Alfred Hanssen

A generalized likelihood ratio test for impropriety of complex signals

A complex random vector is called improper if it is correlated with its complex conjugate. We present a hypothesis test for impropriety based on a generalized likelihood ratio (GLR). This GLR is invariant to linear transformations on the data, including rotation and scaling, because propriety is preserved by linear transformations. More specifically, we show that the GLR is a function of the squared canonical correlations between the data and their complex conjugate. These canonical correlations make up a complete, or maximal, set of invariants for the Hermitian and complementary covariance matrices under linear, but not widely linear, transformation

Numerical test of the weak turbulence approximation to ionospheric Langmuir turbulence

The standard weak Langmuir turbulence approach to explain the artificial plasma line in ionospheric radio modification experiments is examined. We compare solutions of a weak turbulence approximation (WTA) derived from a version of the one-dimensional driven and damped Zakharov system of equations (ZSE) with solutions to the same full ZSE. The electromagnetic pump field is modeled as a long-wavelength parametric driving term. We found that from a certain distance below the O mode reflection level the wave number saturation spectra computed from the WTA agree qualitatively with those from the ZSE for weak driving strengths, in the sense that the number of cascade lines increases with increasing pump strength. However, in general, the number of cascades apparent in the WTA solutions is larger than that predicted from the full ZSE. At higher intensities of the driver the saturation spectra from the ZSE differ from the WTA cascade spectra, in that a truncation of the cascade sets in, with a subsequent filling in of the bands between the cascades. This truncation takes place far before the ZSE cascade spectra reach the so-called “Langmuir condensate,”; contrary to earlier conjectures based mainly on dimensional analysis arguments. In the reflection region a qualitatively different process takes place: temporal cycles of large ensembles of localized events; nucleation of cavitons, collapse, and burnout constitute the basic elements of the turbulence in this region of space. No WTA exists for this region. Our findings are discussed with respect to the experiments performed at Arecibo and Tromso, the conclusion being that the ZSE yields results closer to observations than does the WTA, in all regions of space.

EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution

The heavy-tailed multivariate normal inverse Gaussian (MNIG) distribution is a recent variance-mean mixture of a multivariate Gaussian with a univariate inverse Gaussian distribution. Due to the complexity of the likelihood function, parameter estimation by direct maximization is exceedingly difficult. To overcome this problem, we propose a fast and accurate multivariate expectation-maximization (EM) algorithm for maximum likelihood estimation of the scalar, vector, and matrix parameters of the MNIG distribution. Important fundamental and attractive properties of the MNIG as a modeling tool for multivariate heavy-tailed processes are discussed. The modeling strength of the MNIG, and the feasibility of the proposed EM parameter estimation algorithm, are demonstrated by fitting the MNIG to real world hydrophone data, to wideband synthetic aperture sonar data, and to multichannel radar sea clutter data.

A theory of polyspectra for nonstationary stochastic processes

Harmonizable processes constitute an important class of nonstationary stochastic processes. We present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic quantities: the nth-order moment function, the nth-order time-frequency polyspectrum, the nth-order ambiguity function, and the nth-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous nth-order moment and the conventional nth-order stationary polyspectrum, respectively. All quantities except the nth-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order n, which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels.

Stochastic multipulse-PAM: a subspace modulation technique with diversity

In this paper, we introduce a non-conventional communication technique appropriately named Stochastic Multipulse-PAM (SM-PAM). SM-PAM systems operate by transmission of stochastic processes residing in low-dimensional subspaces. The transmitted signal is thus inherently stochastic, which is beneficial from a low-probability of intercept transmission point of view. The underlying structure of the transmission is such that the information is actually coded by the subspaces themselves, rather than by fixed coordinates in a preassigned basis, as is the case for conventional communication. We introduce a Fourier-based version of SM-PAM, and show that intersymbol interference (ISI) may readily be mitigated by employing a zero-forcing strategy and the insertion of a cyclic prefix in the transmitted vector. A substantial part of this paper is devoted to the derivation of matched subspace detectors for the transmitted SM-PAM waveform, for various classes of interference. We carry out a performance evaluation of SM-PAM, which shows that a proper energy scaling of the transmitted stochastic symbol vector has positive effects on the performance of the system. Various interesting scenarios are demonstrated by numerical simulations. The proposed multidimensional, stochastic, broadband, non-coherent communication technique buys security and diversity at the expense of spectral efficiency. We believe that SM-PAM may be important for non-standard applications where high data rates are not mandatory, but where security and interference rejection is imperative.

Radiation from electromagnetically driven Langmuir turbulence

A two-level model for the interaction between the electromagnetic pump wave and the electrostatic turbulence is formulated for ionospheric radio modification experiments. On the local level, the Zakharov equations, or similar models, apply. The interaction with the global electromagnetic level is represented by a second-order current density averaged over the local spatial variable. The energy exchange between the local and global level is represented by a Joulean product involving this second-order current density. The global generation problem is solved in the simplest cases. The escaping energy flux in the sideband ω is shown to be represented as a folding between the power spectrum of the local source and a squared Airy function. This power spectrum has been calculated from numerical simulations of electromagnetically driven Langmuir turbulence, using a one-dimensional model of the Zakharov type, for varying values of the parameters. For parameters in the cascade range, narrow line structured spectra were obtained, while for parameters in the cavitation range, very broad featureless spectra were obtained. Comparison with recent experimental stimulated electromagnetic emissions data did not confirm a signature for the existence of cavitation in the experiments.

Multidimensional multiplier spectral estimation

Abstract We propose a very simple generalization of the one-dimensional multitaper spectral estimation technique to the multi-dimensional case. Multidimensional tapers are constructed from products of one-dimensional taper elements, or formally a outer products of lower dimensional tapers. The two-dimensional case is treated in some detail, and several properties of the novel tapers are listed.

The normal inverse Gaussian distribution: a versatile model for heavy-tailed stochastic processes

The normal inverse Gaussian (NIG) distribution is a recent flexible closed form distribution that may be applied as a model of heavy-tailed processes. The NIG distribution is completely specified by four real valued parameters that have natural interpretations in terms of the shape of the resulting probability density function. By choosing the parameters appropriately, one can describe a wide range of shapes of the distribution. We discuss several of the desirable properties of the NIG distribution. In particular, we discuss the cumulant generating function and the cumulants of the NIG-variables. A particularly important property is that the NIG distribution is closed under convolution. Finally, we derive a set of very simple yet accurate estimators of the NIG parameters. Our estimators differ fundamentally from estimators suggested by other authors in that our estimators take advantage of the surprisingly simple structure of the cumulant generating function.

The Hilbert space geometry of the Rihaczek distribution for stochastic analytic signals

The Rihaczek distribution for stochastic signals is a time- and frequency-shift covariant bilinear time-frequency distribution (TFD) based on the Crame/spl acute/r-Loe/spl grave/ve spectral representation for a harmonizable process. It is a complex Hilbert space inner product (or cross correlation) between the time series and its infinitesimal stochastic Fourier generator. To this inner product, we may attach an illuminating geometry, wherein the cosine squared of the angle between the time series and its infinitesimal stochastic Fourier generator is given by the Rihaczek distribution. The Rihaczek distribution also determines a time-varying Wiener filter for estimating a time series from its infinitesimal stochastic Fourier generator and measures the resulting error covariance. We propose a factored kernel to construct estimators of the Rihaczek distribution that are contained in Cohens class of bilinear TFDs.

Multitaper estimators of polyspectra

In theory, polyspectra can be applied to solve many important problems in signal processing and data analysis. In practice, however, one has been discouraged by the poor statistical properties of most polyspectral estimators. In this paper, we extend Thomsons original multitaper power (and bispectral) estimator to polyspectra of arbitrary order, and to any orthonormal family of tapers. The multitaper polyspectral estimators we derive have favorable statistical properties, and they are well suited for the analysis of short data segments. We derive useful expressions for the first and second moments of the estimator. It is shown that a quantity we call the total polyspectral window is the core quantity for describing and understanding the statistical properties of the proposed estimators. Leakage can be a dominant effect in polyspectral estimation. We thus extend Thomsons adaptive power spectral leakage reduction scheme, to polyspectral estimators of arbitrary order. Based on an extensive Monte Carlo simulation of the bispectrum for four different stochastic processes (one Gaussian, and three non-Gaussian), we conduct a comparison between four conventional bispectral estimators and two multitaper bispectral estimators. Our simulations show that the estimator of choice is the adaptive multitaper polyspectral estimator using discrete prolate spheroidal sequences. Finally, we apply the estimators to Volterra model system identification based on a real data set from an offshore platform experiment. The system identification requires polyspectra up to and including order four, and we show that the adaptive multitaper polyspectral estimator outperforms the conventional estimators with respect to modeling accuracy.