Modulo Sampling: Performance Guarantees in The Presence of Quantization

TL;DR

This paper analyzes the performance of modulo ADCs with quantization, providing conditions under which they outperform conventional ADCs in terms of MSE, and introduces algorithms for signal recovery with and without additional folding information.

Contribution

The paper introduces new recovery algorithms for modulo ADCs, establishes performance guarantees under specific oversampling and quantization conditions, and compares their MSE performance to conventional ADCs.

Findings

Modulo ADC with folding info achieves lower MSE than conventional ADC.

Oversampling factor > 3 and resolution > 3 bits suffice for effective unfolding.

Modulo ADC's MSE scales as 1/OF^3, outperforming the 1/OF scaling of conventional ADC.

Abstract

In this paper, we investigate the relationship between the dynamic range and quantization noise power in modulo analog-to-digital converters (ADCs). Two modulo ADC systems are considered: (1) a modulo ADC which outputs the folded samples and an additional 1-bit folding information signal, and (2) a modulo ADC without the 1-bit information. A recovery algorithm that unfolds the quantized modulo samples using the extra 1-bit folding information is analyzed. Using the dithered quantization framework, we show that an oversampling factor of $\mathrm{OF} > 3$ and a quantizer resolution of $b > 3$ are sufficient conditions to unfold the modulo samples. When these conditions are met, we demonstrate that the mean squared error (MSE) performance of modulo ADC with an extra 1-bit folding information signal is better than that of a conventional ADC with the same number of bits used for amplitude quantization. Since folding information is typically not available in modulo ADCs, we also propose and analyze a recovery algorithm based on orthogonal matching pursuit (OMP) that does not require the 1-bit folding information. In this case, we prove that $\mathrm{OF} > 3$ and $b > 3 + \log_2(\delta)$ for some $\delta > 1$ are sufficient conditions to unfold the modulo samples. For the two systems considered, we show that, with sufficient number of bits for amplitude quantization, the mean squared error (MSE) of a modulo ADC is $\mathcal{O}\left(\frac{1}{\mathrm{OF}^3}\right)$ whereas that of a conventional ADC is only $\mathcal{O}\left(\frac{1}{\mathrm{OF}}\right)$. We extend the analysis to the case of simultaneous acquisition of weak and strong signals occupying different frequency bands. Finally, numerical results are presented to validate the derived performance guarantees.