  Studies of sparse representation of deterministic signals have been well
developed. Amongst there exists one called adaptive Fourier decomposition (AFD)
established through adaptive selections of the parameters defining a
Takenaka-Malmquist system in one-complex variable. The AFD type algorithms give
rise to sparse representations of signals of finite energy. The multivariate
generalization of AFD is one called pre-orthogonal AFD (POAFD), the latter
being established with the context Hilbert space possessing a dictionary. The
purpose of the present study is to generalize both AFD and POAFD to random
signals. We work on two types of random signals. One is those expressible as
the sum of a deterministic signal with an error term such as a white noise; and
the other is, in general, as mixture of several classes of random signals
obeying certain distributive law. In the first part of the paper we develop an
AFD type sparse representation for one-dimensional random signals by making use
analysis of one complex variable. In the second part, without complex analysis,
we treat multivariate random signals in the context of stochastic Hilbert space
with a dictionary. Like in the deterministic signal case the established random
sparse representations are powerful tools in practical signal analysis.
