# Above the Nyquist Rate, Modulo Folding Does Not Hurt

arXiv: https://arxiv.org/abs/1903.12289 · HTML: https://mlsanity.com/paper/1903.12289/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2019.

## Built on by (most cited first, 9 of 9)

- [UNO: Unlimited Sampling Meets One-Bit Quantization](https://mlsanity.com/paper/2301.10155/) (2022)
- [Compressed Sensing Based Residual Recovery Algorithms and Hardware for Modulo Sampling](https://mlsanity.com/paper/2412.12724/) (2024)
- [Spiked Covariance Estimation from Modulo-Reduced Measurements](https://mlsanity.com/paper/2110.01150/) (2021)
- [Blind-Adaptive Quantizers](https://mlsanity.com/paper/2409.04077/) (2024)
- [Modulo Sampling: Performance Guarantees in The Presence of Quantization](https://mlsanity.com/paper/2501.01506/) (2025)
- [mSQUID: Model-Based Leanred Modulo Recovery at Low Sampling Rates](https://mlsanity.com/paper/2510.18729/) (2025)
- [One-Bit-Aided Modulo Sampling for DOA Estimation](https://mlsanity.com/paper/2309.04901/) (2023)
- [1-Bit Unlimited Sampling Beyond Fourier Domain: Low-Resolution Sampling of Quantization Noise](https://mlsanity.com/paper/2508.19408/) (2025)
- [Fold First, Detect Directly: Communication Symbol Detection Without Unfolding for Low-Bitrate Modulo-ADCs](https://mlsanity.com/paper/2609.11298/) (2026)

## Abstract (arXiv)

We consider the problem of recovering a continuous-time bandlimited signal from the discrete-time signal obtained from sampling it every Ts seconds and reducing the result modulo Delta, for some Delta>0. For Delta=∞ the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible provided that the sampling rate 1/Ts 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 π 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.