We consider the problem of recovering a continuous-time bandlimited signal from the discrete-time signal obtained from sampling it every $T_s$ seconds and reducing the result modulo Δ, for some Δ>0. For Δ=∞ the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible provided that the sampling rate 1/ $T_s$ exceeds the so-called Nyquist rate. Recent work by Bhandari et al. has shown that for any Δ>0 perfect reconstruction is still possible if the sampling rate exceeds the Nyquist rate by a factor of πe. In this letter we improve upon this result and show that for finite energy signals, perfect recovery is possible for any Δ>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.

Systems that capture and process analog signals must first acquire them through an analog-to-digital converter. While subsequent digital processing can remove statistical correlations present in the acquired data, the dynamic range of the converter is typically scaled to match that of the input analog signal. The present paper develops an approach for analog-to-digital conversion that aims at minimizing the number of bits per sample at the output of the converter. This is attained by reducing the dynamic range of the analog signal by performing a modulo operation on its amplitude, and then quantizing the result. While the converter itself is universal and agnostic of the statistics of the signal, the decoder operation on the output of the quantizer can exploit the statistical structure in order to unwrap the modulo folding. The performance of this method is shown to approach information theoretical limits, as captured by the rate-distortion function, in various settings. An architecture for modulo analog-to-digital conversion via ring oscillators is suggested, and its merits are numerically demonstrated.

This paper provides a simplified proof for the existence of nested lattice codebooks allowing to achieve the capacity of the additive white Gaussian noise channel, as well as the optimal rate-distortion tradeoff for a Gaussian source. The proof is self-contained and relies only on basic probabilistic and geometrical arguments. An ensemble of nested lattices that is different, and more elementary, than the one used in the previous proofs is introduced. This ensemble is based on lifting different subcodes of a linear code to the Euclidean space using Construction A. In addition to being simpler, the analysis is less sensitive to the assumption that the additive noise is Gaussian. In particular, for additive ergodic noise channels, it is shown that the achievable rates of the nested lattice coding scheme depend on the noise distribution only via its power. Similarly, the nested lattice source coding scheme attains the same rate-distortion tradeoff for all ergodic sources with the same second moment.

Integer-Forcing (IF) is a new framework, based on compute-and-forward, for decoding multiple integer linear combinations from the output of a Gaussian multiple-input multiple-output channel. This paper applies the IF approach to arrive at a new low-complexity scheme, IF source coding, for distributed lossy compression of correlated Gaussian sources under a minimum mean squared error distortion measure. All encoders use the same nested lattice codebook. Each encoder quantizes its observation using the fine lattice as a quantizer and reduces the result modulo the coarse lattice, which plays the role of binning. Rather than directly recovering the individual quantized signals, the decoder first recovers a full-rank set of judiciously chosen integer linear combinations of the quantized signals, and then inverts it. In general, the linear combinations have smaller average powers than the original signals. This allows to increase the density of the coarse lattice, which in turn translates to smaller compression rates. We also propose and analyze a one-shot version of IF source coding that is simple enough to potentially lead to a new design principle for analog-to-digital converters that can exploit spatial correlations between the sampled signals.

Consider two sequences of n independent and identically distributed fair coin tosses, X = (X1,...,Xn) and Y = (Y1,...,Yn), which are ρ-correlated for each j, i.e. P[Xj = Yj] = 1+ρ/2. We study the question of how large (small) the probability P[X ∈ A; Y ∈ B] can be among all sets A;B ⊂ {0,1} of a given cardinality. For sets |A| |B| = Θ(2n) it is well known that the largest (smallest) probability is approximately attained by concentric (anticoncentric) Hamming balls, and this can be proved via the hypercontractive inequality (reverse hypercontractivity). Here we consider the case of |A| |B| = 2Θ(n). By applying a recent extension of the hypercontractive inequality of Polyanskiy-Samorodnitsky (J. Functional Analysis, 2019), we show that Hamming balls of the same size approximately maximize P[X ∈ A; Y ∈ B] in the regime of ρ → 1. We also prove a similar tight lower bound, i.e. show that for ρ → 0 the pair of opposite Hamming balls approximately minimizes the probability P[X ∈ A; Y ∈ B].

For any channel $P_{Y|X}$ the strong data processing constant is defined as the smallest number $\eta _{KL}\in [{0,1}]$ such that $I(U;Y)\le \eta _{KL} I(U;X)$ holds for any Markov chain $U-X-Y$ . It is shown that the value of $\eta _{KL}$ is given by that of the best binary-input subchannel of $P_{Y|X}$ . The same result holds for any $f$ -divergence, verifying a conjecture of Cohen, Kemperman and Zbaganu (1998).