Perfect Reconstruction Schemes for Sampling Piecewise Sinusoidal Signals

Consider sampling a signal that is piecewise sinusoidal. Classical sampling theory does not enable a perfect reconstruction of the continuous time signal since the band is not limited (C.E. Shannon, 1949). However, we show that it is still possible to recover all the parameters of the sinusoids and the exact locations of the discontinuities using the annihilating filter method and recently developed Finite Rate of Innovation (FRI) sampling schemes (M. Vetterli et al., 2002) (P.L. Dragotti et al., 2005). Moreover, we show that there is a tradeoff between the number of sinusoids per piece and the proximity of the discontinuities in order to have a unique solution. This result recalls a sort of uncertainty principle