mSQUID: Model-Based Leanred Modulo Recovery at Low Sampling Rates

Modulo sampling enables acquisition of signals with unlimited dynamic range by folding the input into a bounded interval prior to sampling, thus eliminating the risk of signal clipping and preserving information without requiring highresolution ADCs. While this enables low-cost hardware, the nonlinear distortion introduced by folding presents recovery challenges, particularly under noise and quantization. We propose a model-based deep unfolding network tailored to this setting, combining the interpretability of classical compress sensing (CS) solvers with the flexibility of learning. A key innovation is a soft-quantization module that encodes the modulo prior by guiding the solution toward discrete multiples of the folding range in a differentiable and learnable way. Our method, modulo soft-quantized unfolded iterative decoder (mSQUID), achieves superior reconstruction performance at low sampling rates under additive Gaussian noise. We further demonstrate its utility in a challenging case where signals with vastly different amplitudes and disjoint frequency bands are acquired simultaneously and quantized. In this scenario, classical sampling often struggles due to weak signal distortion or strong signal clipping, while our approach is able to recover the input signals. Our method also offers significantly reduced runtimes, making it suitable for real-time, resource-limited systems.