Above the Nyquist Rate, Modulo Folding Does Not Hurt

We consider the problem of recovering a continuous-time bandlimited signal from the discrete-time signal, obtained from sampling it every T-s seconds and reducing the result modulo Delta, for some Delta >0. For Delta =infty, the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible, provided that the sampling rate 1/T-s exceeds the so-called Nyquist rate. Recent work by Bhandari et al. has shown that for any Delta >0 perfect reconstruction is still possible, if the sampling rate exceeds the Nyquist rate by a factor of pi e. In this letter, we improve upon this result and show that for finite energy signals, perfect recovery is possible for any Delta >0 and any sampling rate above the Nyquist rate. Thus, modulo folding does not degrade the signal, provided that the sampling rate exceeds the Nyquist rate. This claim is proved by establishing a connection between the recovery problem of a discrete-time signal from its modulo reduced version and the problem of predicting the next sample of a discrete-time signal from its past, and leveraging the fact that for a bandlimited signal the prediction error can be made arbitrarily small.