Fold First, Detect Directly: Communication Symbol Detection Without Unfolding for Low-Bitrate Modulo-ADCs
Modulo-folding ADCs reduce power consumption by restricting the dynamic range of the sampled signal before quantization, at the cost of an unfolding step needed to recover the true samples before any further processing. We show that this unfolding step can be skipped entirely for symbol detection, even in a realistic, oversampled setting where additive noise is present before the modulo operator and becomes correlated as it passes through the receiver's front-end filter, and separate quantization noise is further introduced due to the ADC. We show that a specific residual, formed from the folded, quantized observations and a candidate symbol hypothesis, exactly cancels the unknown integer wrap introduced by folding, so that the likelihood of a hypothesis is the density of the folded noise evaluated at that residual. Starting from this exact likelihood an intractable lattice sum over all integer wrap vectors, we show that the wrap vector is ternary and sparse with high probability whenever the folding threshold exceeds the noise standard deviation, so that at a threshold-to-noise ratio of three or more the sum is well approximated by a single Gaussian term. The resulting Mahalanobis maximum-likelihood detector works directly on the folded, quantized samples, and a block-structured search keeps detection tractable for long symbol sequences. Simulations confirm that in this regime our detector tracks the accuracy of a conventional, non-folding ADC closely across a wide range of SNRs, while the unfolding-based baselines need substantially higher oversampling to reach comparable accuracy.