The problem of estimating the parameters of biased and exponentially-damped multi-sinusoidal signals is addressed in this paper by a finite-time identification scheme based on Volterra integral operators. These parameters are the amplitudes, frequencies, initial phase angles, damping factors and the offset. The proposed strategy entails the design of a new kind of kernel function that, compared to existing ones, allows for the identification of the initial conditions of the signal-generator system. The worst-case behavior of the proposed algorithm in the presence of bounded additive disturbances is fully characterized by Input-to-State Stability arguments. Numerical examples including the comparisons with some existing tools are reported to show the effectiveness of the proposed methodology.