Measuring impropriety in complex and real representations

So-called improper signals, i.e., signals which are correlated with their complex conjugates, can occur in many signal processing applications such as communication systems, medical imaging, audio and speech processing, analysis of oceanographic data, and many more. Being aware of potential impropriety can be crucial whenever we model signals as complex random quantities since an appropriate treatment of improper signals, e.g., by widely linear filtering, can significantly improve the system performance. After a brief introduction into the fundamentals of improper signals, this article focuses on the problem of quantifying the impropriety of complex random vectors and gives a survey of various impropriety measures in both the composite real representation and the augmented complex formulation. Unlike in previous publications, these two frameworks are presented side by side to reveal the differences and common points between them. Moreover, their applicability is compared in several practical examples. As additional aspects, we consider the problem of testing for impropriety based on measurement data, and the differential entropy of Gaussian vectors as an impropriety measure in information theoretic studies. The article includes a tutorial-style introduction, a collection of important formulae, a comparison of various mathematical approaches, as well as some new reformulations.

Measuring Impropriety in Complex and Real Representations

This work deals with the modelling of multiple and structured oscillatory phenomena. The goal of the thesis is to show how stochastic oscillations can be modelled, and define their elliptical structures as a special class of bivariate time-dependant variation. The central part of the research is the introduction of new multivariate elliptical models and the review of existing definitions. The findings are presented in a table, where the classification is made based on whether the definitions are random or deterministic and whether they are defined in time or frequency domains. The previously introduced ellipse definitions for stochastic processes that have been described in the literature are limited to the frequency domain only. The main contribution of this work is in adding to existing time domain models by defining the description of the autocovariance ellipse and the forecast ellipse. Both of these definitions are non-random. The ellipses are defined from either the autocovariance or the forecast functions of the process as one moves forward in lag-time or forecast-time. In order to illustrate these theoretical concepts and show the usefulness of the new definition we investigate these concepts using a parametric model. Univariate and bivariate, real-valued and complex-valued models are considered, and their representation discussed. The richest model proposed is that of a complex-valued bivariate autoregressive process of order one and this is based on modelling using affine transformation matrices. This model results in a stochastic oscillation and the elliptical definitions proposed are explored in this context. The actual behaviour of the proposed stochastic process is also illustrated on simulated data. Some limitations of this approach are discussed and extensions of this model are presented.

Modelling Multiple and Structured Oscillatory Phenomena

Widely Linear Estimators and Adaptive Filters for Real-Valued Quantities in Complex-Valued Environments

Individuals from all related scientific disciplines and specialties are encouraged to participate to provide insight into issues pertinent to the area of amplification and signal processing and to formulate the future directions of hearing aid research and development. Scientific abstracts emphasizing current research findings are due March 15,1997. The conference format will include both podium presentations and poster sessions, with considerable time allotted for audience discussion. Abstracts will be peer reviewed for scientific merit and relevance. Exhibit space will be available.

G. Johnson

Papers

The Complex (and Circular) Argument Continues