We address a parametric joint detection-estimation problem for discrete signals of the form x(t) = ΣN n=1αne-iλnt + ϵt, t ∈ N, with an additive noise represented by independent centered complex random variables ϵt. The distributions of ϵt 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., frequencies λn, their number N, and complex coefficients αn. For example, one of considered classes of noise is the following: ϵt are independent identically distributed random variables with E(ϵt)=0 and E𝔼(|ϵt|ln|ϵt|) < ∞. 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.