  This paper introduces the class of ambiguity sparse processes, containing
subsets of popular nonstationary time series such as locally stationary,
cyclostationary and uniformly modulated processes. The class also contains
aggregations of the aforementioned processes. Ambiguity sparse processes are
defined for a fixed sampling regime, in terms of a given number of sample
points and a fixed sampling period. The framework naturally allows us to treat
heterogeneously nonstationary processes, and to develop methodology for
processes that have growing but controlled complexity with increasing sample
sizes and shrinking sampling periods. Expressions for the moments of the sample
ambiguity function are derived for ambiguity sparse processes. These properties
inspire an Empirical Bayes shrinkage estimation procedure. The representation
of the covariance structure of the process in terms of a time-frequency
representation is separated from the estimation of these second order
properties. The estimated ambiguity function is converted into an estimate of
the time-varying moments of the process, and from these moments, any bilinear
representation can be calculated with reduced estimation risk. Any of these
representations can be used to understand the time-varying spectral content of
the signal. The choice of representation is discussed. Parameters of the
shrinkage procedure quantify the performance of the proposed estimation.
