# Above the Nyquist Rate, Modulo Folding Does Not Hurt

## TL;DR

This paper proves that for bandlimited signals, sampling above the Nyquist rate allows perfect recovery from modulo-reduced samples, showing modulo folding does not impair signal reconstruction.

## Contribution

It improves previous bounds by demonstrating perfect recovery is possible at any rate above Nyquist for finite energy signals, regardless of modulo reduction.

## Key findings

- Modulo folding does not affect perfect recovery above Nyquist rate.
- Recovery is achieved by predicting next samples from past data.
- The prediction error can be made arbitrarily small for bandlimited signals.

## Abstract

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.