On the capacity of the dither-quantized Gaussian channel

This paper studies the capacity of the peak-and-average-power-limited Gaussian channel when its output is quantized using a dithered uniform quantizer of step size ∆. It is shown that the capacity of this channel tends to that of the unquantized Gaussian channel when ∆ tends to zero, and it tends to zero when ∆ tends to infinity. In the low signal-to-noise ratio (SNR) regime, it is shown that, when the peak-power constraint is absent, the lowSNR asymptotic capacity is equal to that of the unquantized channel irrespective of ∆. Furthermore, an expression for the low-SNR asymptotic capacity for finite peak-to-averagepower ratios is given and evaluated in the lowand high-resolution limit. It is demonstrated that, in this case, the low-SNR asymptotic capacity converges to that of the unquantized channel when ∆ tends to zero, and it tends to zero when ∆ tends to infinity. Comparing these results with achievability results for (undithered) 1-bit quantization, it is observed that the dither reduces capacity in the low-precision limit, and it reduces the low-SNR asymptotic capacity unless the peak-to-average-power ratio is unbounded.