Shannon Meets Nyquist: Capacity of Sampled Gaussian Channels

TL;DR

This paper analyzes how sub-Nyquist sampling affects the capacity of Gaussian channels and proposes optimal sampling strategies that maximize capacity by focusing on high SNR frequencies, revealing fundamental tradeoffs.

Contribution

It derives the capacity of sampled Gaussian channels under various sub-Nyquist sampling methods and establishes the optimal sampling structures for maximizing capacity.

Findings

Optimal sampling extracts high SNR frequencies to maximize capacity.

Sampling with filters and filter banks can suppress aliasing and noise.

Capacity-maximizing filters are equivalent to MSE-minimizing filters.

Abstract

We explore two fundamental questions at the intersection of sampling theory and information theory: how channel capacity is affected by sampling below the channel's Nyquist rate, and what sub-Nyquist sampling strategy should be employed to maximize capacity. In particular, we derive the capacity of sampled analog channels for three prevalent sampling strategies: sampling with filtering, sampling with filter banks, and sampling with modulation and filter banks. These sampling mechanisms subsume most nonuniform sampling techniques applied in practice. Our analyses illuminate interesting connections between under-sampled channels and multiple-input multiple-output channels. The optimal sampling structures are shown to extract out the frequencies with the highest SNR from each aliased frequency set, while suppressing aliasing and out-of-band noise. We also highlight connections between undersampled channel capacity and minimum mean-squared error (MSE) estimation from sampled data. In particular, we show that the filters maximizing capacity and the ones minimizing MSE are equivalent under both filtering and filter-bank sampling strategies. These results demonstrate the effect upon channel capacity of sub-Nyquist sampling techniques, and characterize the tradeoff between information rate and sampling rate.