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\begin{document}

\title{Sparse Representations of Random Signals}



\author[1]{Tao Qian}%
\affil[1]{Macau University of Science and Technology}%


\vspace{-1em}



  
  \date{May 23, 2024}


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\begin{abstract}
Sparse (fast) representations of deterministic signals have been well
studied. Among other types there exists one called adaptive Fourier
decomposition (AFD) for functions in analytic Hardy spaces. Through the
Hardy space decomposition of the \$L\^{}2\$ space the AFD algorithm also
gives rise to sparse representations of signals of finite energy. To
deal with multivariate signals the general Hilbert space context comes
into play. The multivariate counterpart of AFD in general Hilbert spaces
with a dictionary has been named pre-orthogonal AFD (POAFD). In the
present study we generalize AFD and POAFD to random analytic signals
through formulating stochastic analytic Hardy spaces and stochastic
Hilbert spaces. To analyze random analytic signals we work on two
models, both being called stochastic AFD, or SAFD in brief. The two
models are respectively made for (i) those expressible as the sum of a
deterministic signal and an error term (SAFDI); and for (ii) those from
different sources obeying certain distributive law (SAFDII). In the
later part of the paper we drop off the analyticity assumption and
generalize the SAFDI and SAFDII to what we call stochastic Hilbert
spaces with a dictionary. The generalized methods are named as
stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and
SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed
stochastic POAFD algorithms offer powerful tools to approximate and thus
to analyze random signals.%
\end{abstract}%



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