  We address a parametric joint detection-estimation problem for discrete
signals of the form $x(t) = \sum_{n=1}^{N} \alpha_n e^{-i \lambda_n t } +
\epsilon_t$, $t \in \mathbb{N}$, with an additive noise represented by
independent centered complex random variables $\epsilon_t$. The distributions
of $\epsilon_t$ are assumed to be unknown, but satisfying various sets of
conditions. We prove that in the case of a heavy-tailed noise it is possible to
construct asymptotically strongly consistent estimators for the unknown
parameters of the signal, i.e., the frequencies $\lambda_n$, their number $N$,
and complex amplitudes $\alpha_n$. For example, one of considered classes of
noise is the following: $\epsilon_t$ are independent identically distributed
random variables with $\mathbb{E} (\epsilon_t) = 0$ and $\mathbb{E}
(|\epsilon_t| \ln |\epsilon_t|) < \infty$. The construction of estimators is
based on detection of singularities of anti-derivatives for $Z$-transforms and
on a two-level selection procedure for special discretized versions of
superlevel sets. The consistency proof relies on the convergence theory for
random Fourier series. We discuss also decaying signals and the case of
infinite number of frequencies.
