On the dither-quantized Gaussian channel at low SNR

We study the capacity of the peak-and-average-power-limited Gaussian channel when its output is quantized using a dithered, infinite-level, uniform quantizer of step size Δ. We focus on the low signal-to-noise-ratio (SNR) regime, where communication at low spectral efficiencies takes place. We show that, when the peak-power constraint is absent, the low-SNR asymptotic capacity is equal to that of the unquantized channel irrespective of Δ. We further derive an expression for the low-SNR asymptotic capacity for finite peak-to-average-power ratios and evaluate it in the low- and high-resolution limit. We demonstrate that, in this case, the low-SNR asymptotic capacity converges to that of the unquantized channel when Δ tends to zero, and it tends to zero when Δ tends to infinity.