Modulo Sampling: Performance Guarantees in The Presence of Quantization

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.