... In this paper we investigate linear timevarying filters characterized by temporal and spectraldomain parameters. Existence and uniqueness properties are presented for four classes of parametric timevarying filters: (1) the rational class; (2) the rational adjoint class; (3) the wellknown ARMA c ..."
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In this paper we investigate linear timevarying filters characterized by temporal and spectraldomain parameters. Existence and uniqueness properties are presented for four classes of parametric timevarying filters: (1) the rational class; (2) the rational adjoint class; (3) the wellknown ARMA class; and (4) the ARMA adjoint class. In previous work on nonstationary processes [1], we presented membership conditions on the Green’s function for each of these classes. These conditions were used to determine when minimumorder parameterizations are unique and to give precise conditions under which a unique minimumorder filter is a member of one or more of these classes. In this paper, we present these results in a system theoretic framework. 1.
Uncertainty and Concentration Inequalities for Nonstationary Random Processes and TimeFrequency Energy Spectra
"... We provide various sharp uncertainty inequalities for nonstationary random processes. These inequalities relate the temporal and spectral energy concentration of a process to the "effective rank" of its correlation operator. Similar inequalities are established for the timefrequency (TF) ..."
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We provide various sharp uncertainty inequalities for nonstationary random processes. These inequalities relate the temporal and spectral energy concentration of a process to the "effective rank" of its correlation operator. Similar inequalities are established for the timefrequency (TF) concentration of a wide range of TF energy spectra. We further identify TF energy spectra that feature maximum TF concentration and illustrate our results for the practically important class of underspread random processes.
Dynamics of structures coupled with elastic media a review of numerical models and methods
, 2013
"... This paper reviews issues and developments in the field of structureenvironment interaction problems, in which the environment is an elastic body, possibly unbounded. It covers in particular the fields of soilstructure interaction, groundborne noise and vibration emitted by transportation systems ..."
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This paper reviews issues and developments in the field of structureenvironment interaction problems, in which the environment is an elastic body, possibly unbounded. It covers in particular the fields of soilstructure interaction, groundborne noise and vibration emitted by transportation systems and wave diffraction by obstacles in an elastic medium. The general setting for a linear bounded structure coupled to an unbounded linear environment through a bounded interface is first recalled, and the domain decomposition technique classically used for its description is put up. Extensions for the cases of nonlinear structure and uncertain environment are then discussed. Finally, the ongoing research in the fields of unbounded interface, moving interface, and multiple interfaces are reviewed and summarized.