We present a novel variational model for the additive decomposition of 1D noisy signals. The model relies on sparsifying fractional-order derivatives of the sought-for components to capture intricate signal structures. To efficiently solve the resulting optimization problem, an alternating direction method of multipliers-based algorithm is developed. Furthermore, a bilevel optimization framework is proposed to automatically select “optimal” values of all the free parameters in the model, including the orders of the sparsified fractional derivatives. Preliminary results validate the effectiveness of the proposed approach in accurately decomposing noisy signals, even in the presence of abrupt changes.