Patrik Wahlberg

Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes

This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohens class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silvermans 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 eigenvalue-eigenfunction 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

Methods for robust clustering of epileptic EEG spikes

The authors investigate algorithms for clustering of epileptic electroencephalogram (EEG) spikes. Such a method is useful prior to averaging and inverse computations since the spikes of a patient often belong to a few distinct classes. Data sets often contain outliers, which makes algorithms with robust performance desirable. The authors compare the fuzzy C-means (FCM) algorithm and a graph-theoretic algorithm. They give criteria for determination of the correct level of outlier contamination. The performance is then studied by aid of simulations, which show good results for a range of circumstances, for both algorithms. The graph-theoretic method gave better results than FCM for simulated signals. Also, when evaluating the methods on seven real-life data sets, the graph-theoretic method was the better method, in terms of closeness to the manual assessment by a neurophysiologist. However, there was some discrepancy between manual and automatic clustering and the authors suggest as an alternative method a human choice among a limited set of automatically obtained clusterings. Furthermore, the authors evaluate geometrically weighted feature extraction and conclude that it is useful as a supplementary dimension for clustering.

Spectral Relations for Multidimensional Complex Improper Stationary and (Almost) Cyclostationary Processes

We study continuous-time multidimensional wide- sense stationary (WSS) and (almost) cyclostationary processes in the frequency domain. Under the assumption that the correlation function is uniformly continuous, we prove the existence of a unique sequence of spectral measures, which coincide with the restrictions to certain subdiagonals of the spectral measure in the strongly harmonizable case. Moreover, the off-diagonal measures are absolutely continuous with respect to the diagonal measure. As a consequence, for strongly harmonizable scalar improper almost cyclostationary processes, we obtain representation formulas for the components of the complementary spectral measure and the off-diagonal components of the spectral measure, in terms of the diagonal component of the spectral measure. We apply these results to analytic signals, which produces sufficient conditions for propriety for almost cyclostationary analytic signals.

The equality of the homogeneous and the Gabor wave front set

ABSTRACT We prove that Hörmander’s global wave front set and Nakamura’s homogeneous wave front set of a tempered distribution coincide. In addition, we construct a tempered distribution with a given wave front set, and we develop a pseudodifferential calculus adapted to Nakamura’s homogeneous wave front set.

Schrödinger-type propagators, pseudodifferential operators and modulation spaces

We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of non-degenerate generalized quadratic forms that includes Schr¨odinger propagators and pseudodifferential operators. As a byproduct, we obtain a characterization of all exponents p, q, r1, r2, t1, t2 ∈ [1,∞] of modulation spaces such that a symbol in Mp,q(R2d) gives a pseudodifferential operator that is continuous from Mr1,r2 (Rd) into Mt1,t2 (Rd)

On Wiener filtering of certain locally stationary stochastic processes

We study linear minimum mean squared error filters for continuous-time second-order stochastic processes that are locally stationary in Silvermans sense. We show that the optimal filter is rarely locally stationary even when the covariance functions have Gaussian shape. Using Mehlers formula we derive series expansions of the filter kernel for locally stationary covariances that are determined by Gaussians.

Microlocal properties of Shubin pseudodifferential and localization operators

We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.

Locally stationary harmonizable complex improper stochastic processes

This article concerns continuous-time second-order complex-valued improper stochastic processes that are harmonizable and locally stationary in Silvermans sense. We study necessary and sufficient conditions for the property of local stationarity in the time and frequency domains. A sufficient condition by Silverman is generalized and extended to the improper case. We obtain a result on the absolute continuity of the complementary spectral measure with respect to the spectral measure, which is related to a spectral characterization of improper wide-sense stationary processes.

The Wigner distribution of Gaussian weakly harmonizable stochastic processes

The paper treats the Wigner distribution of scalar-valued stochastic processes defined on ℝd. We show that if the process is Gaussian and weakly harmonizable then a stochastic Wigner distribution is well defined. The special case of stationary processes is studied, in which case the Wigner distribution is weakly stationary in the time variable and the variance is equal to the deterministic Wigner distribution of the covariance function.

Optimal time-frequency kernels for spectral estimation of locally stationary processes

This paper investigates the mean square error optimal time-frequency kernel for estimation of the Wigner-Ville spectrum of a certain class of nonstationary processes. The class of locally stationary processes have a simplified covariance structure which facilitates analysis. We give a formula for the optimal kernel in the ambiguity domain and conditions that are sufficient for the optimal time-frequency kernel to he a continuous function, decaying at infinity.