Widely linear Markov signals
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RESEARCH
Open Access
Widely linear Markov signals ´ Navarro-Moreno, Rosa M Fern´andez-Alcal´a* and Juan C Ruiz-Molina Juan A Espinosa-Pulido, Jesus
Abstract The insufficiency to guarantee the existence of a state-space representation of the classical wide-sense Markov condition for improper complex-valued signals is shown and a generalization is suggested. New characterizations for wide-sense Markov signals which are based either on second-order properties or on state-space representations are studied in a widely linear setting. Moreover, the correlation structure of such signals is revealed and interesting results on modeling in both the forwards and backwards time directions are proved. As an application we give some recursive estimation algorithms obtained from the Kalman filter. The performance of the proposed results is illustrated in a numerical example in the areas of estimation and simulation. Keywords: Modeling, Wide-sense Markov signals, Widely linear processing
1 Introduction Markov signals are characterized by the condition that future development of these signals depends only on current states and not their history up to that time. In general, Markov processes are easier to model and analyze, and they do include interesting applications. Among others, estimation and detection are areas of signal processing where this kind of process has provided efficient solutions (see, e.g., [1,2]). Non-Markov processes in which the future state of a process depends on its whole history are generally harder to analyze mathematically [3]. In linear minimum-mean square error (MMSE) estimation theory, when the processes under consideration are not Gaussian, the classes of stochastic processes which are of practical importance are wide-sense Markov (WSM) processes. The concept of WSM signal is easier to check than the condition of (strictly) Markov since it involves only second-order characteristics [4]. In general, WSM processes (with the exception of Gaussian processes) are not Markov in the strict sense. The equivalence between the WSM condition and the state-space representation for the signal is really what makes WSM signals especially attractive in signal processing [1]. Widely linear (WL) processing is an emerging research area in the complex-valued signal analysis which gives significant performance gains with respect to strictly linear (SL) processing (excellent account of the topic *Correspondence: [email protected] Department of Statistics and Operations Research, University of Ja´en, Campus Las Lagunillas, 23071 Ja´en, Spain
and the literature can be found in [5,6]). It has proved to be a more useful approach than SL processing since complex-valued random signals are in general improper (i.e., they are correlated with their complex conjugates). Thus, the improper nature of most signals forces us to consider the so-called augmented statistics to entirely describe their second-order properties. Using augmented statistics means incorporating in the analysis the information supplied by the complex conjugate of the
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